Optimal Control in Financial Markets for the Uncertain Volatility Model

Abstract

This paper generalizes the well-known Black–Scholes model, specifically the uncertain volatility model. To calculate the fair price range of a payment obligation, Hamilton–Jacobi–Bellman equations are derived and transformed into nonlinear heat equations with boundary conditions. Theorems are proven stating that, for a certain class of payment obligations, solutions to nonlinear heat equations satisfy the linear heat equations. A computational example using real data is provided.

1. Introduction

The first model to describe the evolution of stock prices was the Bachelier linear model [1]. One of the fundamental shortcomings of the Bachelier model is the fact that stock prices can take negative values. In [2], Samuelson proposed to describe stock prices by geometric Brownian motion. In other words, Samuelson proposed to consider that the logarithms of stock prices obey a Bachelier-type linear model. In [3], Black and Scholes proposed a standard diffusion model of the (B,S)-market and obtained a formula for the fair value of European-type call options. A significant shortcoming of this model is the fact that the coefficients of drift, interest rate and volatility are constant, although in reality they change over time. In this regard, the so-called “volatility smile” effect should be noted, which is a fact that is not explained by the standard diffusion model, which has led to its various generalizations and improvements. The essence of the “volatility smile” effect is as follows. The Black and Scholes formula provides an explicit relationship between the fair price and volatility, the expiration date, and the contract price. We can look at actual option prices in financial markets and compare them with theoretical values. We can then use this equation to calculate the implied volatility, which depends on the expiration date and the contract price. The following has been experimentally established:

• For a fixed contract price, implied volatility changes with the expiration date.
• For a fixed expiration date, implied volatility changes with the contract price, being a downward-convex function, which explains the name “volatility smile”.

To account for the first observed effect, Merton proposed considering drift and volatility as functions of time in the standard model [4], and such schemes are indeed used in financial markets, especially in the settlement of American-style options.

To take into account the second observed effect, a wide variety of complications of the standard model are introduced. In [5], models of the “diffusion with jumps” type are presented. In [6], models with stochastic interest rates are presented. In [7,8], the Dupire model is proposed, in which the drift is a function of time, and volatility depends on the time and the cost of the risky asset. In the mentioned works, Dupire also showed that the ideas of arbitrage-free market conditions and market completeness make it possible to estimate the unknown volatility using knowledge of the actually observed prices of standard European-type call options with fixed exercise time and exercise price. In [9,10,11,12,13], models of stochastic volatility are presented. In [14], the Heston stochastic volatility model is proposed. In [15,16,17,18,19,20,21], various numerical methods for calculating option prices in the Heston model are proposed. In [15], a tree method is proposed for calculating the prices of American options in the Heston model. In [16], a hybrid method for estimating option prices in the Heston model is proposed. In [17], a Monte Carlo method for calculating the prices of American options in the Heston model is proposed. In [18], a numerical method for calculating the prices of barrier options in the Heston model is constructed. In [19], finite-difference methods for calculating option prices in the Heston model are presented, as well as algorithms for estimating the model parameters. In [20], an algorithm for calculating option prices in the Heston model using artificial neural networks is presented. In [21], a numerical implementation of European-type option prices in the Heston model using integral transformations is proposed. In [22,23,24,25], models of uncertain volatility are considered. In particular, [22] examines two models of uncertain volatility and presents formulas for calculating the fair value range of a European-style option.

This article considers a model of uncertain volatility, a generalization of the well-known Black–Scholes model. For the uncertain volatility model, the upper and lower prices of the payment obligation are represented as a submartingale and supermartingale, respectively. The Hamilton–Jacobi–Bellman equations are derived for calculating these. The resulting equations are transformed into nonlinear heat equations with a boundary condition. Theorems are proven stating that if the boundary condition is continuous and convex function, then the solutions of the nonlinear heat equations are convex functions satisfying the linear heat equations. The central convex-reduction result is classical in the uncertain volatility and G-expectation literature [23,24,25]. A novel aspect is the method of proof, based on a discrete Rademacher construction and a backward-induction argument. This methodology preserves convexity at each discrete step, while a Taylor expansion generates the quadratic (variance) term that results in the nonlinear heat equation. By taking the limit, a viscosity solution is obtained, which represents the upper and lower prices. To assess the accuracy of the uncertain volatility model, its premium estimates for 48 call options on the S&P 100 exchange-traded fund from January 1991 to June 1997 are tested and compared.

2. Black–Scholes Model

Let $W\left(t\right)$ is a standard Wiener process on a stochastic basis $\langle C[0,T],{\left(F_t^W\right)}_{t\ge 0},F_T^W,P^W\rangle$. Let us consider the equation for the discounted stock price process:

$$dS\left(t\right)=S\left(t\right)\sigma dW\left(t\right)$$

(1)

with initial condition $S\left(0\right)=S_0$. A process $Y\left(t\right)=E(f\left(S\left(T\right)\right)\mid F_t^W)$, where the function $f\left(x\right)\ge 0$ and the mathematical expectation $E\left(f\right(S\left(T\right)\left)\right)$ is finite, is the portfolio capital. Capital can be represented as a function of time and asset value $Y\left(t\right)=V(t,S(t\left)\right),$ where the function $V(t,x)\in C^{1,2}$ has the form $V(t,x)=E_{t,x}\left(f\left(S\left(T\right)\right)\right).$ This mathematical expectation $E_{t,x}\left(f\left(S\left(T\right)\right)\right)$ is calculated under the condition that the value of the Markov random process $S\left(t\right)$ at time t is equal to x. This function is a solution to the Cauchy problem [26]:

$${\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{{\sigma }^2}{2}}{\displaystyle \frac{{\partial }^{2}V}{\partial x^2}}=0,V(T,x)=f\left(x\right).$$

(2)

The solution to this equation is known [26]:

$$V(t,x)={\displaystyle \frac{1}{\sqrt{2\pi }}}{\int }_{-\infty }^{+\infty }f\left(x\exp \left(-{\displaystyle \frac{{\sigma }^2}{2}}(T-t)+\sigma \sqrt{T-t}y\right)\right)\exp \left(-{\displaystyle \frac{y^2}{2}}\right)dy.$$

