Explicit Formulas for Hedging Parameters of Perpetual American Options with General Payoffs: A Mellin Transform Approach

Abstract

Risk is often the most concerning factor in financial transactions. Option Greeks provide valuable insights into the risks inherent in option trading and serve as tools for risk mitigation. Traditionally, scholars compute option Greeks through extensive calculations. This article introduces an alternative method that bypasses conventional derivative computation using the Mellin transform and its properties. Specifically, we derive Greeks for perpetual American options with general payoffs, represented as piecewise linear functions, and examine higher-order risk metrics for practical implications. Examples are provided to illustrate the effectiveness of our approach, offering a novel perspective on calculating and interpreting option Greeks.

1. Introduction

Options are financial derivatives that offer the right, but not the obligation, to buy or sell an underlying asset at a predetermined price on, or before, a specified date [1] (p. 31). This specified date is known as the expiry date, or date of maturity, and the agreed-upon price is known as the strike price. The call option gives the holder the right, but not the obligation, to buy the underlying asset for the strike price on, or before, the expiry date. The put option gives the holder the right, but not the obligation, to sell the underlying asset for the strike price on, or before, the expiry date. The option is said to be European if the contract can only be exercised on the date of maturity [1] (p. 31). The European call option affords the holder an ability to generate profit in the future if the market rises. We showcase this by means of some arithmetic. For instance, if a European call option were purchased today, allowing the buyer to acquire a share from an investor for USD 50 in six months, even though the current stock price is USD 45, the call option holder can realise a profit if the share price rises above USD 50 after six months. Here is how the process unfolds: If, after six months of the option purchase, the market price of the share has exceeded USD 50, the investor has the option to exercise their call option on the date of expiration. This grants them the ability to purchase the stock from the option writer at the pre-agreed price of USD 50. Once in possession of the stock, the investor can capitalise on its current market value, which is now higher than USD 50, by selling it in the market. Evidently, this is a strategy investors use to generate profits because the investor sells a share on the market for more than he purchased it from the option writer. In fact, the call option remains a very handsome investment vehicle for the reason that the profit is potentially infinite. Furthermore, the capital requirement can be substantially decreased as investors are only required to acquire the less expensive option contract instead of buying the actual stock. They can then trade the value of the option contract itself. For instance, in the case of a call option, as the price of the underlying asset increases, the value of the call option also rises. At this point, the investor may opt to sell the call option currently in their possession. This presents them with the opportunity to capitalise on the growth of the specified stock at a more affordable cost. The opposite holds for put options; they enable investors to protect against losses, and this can be achieved at amounts considerably lower than the cost of a stock. With these strategies in mind, it becomes apparent that a call option has a bullish buyer and a bearish seller, while the put option has a bearish buyer and a bullish seller.

As an option provides an investor with the ability to exercise after purchase, this privilege comes at a cost known as the premium. The premium represents the price of an option [1] (p. 843). Although the origins of options can be traced back to ancient times [2], in antiquity, the premium had yet to be derived mathematically. Hence, despite the long-standing utilisation of options spanning millennia, their precise valuation remained elusive. It was not until 1973 when Fischer Black and Myron Scholes introduced the Nobel prize-winning “Black–Scholes PDE” (see Table of Abbreviations), which revolutionised the field by providing a comprehensive framework for option pricing [3]. We also highlight that Robert Merton independently produced the same equation in 1973, and was also awarded the Nobel Prize for his achievement.

Upon achieving accurate pricing for options, the financial markets saw a rational incentive for institutions to engage in OTC trading of these financial instruments. Nevertheless, this endeavour is not without its inherent risks. For instance, a financial institution that offers a customised option to a client in OTC markets faces the challenge of effectively handling their risk exposure. When the option aligns with those actively traded on an exchange or in the OTC market, the institution can offset its exposure by acquiring a corresponding option. However, when the option is tailored specifically to meet a client’s unique requirements and does not correspond to standard exchange-traded products, mitigating the exposure becomes a more complex task. So, if one trades an option, they inherently expose themself to risk.

Counter-intuitively, and as one may know, the importance of hedging risk is arguably more important than correctly pricing an option [4] (p. 90). One might even add that hedging the risk is more important than the cost of any financial transaction. We acknowledge that this is a startling supposition. However, the rationale is that precise hedging either minimises or can totally eradicate forthcoming uncertainty. Precise hedging results in a predetermined profit or loss at the point of contact of a purchase or a sale. Conversely, if your hedging is imprecise, then, to a reasonable extent, the initial contract price becomes inconsequential, or irrelevant, because future uncertainty can potentially overshadow any initial gains.

What tool, then, does an investor use for option sensitivity analysis to offset one’s incurred risks? Enter the “Greeks”. Each Greek measures a different dimension of the risk in an option position [1] (p. 417). More simply, they are defined as an option’s hedge parameters. The trader’s objective is to skilfully manage these Greek metrics to ensure all risks remain within acceptable bounds. Further exposition of each individual Greek will be defined throughout this paper.

Now that the importance of the Greeks has been established, the following question still remains: How does one solve for the Greeks? The literature presents various formulations and methods. We will only enumerate a few such methods here. Firstly, the Greeks can be solved by direct computation of the first derivative of the Black–Scholes formula. In fact, the Greeks were first revealed by this very method [3]. Other sources that derive the option Greeks using this standard, logical approach include well-known textbooks that cover financial derivatives [1,5,6].

In contradistinction to the Black–Scholes analytical option formulas, obtaining analytical formulas for exotic options and their associated Greeks can be either challenging or impossible, especially when dealing with complex payoffs or underlying models that prevent straightforward mathematical solutions. This complexity arises amongst numerous types of options, particularly those governed by stochastic volatility and jump-diffusion models. If the option pricing function happens to be less tractable or the option pricing function does not have a closed-form solution, one must resort to other methods of computation. One method involves finding the option price using a binomial tree, first introduced by [7], and then taking the first derivatives. One takes the first derivative of the option price with respect to a certain option parameter. This is for both the call and put option. However, this was shown to be a flawed approach by [8]. Utilising tree methods for computing partial derivatives may be hindered by the inherent characteristics of the binomial discretisation they incorporate. Ref. [9] developed another approach by noting that for options where the strike price matches the forward price of the underlying asset, it is observed that the option’s value shows an almost linear relationship with volatility. Leveraging this linear characteristic, they derived remarkably precise approximations of option values and key risk factors, including hedge ratio, convexity, time decay, and volatility sensitivity. Interestingly, ref. [10] recommends a completely different approach to option selection by actually disregarding the Greeks altogether and proposing that an option should be viewed not as risky but also as a function of premium cost against units of time. Ref. [11] introduce a novel probabilistic approach for the numerical calculation of the Greek letters under the Black–Scholes formulation. Their methodology relies on the integration-by-parts formula, a fundamental concept at the heart of variational stochastic calculus theory, as developed in the field of Malliavin calculus. Ref. [12] had a very similar approach, but the volatility in the Black–Scholes formula was assumed to be stochastic, not constant. Finally, when no analytical solutions are available for pricing an exotic option, one can turn to the widely adopted Monte Carlo simulation method as an alternative approach. The literature on this topic is quite vast [13,14,15,16,17].

However, we redirect our focus towards computing the Greeks through the conventional means of calculating the first derivative. In this approach, we calculate the first derivative of an analytical solution of an option. As discussed, computation and even complex integrals are required for such computations [18]. For instance, the hedge ratio “delta” of an option that has been priced using the Black–Scholes PDE can be computed using two distinct methods. One approach involves finding the partial derivative of the option pricing formula with respect to the underlying asset price, employing the chain rule. Alternatively, the hedge ratio can be determined by differentiating the original formula, which represents the option’s worth as a discounted expectation under the risk-neutral framework. The former approach necessitates the calculation of various intricate partial derivatives, including the derivative of the standard normal distribution function. Meanwhile, the latter method entails the derivative of an integral arising from the discounted risk-neutral expectation. Nota bene, both can be challenging to comprehend. We have termed this approach “brute-force” calculation. The aim of this paper is then to bypass the practice of brute force to compute the Greeks. Our approach is to employ properties of what is known as the Mellin transform. The Mellin transform will be defined and expounded upon in the later sections of this article.

It appears that once the analytical formula for each Greek was derived, most turned their sights to other, more pressing issues, that is, solving for the Greeks of options that do not have closed-form solutions. The issue of time-consuming calculations has primarily garnered recognition from a limited number of individuals, resulting in a lack of comprehensive efforts to tackle this problem. However, Ref. [19] accepted the challenge of computing the Greeks via the first derivative but simplified the process by identifying some neat identities that reduced the amount of algebraic manipulation. Ref. [20] appears to have discovered a similar approach at around the same time. Lastly, ref. [21] bypassed the need for derivative calculation altogether by using the Mellin transform. Rodrigo and Mamon applied this technique to solving for the Greeks of European options with general payoffs, both with and without jumps in the underlying asset price dynamics. This paper draws inspiration from [21], but with the difference of computing the Greeks of perpetual American options without jumps, not European options with jumps. A jump process is a stochastic process for a variable involving jumps in the value of the variable [1]. The author of [22], too, used the Mellin transform to re-derive the solution to the European put option seen in [23] when the return is discontinuous, and his approach of deriving a separable first-order linear DE in order to obtain a solution was almost identical to [24]. With that, Frontczak also derived the common Greeks for options through taking the derivative of the option pricing function in the conventional sense. We aim to improve the method proposed by Frontczak by removing the need to explicitly solve for the first derivative when computing the Greeks.

In prior work undertaken by [21], Rodrigo and Mamon utilised the Mellin transform to circumvent the need for cumbersome calculations when solving for the Greeks of European options based upon the Black–Scholes framework. This paper serves as an extension of their approach, albeit with a distinctive focus on computing the Greeks for perpetual American options featuring general payoffs, rather than European options with general payoffs. Furthermore, in [25], a classification system was introduced that covers four distinct categories of perpetual American options under the Black–Scholes framework. In the following sections, we undertake the calculation of Greeks specific to each one of these categories of American options. The key distinction of this paper, setting it apart from [25], lies in our use of the Mellin transform to compute derivatives of the option pricing function rather than the pricing function itself. We utilise the properties of the Mellin transform to solve for derivatives, whereas ref. [25] employed the definition of the Mellin transform solely for deriving the option pricing function.

The structure of the paper is as follows. In Section 2, we lay the theoretical foundation for option pricing. Section 3 introduces the Mellin transform and presents a preliminary lemma that provides the solution to a non-homogeneous linear ODE, which is crucial for the results in Section 5. In Section 4, we define the classes of perpetual American options, enabling the categorisation of payoffs and facilitating our derivations in Section 5. The main contribution of this paper is presented in Section 6, where we derive the Greeks for perpetual American options. Section 7 focuses on analytical validation, ensuring the applicability of the explicit formulas derived. Finally, Section 8 and Section 9 offer a discussion of the results and their implications, followed by concluding remarks.

2. Formulation of Perpetual American Option Pricing Problem with General Payoffs

Firstly, assume that the option price is dependent on the asset price under a risk-neutral probability measure and that the asset is governed by the stochastic differential equation

$$\mathrm{d}{S}_t=(r-D)S_t\mathrm{d}t+\sigma S_t\mathrm{d}{W}_t,$$

(1)

where $S=\left\{S_t:t\ge 0\right\}$ is the price process of the underlying asset, and $W=\left\{W_t:t\ge 0\right\}$ is a Wiener process (with respect to the risk neutral measure). The risk-free rate r and the volatility $\sigma$ are assumed to be positive constants. The dividend yield D is assumed to be non-negative, that is $r,\sigma >0$ and $D\ge 0$. Let the European option pricing function at time t be denoted by $V\left(t\right)$. Then, it is well attested [1,26,27] that $V\left(t\right)=v(S_t,t)$, where the option pricing function $v=v(x,t)$ satisfies the Black–Scholes PDE

$${\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{1}{2}}{\sigma }^2{x}^2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+(r-D)x{\displaystyle \frac{\partial v}{\partial x}}-rv=0,x\ge 0,0\le t<T,$$

(2)

with the final condition $v(x,T)=g\left(x\right)$ for $x\ge 0$. The payoff function $g:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$, where ${\mathbb{R}}_{\ge 0}=[0,\infty )$, is a simple contingent claim. A simple contingent claim can be conceived of as the option’s payoff on expiration.

An option is said to be perpetual if the expiration date is infinitely far away [26] (p. 13). Options in this category are also called perpetual derivatives [1] (p. 843). European options can never be perpetual derivatives since they would never be exercised. Therefore, a perpetual derivative must provide the holder with the ability to exercise the contract at any point in time. The American option serves this very purpose. The American option can be exercised at any time during its life [1] (p. 827). The perpetual American option is, therefore, an option that can be exercised at any point in time and, at the same time, does not have an expiration date. Hence, the perpetual American option pricing function can be derived from a timeless variation of (2), specifically when $T\to \infty$. With that said, the perpetual American option pricing function v satisfies (3), as illustrated below

$${\displaystyle \frac{1}{2}}{\sigma }^2{x}^2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+(r-D)x{\displaystyle \frac{\partial v}{\partial x}}-rv=0,x\ge 0,$$

(3)

where the proposed option pricing function v now exclusively depends on the stock price x. This ordinary differential Equation (3) possesses the characteristics of being homogeneous and linear.