For example, in financial mathematics, the problem of calculating the fair price of a European call option with a contract price K and an interest rate r on a risk-free asset is relevant. For this problem the function $f\left(x\right)=\exp (-rT){(x-K)}^{+}$, where ${\left(x\right)}^{+}=max(x,0)$. Function $V(t,x)=x\mathrm{\Phi }\left(d_1(t,x)\right)-K{e}^{-r(T-t)}\mathrm{\Phi }\left(d_2(t,x)\right)$, process $Y\left(t\right)$ and fair price for a European call option $C=Y\left(0\right)=V(0,S_0)$. As a result, the fair price is calculated using the formula $C=S_0\mathrm{\Phi }\left(d_1(0,S_0)\right)-K{e}^{-rT}\mathrm{\Phi }\left(d_2(0,S_0)\right),$ where $\mathrm{\Phi }\left(x\right)$ denotes the standard normal distribution function, $d_1(t,x)={\displaystyle \frac{\ln (x/K)+(r+{\sigma }^2/2)(T-t)}{\sigma \sqrt{T-t}}},$ $d_2(t,x)=d_1(t,x)-\sigma \sqrt{T-t}.$

3. Heston’s Model

Heston’s model considers a two-dimensional Wiener process $W\left(t\right)=\left(\begin{array}{c}{W}_1\left(t\right)\\ W_2\left(t\right)\end{array}\right)$ with $dW\left(t\right)\in N\left(\begin{array}{cc}0,& \left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right)dt\end{array}\right).$ The equations for the discounted stock price and the volatility processes are as follows:

$$\left\{\begin{array}{cc}\hfill dS\left(t\right)& =S\left(t\right)\sqrt{\sigma \left(t\right)}d{W}_1\left(t\right)\hfill \\ \hfill d\sigma \left(t\right)& =k\left(\theta -\sigma \left(t\right)\right)dt+\nu \sqrt{\sigma \left(t\right)}d{W}_2\left(t\right)\hfill \end{array}\right.$$

(3)

where $S\left(0\right)=S_0,\sigma \left(0\right)={\sigma }_0.$ The quantities $k,\theta ,\nu >0$ are the model parameters. If $2k\theta \ge {\nu }^2$ and ${\sigma }_0>0$, then the process $\sigma \left(t\right)$ is strictly positive, which is important since volatility is non-negative.

The following interpretation can be given to the parameters of the stochastic dispersion process. Denoting $F\left(t\right)=E\sigma \left(t\right)$, we obtain the equation $F\left(t\right)={\sigma }_0+k{\int }_0^t(\theta -F\left(s\right))ds,$ the solution of which is $F\left(t\right)=\theta +({\sigma }_0-\theta )e^{-kt}.$ Thus, the process $\sigma \left(t\right)$ has the property of returning to a level $\theta$, called the long-run average. The parameter k specifies the speed of return to $\theta$. The parameter $\nu$ is called the “volatility of volatility” and it controls how strong the volatility swings are.

Let us consider $F^2\left(t\right)=E{\sigma }^2\left(t\right)$. The application of the Ito formula allows us to write an equation $d{F}^2=(-2k)e^{-kt}({\sigma }_0-\theta )(({\sigma }_0-\theta )e^{-kt}+\theta )dt,$ the solution of which

$$F^2\left(t\right)={\displaystyle \frac{1}{2k}}\left(F\left(t\right)\left(2k\right)\theta +{\nu }^2-(2k{F}^2\left(0\right)+(F\left(0\right)\left(2k\right)\theta +{\nu }^2)\exp (-2kt)\right)$$

allows us to find the volatility dispersion $D\sigma \left(t\right)=F^2\left(t\right)-{\left(F\left(t\right)\right)}^2$, which at $k\to \infty$ tends to zero and “reduces” the Heston model to the Black–Scholes model.

For the Heston model, we consider the process $Y\left(t\right)=E(f\left(S\left(T\right)\right)\mid F_t^W)$, where the function $f\left(x\right)\ge 0$ and the mathematical expectation $E\left(f\right(S\left(T\right)\left)\right)$ is finite, which can be represented by the equality: $Y\left(t\right)=V(t,S(t),\sigma (t\left)\right)$. Note that we need a function of three variables. The function $V(t,x,y)\in C^{1,2,2}$ is the conditional mathematical expectation: $V(t,x,y)=E_{t,x,y}\left(f\left(S\left(T\right)\right)\right)$, is the solution to the Cauchy problem [14]:

$${\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{\partial V}{\partial y}}k(\theta -y)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}V}{\partial x^2}}{x}^{2}y+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}V}{\partial y^2}}{\nu }^{2}y+{\displaystyle \frac{{\partial }^{2}V}{\partial x\partial y}}xy\nu \rho =0,V(T,x,y)=f\left(x\right).$$

(4)

To obtain an approximate solution to Equation (4), we apply the standard time-stepping scheme for the Heston PDE [27], for which we divide time into segments of equal length, equal to $h={\displaystyle \frac{T}{N}}$. In accordance with this division, we represent Equation (4) as a system of elliptic partial differential equations with respect to spatial variables (the standard time-stepping scheme for the Heston PDE) $V(i-1,x,y)=$$=h\left(V_y(i,x,y)k(\theta -y)+{\displaystyle \frac{1}{2}}{V}_{xx}(i,x,y)x^{2}y+{\displaystyle \frac{1}{2}}{V}_{yy}(i,x,y){\nu }^{2}y+V_{xy}(i,x,y)xy\nu \rho \right)+V(i,x,y),$$V(N,x,y)=f\left(x\right).$ The notation $V(i,x,y)=V(ih,x,y)$ is used. In addition to this method, there are other methods for calculating the function V, one of them was proposed directly by Heston [14] for the already considered problem of calculating the fair price of a European call option.

Let us consider a call option for the Heston model similar to how we did for the Black–Scholes model. Let us state a theorem.

Theorem 1

([14]). The price at time t of a call option with contract price K and exercise time T is equal:

$$\begin{array}{cc}\hfill V(t,x,y)& =\exp (-r(T-t\left)\right)F(t,\ln x,y)\hfill \\ \hfill F(t,x,y)& =e^x\tilde{\Pi }(t,x,y)-K\Pi (t,x,y)\hfill \\ \hfill \Pi (t,x,y)& ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }Re\left({\displaystyle \frac{e^{-iu\ln K}\varphi (t,x,y,u)}{iu}}\right)du\hfill \\ \hfill \tilde{\Pi }(t,x,y)& ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }Re\left({\displaystyle \frac{e^{-iu\ln K}\tilde{\varphi }(t,x,y,u)}{iu}}\right)du\hfill \\ \hfill \varphi (t,x,y,u)& =\exp \left(C\right(T-t,u)+D(T-t,u)y+iux)\hfill \\ \hfill \tilde{\varphi }(t,x,y,u)& ={\displaystyle \frac{\varphi (t,x,y,u-i)}{\varphi (t,x,y,-i)}}\hfill \\ \hfill C(\tau ,u)& =riu\tau +{\displaystyle \frac{k\theta }{{\nu }^2}}\left((k-\rho \nu iu-d\left(u\right))\tau -2\ln \left({\displaystyle \frac{1-g\left(u\right)e^{-d\left(u\right)\tau }}{1-g\left(u\right)}}\right)\right)\hfill \\ \hfill D(\tau ,u)& ={\displaystyle \frac{k-\rho \nu iu-d\left(u\right)}{{\nu }^2}}\left({\displaystyle \frac{1-e^{-d\left(u\right)\tau }}{1-g\left(u\right)e^{-d\left(u\right)\tau }}}\right)\hfill \\ \hfill d\left(u\right)& =\sqrt{{(\rho \nu iu-k)}^2+{\nu }^2(iu+u^2)}\hfill \\ \hfill g\left(u\right)& ={\displaystyle \frac{\rho \nu iu-k+d\left(u\right)}{\rho \nu iu-k-d\left(u\right)}}\hfill \end{array}$$

(5)

Let us split the interval $[0,T]$ into N parts with a step $h={\displaystyle \frac{T}{N}}$ and consider the Euler discretization of the Heston model. Euler scheme [19] for the variance is $\sigma \left(nh\right)=\sigma \left((n-1)h\right)+{\int }_{(n-1)h}^{nh}k(\theta -\sigma \left(u\right))du+{\int }_{(n-1)h}^{nh}\nu \sqrt{\sigma \left(u\right)}d{W}_2\left(u\right).$ Euler discretization approximates the integrals as ${\int }_{(n-1)h}^{nh}k(\theta -\sigma \left(u\right))du\approx k\left(\theta -\sigma \left(\right(n-1\left)h\right)\right)h,$ ${\int }_{(n-1)h}^{nh}\nu \sqrt{\sigma \left(u\right)}d{W}_2\left(u\right)\approx \nu \sqrt{\sigma \left(\right(n-1\left)h\right)}\left(W_2\left(nh\right)-W_2\left((n-1)h\right)\right)\stackrel{d}{=}\nu \sqrt{\sigma \left(\right(n-1\left)h\right)}\sqrt{h}{\delta }_n^2,$ where ${\delta }_n^2$ is a standard normal random variable. Euler scheme for the stock price is $S\left(nh\right)=S\left((n-1)h\right)+{\int }_{(n-1)h}^{nh}S\left(u\right)\sqrt{\sigma \left(u\right)}d{W}_1\left(u\right).$ Euler discretization approximates the integral as

$$\begin{array}{c}\hfill {\int }_{(n-1)h}^{nh}S\left(u\right)\sqrt{\sigma \left(u\right)}d{W}_1\left(u\right)\approx S\left((n-1)h\right)\sqrt{\sigma \left(\right(n-1\left)h\right)}\left(W_1\left(nh\right)-W_1\left((n-1)h\right)\right)\stackrel{d}{=}\\ \hfill S\left((n-1)h\right)\sqrt{\sigma \left(\right(n-1\left)h\right)}\sqrt{h}{\delta }_n^1,\end{array}$$

where ${\delta }_n^1$ is a standard normal random variable.

Let us denote

$$S\left((n-1)h\right)=S_{n-1},S\left(nh\right)=S_n,\sigma \left((n-1)h\right)={\sigma }_{n-1},\sigma \left(nh\right)={\sigma }_n$$

and consider the discrete Heston model:

$$\left\{\begin{array}{c}\Delta S_n=S_{n-1}\sqrt{{\sigma }_{n}h}{\delta }_n^1\hfill \\ \Delta {\sigma }_n=k(\theta -{\sigma }_{n-1})h+\nu \sqrt{{\sigma }_{n-1}h}{\delta }_n^2\hfill \end{array}\right.$$

(6)

$n=1,\dots ,N$.

Let us consider two implementations of model (6).

• Sequences ${\left({\delta }_n^1\right)}_{n=1}^N$ and ${\left({\delta }_n^2\right)}_{n=1}^N$ consist of independent Rademacher random variables with $\mathrm{cov}({\delta }_n^1,{\delta }_n^2)=\rho$.
• The sequence $\left(\begin{array}{c}{\delta }_n^1\\ {\delta }_n^2\end{array}\right)$ consists of independent random vectors distributed according to a two-dimensional normal law with zero mathematical expectation and a covariance matrix $\left(\begin{array}{cc}1& \rho \\ \rho & 1\end{array}\right).$

Note that $E{\sigma }_N=\theta +({\sigma }_0-\theta ){(1-kh)}^N$ and ${lim}_{N\to \infty }E{\sigma }_N=E\sigma \left(T\right)$, where $E\sigma \left(T\right)=\theta +({\sigma }_0-\theta )\exp (-kT)$. Therefore with a given accuracy $\epsilon$ the number of partition points N can be determined according to the following formula: $N=inf\{n=1,2,\dots :|E{\sigma }_n-E\sigma \left(T\right)|<\epsilon \}$.

As before, we are now interested in a discrete process $Y_n=E(f\left(S_N\right)\mid F_n)$, where the function $f\left(x\right)\ge 0$ and the mathematical expectation $E\left(f\left(S_N\right)\right)$ is finite. The process $Y_n$ can be represented as $Y_n=V(n,S_n,{\sigma }_n)$, where the function V is the conditional mathematical expectation $V(n,x,y)=E(f\left(S_N\right)\mid S_n=x,{\sigma }_n=y)$. The discrete model allows calculating conditional mathematical expectations using the Monte Carlo method. For any discrete time n it is necessary to reproduce the required number of trajectory residuals; for each trajectory residual, it is necessary to generate random variables ${\delta }_i^1,{\delta }_i^2$. Computationally, the first option is preferable to the second.