Furthermore, we can add another piece of information into this formulation to zero in on an analytical solution of the perpetual American option pricing function. The flexibility to exercise an American perpetual option at any time in the future suggests the existence of an optimal exercise point, since the function v is only dependent on the stock price x. In other words, there is a particular stock price that dictates whether or not it is profitable to exercise a perpetual American option. When this boundary is crossed, investors opt for exercising their option. One may intuitively justify the existence of such a stock price, positing that an investor would naturally elect to exercise the option when it possesses a substantial in-the-money value. This pivotal threshold is referred to as the optimal exercise price [26] (p. 251). We denote the optimal exercise price by $S^{\ast }$. By extension, there also exists the optimal exercise region, which is the collection of all the optimal exercise prices over the life of the American option. We denote the exercise region by R. The search for the solution of an unknown but critical value such as the optimal exercise price is known as a free boundary problem [26] (p. 252).

To continue further, we need to define more concepts. We begin by defining the support. Consider a function $g:[0,\infty )\to [0,\infty )$; the support is defined by $supp\left(g\right)=\{x:g(x)>0\}$. In essence, the support of an option delineates the domain where the underlying asset generates a positive payoff, as elucidated by [25]. For instance, consider the call option, whose payoff function is $g\left(x\right)={(x-K)}^{+}$. As a result, the support of the call option is $supp\left(g\right)=(K,\infty )$. Denote the usual indicator function of a set A by ${\chi }_A$, i.e., ${\chi }_A\left(x\right)=1$ if $x\in A$ and ${\chi }_A\left(x\right)=0$ if $x\notin A$.

Table 1. Payoff functions.

Type	Payoff Function	Support
Put	${(K-x)}^{+}$	$[0,K)$
Bear spread	${(K_2-x)}^{+}-{(K_1-x)}^{+}$	$[0,K_2)$
Call	${(x-K)}^{+}$	$(K,\infty )$
Bull spread	${(x-K_1)}^{+}-{(x-K_2)}^{+}$	$(K_1,\infty )$
Digital call	${\chi }_{[K,\infty )}\left(x\right)$	$[K,\infty )$
Asset-or-nothing call	$x{\chi }_{[K,\infty )}\left(x\right)$	$[K,\infty )$
Butterfly spread	${(x-K_1)}^{+}+{(x-K_3)}^{+}-2{(x-K_2)}^{+}$	$(K_1,K_3)$
	$-K_1-K_3+2K_2=0$
Iron condor	${(x-K_1)}^{+}-{(x-K_2)}^{+}-{(x-K_3)}^{+}+{(x-K_4)}^{+}$	$(K_1,K_4)$
	$-K_1+K_2+K_3-K_4=0$
Straddle	${(x-K)}^{+}+{(K-x)}^{+}$	$[0,K)\cup (K,\infty )$
Strip	${(x-K)}^{+}+2{(K-x)}^{+}$	$[0,K)\cup (K,\infty )$
Strap	${(K-x)}^{+}+2{(x-K)}^{+}$	$[0,K)\cup (K,\infty )$
Strangle	${(x-K_2)}^{+}+{(K_1-x)}^{+}$	$[0,K_1)\cup (K_2,\infty )$

provides a list of payoff functions and their respective support.

Additionally, if $v\left(x\right)=x-K$ for $S^{\ast }<x<\infty$, then $R=(S^{\ast },\infty )$. That is to say, it is optimal to exercise the option in the region $R=(S^{\ast },\infty )$. Therefore, the investor’s objective is to identify R in such a way that $R\subset supp\left(g\right)$, as the goal is to ensure that the exercise region leads to a positive payoff. In the case of the call option, $R=(S^{\ast },\infty )\subset (K,\infty )=supp\left(g\right)$. On the contrary, if $x\notin R$, then x must be in a region where it is not optimal to exercise the option. This region is known as the continuation region. The continuation region is expressed mathematically as $x\in R^c=[0,\infty )\setminus R$. When $x\in R^c$, then the function v satisfies (3). This is true not just in the case of a call, but any of the payoffs in Table 1.

It is worth noting that the analytical solution for the boundary point $S^{\ast }$ of a perpetual American option is independent of an investor’s risk aversion or risk-seeking behaviour. The reason for this is the following: in the case of a perpetual American option, the optimal exercise price is the point at which the option pricing function is equal to the payoff function of the underlying contingent claim. And importantly, this characteristic is not influenced by the unique traits or behaviours of individual investors. Unfortunately, there is no way of knowing the optimal exercise boundary a priori [27] (p. 129). Therefore, to find a solution for the optimal exercise boundary becomes an indispensable facet in the process of ascertaining the premium associated with an American option.

Tying everything together, we have seen that if $x\in R^c$, it is not optimal to exercise the option. So, we have

$${\displaystyle \frac{1}{2}}{\sigma }^2{x}^2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+(r-D)x{\displaystyle \frac{\partial v}{\partial x}}-rv=0,x\in R^c.$$

(4)

Thus, for any $x\ge 0$, v can be embedded into the linear ordinary differential equation

$${\displaystyle \frac{1}{2}}{\sigma }^2{x}^2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+(r-D)x{\displaystyle \frac{\partial v}{\partial x}}-rv=F\left(x\right)={\chi }_R\left(x\right)G\left(x\right),x\ge 0,$$

(5)

and $G=G\left(x\right)$ is a function that is yet to be determined but is dependent on the boundary condition

$$v\left(x\right)={\left.g\right|}_{supp\left(g\right)}\left(x\right),x\in \partial R.$$

(6)

Here, $\partial R$ denotes the boundary of the set R relative to the set $[0,\infty )$. Additionally, $\partial R$ represents the point of contact of the option pricing function and the option’s value at expiry on a payoff diagram. After G is obtained, we determine the exercise region R that will maximise the option pricing function $v(x;R)$ for each fixed $x\in R^c$. The idea behind this is that we wish to maximise the value of the perpetual American option value among all possible optimal exercise prices. Note that the value of each $S^{\ast }$ has already been derived in [25].

3. Preliminary Results

3.1. Mellin Transform and Its Properties

The fundamental building blocks of the Mellin transform and its association with the pricing of options based on the Black–Scholes PDE can be found in several articles [22,24,28,29,30]. Nonetheless, to ensure clarity in presenting the results of this paper, we offer a foundational overview of the perpetual American option pricing problem, the Mellin transform, and its application to the Black–Scholes framework.

We commence by introducing the Mellin transform. Consider a function $f:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$. The Mellin transform of f is defined as

$$\mathcal{M}\{f\left(x\right);\xi \}=\widehat{f}\left(\xi \right)={\int }_0^{\infty }{x}^{\xi -1}f\left(x\right)\mathrm{d}x,$$

(7)

provided that the improper integral converges at the complex number $\xi$. By definition, it can be seen that $x=[0,\infty )$. The function f does not need to be smooth, but f needs to be integrable. As we will demonstrate, the Mellin transform proves to be valuable in solving the Black–Scholes PDE due to several properties it exhibits when applied to derivative functions. To explore these properties, we first introduce the identity operator, denoted as id, which essentially has no transformative effect when applied to a function. Combining the identity operator with the Mellin transform leads to the following relationships:

$$\begin{array}{c}\widehat{\left(id{f}^{\prime }\right)}\left(\xi \right)=-\xi \widehat{f}\left(\xi \right),\widehat{\left({\mathrm{id}}^2{f}^{″}\right)}\left(\xi \right)=\xi (\xi +1)\widehat{f}\left(\xi \right),\hfill \\ \widehat{f^{\prime }}\left(\xi \right)=-(\xi -1)\widehat{f}(\xi -1),\widehat{f^{″}}\left(\xi \right)=(\xi -1)(\xi -2)\widehat{f}(\xi -2),\hfill \\ \widehat{\left({\displaystyle \frac{f}{\mathrm{id}}}\right)}\left(\xi \right)={\int }_0^{\infty }{x}^{\xi -1}{\displaystyle \frac{f\left(x\right)}{x}}\mathrm{d}x={\int }_0^{\infty }{x}^{(\xi -1)-1}f\left(x\right)\mathrm{d}x=\widehat{f}(\xi -1).\hfill \end{array}$$

These properties can be extended to the nth derivative through an induction argument, resulting in the general expressions

$$\begin{array}{c}\widehat{\left({\mathrm{id}}^n{f}^{\left(n\right)}\right)}\left(\xi \right)={(-1)}^n\left[{\displaystyle \prod _{j=0}^{n-1}}(\xi +j)\right]\widehat{f}\left(\xi \right),\widehat{f^{\left(n\right)}}\left(\xi \right)={(-1)}^n\left[{\displaystyle \prod _{j=1}^n}(\xi -j)\right]\widehat{f}(\xi -n),\\ \widehat{\left({\displaystyle \frac{f}{{\mathrm{id}}^n}}\right)}\left(\xi \right)=\widehat{f}(\xi -n).\end{array}$$

We can define $x^{\overline{n}}={\prod }_{k=0}^{n-1}(x+k)$ as a higher factorial sequence and $x^{\underline{n}}={\prod }_{k=0}^{n-1}(x-k)$ as a lower factorial sequence [31]. We can see that

$$\begin{array}{cc}\hfill {(\xi -1)}^{\underline{n}}& =\prod _{j=0}^{n-1}(\xi -1-j)\hfill \\ \hfill & =(\xi -1)(\xi -2)\dots (\xi -n)\hfill \\ \hfill & =\prod _{j=1}^n(\xi -j).\hfill \end{array}$$

(8)

Then, the above becomes

$$\begin{array}{c}\widehat{\left({\mathrm{id}}^n{f}^{\left(n\right)}\right)}\left(\xi \right)={(-1)}^n{\left(\xi \right)}^{\overline{n}}\widehat{f}\left(\xi \right),\widehat{f^{\left(n\right)}}\left(\xi \right)={(-1)}^n{(\xi -1)}^{\underline{n}}\widehat{f}(\xi -n),\\ \widehat{\left({\displaystyle \frac{f}{{\mathrm{id}}^n}}\right)}\left(\xi \right)=\widehat{f}(\xi -n).\end{array}$$

(9)

For the sake of completeness, we define the operator $U_k$ and its powers as $U_{k}g={\mathrm{id}}^k{g}^{\left(k\right)}$, and in addition, define ${id}^0\left(x\right)=1$ and $f^{\left(0\right)}\left(x\right)=f\left(x\right)$. The significance and utility of the Mellin properties (9) arise from their ability to streamline the calculation of derivatives, eliminating the need for cumbersome algebraic manipulations. This will become evident in practical applications, particularly when calculating delta and other Greeks. Thus, familiarity with the properties of the Mellin transform is significant. Moreover, additional valuable characteristics of the Mellin transform arise from what is known as the Mellin convolution. The Mellin convolution of $f:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ and $g:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$ is

$$(f\ast g)\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{1}{y}}f\left({\displaystyle \frac{x}{y}}\right)g\left(y\right)\mathrm{d}y.$$

(10)

An implication of the Mellin convolution (10) is the convolution property. The convolution property is expressed mathematically as

$$\widehat{(f\ast g)}\left(\xi \right)=\widehat{f}\left(\xi \right)\widehat{g}\left(\xi \right).$$

(11)

Remark 1.

One can draw from (11) the conclusion that

$$\left(f\ast g\right)\left(x\right)={\mathcal{M}}^{-1}\{\mathcal{M}\{f\left(x\right);\xi \}\mathcal{M}\{g\left(x\right);\xi \};x\}.$$

In other words, if one inverts the product of two Mellin transforms, we arrive at the convolution of those two functions.

Remark 2.

By inference, the equality

$$\left({\displaystyle \frac{f}{{id}^n}}\ast g\right)\left(x\right)=\left(f\ast {\displaystyle \frac{g}{{id}^n}}\right)\left(x\right)$$

holds for general f and g.

The fundamental property (11) will play a crucial role in substantiating various results throughout this work. In summary, leveraging the properties of the Mellin transform, as delineated by (9) and (11), enables us to solve Black–Scholes-type PDEs without the need for conventional techniques typically employed in such endeavours. To illustrate this capability, we introduce our first lemma.

3.2. Perpetual Analogue of the Black–Scholes Kernel

Lemma 1.

Given the non-homogeneous Euler ordinary differential equation

$$c_2{x}^2{\displaystyle \frac{{\mathrm{d}}^{2}v}{\mathrm{d}{x}^2}}+c_{1}x{\displaystyle \frac{\mathrm{d}v}{\mathrm{d}x}}+c_{0}v=F\left(x\right),x\ge 0,$$

(12)

for some $F:{\mathbb{R}}_{\ge 0}\to \mathbb{R}$, and $c_2>0,c_1\in \mathbb{R}$ and $c_0<0$, then

$$v\left(x\right)=\left(\mathcal{K}\ast F\right)\left(x\right)=-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{\int }_0^x{y}^{{\alpha }_2-1}F\left(y\right)\mathrm{d}y-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{\int }_x^{\infty }{y}^{{\alpha }_1-1}F\left(y\right)\mathrm{d}y,$$

where

$$\begin{array}{c}\mathcal{K}\left(x\right)=-{\displaystyle \frac{1}{\rho }}\left[x^{-{\alpha }_1}{\chi }_{[0,1]}\left(x\right)+x^{-{\alpha }_2}{\chi }_{(1,\infty )}\left(x\right)\right],\\ \rho =c_2\left({\alpha }_2-{\alpha }_1\right),\\ {\alpha }_1={\displaystyle \frac{\left(c_1-c_2\right)-{\left[{\left(c_1-c_2\right)}^2-4c_2{c}_0\right]}^{1/2}}{2c_2}},\\ {\alpha }_2={\displaystyle \frac{\left(c_1-c_2\right)+{\left[{\left(c_1-c_2\right)}^2-4c_2{c}_0\right]}^{1/2}}{2c_2}},\end{array}$$

(13)

is a solution to (12). Moreover, ${\alpha }_2>0$, and if $c_1\le -c_0,$ then ${\alpha }_1\le -1$.