The fair price can be calculated using recurrence formulas that use a sequence of Bellman functions $V(i,x,y)$, related by the Bellman equation $V(i-1,x,y)=EV(i,x+\Delta S_i,y+\Delta {\sigma }_i)$. We apply the second-order Taylor formula and discard terms of order of smallness $h^2$, as a result we obtain the following:

$$\begin{array}{cc}\hfill V(i-1,x,y)& =EV(i,x+\Delta S_i,y+\Delta {\sigma }_i)=\hfill \\ & =V(i,x,y)+V_x(i,x,y)E\Delta S_i+V_y(i,x,y)E\Delta {\sigma }_i+{\displaystyle \frac{1}{2}}{V}_{xx}(i,x,y)E{(\Delta S_i)}^2+\hfill \\ & V_{xy}(i,x,y)E\Delta S_{i}E\Delta {\sigma }_i+{\displaystyle \frac{1}{2}}{V}_{yy}(i,x,y)E{(\Delta {\sigma }_i)}^2=\hfill \\ & =h\left(V_y(i,x,y)k(\theta -y)+{\displaystyle \frac{1}{2}}{V}_{xx}(i,x,y)x^{2}y+{\displaystyle \frac{1}{2}}{V}_{yy}(i,x,y){\nu }^{2}y+V_{xy}(i,x,y)xy\nu \rho \right),\hfill \\ & V(N,x,y)=f\left(x\right).\hfill \end{array}$$

This system of equations is completely identical to the system of equations for the standard time-stepping scheme for the Heston PDE, and is the same for both the first and the second variants.

This fact allows us to assert that both the first and the second variants of the discrete Heston model are equally good approximations of the continuous Heston model. Specifically, the approximation error when calculating the function $V(t,x,y)$ in the discrete model and in the standard time-stepping scheme for the Heston PDE is the same and is of the order of h provided that the system of differential equations is solved with the required accuracy and the number of iterations in the Monte Carlo method is of the order of ${\displaystyle \frac{1}{h^2}}$. It should be noted that the Monte Carlo method in its binary implementation has undeniable advantages over other computational methods, since at each iteration it requires simple calculations and the generation of two random variables uniformly distributed over the interval $[0,1]$. It should also be noted that in the Rademacher scheme calculations of processes take place on binary trees and parallel algorithms [28] can be applied. Therefore the Rademacher scheme is numerically advantageous. In the next section on the uncertain volatility model the discrete Rademacher scheme will be central to the new proof and the discrete time-stepping idea will reappear in the uncertain volatility setting.

4. Uncertain Volatility Model

Let us consider the Banach space of bounded functions $l_{\infty }\left([0,T]\right)$, probability space $\langle l_{\infty },B\left(l_{\infty }\right),P\rangle$. The set ${\Sigma }_t=\{x\in R^{+}:\exists (\sigma \in \Sigma )\wedge (\sigma \left(t\right)=x)\},\Sigma \in l_{\infty }$ is compact for every $t\in [0,T]$. Based on any probability measure $P\in \mathcal{P}$ one can define a measure $P_{\Sigma }\left(A\right)=\left\{\begin{array}{cc}P(A·\Sigma )/P(\Sigma ),\hfill & \mathrm{if}P(\Sigma )\ne 0\hfill \\ 0,\hfill & \mathrm{if}P(\Sigma )=0\hfill \end{array}\right.$. Let us denote the family of such measures by ${\mathcal{P}}_{\Sigma }$.

Let us consider independent random processes, each defined on its own stochastic basis $\langle C[0,T],{\left(F_t^W\right)}_{t\ge 0},F_T^W,P^W\rangle$ and $\langle l_{\infty }\left([0,T]\right),{\left(F_t^X\right)}_{t\ge 0},F_T^X,P_{\Sigma }\rangle .$ Let us consider the equation for the discounted stock price process $dS\left(t\right)=S\left(t\right)X\left(t\right)dW\left(t\right)$ with initial conditions $S\left(0\right)=S_0,X\left(0\right)=X_0.$ This process is a martingale with respect to filtration ${(F_t^W\times F_t^X)}_{t\ge 0}$ and any measure from $P^W\times {\mathcal{P}}_{\Sigma }$. It follows from the compactness of ${\Sigma }_t$ and uniform boundedness of $X\left(t\right)$ on $[0,T]$.

The problem is to calculate the submartingale $Y\left(t\right)={sup}_{Q\in P^W\times {\mathcal{P}}_{\Sigma }}{E}_{Q}f(S_T\mid F_t^W)$. Let $m\left(t\right)$ ($m\left(t\right)\ne 0$) be the lower envelope, and $M\left(t\right)$ the upper envelope of the family of slices ${\Sigma }_t$. Let us consider the probability space $\langle \left[m\right(t),M(t\left)\right],B\left(\right[m\left(t\right),M\left(t\right)\left]\right),P\left(t\right)\rangle$. For every $t\in [0,T]$, the set ${\Sigma }_t=[m\left(t\right),M\left(t\right)]$ is a nonempty, closed (and hence compact) interval, and the functions $t\to m\left(t\right),M\left(t\right)$ are measurable. To avoid ambiguity in cases when the supremum or infimum over measures $P^T\times P_{\Sigma }$ may not be attained in classical (strong) controls we note that the supremum can be approximated by piecewise-constant volatility processes $y^N\left(t\right)\in [m\left(t\right),M\left(t\right)]$. To calculate the submartingale $Y\left(t\right)$ we use the Hamilton–Jacobi–Bellman equation ${\displaystyle \frac{\partial V}{\partial t}}+{\displaystyle \frac{x^2}{2}}{sup}_{P\in \mathcal{P}\left(t\right)}E{\displaystyle \frac{{\partial }^2\overline{V}}{\partial x^2}}=0,$ which is transformed into a nonlinear heat equation

$${\displaystyle \frac{\partial \overline{V}}{\partial t}}+{\displaystyle \frac{x^2}{2}}\left[{\displaystyle \frac{M^2\left(t\right)-m^2\left(t\right)}{2}}\left|{\displaystyle \frac{{\partial }^2\overline{V}}{\partial x^2}}\right|+{\displaystyle \frac{M^2\left(t\right)+m^2\left(t\right)}{2}}{\displaystyle \frac{{\partial }^2\overline{V}}{\partial x^2}}\right]=0$$

(7)

with a boundary condition $\overline{V}(T,x)=f\left(x\right)$.

Theorem 2

([29]). If $f\left(x\right)$ is a continuous function, then there exists a unique viscosity solution of Equation (7) $Y\left(t\right)=\overline{V}(t,x)$, which is a submartingale.

Let us consider a discrete model that arises as a result of dividing an interval $[0,T]$ into N equal parts: $S_n^N=S_{n-1}^N\left(1+\sqrt{{\displaystyle \frac{T}{N}}}{X}_{n-1}^N{\epsilon }_n\right),$ where ${\epsilon }_n$ are independent Rademacher random variables, $S^N$ is a discrete version of the process S, and $X^N$ is a discrete version of the process X. For this model, we consider a sequence of functions $V_n^N\left(x\right)={max}_{m_n^N\le X_n^N\le M_n^N,\dots ,m_N^N\le X_N^N\le M_N^N}\mathbb{E}\left(f\left(S_N^N\right)\mid S_n^N=x\right),$ where $m_n^N=m\left(t_n\right)$, $M_n^N=M\left(t_n\right)$, $t_n={\displaystyle \frac{nT}{N}}$, for which the recurrence equation is determined: $V_n^N\left(x\right)={max}_{m_n^N\le y\le M_n^N}\mathbb{E}{V}_{n+1}^N\left(x\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}{\epsilon }_n\right)\right),V_N^N\left(x\right)=f\left(x\right).$ The family of uniformly bounded functions $V_n^N\left(x\right)$ is equicontinuous.

Lemma 1.

If function $V_{n+1}^N$ is convex, then function $V_n^N$ is also convex.

Proof. 

Let us consider for each $x_1,x_2$ and $\alpha \in [0,1]$

$$\begin{array}{cc}\hfill V_n^N(\alpha x_1+(1-\alpha )x_2)& =\underset{m_n^N\le y\le M_n^N}{max}\mathbb{E}{V}_{n+1}^N\left((\alpha x_1+(1-\alpha )x_2)\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}{\epsilon }_n\right)\right)=\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\underset{m_n^N\le y\le M_n^N}{max}(V_{n+1}^N\left((\alpha x_1+(1-\alpha )x_2)\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)+\hfill \\ \hfill & +V_{n+1}^N\left((\alpha x_1+(1-\alpha )x_2)\left(1-y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right))\le \hfill \\ \hfill & {\displaystyle \frac{1}{2}}\underset{m_n^N\le y\le M_n^N}{max}(\alpha V_{n+1}^N\left(x_1\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)+\hfill \\ \hfill & +(1-\alpha )V_{n+1}^N\left(x_2\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)+\hfill \\ \hfill & +\alpha V_{n+1}^N\left(x_1\left(1-y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)+\hfill \\ \hfill & +(1-\alpha )V_{n+1}^N\left(x_2\left(1-y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right))=\hfill \\ \hfill & =\underset{m_n^N\le y\le M_n^N}{max}(\alpha \mathbb{E}{V}_{n+1}^N\left(x_1\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)+\hfill \\ \hfill & +(1-\alpha )\mathbb{E}{V}_{n+1}^N\left(x_2\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right))\le \hfill \\ \hfill & \alpha \underset{m_n^N\le y\le M_n^N}{max}\mathbb{E}{V}_{n+1}^N\left(x_1\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)+\hfill \\ \hfill & +(1-\alpha )\underset{m_n^N\le y\le M_n^N}{max}\mathbb{E}{V}_{n+1}^N\left(x_2\left(1+y\sqrt{{\displaystyle \frac{T}{N}}}\right)\right)=\hfill \\ \hfill & =\alpha V_n^N\left(x_1\right)+(1-\alpha )V_n^N\left(x_2\right).\hfill \end{array}$$

   □

Theorem 3.

If $f\left(x\right)$ is a continuous and convex function, then the solution to the nonlinear heat Equation (7) is a convex function satisfying the linear heat equation:

$${\displaystyle \frac{\partial \overline{V}}{\partial t}}+{\displaystyle \frac{x^2{M}^2\left(t\right)}{2}}{\displaystyle \frac{{\partial }^2\overline{V}}{\partial x^2}}=0.$$

(8)

Proof .

Using backward induction and Lemma 1, it is easy to prove that the sequence $V_n^N\left(x\right)$ is a sequence of convex functions. Using the second-order Taylor formula, we can write an approximate equation for this sequence:

 

$$\begin{array}{cc}\hfill {\tilde{V}}_n^N\left(x\right)& ={\tilde{V}}_{n+1}^N\left(x\right)+{\displaystyle \frac{T}{2N}}\underset{m_n^N\le y\le M_n^N}{max}{y}^2{\left({\tilde{V}}_{n+1}^N\left(x\right)\right)}^{″}\hfill \\ \hfill & ={\tilde{V}}_{n+1}^N\left(x\right)+{\displaystyle \frac{T}{2N}}\left[{\displaystyle \frac{{\left(M_n^N\right)}^2-{\left(m_n^N\right)}^2}{2}}\left|{\left({\tilde{V}}_{n+1}^N\left(x\right)\right)}^{″}\right|+{\displaystyle \frac{{\left(M_n^N\right)}^2+{\left(m_n^N\right)}^2}{2}}{\left({\tilde{V}}_{n+1}^N\left(x\right)\right)}^{″}\right],\hfill \\ \hfill {\tilde{V}}_N^N\left(x\right)& =f\left(x\right).\hfill \end{array}$$

Let us define right semi-continuous piecewise constant functions:

$${\tilde{V}}^N(t,x)={\tilde{V}}_{n-1}^N\left(x\right),(n-1){\displaystyle \frac{T}{N}}\le t<n{\displaystyle \frac{T}{N}},V^N(t,x)=V_{n-1}^N\left(x\right),(n-1){\displaystyle \frac{T}{N}}\le t<n{\displaystyle \frac{T}{N}}.$$

The first one is an approximate solution of the Hamilton–Jacobi–Bellman equation, therefore ${lim}_{N\to \infty }{\tilde{V}}^N(t,x)=V(t,x)$ for any x.