See [25] for proof. The function $\mathcal{K}$ is known the perpetual analogue of the Black–Scholes kernel first introduced in [29]. The perpetual analogue of the kernel emerges naturally during the solution process outlined in [25], and that is why it is specific to perpetual American options. Due to the time independence of the ODE, applying the Mellin transform converts the equation into a linear, separable differential equation, allowing us to define the kernel on one side of the equation.

We now have the requisite tools in order to price a perpetual American option with a general payoff. We apply Lemma 1 to the non-homogeneous perpetual American option ODE (5). The resulting function

$$v\left(x\right)=\left(\mathcal{K}\ast F\right)\left(x\right)=-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{\int }_0^x{y}^{{\alpha }_2-1}F\left(y\right)\mathrm{d}y-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{\int }_x^{\infty }{y}^{{\alpha }_1-1}F\left(y\right)\mathrm{d}y,$$

(14)

where

$$\begin{array}{c}\mathcal{K}\left(x\right)=-{\displaystyle \frac{1}{\rho }}\left[x^{-{\alpha }_1}{\chi }_{[0,1]}\left(x\right)+x^{-{\alpha }_2}{\chi }_{(1,\infty )}\left(x\right)\right],\\ \rho ={\displaystyle \frac{1}{2}}{\sigma }^2\left({\alpha }_2-{\alpha }_1\right),\\ {\alpha }_1={\displaystyle \frac{\left(r-D-\frac{{\sigma }^2}{2}\right)-{\left[{\left(r-D-\frac{{\sigma }^2}{2}\right)}^2+2{\sigma }^{2}r\right]}^{1/2}}{{\sigma }^2}},\\ {\alpha }_2={\displaystyle \frac{\left(r-D-\frac{{\sigma }^2}{2}\right)+{\left[{\left(r-D-\frac{{\sigma }^2}{2}\right)}^2+2{\sigma }^{2}r\right]}^{1/2}}{{\sigma }^2}},\end{array}$$

is a solution to (5). Additionally, ${\alpha }_1\le -1$ and ${\alpha }_2>0$ remain valid. Moreover, ${\alpha }_1\le -1$ and ${\alpha }_2>0$ consistently yield real solutions for all values of r, D, and $\sigma$, as each parameter in (2) is defined to be either positive or greater than zero, leading to a positive discriminant. This holds true even in the case of the perpetual Black–Scholes PDE. Notice that for ${(r-D-{\displaystyle \frac{{\sigma }^2}{2}})}^2+2{\sigma }^{2}r<0$, only r can satisfy this condition. Consequently, we must impose $r>0$. Financially, this implies that perpetual American options can only be priced in our derivation when the interest rate is positive.

This expression does not fully represent the contract payoff function for a generalised perpetual American option, since we have yet to maximise $S^{\ast }$. First, it is imperative to specify the particular option payoff structure (as detailed in Table 1). This acts as a sequitur for the forthcoming Section 4.

Example 1.

Suppose $F\left(x\right)={\chi }_R\left(x\right)G\left(x\right)$ for some function $G=G\left(x\right)$ and some set $R\subset {\mathbb{R}}_{\ge 0}$. Then,

$$v\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right)F\left(y\right)\mathrm{d}y={\int }_R{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right)G\left(y\right)\mathrm{d}y.$$

Alternatively, by our knowledge of $\mathcal{K}$ in (13), we have

$$v\left(x\right)=-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{\int }_{[0,x]\cap R}{y}^{{\alpha }_2-1}G\left(y\right)\mathrm{d}y-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{\int }_{[x,\infty )\cap R}{y}^{{\alpha }_1-1}G\left(y\right)\mathrm{d}y.$$

The function G is to be determined such that

$$\begin{array}{cc}\hfill {g|}_{supp\left(g\right)}\left(x\right)& =v\left(x\right)\hfill \\ \hfill & =-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{\int }_{[0,x]\cap R}{y}^{{\alpha }_2-1}G\left(y\right)\mathrm{d}y-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{\int }_{[x,\infty )\cap R}{y}^{{\alpha }_1-1}G\left(y\right)\mathrm{d}y,x\in \partial R.\hfill \end{array}$$

Lemma 2.

The nth derivative of the perpetual Black–Scholes kernel is given by

$${\mathcal{K}}^{\left(n\right)}\left(x\right)={\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}[{\alpha }_1^{\overline{n}}{x}^{-{\alpha }_1-n}{\chi }_{(0,1)}\left(x\right)+{\alpha }_2^{\overline{n}}{x}^{-{\alpha }_2-n}{\chi }_{(1,\infty )}\left(x\right)],$$

(15)

where $x^{\overline{n}}={\prod }_{k=0}^{n-1}(x+k)$. By extension,

$$\left({id}^n{\mathcal{K}}^n\right)\left(x\right)={\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}[{\alpha }_1^{\overline{n}}{x}^{-{\alpha }_1}{\chi }_{(0,1)}\left(x\right)+{\alpha }_2^{\overline{n}}{x}^{-{\alpha }_2}{\chi }_{(1,\infty )}\left(x\right)].$$

See Appendix A.1 for proof.

Remark 3.

By a simple substitution, we see that (15) becomes

$${\mathcal{K}}^{\left(n\right)}\left({\displaystyle \frac{x}{y}}\right)={\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}\left[{\alpha }_1^{\overline{n}}{x}^{-{\alpha }_1-n}{y}^{{\alpha }_1+n}{\chi }_{(0,1)}\left({\displaystyle \frac{x}{y}}\right)+{\alpha }_2^{\overline{n}}{x}^{-{\alpha }_2-n}{y}^{{\alpha }_2+n}{\chi }_{(1,\infty )}\left({\displaystyle \frac{x}{y}}\right)\right].$$

There are two cases, either $x<y$ or $x>y$ ($x\ne y$ because ${\mathcal{K}}^{\left(n\right)}\left(z\right)$ for $n\ge 1$ does not exist at $z=1$). By extension, the above can be written as

$${\mathcal{K}}^{\left(n\right)}\left({\displaystyle \frac{x}{y}}\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}{\alpha }_1^{\overline{n}}{x}^{-{\alpha }_1-n}{y}^{{\alpha }_1+n},\hfill & \mathit{if}x<y,\hfill \\ {\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}{\alpha }_2^{\overline{n}}{x}^{-{\alpha }_2-n}{y}^{{\alpha }_2+n},\hfill & \mathit{if}x>y.\hfill \end{array}\right.$$

Lemma 3.

The partial derivative of $\mathcal{K}$ with respect to $c_k$ is

$${\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left(x\right)={\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}\left[{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\right]{\chi }_{(0,1)}+{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}\left[{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(x\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\right]{\chi }_{(1,\infty )},$$

(16)

for $x\in {\mathbb{R}}_{\ge 0}$. See Appendix A.2 for proof.

4. Classes of Payoff Functions

It was previously mentioned that in order to arrive at a more precise solution for the option pricing function, denoted v in (14), it becomes essential to specify the class to which the option belongs. Let $a,b>0$. As expounded in [25], each payoff function could be categorised into one of the four following classes:

• Class (a): $[0,b)$ or $[0,b]$, i.e., the support of g is a bounded interval containing ‘0’.
• Class (b): $(a,\infty )$, i.e., the support of g is an infinite interval not containing ‘0’.
• Class (c): $(a,b)$ or $(a,b]$, (if $a<b)$ i.e., the support of g is a bounded interval not containing ‘0’.
• Class (d): $[0,a)\cup (b,\infty )$ or $[0,a]\cup (b,\infty )$ (if $a<b)$ or $[0,a)\cup [b,\infty )$ (if $a<b)$, i.e., the support of g is the disjoint union of a bounded interval containing ‘0’ and an infinite interval not containing ‘0’.

As a natural consequence of these classifications, it becomes feasible to categorise the payoff functions delineated in Table 1. Specifically, puts and bear spreads fall within class (a), while calls, bull spreads, digital calls and asset-or-nothing calls belong to class (b). Class (c) encompasses butterfly spreads and iron condors, and finally, class (d) comprises straddles, strips, straps and strangles.

Remark 4.

The perpetual kernel $\mathcal{K}$ directly influences the solution for each v defined by a given class. As demonstrated in [25], the perpetual kernel determines the limits of the integrals in (14), influencing whether either of the integrals vanishes or remains, depending on the class of the option specified. This relationship between the perpetual kernel and perpetual American options extends to the calculation of their derivatives, enabling the determination of the Greeks.

5. Partial Derivatives of the Option Pricing Function

Each partial derivative below has a formal proof given in Appendix B.

Lemma 4

(Partial derivatives of the option pricing function). Recall (14). Let $k\in \{0,1,2\}$. We have the following first-order partial derivatives:

(i)

$$\begin{array}{cc}\hfill v^{\left(n\right)}\left(x\right)& ={\displaystyle \frac{1}{x^n}}\left(\left({id}^n{\mathcal{K}}^{\left(n\right)}\right)\ast \left({\chi }_{R}G\right)\right)\left(x\right)\hfill \\ \hfill & =\left({\displaystyle \frac{\mathcal{K}}{{id}^n}}\ast {\left({\chi }_{R}G\right)}^{\left(n\right)}\right)\left(x\right)\hfill \\ \hfill & =\left({\mathcal{K}}^{\left(n\right)}\ast {\displaystyle \frac{\left({\chi }_{R}G\right)}{{id}^n}}\right)\left(x\right).\hfill \end{array}$$

(ii)

$${\displaystyle \frac{{\partial }^{n}v}{\partial c_k^n}}\left(x\right)=\sum _{j=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)\left({\displaystyle \frac{{\partial }^{n-j}\mathcal{K}}{\partial c_k^{n-j}}}\ast {\displaystyle \frac{{\partial }^j\left({\chi }_{R}G\right)}{\partial c_k^j}}\right)\left(x\right).$$

(iii)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{n+1}v}{\partial c_k\partial x^n}}\left(x\right)& =\left({\displaystyle \frac{{\partial }^{n+1}\mathcal{K}}{\partial c_k\partial x^n}}\ast \left({\displaystyle \frac{{\chi }_{R}G}{{\mathrm{id}}^n}}\right)\right)\left(x\right)+\left({\displaystyle \frac{{\partial }^n\mathcal{K}}{\partial c_k^n}}\ast \left({\displaystyle \frac{\frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}{{\mathrm{id}}^n}}\right)\right)\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}\left({id}^n{\displaystyle \frac{{\partial }^{n+1}\mathcal{K}}{\partial c_k\partial x^n}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{x^n}}\left({id}^n{\displaystyle \frac{{\partial }^n\mathcal{K}}{\partial c_k^n}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right).\hfill \end{array}$$

(iv)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}v}{\partial c_k\partial c_{k^{\prime }}}}\left(x\right)=& \left({\displaystyle \frac{{\partial }^2\mathcal{K}}{\partial c_k\partial c_{k^{\prime }}}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_{k^{\prime }}}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)\hfill \\ \hfill & +\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_{k^{\prime }}}}\right)\left(x\right)+\left(\mathcal{K}\ast {\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_k\partial c_{k^{\prime }}}}\right)\left(x\right).\hfill \end{array}$$

It is through Lemma 4 that we can derive the Greeks of American option pricing functions, since the Greeks are defined as the derivatives, or partial derivatives, of options.

Example 2.

Let us attempt to simplify Lemma 4(i).