For the second ${lim}_{N\to \infty }(V^N(t,x)-{\tilde{V}}^N(t,x))=0$ for any x.

The presented discrete scheme is monotone, consistent, and stable. Hence, the Barles–Souganidis convergence theorem guarantees that the numerical solution converges locally uniformly on compact subsets of $[0,T]\times (0,\infty )$ to the unique viscosity solution [30].

Therefore for the second ${lim}_{N\to \infty }{V}^N(t,x)=\overline{V}(t,x)$. Function $\overline{V}(t,x)$ is convex, because $V_n^N\left(x\right)$ is uniformly Lipschitz on compacts and applying Arzel‘a-Ascoli [31] is sufficient to ensure the limit remains convex.

Note 1.

The reduction from the nonlinear to the linear heat equation for convex payoffs relies on a comparison principle for degenerate parabolic equations (e.g., Crandall–Lions [29]).

Note 2.

For convex function ${\displaystyle \frac{{\partial }^2\overline{V}}{\partial x^2}}\ge 0$. This allows us to remove the absolute value operator.

Let us consider the problem of calculating a supermartingale

$$Y\left(t\right)=\underset{\mathcal{Q}\in P^W\times {\mathcal{P}}_{\Sigma }}{inf}{E}_{\mathcal{Q}}(f\left(S_T\right)\mid F_t^W).$$

To calculate a supermartingale $Y\left(t\right)$ we use the Hamilton–Jacobi–Bellman equation ${\displaystyle \frac{\partial \underline{V}}{\partial t}}+{\displaystyle \frac{x^2}{2}}{inf}_{P\in \mathcal{P}\left(t\right)}E{\displaystyle \frac{{\partial }^2\underline{V}}{\partial x^2}}=0,$ which is transformed into a nonlinear heat equation

$${\displaystyle \frac{\partial \underline{V}}{\partial t}}+{\displaystyle \frac{x^2}{2}}\left[-{\displaystyle \frac{M^2\left(t\right)-m^2\left(t\right)}{2}}\left|{\displaystyle \frac{{\partial }^2\underline{V}}{\partial x^2}}\right|+{\displaystyle \frac{M^2\left(t\right)+m^2\left(t\right)}{2}}{\displaystyle \frac{{\partial }^2\underline{V}}{\partial x^2}}\right]=0$$

(9)

with the boundary condition $\underline{V}(T,x)=f\left(x\right)$.    □

Theorem 4

([29]). If $f\left(x\right)$ is a continuous function, then there exists a unique viscosity solution of Equation (9) $Y\left(t\right)=\underline{V}(t,x)$, which is a supermartingale.

Theorem 5.

If $f\left(x\right)$ is a continuous and convex function, then the solution to the nonlinear heat Equation (9) is a convex function satisfying the linear heat equation:

$${\displaystyle \frac{\partial \underline{V}}{\partial t}}+{\displaystyle \frac{x^2{m}^2\left(t\right)}{2}}{\displaystyle \frac{{\partial }^2\underline{V}}{\partial x^2}}=0.$$

(10)

The proof of Theorem 5 is similar to the proof of Theorem 3.

Note 3.

If $f\left(x\right)=\exp (-rT){(x-K)}^{+}$, $m\left(t\right)=\underline{\sigma }$, $M\left(t\right)=\overline{\sigma }$, then $C\in \left[\underline{C},\overline{C}\right]$, where

$$\begin{array}{cc}\hfill \underline{C}& =S_0\mathrm{\Phi }\left({\displaystyle \frac{\ln \left(\frac{S_0}{K}\right)+T\left(r+\frac{{\underline{\sigma }}^2}{2}\right)}{\underline{\sigma }\sqrt{T}}}\right)-K\exp (-rT)\mathrm{\Phi }\left({\displaystyle \frac{\ln \left(\frac{S_0}{K}\right)+T\left(r-\frac{{\underline{\sigma }}^2}{2}\right)}{\underline{\sigma }\sqrt{T}}}\right),\hfill \\ \hfill \overline{C}& =S_0\mathrm{\Phi }\left({\displaystyle \frac{\ln \left(\frac{S_0}{K}\right)+T\left(r+\frac{{\overline{\sigma }}^2}{2}\right)}{\overline{\sigma }\sqrt{T}}}\right)-K\exp (-rT)\mathrm{\Phi }\left({\displaystyle \frac{\ln \left(\frac{S_0}{K}\right)+T\left(r-\frac{{\overline{\sigma }}^2}{2}\right)}{\overline{\sigma }\sqrt{T}}}\right).\hfill \end{array}$$

When $m\left(t\right)=M\left(t\right)=\sigma$ the interval collapses and both bounds coincide with Black–Scholes.

Note 4.

If $m\left(t\right)=\underline{\sigma },M\left(t\right)=\overline{\sigma }$, then we will say that we are dealing with the first model of uncertain volatility. If $m\left(t\right)=X_0\exp \left(\underline{\sigma }t\right),M\left(t\right)=X_0\exp \left(\overline{\sigma }t\right)$, then we will say that we are dealing with the second model of uncertain volatility.

So, the main contributions are the following: (i) discrete Rademacher construction, (ii) discrete-time convexity preservation, (iii) convergence to viscosity solution, (iv) empirical illustration. In the next section the methods of parameter estimation will be presented.