Class (a) payoffs

For any class (a) function, we saw in [25] that

$$-{\displaystyle \frac{1}{\rho }}{\int }_0^S{y}^{{\alpha }_2-1}G\left(y\right)\mathrm{d}y={\left.S^{{\alpha }_2}g\right|}_{supp\left(g\right)}\left(S\right),$$

(17)

which implies

$${\int }_0^S{y}^{{\alpha }_2-1}G\left(y\right)\mathrm{d}y=-{\left.\rho S^{{\alpha }_2}g\right|}_{supp\left(g\right)}\left(S\right).$$

Differentiating with respect to S,

$$S^{{\alpha }_2-1}G\left(S\right)=-{\left.\rho {\alpha }_2{S}^{{\alpha }_2-1}g\right|}_{supp\left(g\right)}\left(S\right)-\rho S^{{\alpha }_2}{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(S\right),$$

or

$$G\left(S\right)=-{\left.\rho {\alpha }_{2}g\right|}_{supp\left(g\right)}\left(S\right)-\rho S{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(S\right).$$

Replacing S by y,

$$G\left(y\right)=-{\left.\rho {\alpha }_{2}g\right|}_{supp\left(g\right)}\left(y\right)-\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right).$$

(18)

Since the option is in class (a), we know that $x\in R=(0,S^{\ast })$, and $x\in R^c=[S^{\ast },\infty ]$. If $x\ge S^{\ast }$ and $y\le S^{\ast }$, then $y\le S^{\ast }\le x$, so ${\displaystyle \frac{x}{y}}\ge 1$. Hence, by (15) and (18), Lemma 4(i) becomes

$$\begin{array}{cc}\hfill v^{\left(n\right)}\left(x\right)& ={\displaystyle \frac{1}{x^n}}\left(\left({id}^n{\mathcal{K}}^{\left(n\right)}\right)\ast \left({\chi }_{R}G\right)\right)\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_0^{\infty }{\displaystyle \frac{1}{y}}{\left({\displaystyle \frac{x}{y}}\right)}^n{\mathcal{K}}^{\left(n\right)}\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_0^{S^{\ast }}{\displaystyle \frac{1}{y}}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}[{\alpha }_1^{\overline{n}}{\left({\displaystyle \frac{x}{y}}\right)}^{-{\alpha }_1}{\chi }_{(0,1)}\left({\displaystyle \frac{x}{y}}\right)\hfill \\ \hfill & +{\alpha }_2^{\overline{n}}{\left({\displaystyle \frac{x}{y}}\right)}^{-{\alpha }_2}{\chi }_{(1,S^{\ast })}\left({\displaystyle \frac{x}{y}}\right)]G\left(y\right)\mathrm{d}y\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_0^{S^{\ast }}{\displaystyle \frac{1}{y}}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}{\alpha }_2^{\overline{n}}{\left({\displaystyle \frac{x}{y}}\right)}^{-{\alpha }_2}G\left(y\right)\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_2-n}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}{\alpha }_2^{\overline{n}}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}[-{\left.\rho {\alpha }_{2}g\right|}_{supp\left(g\right)}\left(y\right)\hfill \\ \hfill & -\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)]\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_2-n}{(-1)}^n{\alpha }_2^{\overline{n}}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}\left[{\left.{\alpha }_{2}g\right|}_{supp\left(g\right)}\left(y\right)+y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\right]\mathrm{d}y.\hfill \end{array}$$

If $x<S^{\ast }$, then the nth derivative of the option function in class (a) is

$$v^{\left(n\right)}\left(x\right)={\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\left(n\right)}\left(x\right).$$

We can now specify the payoff of an option within class (a). For a put, it is known that $g\left(x\right)={(K-x)}^{+},supp\left(g\right)=[0,K)$, and $R=[0,S^{\ast })$, where $0<S^{\ast }<K$, with $S^{\ast }={\displaystyle \frac{{\alpha }_{2}K}{{\alpha }_2+1}}$. Then, ${\left.g\right|}_{supp\left(g\right)}\left(S\right)=$ $K-S$. We see that

$$\begin{array}{cc}\hfill v^{\left(n\right)}\left(x\right)& =x^{-{\alpha }_2-n}{(-1)}^n{\alpha }_2^{\overline{n}}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}\left[{\alpha }_2(K-y)+y(-1)\right]\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_2-n}{(-1)}^n{\alpha }_2^{\overline{n}}{\int }_0^{S^{\ast }}{\alpha }_{2}K{y}^{{\alpha }_2-1}-{\alpha }_2{y}^{{\alpha }_2}-y^{{\alpha }_2}\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_2-n}{(-1)}^n{\alpha }_2^{\overline{n}}{S^{\ast }}^{{\alpha }_2}(K-S^{\ast }).\hfill \end{array}$$

(19)

If $x<S^{\ast }$, then the nth derivative of the option function for a put is 

$$v^{\left(n\right)}\left(x\right)={(K-x)}^{\left(n\right)}.$$

Altogether, we have

$$v^{\left(n\right)}\left(x\right)=\left\{\begin{array}{cc}{(K-x)}^{\left(n\right)},\hfill & \mathit{if}0<x<S^{\ast },\hfill \\ {(-1)}^n{\alpha }_2^{\overline{n}}{x}^{-{\alpha }_2-n}{S}^{\ast {\alpha }_2}(K-S^{\ast }),\hfill & \mathit{if}x>S^{\ast }.\hfill \end{array}\right.$$

(20)

Remark 5.

If we were to have only specified that we are in possession of a class (a) option, we assume the existence of ${\left(g{|}_{supp\left(g\right)}\right)}^{\left(n\right)}\left(x\right)$ where $0<x<S^{\ast }$. However, once the option payoff is specified, this assumption may not hold. For instance, the bear spread is not differentiable at $x=K_1$. A similar remark holds for all derivatives throughout this article.

Example 3.

Let us attempt to simplify Lemma 4(i) once again.

Class (b) payoffs

Assume that we have a class (b) option. From [25], we saw that

$$-{\displaystyle \frac{1}{\rho }}{\int }_S^{\infty }{y}^{{\alpha }_1-1}G\left(y\right)\mathrm{d}y={\left.S^{{\alpha }_1}g\right|}_{supp\left(g\right)}\left(S\right).$$

(21)

This implies that

$$\begin{array}{cc}\hfill {\int }_S^{\infty }{y}^{{\alpha }_1-1}G\left(y\right)\mathrm{d}y& =-\rho S^{{\alpha }_1}{g|}_{supp\left(g\right)}\left(S\right).\hfill \end{array}$$

(22)

Differentiating with respect to S,

$$-S^{{\alpha }_1-1}G\left(S\right)=-\rho {\alpha }_1{S}^{{\alpha }_1-1}{g|}_{supp\left(g\right)}\left(S\right)-\rho S^{{\alpha }_1}{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(S\right),$$

or

$$G\left(S\right)=\rho {\alpha }_1{g|}_{supp\left(g\right)}\left(S\right)+\rho S{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(S\right).$$

Replacing S by y,

$$G\left(y\right)={\left.\rho {\alpha }_{1}g\right|}_{supp\left(g\right)}\left(y\right)+\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right).$$

(23)

We use similar logic to that illustrated in the previous Example 2. Since the option is in class (b), we know that $x\in R^c=[0,S^{\ast })$. If $x\le S^{\ast }$, and $y\ge S^{\ast }$, then $y\ge S^{\ast }\ge x$. So, we obtain ${\displaystyle \frac{x}{y}}\le 1$. Hence, by (15) and (23), we see that

$$\begin{array}{cc}\hfill v^{\left(n\right)}\left(x\right)& ={\displaystyle \frac{1}{x^n}}\left(\left({id}^n{\mathcal{K}}^{\left(n\right)}\right)\ast \left({\chi }_{R}G\right)\right)\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_0^{\infty }{\displaystyle \frac{1}{y}}{\left({\displaystyle \frac{x}{y}}\right)}^n{\mathcal{K}}^{\left(n\right)}\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_{S^{\ast }}^{\infty }{\displaystyle \frac{1}{y}}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}[{\alpha }_1^{\overline{n}}{\left({\displaystyle \frac{x}{y}}\right)}^{-{\alpha }_1}{\chi }_{(0,1)}\left({\displaystyle \frac{x}{y}}\right)\hfill \\ \hfill & +{\alpha }_2^{\overline{n}}{\left({\displaystyle \frac{x}{y}}\right)}^{-{\alpha }_2}{\chi }_{(1,\infty )}\left({\displaystyle \frac{x}{y}}\right)]G\left(y\right)\mathrm{d}y\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_{S^{\ast }}^{\infty }{\displaystyle \frac{1}{y}}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}{\alpha }_1^{\overline{n}}{\left({\displaystyle \frac{x}{y}}\right)}^{-{\alpha }_1}G\left(y\right)\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_1-n}{\displaystyle \frac{1}{\rho }}{(-1)}^{n+1}{\alpha }_1^{\overline{n}}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}[{\left.\rho {\alpha }_{1}g\right|}_{supp\left(g\right)}\left(y\right)\hfill \\ \hfill & +\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)]\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_1-n}{(-1)}^{n+1}{\alpha }_1^{\overline{n}}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}\left[{\left.{\alpha }_{1}g\right|}_{supp\left(g\right)}\left(y\right)+y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\right]\mathrm{d}y.\hfill \end{array}$$

If $x>S^{\ast }$, then the nth derivative of the option function for a class (b) option is

$$v^{\left(n\right)}\left(x\right)={\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\left(n\right)}\left(x\right).$$

Assume now that the option is a call. We know $supp\left(g\right)=(K,\infty ),$ and the payoff is ${g|}_{supp\left(g\right)}\left(x\right)={(x-K)}^{+}$, where $K>0$, and $S^{\ast }={\displaystyle \frac{{\alpha }_{1}K}{{\alpha }_1+1}}$.

$$\begin{array}{cc}\hfill v^{\left(n\right)}\left(x\right)& =x^{-{\alpha }_1-n}{(-1)}^{n+1}{\alpha }_1^{\overline{n}}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}\left[{\alpha }_1(y-K)+y\left(1\right)\right]\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_1-n}{(-1)}^{n+1}{\alpha }_1^{\overline{n}}{\int }_{S^{\ast }}^{\infty }{\alpha }_1{y}^{{\alpha }_1}-{\alpha }_{1}K{y}^{{\alpha }_1-1}+y^{{\alpha }_1}\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_1-n}{(-1)}^n{\alpha }_1^{\overline{n}}{S^{\ast }}^{{\alpha }_1}(S^{\ast }-K).\hfill \end{array}$$

If $x>S^{\ast }$, then the nth derivative of the option function for a call is

$$v^{\left(n\right)}\left(x\right)={(x-K)}^{\left(n\right)}.$$

Altogether, we have

$$v^{\left(n\right)}\left(x\right)=\left\{\begin{array}{cc}{(-1)}^n{\alpha }_1^{\overline{n}}{x}^{-{\alpha }_1-n}{S}^{\ast {\alpha }_1}(S^{\ast }-K),\hfill & \mathit{if}0<x<S^{\ast },\hfill \\ {(x-K)}^{\left(n\right)},\hfill & \mathit{if}x>S^{\ast }.\hfill \end{array}\right.$$

(24)

Remark 6.

We make the point that we have not included $x=S^{\ast }$ in either one of the cases because it may not actually exist. Take, for instance, the asset-or-nothing call. The derivative does not exist at $x=S^{\ast }$.

Example 4.

Let us attempt to simplify Lemma 4(ii).

Class (a) payoffs

Differentiating (18), we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial G}{\partial c_k}}\left(y\right)& =-\left({\displaystyle \frac{\partial \rho }{\partial c_k}}{\alpha }_2+\rho {\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}\right){g|}_{supp\left(g\right)}\left(y\right)-{\displaystyle \frac{\partial \rho }{\partial c_k}}y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\left[-{\left.\rho {\alpha }_{2}g\right|}_{supp\left(g\right)}\left(y\right)-\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\right]-\rho {\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)-\rho {\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right).\hfill \end{array}$$

(25)

We can simplify Lemma 4(ii) using (25). Assume that $n=1$ so that Lemma 4(ii) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =\sum _{j=0}^1\left({\displaystyle \genfrac{}{}{0pt}{}{1}{j}}\right)\left({\displaystyle \frac{{\partial }^{1-j}\mathcal{K}}{\partial c_k^{1-j}}}\ast {\displaystyle \frac{{\partial }^j\left({\chi }_{R}G\right)}{\partial c_k^j}}\right)\left(x\right)\hfill \\ \hfill & =\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+\left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right).\hfill \end{array}$$

Since the option has been identified as being in class (a), then $R=\left[0,S^{\ast }\right)$, and so each convolution has the form

$$\begin{array}{c}\hfill \left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\left(y\right)\mathrm{d}y={\int }_0^{S^{\ast }}{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial G}{\partial c_k}}\left(y\right)\mathrm{d}y,\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{1}{y}}{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y={\int }_0^{S^{\ast }}{\displaystyle \frac{1}{y}}{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left({\displaystyle \frac{x}{y}}\right)G\left(y\right)\mathrm{d}y.\end{array}$$

(27)

If $0<y<S^{\ast }$, and since $S^{\ast }\le x$, then $y<x$ and ${\displaystyle \frac{x}{y}}>1$. Then, from Lemmas 2 and 3,

$$\mathcal{K}\left({\displaystyle \frac{x}{y}}\right)=-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{y}^{{\alpha }_2},{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left({\displaystyle \frac{x}{y}}\right)={\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{y}^{{\alpha }_2}\left[{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left({\displaystyle \frac{x}{y}}\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\right].$$

By substitution, (26) and (27) become

$$\begin{array}{cc}\hfill \left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)& ={\int }_0^{S^{\ast }}{\displaystyle \frac{1}{y}}(-1){\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{y}^{{\alpha }_2}\left[{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)-\rho {\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)\right]\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}\left[-{\displaystyle \frac{1}{{\rho }^2}}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)\right]\mathrm{d}y,\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)& ={\int }_0^{S^{\ast }}{\displaystyle \frac{1}{y}}{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_2}{y}^{{\alpha }_2}\left[{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left({\displaystyle \frac{x}{y}}\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\right]G\left(y\right)\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}[{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(x\right)G\left(y\right)-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(y\right)G\left(y\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{{\rho }^2}}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)]\mathrm{d}y.\hfill \end{array}$$

(29)

Then, by (17), (18), (28), and (29),

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =x^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}[{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(x\right)G\left(y\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(y\right)G\left(y\right)]\mathrm{d}y\hfill \\ \hfill & ={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right){\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}G\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}\left[{\left.g\right|}_{supp\left(g\right)}\left(y\right)-{\displaystyle \frac{1}{\rho }}log\left(y\right)G\left(y\right)\right]\mathrm{d}y\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}\left[{\left.g\right|}_{supp\left(g\right)}\left(y\right)-{\displaystyle \frac{1}{\rho }}log\left(y\right)G\left(y\right)\right]\mathrm{d}y\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}\{{\left.g\right|}_{supp\left(g\right)}\left(y\right)-{\displaystyle \frac{1}{\rho }}log\left(y\right)[-{\left.\rho {\alpha }_{2}g\right|}_{supp\left(g\right)}\left(y\right)\hfill \\ \hfill & -\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\left]\right\}\mathrm{d}y\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\alpha }_2{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}log\left(y\right)y^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2}log\left(y\right){\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\mathrm{d}y\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\alpha }_2{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}log\left(y\right)y^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\left[y^{{\alpha }_2}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]}_{y=0}^{y=S^{\ast }}\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}\left[{\alpha }_2{y}^{{\alpha }_2-1}log\left(y\right)+y^{{\alpha }_2-1}\right]{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\alpha }_2{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}log\left(y\right)y^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\left[y^{{\alpha }_2}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]}_{y=0}^{y=S^{\ast }}\hfill \\ \hfill & -{\alpha }_2{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\int }_0^{S^{\ast }}{y}^{{\alpha }_2-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{\left[y^{{\alpha }_2}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]}_{y=0}^{y=S^{\ast }}.\hfill \end{array}$$

(30)

We must now consider how we evaluate the expression when $y=0$. By L’Hôpital’s rule,

$$\begin{array}{cc}\hfill \underset{y\to 0^{+}}{lim}\left[y^{{\alpha }_2}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]& =\underset{y\to 0^{+}}{lim}{\displaystyle \frac{log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)}{y^{-{\alpha }_2}}}\hfill \\ \hfill & =\underset{y\to 0^{+}}{lim}{\displaystyle \frac{\frac{1}{y}{\left.g\right|}_{supp\left(g\right)}\left(y\right)+log\left(y\right){\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)}{-{\alpha }_2{y}^{-{\alpha }_2-1}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{-{\alpha }_2}}\underset{y\to 0^{+}}{lim}[{\left.g\right|}_{supp\left(g\right)}\left(y\right)y^{{\alpha }_2}\hfill \\ \hfill & +y^{{\alpha }_2+1}log\left(y\right){\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)]\hfill \\ \hfill & =0,\hfill \end{array}$$

since ${\alpha }_2>0$. So, (30) simplifies to

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}log\left(x\right)+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}{S^{\ast }}^{{\alpha }_2}log\left(S^{\ast }\right){\left.g\right|}_{supp\left(g\right)}\left(S^{\ast }\right)\hfill \\ \hfill & =-{\left.S^{\ast {\alpha }_2}g\right|}_{supp\left(g\right)}\left(S^{\ast }\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right].\hfill \end{array}$$

(31)

Assume now that the option is a put. We know $supp\left(g\right)=[0,K),$ and ${g|}_{supp\left(g\right)}\left(x\right)=K-x$, where $K>0$, and with $S^{\ast }={\displaystyle \frac{{\alpha }_{2}K}{{\alpha }_2+1}}$. Then, (31) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =-S^{\ast {\alpha }_2}(K-S^{\ast }){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right].\hfill \end{array}$$

(32)

Assume that we have a bear spread. We know $supp\left(g\right)=[0,K_2),$ and ${g|}_{supp\left(g\right)}\left(x\right)={\left(K_2-x\right)}^{+}-{\left(K_1-x\right)}^{+}$, where $K>0$. If $S^{\ast }=K_1$, or $S^{\ast }={\displaystyle \frac{{\alpha }_2{K}_2}{{\alpha }_2+1}}$, then (31) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =-S^{\ast {\alpha }_2}\left(K_1-K_2\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right].\hfill \end{array}$$

(33)

Example 5.

Let us attempt to simplify Lemma 4(ii) once again.

Class (b) payoffs

Assume that we have a class (b) option. Differentiating (23),

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial G}{\partial c_k}}\left(y\right)& =\left({\displaystyle \frac{\partial \rho }{\partial c_k}}{\alpha }_1+\rho {\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}\right){g|}_{supp\left(g\right)}\left(y\right)+{\displaystyle \frac{\partial \rho }{\partial c_k}}y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\left[{\left.\rho {\alpha }_{1}g\right|}_{supp\left(g\right)}\left(y\right)+\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\right]+\rho {\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)+\rho {\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right).\hfill \end{array}$$

(34)

We can simplify Lemma 4(ii) using (34). Assume that $n=1$ so that Lemma 4(ii) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =\sum _{j=0}^1\left({\displaystyle \genfrac{}{}{0pt}{}{1}{j}}\right)\left({\displaystyle \frac{{\partial }^{1-j}\mathcal{K}}{\partial c_k^{1-j}}}\ast {\displaystyle \frac{{\partial }^j\left({\chi }_{R}G\right)}{\partial c_k^j}}\right)\left(x\right)\hfill \\ \hfill & =\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+\left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right).\hfill \end{array}$$

(35)

Since the option has been identified as being in class (b), then $R=\left[S^{\ast },\infty \right)$, and so each convolution has the form

$$\begin{array}{c}\hfill \left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\left(y\right)\mathrm{d}y={\int }_{S^{\ast }}^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial G}{\partial c_k}}\left(y\right)\mathrm{d}y,\end{array}$$

(36)

and

$$\begin{array}{c}\hfill \left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)={\int }_0^{\infty }{\displaystyle \frac{1}{y}}{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y={\int }_{S^{\ast }}^{\infty }{\displaystyle \frac{1}{y}}{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left({\displaystyle \frac{x}{y}}\right)G\left(y\right)\mathrm{d}y.\end{array}$$

(37)

If $y>S^{\ast }$, and since $x\le S^{\ast }$, then $y>x$ and ${\displaystyle \frac{x}{y}}<1$. Then, from Lemma 2 and Lemma 3,

$$\mathcal{K}\left({\displaystyle \frac{x}{y}}\right)=-{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{y}^{{\alpha }_1},{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left({\displaystyle \frac{x}{y}}\right)={\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{y}^{{\alpha }_1}\left[{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left({\displaystyle \frac{x}{y}}\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\right].$$

So, we obtain

$$\begin{array}{cc}\hfill \left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)& ={\int }_{S^{\ast }}^{\infty }{\displaystyle \frac{1}{y}}(-1){\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{y}^{{\alpha }_1}\left[{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)+\rho {\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)\right]\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}\left[-{\displaystyle \frac{1}{{\rho }^2}}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)\right]\mathrm{d}y,\hfill \end{array}$$

(38)

and

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)& ={\int }_{S^{\ast }}^{\infty }{\displaystyle \frac{1}{y}}{\displaystyle \frac{1}{\rho }}{x}^{-{\alpha }_1}{y}^{{\alpha }_1}\left[{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left({\displaystyle \frac{x}{y}}\right)+{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial \rho }{\partial c_k}}\right]G\left(y\right)\mathrm{d}y\hfill \\ \hfill & =x^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}[{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)G\left(y\right)-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(y\right)G\left(y\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{{\rho }^2}}{\displaystyle \frac{\partial \rho }{\partial c_k}}G\left(y\right)]\mathrm{d}y.\hfill \end{array}$$

(39)

Combining (21), (23), (38), and (39), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =x^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}[{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)G\left(y\right)-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(y\right)G\left(y\right)\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{g|}_{supp\left(g\right)}\left(y\right)]\mathrm{d}y\hfill \\ \hfill & ={\displaystyle \frac{1}{\rho }}{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}G\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}\left[{g|}_{supp\left(g\right)}\left(y\right)+{\displaystyle \frac{1}{\rho }}log\left(y\right)G\left(y\right)\right]\mathrm{d}y\hfill \\ \hfill & =-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}{\left\{g\right|}_{supp\left(g\right)}\left(y\right)+{\displaystyle \frac{1}{\rho }}log\left(y\right)[{\left.\rho {\alpha }_{1}g\right|}_{supp\left(g\right)}\left(y\right)\hfill \\ \hfill & +\rho y{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\left]\right\}\mathrm{d}y\hfill \\ \hfill & =-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}{g|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\alpha }_1{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }log\left(y\right)y^{{\alpha }_1}{\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\mathrm{d}y\hfill \\ \hfill & =-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}{g|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\alpha }_1{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\left[y^{{\alpha }_1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]}_{y=S^{\ast }}^{y=\infty }\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }\left[y^{{\alpha }_1-1}+{\alpha }_1{y}^{{\alpha }_1-1}log\left(y\right)\right]{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & =-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}{g|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\alpha }_1{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\left[y^{{\alpha }_1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]}_{y=S^{\ast }}^{y=\infty }\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{y}^{{\alpha }_1-1}{\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\int }_{S^{\ast }}^{\infty }{\alpha }_1{y}^{{\alpha }_1-1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\mathrm{d}y\hfill \\ \hfill & =-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)\hfill \\ \hfill & -{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\left[y^{{\alpha }_1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]}_{y=S^{\ast }}^{y=\infty }.\hfill \end{array}$$

(40)

We must now consider how we evaluate the expression when $y=\infty$. We see

$$\begin{array}{cc}\hfill \underset{y\to \infty }{lim}\left[y^{{\alpha }_1}log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)\right]& =\underset{y\to \infty }{lim}{\displaystyle \frac{log\left(y\right){\left.g\right|}_{supp\left(g\right)}\left(y\right)}{y^{-{\alpha }_1}}}\hfill \\ \hfill & =\underset{y\to \infty }{lim}{\displaystyle \frac{\frac{1}{y}{\left.g\right|}_{supp\left(g\right)}\left(y\right)+log\left(y\right){\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)}{-{\alpha }_1{y}^{-{\alpha }_1-1}}}\hfill \\ \hfill & ={\displaystyle \frac{1}{-{\alpha }_1}}\underset{y\to \infty }{lim}[{\left.g\right|}_{supp\left(g\right)}\left(y\right)y^{{\alpha }_1}\hfill \\ \hfill & +y^{{\alpha }_1+1}log\left(y\right){\left({\left.g\right|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)]\hfill \\ \hfill & =0,\hfill \end{array}$$

since ${\alpha }_1\le -1.$ Then (40) simplifies to

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =-{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}log\left(x\right)x^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)+{\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}log\left(S^{\ast }\right){S^{\ast }}^{{\alpha }_1}{g|}_{supp\left(g\right)}\left(S^{\ast }\right)\hfill \\ \hfill & ={\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{\left.{S^{\ast }}^{{\alpha }_1}g\right|}_{\mathrm{supp}\left(\mathrm{g}\right)}\left(S^{\ast }\right)\left[log\left(S^{\ast }\right)-log\left(x\right)\right].\hfill \end{array}$$

(41)

Assume now that the option is a call. We know $supp\left(g\right)=(K,\infty ),$ and ${g|}_{supp\left(g\right)}\left(x\right)=x-K$, where $K>0$, and$S^{\ast }={\displaystyle \frac{{\alpha }_{1}K}{{\alpha }_1+1}}$. Then, (41) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& ={\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{S^{\ast }}^{{\alpha }_1}(S^{\ast }-K)\left[log\left(S^{\ast }\right)-log\left(x\right)\right].\hfill \end{array}$$

(42)

Assume now that the option is an asset-or-nothing call. We know that $supp\left(g\right)=[K,\infty ),$ and ${g|}_{supp\left(g\right)}\left(x\right)=x{\mathbf{1}}_{[K,\infty )}\left(x\right)$, where $K>0$, and $S^{\ast }=K$. Then, (41) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& ={\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{S^{\ast }}^{{\alpha }_1}{S}^{\ast }\left[log\left(S^{\ast }\right)-log\left(x\right)\right]\hfill \\ \hfill & ={\displaystyle \frac{\partial {\alpha }_1}{\partial c_k}}{x}^{-{\alpha }_1}{S^{\ast }}^{{\alpha }_1+1}\left[log\left(S^{\ast }\right)-log\left(x\right)\right].\hfill \end{array}$$

(43)

Example 6.

Alternative formulation of Lemma 4(ii).