5. Parameter Estimation

To estimate parameters in the Heston model we may use the maximum likelihood estimation method [32]. Let $R_n={\displaystyle \frac{\Delta S_n}{S_{n-1}}},n=1,\dots ,N.$ From (6) follows that $R_n\sim N(0,{\sigma }_{n-1}h),{\sigma }_n\sim N\left({\sigma }_{n-1}+k(\theta -{\sigma }_{n-1})h,{\nu }^2{\sigma }_{n-1}h\right).$ Further, since ${\delta }_n^1$ and ${\delta }_n^2$ have correlation $\rho$, $R_n$ and ${\sigma }_n$ have that same correlation $\rho$. Based on those properties of $R_n$ and ${\sigma }_n$, the joint probability density function

$$\begin{array}{cc}\hfill F(R_n,{\sigma }_n)& ={\displaystyle \frac{1}{2\pi \nu {\sigma }_{n-1}h\sqrt{1-{\rho }^2}}}\exp [-{\displaystyle \frac{R_n^2}{2{\sigma }_{n-1}h(1-{\rho }^2)}}+{\displaystyle \frac{\rho R_n\left({\sigma }_n-{\sigma }_{n-1}-k(\theta -{\sigma }_{n-1})h\right)}{{\nu }^2{\sigma }_{n-1}^2{h}^2(1-{\rho }^2)}}\hfill \\ & -{\displaystyle \frac{{\left({\sigma }_n-{\sigma }_{n-1}-k(\theta -{\sigma }_{n-1})h\right)}^2}{2{\nu }^2{\sigma }_{n-1}h(1-{\rho }^2)}}].\hfill \end{array}$$

By maximizing the likelihood function $L(r,k,\theta ,\nu ,\rho )={\prod }_{n=1}^{N}F(R_n,{\sigma }_n\mid r,k,\theta ,\nu ,\rho )$ we may estimate parameters $r,k,\theta ,\nu ,\rho$. This algorithm must be used in this regard so that the constraints on the parameters $r\in \mathbb{R}$, $k\ge 0$, $\theta \ge 0$, $\nu \ge 0$ and $-1\le \rho \le 1$ are respected.

For calibration, the bounds $\underline{\sigma }$ and $\overline{\sigma }$ confidence set technology [33] may be used. Let $V={\left({\sigma }_i\right)}_{i=1}^N$ be a sample of historical volatilities. A subsample $D\subset V$ is a confidence set if for all ${\sigma }_i\in V$, $P({\sigma }_i\in D)\ge \eta ,$ where $\eta$ is a confidence probability. The number of elements in a confidence set $\left|D\right|=\lfloor \eta V\rfloor +1=L$. For a sample consisting of one-dimensional data, the confidence set is the interval of minimum length $[\underline{\sigma },\overline{\sigma }]$. To find it, the following algorithm may be applied (Algorithm 1).

Indeed, obtaining any interval containing a given number of points of an ordered sample can be represented as successively removing the left extreme point or the right extreme point from the ordered subset. The algorithm chooses the extreme point to be optimally deleted.

As experience has shown, using a confidence interval $[\underline{\sigma },\overline{\sigma }]$ is much more effective than using historical high $\left({\sigma }_N\right)$ or historical low $\left({\sigma }_1\right)$ volatility because we do not take into account unwanted emissions.

Algorithm 1 Finding the interval of minimum length

Let us assume that elements of the sample V are ordered in ascending order: ${\sigma }_1<\cdots <{\sigma }_N$. Let $i=1$, $j=N$, $k=1$. Update: $$i=\left\{\begin{array}{cc}i+1,\hfill & {\sigma }_{i+1}-{\sigma }_i>{\sigma }_j-{\sigma }_{j-1},\hfill \\ i,\hfill & {\sigma }_{i+1}-{\sigma }_i\le {\sigma }_j-{\sigma }_{j-1};\hfill \end{array}\right.$$ $$j=\left\{\begin{array}{cc}j,\hfill & {\sigma }_{i+1}-{\sigma }_i>{\sigma }_j-{\sigma }_{j-1},\hfill \\ j-1,\hfill & {\sigma }_{i+1}-{\sigma }_i\le {\sigma }_j-{\sigma }_{j-1};\hfill \end{array}\right.$$ $k=k+1$. If $k>N-L$ then set $\underline{\sigma }={\sigma }_i$, $\overline{\sigma }={\sigma }_j$ and stop; else go to step 2. Statement. The algorithm calculates the interval of minimum length.

6. Example 1

Let us consider the S&P100 index for the period from 2 January 1991 to 11 June 1997.

In [32] the parameters for the Heston and Black–Scholes models were estimated using two different methods: the method of moments (MOM) and maximum likelihood estimation (MLE). We used the MLE parameter estimates: $r=0.00064$, $k=0.00657$, $\theta =0.0000647$, $\nu =0.000509$, $\rho =-0.00198$.

For the Black–Scholes model $\sigma =0.00723$, $r=0.0485$.

For the uncertain volatility model parameter estimates for a sample of index values $\underline{\sigma },\overline{\sigma }:\underline{\sigma }=0.005,\overline{\sigma }=0.01$. We used confidence set technology described above, sample size $N=2352$, confidence probability $\eta =0.95$.

The sample of data contains options with expiration dates that were 24 days, 87 days and 115 days into the future. With this expiration separation, in [32] the abilities of the Heston model and the Black–Scholes model are tested to accurately estimate the premiums of short-term, mid-term and long-term options.

We calculated upper prices $\overline{C}$ and lower prices $\underline{C}$ for the first model of uncertain volatility (

Table 1. Comparison of option prices (Part 1).

Expiration	Stock Price	Strike Price	Actual Call	Black–Scholes Call	Heston Call	$\overline{\mathbf{C}}$	$\underline{\mathbf{C}}$
24	425.73	395	30.75	32.55	30.83	30.45	31.49
24	425.73	400	25.88	27.57	25.95	25.55	26.15
24	425.73	405	21.00	22.60	21.23	20.56	21.61
24	425.67	410	16.50	17.56	16.71	16.05	17.02
24	425.68	415	11.88	12.53	12.65	11.62	12.10
24	425.65	420	7.69	7.26	9.09	7.15	8.03
24	425.65	425	4.44	2.37	6.18	3.51	4.56
24	425.68	430	2.10	0.00	3.99	1.73	2.46
24	425.65	435	0.78	0.00	2.39	0.74	0.93
24	425.16	440	0.25	0.00	1.26	0.18	0.56
24	424.78	445	0.10	0.00	0.62	0.09	0.12
24	425.19	450	0.10	0.00	0.32	0.09	0.13
24	425.73	395	30.75	32.55	30.83	30.45	31.49
24	425.73	400	25.88	27.57	25.95	25.55	26.15
24	425.73	405	21.00	22.60	21.23	20.56	21.61
24	425.67	410	16.50	17.56	16.71	16.05	17.02
24	425.68	415	11.88	12.53	12.65	11.62	12.10

$\overline{C}$: upper bound; $\underline{C}$: lower bound.

and

Table 2. Comparison of option prices (Part 2).