We saw in [25] that

$$\widehat{\mathcal{K}}\left(\xi \right)={\left[p\left(\xi \right)\right]}^{-1},p\left(\xi \right)=\sum _{k=0}^2{(-1)}^k{c}_k{\xi }^{\overline{k}},{\displaystyle \frac{\partial p}{\partial c_k}}\left(\xi \right)={(-1)}^k{\xi }^{\overline{k}}.$$

(44)

Hence,

$${\displaystyle \frac{\widehat{{\partial }^n\mathcal{K}}}{\partial c_k^n}}\left(\xi \right)={\displaystyle \frac{{\partial }^n\widehat{\mathcal{K}}}{\partial c_k^n}}\left(\xi \right)={\displaystyle \frac{{\partial }^n}{\partial c_k^n}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}.$$

Then,

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial c_k}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}=-{\left[p\left(\xi \right)\right]}^{-2}{(-1)}^k{\left(\xi \right)}^{\overline{k}}\\ {\displaystyle \frac{{\partial }^2}{\partial c_k^2}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}=(-1)(-2){\left[p\left(\xi \right)\right]}^{-3}{\left[{(-1)}^k{\left(\xi \right)}^{\overline{k}}\right]}^2\\ {\displaystyle \frac{{\partial }^3}{\partial c_k^3}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}=(-1)(-2)(-3){\left[p\left(\xi \right)\right]}^{-4}{\left[{(-1)}^k{\left(\xi \right)}^{\overline{k}}\right]}^3.\end{array}$$

We can generalise to

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^n}{\partial c_k^n}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}& =(-1)(-2)(-3)...(-n){\left[p\left(\xi \right)\right]}^{-1}{\left[p\left(\xi \right)\right]}^{-n}{\left[{(-1)}^k{\left(\xi \right)}^{\overline{k}}\right]}^n\hfill \\ \hfill & ={(-1)}^{n}n!{\left[p\left(\xi \right)\right]}^{-1}{\left[p\left(\xi \right)\right]}^{-n}{\left[{(-1)}^k{\left(\xi \right)}^{\overline{k}}\right]}^n\hfill \\ \hfill & ={(-1)}^{n}n!\widehat{\mathcal{K}}\left(\xi \right){\left[{(-1)}^k{\left(\xi \right)}^{\overline{k}}\right]}^n\widehat{\mathcal{K}}{\left(\xi \right)}^n\hfill \\ \hfill & ={(-1)}^{n}n!\widehat{\mathcal{K}}\left(\xi \right){\left[{(-1)}^k{\left(\xi \right)}^{\overline{k}}\widehat{\mathcal{K}}\left(\xi \right)\right]}^n.\hfill \end{array}$$

(45)

We have

$${\displaystyle \frac{{\partial }^n\mathcal{K}}{\partial c_k^n}}\left(x\right)={(-1)}^{n}n!\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \cdots \ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left(x\right).$$

(46)

Also, the generalised Leibniz rule states that

$${\left(\rho {\alpha }_2\right)}^{\left(n\right)}\left(x\right)=\sum _{l=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right){\displaystyle \frac{{\partial }^{n-l}\rho }{\partial c_k^{n-l}}}\left(x\right){\displaystyle \frac{{\partial }^l{\alpha }_2}{\partial c_k^l}}\left(x\right),$$

and so by (18),

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{n}G}{\partial c_k^n}}\left(y\right)& {=-g|}_{supp\left(g\right)}\left(y\right)\sum _{l=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{l}}\right){\displaystyle \frac{{\partial }^{n-l}\rho }{\partial c_k^{n-l}}}\left(y\right){\displaystyle \frac{{\partial }^l{\alpha }_2}{\partial c_k^l}}\left(y\right)-{\displaystyle \frac{{\partial }^n\rho }{\partial c_k^n}}y{\left({g|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right).\hfill \end{array}$$

If $n=1$, then by (46), we obtain

$${\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\left(x\right)=-\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left(x\right).$$

(47)

and

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial G}{\partial c_k}}\left(y\right)& {=-g|}_{supp\left(g\right)}\left(y\right)\sum _{l=0}^1\left({\displaystyle \genfrac{}{}{0pt}{}{1}{l}}\right){\displaystyle \frac{{\partial }^{1-l}\rho }{\partial c_k^{1-l}}}{\displaystyle \frac{{\partial }^l{\alpha }_2}{\partial c_k^l}}-{\displaystyle \frac{\partial \rho }{\partial c_k}}y{\left({g|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)\hfill \\ \hfill & {=-g|}_{supp\left(g\right)}\left(y\right)\left({\displaystyle \frac{\partial \rho }{\partial c_k}}{\alpha }_2+\rho {\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}\right)-{\displaystyle \frac{\partial \rho }{\partial c_k}}y{\left({g|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right).\hfill \end{array}$$

(48)

Keeping $n=1$, by (47) and (48), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial c_k}}\left(x\right)& =\sum _{j=0}^1\left({\displaystyle \genfrac{}{}{0pt}{}{1}{j}}\right)\left({\displaystyle \frac{{\partial }^{1-j}\mathcal{K}}{\partial c_k^{1-j}}}\ast {\displaystyle \frac{{\partial }^j\left({\chi }_{R}G\right)}{\partial c_k^j}}\right)\left(x\right)\hfill \\ \hfill & =\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+\left(\mathcal{K}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)\hfill \\ \hfill & ={\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & ={\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){[-g|}_{supp\left(g\right)}\left(y\right)\left({\displaystyle \frac{\partial \rho }{\partial c_k}}{\alpha }_2+\rho {\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}\right)\hfill \\ \hfill & -{\displaystyle \frac{\partial \rho }{\partial c_k}}y{\left({g|}_{supp\left(g\right)}\right)}^{\prime }\left(y\right)]\mathrm{d}y.\hfill \end{array}$$

(49)

Example 7.

Simplifying Lemma 4(iii).

By (9) and (46), we see

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{n+1}\widehat{\mathcal{K}}}{\partial c_k\partial x^n}}& ={\displaystyle \frac{{\partial }^n}{\partial x^n}}\left({\displaystyle \frac{\partial \widehat{\mathcal{K}}}{\partial c_k}}\right)\hfill \\ & ={(-1)}^n{(\xi -1)}^n{\displaystyle \frac{\partial \widehat{\mathcal{K}}}{\partial c_k}}(\xi -n)\hfill \\ & ={(-1)}^n{(\xi -1)}^{\underline{n}}{(-1)}^{n}n!\widehat{\mathcal{K}}(\xi -n){\left[\widehat{\left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)}(\xi -n)\right]}^n\hfill \\ & =n!{(-1)}^n{\widehat{\mathcal{K}}}^{\left(n\right)}\left(\xi \right){\left[\widehat{\left({\mathrm{id}}^{k-n}{\mathcal{K}}^{\left(k\right)}\right)}\left(\xi \right)\right]}^n.\hfill \end{array}$$

So,

$${\displaystyle \frac{{\partial }^{n+1}\mathcal{K}}{\partial c_k\partial x^n}}=n!{(-1)}^n\left({\mathcal{K}}^{\left(n\right)}\ast \left({\mathrm{id}}^{k-n}{\mathcal{K}}^{\left(k\right)}\right)\ast \cdots \ast \left({\mathrm{id}}^{k-n}{\mathcal{K}}^{\left(k\right)}\right)\right)\left(x\right).$$

(50)

Hence, by (46) and (50), we can generalise

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{n+1}v}{\partial c_k\partial x^n}}\left(x\right)& ={\displaystyle \frac{1}{x^n}}\left({id}^n{\displaystyle \frac{{\partial }^{n+1}\mathcal{K}}{\partial c_k\partial x^n}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+{\displaystyle \frac{1}{x^n}}\left({id}^n{\displaystyle \frac{{\partial }^n\mathcal{K}}{\partial c_k^n}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{x^n}}{\int }_0^{\infty }{\left({\displaystyle \frac{x}{y}}\right)}^n{\displaystyle \frac{{\partial }^{n+1}\mathcal{K}}{\partial c_k\partial x^n}}\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\displaystyle \frac{1}{x^n}}{\int }_0^{\infty }{\left({\displaystyle \frac{x}{y}}\right)}^n{\displaystyle \frac{{\partial }^n\mathcal{K}}{\partial c_k^n}}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & ={(-1)}^{n}n!{\int }_0^{\infty }{y}^{-n}({\mathcal{K}}^{\left(n\right)}\ast \left({\mathrm{id}}^{k-n}{\mathcal{K}}^{\left(k\right)}\right)\ast \cdots \hfill \\ \hfill & \cdots \ast \left({\mathrm{id}}^{k-n}{\mathcal{K}}^{\left(k\right)}\right))\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{(-1)}^{n}n!{\int }_0^{\infty }{y}^{-n}(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \cdots \hfill \\ \hfill & \cdots \ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right))\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\mathrm{d}y.\hfill \end{array}$$

(51)

When $n=1$, and by (46) and (50), Lemma (4)(iii) gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}v}{\partial c_k\partial x}}\left(x\right)& ={\displaystyle \frac{1}{x}}\left(id{\displaystyle \frac{{\partial }^2\mathcal{K}}{\partial c_k\partial x}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+{\displaystyle \frac{1}{x}}\left(id{\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)\hfill \\ \hfill & =-{\int }_0^{\infty }{y}^{-1}\left({\mathcal{K}}^{\prime }\ast \left({\mathrm{id}}^{k-1}{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\int }_0^{\infty }{y}^{-1}\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\mathrm{d}y.\hfill \end{array}$$

(52)

When $n=2$, and by (46) and (50), Lemma 4(iii) gives

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{3}v}{\partial c_k\partial x^2}}\left(x\right)& ={\displaystyle \frac{1}{x^2}}\left({id}^2{\displaystyle \frac{{\partial }^3\mathcal{K}}{\partial c_k\partial x^2}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+{\displaystyle \frac{1}{x^2}}\left({id}^2{\displaystyle \frac{{\partial }^2\mathcal{K}}{\partial c_k^2}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)\hfill \\ \hfill & =2{\int }_0^{\infty }{y}^{-2}\left({\mathcal{K}}^{\prime \prime }\ast \left({\mathrm{id}}^{k-2}{\mathcal{K}}^{\left(k\right)}\right)\ast \left({\mathrm{id}}^{k-2}{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +2{\int }_0^{\infty }{y}^{-2}\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\mathrm{d}y.\hfill \end{array}$$

(53)

Simplifying Lemma 4(iv)

By (44), we derive that

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^n\widehat{\mathcal{K}}}{\partial c_k^n}}={\displaystyle \frac{{\partial }^n}{\partial c_k^n}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}& ={(-1)}^{n}n!{\left[p\left(\xi \right)\right]}^{-(n+1)}{\left[{(-1)}^k{\xi }^{\overline{k}}\right]}^n.\hfill \end{array}$$

So,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{n+1}}{\partial c_k^n\partial c_{k^{\prime }}}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}& ={\displaystyle \frac{\partial }{\partial c_{k^{\prime }}}}\left[{(-1)}^{n}n!{\left[{(-1)}^k{\xi }^{\overline{k}}\right]}^n{\left[p\left(\xi \right)\right]}^{-n-1}\right]\hfill \\ \hfill & ={(-1)}^{n}n!{\left[{(-1)}^k{\xi }^{\overline{k}}\right]}^n(-n-1){\left[p\left(\xi \right)\right]}^{-n-2}\left[{(-1)}^{k^{\prime }}{\xi }^{\overline{k^{\prime }}}\right].\hfill \end{array}$$

The nth derivative is

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2n}}{\partial c_k^n\partial c_{k^{\prime }}^n}}\left\{{\left[p\left(\xi \right)\right]}^{-1}\right\}& ={(-1)}^{n}n!{\left[{(-1)}^k{\xi }^{\overline{k}}\right]}^n{(-1)}^n(n+1)!{\left[p\left(\xi \right)\right]}^{-(2n+1)}{\left[{(-1)}^{k^{\prime }}{\xi }^{\overline{k^{\prime }}}\right]}^n\hfill \\ \hfill & =(n+1)n{!}^2{\left[{(-1)}^k{\xi }^{\overline{k}}{\left[p\left(\xi \right)\right]}^{-1}{(-1)}^{k^{\prime }}{\xi }^{\overline{k^{\prime }}}{\left[p\left(\xi \right)\right]}^{-1}\right]}^n{\left[p\left(\xi \right)\right]}^{-1}\hfill \\ \hfill & =(n+1)n{!}^2\widehat{\mathcal{K}}\left(\xi \right){\left[\left(\widehat{{\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}}\right)\left(\xi \right)\left(\widehat{{\mathrm{id}}^{k^{\prime }}{\mathcal{K}}^{\left(k^{\prime }\right)}}\right)\left(\xi \right)\right]}^n.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2n}\mathcal{K}}{{\partial }^n{c}_k{\partial }^n{c}_{k^{\prime }}}}\left(x\right)& =(n+1)n{!}^2\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \left({\mathrm{id}}^{k^{\prime }}{\mathcal{K}}^{\left(k^{\prime }\right)}\right)\ast \cdots \right.\hfill \\ \hfill & \left.\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \left({\mathrm{id}}^{k^{\prime }}{\mathcal{K}}^{\left(k^{\prime }\right)}\right)\right)\left(x\right).\hfill \end{array}$$

(54)

When $n=1$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^2\mathcal{K}}{\partial c_k\partial c_{k^{\prime }}}}\left(x\right)=2\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \left({\mathrm{id}}^{k^{\prime }}{\mathcal{K}}^{\left(k^{\prime }\right)}\right)\right)\left(x\right).\end{array}$$

(55)

If $n=1$, then by (46) and (54), Lemma 4(iv) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}v}{\partial c_k\partial c_{k^{\prime }}}}\left(x\right)& =\left({\displaystyle \frac{{\partial }^2\mathcal{K}}{\partial c_k\partial c_{k^{\prime }}}}\ast \left({\chi }_{R}G\right)\right)\left(x\right)+\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_{k^{\prime }}}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\right)\left(x\right)\hfill \\ \hfill & +\left({\displaystyle \frac{\partial \mathcal{K}}{\partial c_k}}\ast {\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_{k^{\prime }}}}\right)\left(x\right)+\left(\mathcal{K}\ast {\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_k\partial c_{k^{\prime }}}}\right)\left(x\right)\hfill \\ \hfill & =2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\ast \left({\mathrm{id}}^{k^{\prime }}{\mathcal{K}}^{\left(k^{\prime }\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^{k^{\prime }}{\mathcal{K}}^{\left(k^{\prime }\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_k}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^k{\mathcal{K}}^{\left(k\right)}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_{k^{\prime }}}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_k\partial c_{k^{\prime }}}}\left(y\right)\mathrm{d}y.\hfill \end{array}$$

(56)

6. Main Result: Greeks for Perpetual American Options with General Payoffs

6.1. First-Order Hedging Parameters

6.1.1. Delta

We now have the capability to derive our first Greek, delta, denoted by $\Delta$. The delta is defined to be the rate of change of an option’s value with respect to the change in the price of the underlying asset [1] (p. 445). Delta is an important measure of a portfolio because it indicates how long or short one is [32]. However, the delta of an option continually changes with the movement of the underlying price and so must be continually monitored. Delta is therefore one of the more popular Greeks. It can be inferred that

$$\Delta \left(x\right)={\displaystyle \frac{\partial v}{\partial x}}\left(x\right).$$

By (20), the simplified delta of a perpetual American put option is

$$\Delta \left(x\right)=\left\{\begin{array}{cc}-1,\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ -{\alpha }_2{x}^{-{\alpha }_2-1}{S}^{\ast {\alpha }_2}(K-S^{\ast }),\hfill & \mathrm{if}x>S^{\ast }.\hfill \end{array}\right.$$

(57)

By (24), the simplified delta of a perpetual American call option is

$$\Delta \left(x\right)=\left\{\begin{array}{cc}-{\alpha }_1{x}^{-{\alpha }_1-1}{S}^{\ast {\alpha }_1}(S^{\ast }-K),\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ 1,\hfill & \mathrm{if}x>S^{\ast }.\hfill \end{array}\right.$$

(58)

6.1.2. Omega

Omega, symbolised as $\Omega$ and often referred to as ‘elasticity’, quantifies the ratio of the option value’s percentage change to the underlying price’s percentage change. This measure of elasticity provides insights into leverage or gearing. Mathematically defined, omega is expressed as

$$\Omega \left(x\right)={\displaystyle \frac{x\Delta \left(x\right)}{v\left(x\right)}}.$$

The calculation of omega is a straightforward process once we have specified the class and payoff of the option. For instance, substitute (57) or (58), and the option pricing function for a class (a) or class (b) option for the numerator and denominator, respectively.

6.1.3. Rho

We now transition our focus to another Greek. Rho, symbolised as $\rho$, is defined as the rate of change of the option price concerning the interest rate [1] (p. 445). An elevated interest rate diminishes the current value of exercising the European call option upon expiration, akin to a reduction in the strike price. Consequently, this leads to an augmentation in the call price. The reverse holds for put options. Mathematically expressed, rho can be defined as

$$\rho \left(x\right)={\displaystyle \frac{\partial v}{\partial r}}\left(x\right).$$

If we juxtapose the perpetual Black–Scholes PDE (3) with (12), we can discern some values of the coefficients. We can identify that $c_0=-r$, $c_1=r-D$, and $c_2={\displaystyle \frac{1}{2}}{\sigma }^2$. This is important as rho can also be articulated as a series of partial derivatives concerning the variables $c_k$. Specifically, it can be expressed as

$$\begin{array}{cc}\hfill \rho & ={\displaystyle \frac{\partial v}{\partial r}}\hfill \\ \hfill & ={\displaystyle \frac{\partial v}{\partial c_1}}-{\displaystyle \frac{\partial v}{\partial c_0}}.\hfill \end{array}$$

(59)

Example 8.

Let us assume that the option is class (a) so that we can utilise the formulas given by the bear spread (33). When $k=1$, the rho for a bear spread is

$$\begin{array}{cc}\hfill \rho & =-S^{\ast {\alpha }_2}\left(K_1-K_2\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_1}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right]\hfill \\ \hfill & +S^{\ast {\alpha }_2}\left(K_1-K_2\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_0}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right]\hfill \\ \hfill & =S^{\ast {\alpha }_2}\left(K_1-K_2\right)x^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right]\left(-{\displaystyle \frac{\partial {\alpha }_2}{\partial c_1}}+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_0}}\right).\hfill \end{array}$$

(60)

6.1.4. Psi

The sensitivity of the option value with respect to dividend yield is measured by psi, symbolically represented by $\psi$, and expressed in mathematical notation as

$$\psi \left(x\right)={\displaystyle \frac{\partial v}{\partial D}}\left(x\right).$$

Just like rho (59), psi can be formulated as a series of partial derivatives. Therefore, psi can be expressed as

$$\psi =-{\displaystyle \frac{\partial v}{\partial c_1}}.$$

Psi is not considered a significant Greek since dividend yield is not a significant factor in the option premium [33] (p. 14).

Example 9.

Through the application of (33) when $k=1$, we gather that the psi for a bear spread option is

$$\begin{array}{cc}\hfill \psi & =S^{\ast {\alpha }_2}\left(K_1-K_2\right){\displaystyle \frac{\partial {\alpha }_2}{\partial c_1}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right].\hfill \end{array}$$

6.1.5. Vega

Vega, $\vartheta$, is defined to be the rate of change in the price of an option with respect to volatility [1] (p. 434). A caveat: vega is the rate of change with respect to implied volatility [33]. Investors are keen to understand the extent to which changes in market volatility will affect the value of their options. This interest arises from the fact that as options approach being in-the-money, the impact of volatility becomes more significant. As a consequence, volatility is one of the more popular risk metrics in option trading. Vega, expressed mathematically, is

$$\vartheta \left(x\right)={\displaystyle \frac{\partial v}{\partial \sigma }}\left(x\right).$$

In a manner analogous to rho and psi, vega can also be formulated as a series of partial derivatives concerning the variables $c_k$. Given that we retain $c_0=-r,$$c_1=r-D,$ and $c_2={\displaystyle \frac{1}{2}}{\sigma }^2$, vega assumes the following expression

$$\begin{array}{cc}\hfill \vartheta & =\sigma {\displaystyle \frac{\partial v}{\partial c_2}}.\hfill \end{array}$$

(61)

Example 10.

One last time for this series of risk parameters, we can employ (43), and so we determine that the vega of an asset-or-nothing call is

$$\begin{array}{cc}\hfill \vartheta & =\sigma {\displaystyle \frac{\partial {\alpha }_1}{\partial c_2}}{x}^{-{\alpha }_1}{S^{\ast }}^{{\alpha }_1+1}\left[log\left(S^{\ast }\right)-log\left(x\right)\right].\hfill \end{array}$$

(62)

6.2. Second-Order Hedging Parameters

6.2.1. Gamma

By extension, we can discover the value of our next risk parameter, gamma, denoted as $\Gamma$. Gamma is defined as the rate of change of delta with respect to the asset price [1] (p. 445). It serves to keep the delta in check. Consequently, it serves as a second-order hedging parameter. In mathematical terms, gamma can be expressed as

$$\begin{array}{cc}\hfill \Gamma \left(x\right)& ={\displaystyle \frac{\partial \Delta }{\partial x}}\left(x\right)\hfill \\ \hfill & ={\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\left(x\right).\hfill \end{array}$$

(63)

By the application of (20) with $n=2$, the gamma of a perpetual American put option is

$$\Gamma \left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ {\alpha }_2({\alpha }_2+1)x^{-{\alpha }_2-2}{S}^{\ast {\alpha }_2}(K-S^{\ast }),\hfill & \mathrm{if}x>S^{\ast }.\hfill \end{array}\right.$$

(64)

and by (24) with $n=2$, the gamma for a perpetual American call option is

$$\Gamma \left(x\right)=\left\{\begin{array}{cc}{\alpha }_1({\alpha }_1+1)x^{-{\alpha }_1-2}{S}^{\ast {\alpha }_1}(S^{\ast }-K),\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ 0,\hfill & \mathrm{if}x>S^{\ast }.\hfill \end{array}\right.$$

(65)

6.2.2. Vera

True to its name, vera is indeed a combination of vega and rho, both mathematically and literally. Vera represents the rate of change of rho with respect to volatility. One would use this risk metric to assess the impact of implied volatility on a rho hedged option. This parameter can be expressed as

$$Vera\left(x\right)={\displaystyle \frac{\partial \rho }{\partial \sigma }}\left(x\right).$$

Proceeding with the calculation of the partial derivative above, we obtain

$${\displaystyle \frac{\partial \rho }{\partial \sigma }}={\displaystyle \frac{\partial }{\partial \sigma }}\left({\displaystyle \frac{\partial v}{\partial c_1}}-{\displaystyle \frac{\partial v}{\partial c_0}}\right)=\sigma {\displaystyle \frac{{\partial }^{2}v}{\partial c_2\partial c_1}}-\sigma {\displaystyle \frac{{\partial }^{2}v}{\partial c_2\partial c_0}}.$$

(66)

We can use Lemma 4(iv) so that (56) or vera for any class of option becomes

$$\begin{array}{cc}\hfill Vera\left(x\right)& =2\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\ast \left(\mathrm{id}{\mathcal{K}}^{\prime }\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left(\mathrm{id}{\mathcal{K}}^{\prime }\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_1}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_2\partial c_1}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -2\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\ast \mathcal{K}\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & -\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \mathcal{K}\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_0}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & -\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_2\partial c_0}}\left(y\right)\mathrm{d}y.\hfill \end{array}$$

6.2.3. Vanna

Vanna is defined as the derivative of delta with respect to volatility [34] (p. 27). Vanna does not have its own symbol. Nonetheless, we express vanna in terms of other variables as

$$Vanna\left(x\right)={\displaystyle \frac{\partial \Delta }{\partial \sigma }}\left(x\right).$$

By extension, vanna is revealed to be

$${\displaystyle \frac{\partial \Delta }{\partial \sigma }}={\displaystyle \frac{\partial }{\partial \sigma }}\left({\displaystyle \frac{\partial v}{\partial x}}\right)=\sigma {\displaystyle \frac{{\partial }^{2}v}{\partial c_2\partial x}}.$$

(67)

As (67) implies, vanna is a parameter an investor would be interested in if they wish to keep either delta or vega hedged. By (51), we discover that for any class of option,

$$\begin{array}{cc}\hfill Vanna\left(x\right)& =-{\int }_0^{\infty }{y}^{-1}\left({\mathcal{K}}^{\prime }\ast \left(\mathrm{id}{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & -{\int }_0^{\infty }{y}^{-1}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y.\hfill \end{array}$$

6.2.4. Volga

The quantity describing the second-order partial derivative with respect to implied volatility is known as volga or vomma [34] (p. 48). Options furthest in-the-money and out-of-the-money have the highest volga. The reason for this is because vanna has a near-linear relationship with volatility when the option is at-the-money. An investor is interested in a high positive volga if there are long options, and interested in a negative volga when there are short options. Just like vanna, volga does not possess its own symbol. Therefore, volga is

$$Volga\left(x\right)={\displaystyle \frac{\partial \vartheta }{\partial \sigma }}\left(x\right).$$

(68)

We can decompose (68) into

$${\displaystyle \frac{\partial \vartheta }{\partial \sigma }}={\displaystyle \frac{\partial }{\partial \sigma }}\left(\sigma {\displaystyle \frac{\partial v}{\partial c_2}}\right)={\displaystyle \frac{\partial v}{\partial c_2}}+{\sigma }^2{\displaystyle \frac{{\partial }^{2}v}{\partial c_2^2}}={\displaystyle \frac{1}{\sigma }}\vartheta +{\sigma }^2{\displaystyle \frac{{\partial }^{2}v}{\partial c_2^2}}.$$

(69)

We let $k=k^{\prime }=2$ in Lemma 4(iv) (and use (56)), and incorporate the definition of vega (61) so that we can express vanna for a call option as

$$\begin{array}{cc}\hfill Volga\left(x\right)& ={\displaystyle \frac{\partial {\alpha }_1}{\partial c_2}}{x}^{-{\alpha }_1}{S^{\ast }}^{{\alpha }_1+1}\left[log\left(S^{\ast }\right)-log\left(x\right)\right]\hfill \\ \hfill & +2{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_2^2}}\left(y\right)\mathrm{d}y.\hfill \\ \hfill & ={\displaystyle \frac{\partial {\alpha }_1}{\partial c_2}}{x}^{-{\alpha }_1}{S^{\ast }}^{{\alpha }_1+1}\left[log\left(S^{\ast }\right)-log\left(x\right)\right]\hfill \\ \hfill & +2{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +2{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\sigma }^2{\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_2^2}}\left(y\right)\mathrm{d}y.\hfill \end{array}$$

6.3. Third-Order Hedging Parameters

6.3.1. Speed

Speed is the measure of the rate of change of gamma with respect to the underlying price of the asset [34] (p. 36). So,

$$Speed\left(x\right)={\displaystyle \frac{\partial \Gamma }{\partial x}}\left(x\right).$$

Speed is a useful measure when the gamma is at its maximum. Upon simplification, we obtain

$${\displaystyle \frac{\partial \Gamma }{\partial x}}={\displaystyle \frac{{\partial }^{3}v}{\partial x^3}}.$$

With reference to (20) and noticing that $n=3$ in the case of speed, we see that

$$Speed\left(x\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ -{\alpha }_2({\alpha }_2+1)({\alpha }_2+2)x^{-{\alpha }_2-3}{S}^{\ast {\alpha }_2}(K-S^{\ast }),\hfill & \mathrm{if}x>S^{\ast },\hfill \end{array}\right.$$

for a put option. For a call option (24), we derive that

$$Speed\left(x\right)=\left\{\begin{array}{cc}-{\alpha }_1({\alpha }_1+1)({\alpha }_1+2)x^{-{\alpha }_1-3}{S}^{\ast {\alpha }_1}(S^{\ast }-K),\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ 0,\hfill & \mathrm{if}x>S^{\ast }.\hfill \end{array}\right.$$

6.3.2. Zomma

Zomma is the rate of change of gamma with respect to implied volatility [34] (p. 35). In order to maintain a gamma-hedged portfolio, the trader must monitor fluctuations in volatility to ensure the continued effectiveness of gamma hedging. Zomma is defined as

$$Zomma\left(x\right)={\displaystyle \frac{\partial \Gamma }{\partial \sigma }}\left(x\right).$$