Expiration	Stock Price	Strike Price	Actual Call	Black–Scholes Call	Heston Call	$\overline{\mathbf{C}}$	$\underline{\mathbf{C}}$
24	425.65	420	7.69	7.26	9.09	7.15	8.03
24	425.65	425	4.44	2.37	6.18	3.51	4.56
24	425.68	430	2.10	0.00	3.99	1.73	2.46
24	425.65	435	0.78	0.00	2.39	0.74	0.93
24	425.16	440	0.25	0.00	1.26	0.18	0.56
24	424.78	445	0.10	0.00	0.62	0.09	0.12
24	425.19	450	0.10	0.00	0.32	0.09	0.13
87	425.73	380	46.75	52.04	46.17	46.70	47.19
87	425.73	385	42.00	47.12	41.40	41.38	43.88
87	425.73	390	37.50	42.21	36.75	36.98	43.07
87	425.73	395	33.00	37.29	32.25	32.08	36.66
87	425.73	400	28.50	32.37	27.96	27.99	29.12
87	425.73	405	24.13	27.43	23.91	22.99	26.19
87	425.26	410	20.38	21.99	19.81	19.49	20.15
87	425.86	415	16.13	17.58	16.82	14.13	18.31
87	425.68	420	12.82	12.41	13.63	11.56	13.62
87	425.42	425	9.32	7.54	10.81	8.67	11.67
87	425.62	430	6.51	3.93	8.60	6.01	8.11
87	425.82	435	4.51	1.48	6.74	4.01	6.07
87	425.68	440	2.75	0.14	5.09	2.05	3.88
87	425.75	445	1.60	0.00	3.82	1.23	2.06
87	425.78	450	0.85	0.00	2.81	0.74	1.03
87	425.39	455	0.44	0.00	1.97	0.31	0.71
115	425.73	380	47.25	54.05	46.50	46.93	49.23
115	425.73	390	38.13	44.27	37.31	37.94	39.14
115	425.73	400	29.38	34.48	28.82	28.94	38.05
115	425.73	410	21.19	24.63	21.28	20.94	22.30
115	425.41	420	13.88	14.50	14.77	13.39	15.31
115	425.63	430	8.13	6.18	9.93	8.01	10.15
115	425.28	440	3.88	1.15	6.16	2.64	4.18
115	425.13	450	1.50	0.00	3.64	1.21	1.73

$\overline{C}$: upper bound; $\underline{C}$: lower bound.

). The value $\overline{C}=\overline{V}(0,S_0)$, where $\overline{V}(t,x)$ is the solution to problem (8) for $M\left(t\right)=\overline{\sigma }$. The value $\underline{C}=\underline{V}(0,S_0)$, where $\underline{V}(t,x)$ is the solution to problem (10) for $m\left(t\right)=\underline{\sigma }$.

In [32] the root-mean-square error (RMSE) is used to compare the Black–Scholes model’s estimates and the Heston model’s estimates to the actual premiums. We used RMSE to compare the values ${\displaystyle \frac{\overline{C}+\underline{C}}{2}}$ to the actual premiums. The uncertain volatility model provides estimates that are closer than the Black–Scholes and the Heston models’ estimates to actual transaction data (

Table 3. RMSE results.

	24 Days	87 Days	115 Days
Black–Scholes model	1.28	3.11	4.12
Heston model	1.13	1.90	2.13
Uncertain volatility model	0.02	0.66	0.14

).

7. Example 2

Let us consider the second model of uncertain volatility. Let $\underline{\sigma }=-0.1,\overline{\sigma }=0.2,$$X_0=0.1,r=0.1,S_0=K=5,T=1.$ The Figure 1 shows the graphs of the dependence $\overline{V}(0,x),\underline{V}(0,x)$ and $V(0,x)$ on x, where $\overline{V}(t,x)$ is the solution to problem (8); $\underline{V}(t,x)$ is the solution to problem (10); $V(t,x)$ is the solution to problem (1) for $\sigma =X_0.$ The solid line shows the graph of dependence of $V(t,x)$ on x. The dot line shows the graph of dependence of $\overline{V}(0,x)$ on x. The dash line shows the graph of dependence of $\underline{V}(t,x)$ on x.

8. Results

This article examines a model of uncertain volatility, a generalization of the well-known Black–Scholes model. Theorems are proven that allow us to estimate the fair price range for an important class of payoff functions. An example of calculating the fair price range for a European call option is given.

9. Discussion

This article is a continuation of [22], which examined two models of uncertain volatility and presented three computational methods for calculating the fair value range of a European-style option. The first method is based on solving the Hamilton–Jacobi–Bellman equations using difference schemes. The second is the Monte Carlo method, which is based on modeling the original stock price process. The third is the tree method, which is based on approximating the original continuous model with a discrete model and obtaining recurrence formulas on a binary tree.

This article proves theorems stating that if the boundary condition is a continuous and convex function, then the solutions to the nonlinear heat equations are convex functions satisfying the linear heat equations. While the central convex-reduction result is classical in the uncertain volatility and G-expectation literature [23,24,25], the novelty of this work is the method of proof, based on discrete Rademacher construction and backward-induction argument. This method ensures that convexity is preserved at each discrete step, and a Taylor expansion produces the quadratic (variance) term that leads to the nonlinear heat equation. Taking the limit then yields the viscosity solution representing the upper and lower prices.

For an important special case, formulas for calculating the fair value range are obtained, which are analogous to the Black–Scholes formula. Calculations using real data are presented, demonstrating the effectiveness of uncertain volatility models compared to the Black–Scholes and Heston models for calculating fair prices for European-style options.

A promising area for future research is the development of a dynamic neural network implementation of uncertain volatility models, e.g., surrogate modeling of option price functionals under bounded volatility, or learning volatility bounds from data within a robust framework.