Using the definition of gamma (63), we see that

$${\displaystyle \frac{\partial \Gamma }{\partial \sigma }}={\displaystyle \frac{{\partial }^{3}v}{\partial \sigma \partial x^2}}=\sigma {\displaystyle \frac{{\partial }^{3}v}{\partial c_2\partial x^2}}.$$

By (53), we obtain

$$\begin{array}{cc}\hfill Zomma\left(x\right)& =2{\int }_0^{\infty }{y}^{-2}\left({\mathcal{K}}^{″}\ast {\mathcal{K}}^{″}\ast {\mathcal{K}}^{″}\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +2{\int }_0^{\infty }{y}^{-2}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right).\hfill \end{array}$$

(70)

6.3.3. Ultima

Ultima is the third derivative of the option value with respect to implied volatility. Consequently, we have a third-order risk metric. An alternative expression for ultima is simply that it is the first derivative of volga with respect to volatility. By definition,

$$Ultima\left(x\right)={\displaystyle \frac{{\partial }^{3}v}{\partial {\sigma }^3}}={\displaystyle \frac{\partial \mathrm{Volga}}{\partial \sigma }}.$$

By multiplying both sides of (69) by $\sigma$, and then differentiating with respect to $\sigma$, we obtain

$$\begin{array}{cc}\hfill Volga\left(x\right)+\sigma {\displaystyle \frac{\partial Volga}{\partial \sigma }}\left(x\right)& ={\displaystyle \frac{\partial \vartheta }{\partial \sigma }}+3{\sigma }^2{\displaystyle \frac{{\partial }^{2}v}{\partial c_2^2}}+{\sigma }^4{\displaystyle \frac{{\partial }^{3}v}{\partial c_2^3}}.\hfill \end{array}$$

Hence,

$$\mathrm{Ultima}\left(x\right)=3\sigma {\displaystyle \frac{{\partial }^{2}v}{\partial c_2^2}}+{\sigma }^3{\displaystyle \frac{{\partial }^{3}v}{\partial c_2^3}}.$$

Using Lemma 4(ii) and (56) with $k=2$, ultima becomes

$$\begin{array}{cc}\hfill \mathrm{Ultima}\left(x\right)& =6\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right)\left({\chi }_{R}G\right)\left(y\right)\mathrm{d}y\hfill \\ \hfill & +6\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\left(\mathcal{K}\ast \left({\mathrm{id}}^2{\mathcal{K}}^{″}\right)\right)\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{\partial \left({\chi }_{R}G\right)}{\partial c_2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +3\sigma {\int }_0^{\infty }{\displaystyle \frac{1}{y}}\mathcal{K}\left({\displaystyle \frac{x}{y}}\right){\displaystyle \frac{{\partial }^2\left({\chi }_{R}G\right)}{\partial c_2^2}}\left(y\right)\mathrm{d}y\hfill \\ \hfill & +{\sigma }^3\sum _{j=0}^3\left({\displaystyle \genfrac{}{}{0pt}{}{3}{j}}\right)\left({\displaystyle \frac{{\partial }^{3-j}\mathcal{K}}{\partial c_2^{3-j}}}\ast {\displaystyle \frac{{\partial }^j\left({\chi }_{R}G\right)}{\partial c_2^j}}\right)\left(x\right).\hfill \end{array}$$

At first glance, the procedure may seem lengthy. Yet, attempting to determine ultima through conventional brute-force methods—such as taking the partial derivative of any option function from any class—would prove considerably more challenging. In fact, it might be deemed nearly impossible without the assistance of computational software.

7. Validation of the Mellin Transform Greeks

The Greeks derived using the Mellin transform are exact solutions, eliminating the need to rely on traditional finite difference methods to validate our results. Instead, we can confirm their accuracy by referencing the previous work of [25], through analytical verification.

Brute-Force Computation of the Greeks

It was seen in [25] that the option pricing function for a put is

$$v\left(x;S^{\ast }\right)=\left\{\begin{array}{cc}K-x,\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ \left(K-S^{\ast }\right){\left({\displaystyle \frac{x}{S^{\ast }}}\right)}^{-{\alpha }_2},\hfill & \mathrm{if}x>S^{\ast },\hfill \end{array}\right.$$

where $S^{\ast }={\displaystyle \frac{{\alpha }_{2}K}{{\alpha }_2+1}}$. We can ascertain the value of delta through the first derivative of v with respect to the stock price x, that is

$$v\left(x;S^{\ast }\right)=\left\{\begin{array}{cc}-1,\hfill & \mathrm{if}0<x<S^{\ast },\hfill \\ -{\alpha }_2{x}^{-{\alpha }_2-1}{S^{\ast }}^{{\alpha }_2}\left(K-S^{\ast }\right),\hfill & \mathrm{if}x>S^{\ast },\hfill \end{array}\right.$$

(71)

where $S^{\ast }={\displaystyle \frac{{\alpha }_{2}K}{{\alpha }_2+1}}$ once again. We see that is identical to (57). While the delta can be derived relatively easily through brute force, computing ${\displaystyle \frac{\partial v}{\partial c_k}}$ quickly becomes more challenging. Furthermore, higher-order derivatives, such as ${\displaystyle \frac{{\partial }^{2n}v}{\partial c_{k^{\prime }}^n\partial c_k^n}}$, may be impractical or even unattainable using conventional methods.

We attempt to derive ${\displaystyle \frac{\partial v}{\partial c_k}}$ when $x>S^{\ast }$. We have

$$\begin{array}{cc}\hfill v\left(x;S^{\ast }\right)& =\left(K-S^{\ast }\right){\left({\displaystyle \frac{x}{S^{\ast }}}\right)}^{-{\alpha }_2},\hfill \end{array}$$

(72)

which is equivalent to

$$\begin{array}{cc}\hfill log\left(v\left(x;S^{\ast }\right)\right)& =log\left(\left(K-S^{\ast }\right)\right)-{\alpha }_{2}log\left(x\right)+{\alpha }_{2}log\left(S^{\ast }\right).\hfill \end{array}$$

(73)

Then,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{v\left(x;S^{\ast }\right)}}{\displaystyle \frac{\partial }{\partial c_k}}v\left(x;S^{\ast }\right)& =-{\displaystyle \frac{1}{K-S^{\ast }}}{\displaystyle \frac{\partial S^{\ast }}{\partial c_k}}-{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(x\right)+{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}log\left(S^{\ast }\right)+{\alpha }_2{\displaystyle \frac{1}{S^{\ast }}}{\displaystyle \frac{\partial S^{\ast }}{\partial c_k}}\hfill \\ \hfill & ={\displaystyle \frac{\partial S^{\ast }}{\partial c_k}}\left({\displaystyle \frac{1}{S^{\ast }-K}}+{\displaystyle \frac{{\alpha }_2}{S^{\ast }}}\right)-{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}[log\left(x\right)-log\left(S^{\ast }\right)].\hfill \end{array}$$

(74)

Recall that $S^{\ast }={\displaystyle \frac{{\alpha }_{2}K}{{\alpha }_2+1}}$, so ${\alpha }_2={\displaystyle \frac{-S^{\ast }}{S^{\ast }-K}}$. So

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{S^{\ast }-K}}+{\displaystyle \frac{{\alpha }_2}{S^{\ast }}}& ={\displaystyle \frac{1}{S^{\ast }-K}}+{\displaystyle \frac{1}{S^{\ast }}}{\displaystyle \frac{-S^{\ast }}{S^{\ast }-K}}\hfill \\ \hfill & ={\displaystyle \frac{1}{S^{\ast }-K}}-{\displaystyle \frac{1}{S^{\ast }-K}}\hfill \\ \hfill & =0.\hfill \end{array}$$

(75)

Hence, (74) becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial c_k}}v\left(x;S^{\ast }\right)& =v\left(x;S^{\ast }\right)\left\{-{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}[log\left(x\right)-log\left(S^{\ast }\right)]\right\}.\hfill \end{array}$$

By (72), we see that the above becomes

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial c_k}}v\left(x;S^{\ast }\right)& =\left(K-S^{\ast }\right){\left({\displaystyle \frac{x}{S^{\ast }}}\right)}^{-{\alpha }_2}\left\{-{\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}[log\left(x\right)-log\left(S^{\ast }\right)]\right\}\hfill \\ \hfill & =-S^{\ast {\alpha }_2}(K-S^{\ast }){\displaystyle \frac{\partial {\alpha }_2}{\partial c_k}}{x}^{-{\alpha }_2}\left[log\left(x\right)-log\left(S^{\ast }\right)\right],\hfill \end{array}$$

(76)

which is identical to (32). All other Greeks can also be determined using this approach, though the process is labour-intensive, ultimately yielding results that align with those obtained through the Mellin transform.

8. Discussion

The findings of this study provide insights into the computation of option Greeks for perpetual American options with general payoffs using the Mellin transform and its properties. Unlike conventional derivative-based methods, our approach bypasses brute-force computation, simplifying the derivation of Greeks without sacrificing accuracy. This aligns with prior work by [21,22], who also applied the Mellin transform to streamline Greek calculations. However, while previous studies focused on European options, our extension to perpetual American options fills a notable gap, broadening the practical applications of Greeks for a wider range of financial products.

The Mellin transform properties offer a computational advantage by enabling direct solutions to Greeks, reducing the need for intensive algebraic manipulation typical of derivative-based methods. This echoes efforts by [19,20], who aimed to simplify these computations but still relied on derivatives. By directly computing Greeks, our method minimises computational costs, making perpetual American options more viable for complex and customised over-the-counter (OTC) contracts. This advancement addresses the growing demand for tailored financial products, helping institutions maintain competitiveness by offering bespoke solutions.

The critical role of Greeks as hedging instruments highlights the broader importance of this study in risk management. As ref. [4] argued, precise hedging often outweighs precise pricing—a principle echoed throughout our findings. Greeks measure sensitivity to various risk factors, such as stock price, interest rate, and implied volatility, and are particularly relevant for perpetual American options, which can be exercised at any time. These options pose unique hedging challenges compared to European options, which are exercised only at maturity. By utilising Greeks, investors can better manage exposure in OTC transactions, as perpetual options provide flexibility in hedging against market movements. This makes derived Greeks vital tools for strategic decision-making, especially in scenarios where effective hedging is paramount.

Our study also highlights the practical application of Greeks in managing tailored financial products. As markets evolve and client demands shift toward individualised contracts, financial institutions face increasing pressure to balance customer needs with robust risk management. Our method offers an efficient way to compute risk metrics for perpetual American options with piecewise linear payoffs. By reducing computational effort and providing exact solutions, institutions can more feasibly offer customised products without compromising risk management strategies.

While the computational advantages of the Mellin transform are clear, this study acknowledges its limitations and outlines potential avenues for further research. The current model assumes a Black–Scholes framework, which does not fully capture market dynamics involving stochastic volatility or jumps. Extending this research to more advanced models, such as those proposed by [23] and explored by [22], would enhance the robustness of our findings. By adapting our method to incorporate stochastic volatility or jump-diffusion characteristics, we could provide broader tools for managing risk in volatile markets. Such advancements would be valuable for pricing exotic options often constrained by traditional methods.

Future research could also explore higher-order Greeks, which address more nuanced risk factors. While this study focuses on primary Greeks (delta, gamma, theta), higher-order Greeks like vanna, vomma, and charm are critical in complex hedging scenarios. Extending our approach to compute these metrics without derivatives could capture more sophisticated risk dynamics, aiding institutions operating in high-risk environments or trading in exotic contracts.

Lastly, our findings have implications for regulators and policymakers. As OTC markets grow, regulatory bodies require effective frameworks for assessing risks in tailored derivatives. Our method’s simplicity and efficiency could help regulators evaluate risk exposure, enhancing oversight and market stability. By providing clear, accessible risk metrics, this approach could prevent systemic risks linked to inadequately hedged positions.

In summary, this study contributes an efficient approach to calculating Greeks for perpetual American options, building on foundational work by [21] and others. By shifting to the Mellin transform, we offer financial institutions a practical way to meet the growing demand for customised products while maintaining strong risk management. This work invites further exploration into complex models and higher-order Greeks, offering promising applications for modern financial markets.

9. Conclusions

Remaining competitive in financial markets often necessitates offering customised OTC products to meet client-specific requirements. Unlike standardised exchange contracts, these tailored instruments present unique challenges in managing risk, necessitating models that reflect realistic risk factors. Here, the ‘Greek’ letters play a pivotal role.

Financial portfolios containing OTC derivatives depend on risk measures such as stock price, interest rate, dividend yield, and implied volatility. Each factor corresponds to a specific risk metric or Greek, with some exceptions like vega. This paper derives Greeks for perpetual American options with general payoffs, bypassing traditional brute-force derivative computations. Using the Mellin transform, we eliminate the labour-intensive processes typically involved, presenting an efficient alternative.

Accurate hedging is arguably more critical than precise option pricing. Effective hedging minimises or eliminates uncertainty, ensuring a predictable outcome at contract execution. Conversely, imprecise hedging renders initial pricing less relevant, as future uncertainties can overshadow potential gains. This principle can be illustrated through a thought experiment: a million riskless bets of one dollar are preferable to a single risky bet of one million dollars. The former guarantees returns, while the latter does not.

As emphasised by [4], understanding an option’s risk is more important than knowing its exact price. This underscores the significance of Greeks in crafting effective investment strategies. In this work, we derive Greeks for perpetual American options with general payoffs and demonstrate their utility for investors.