#### 5.1. Entropy in Option Pricing

Back in 1996, Buchen and Kelly tackled the general problem of how to extract an asset probability distribution from limited and incomplete market information available on an options exchange. The implementations of the MEP in the cases of simple call and put options were showed as below. They assumed that the underlying asset pays no dividends, and they set up usual assumptions for European options in an equilibrium market, where the usual assumptions including no transaction expenses and no taxes, no arbitrage opportunities. The Maximum Entropy Distribution (MED) is given in Equation (34) with constraint functions

c_{i}(

x):

where the

${\lambda}_{i}$ is the Lagrange multiplier.

Let

i index the strike prices

K_{i}, then for a call option:

and for a put option:

The function D(T) denotes the non-stochastic present-value discount factor to time T and acts only as a multiplicative constant. For example, we can choose $D\left(T\right)={e}^{-rT}$ with a constant risk-free rate r or some other suitable representation such as the price of a bond with face value $1 and maturing at time T.

They developed an expectation pricing model and a set of option prices at different strikes. Those data were insufficient to determine the distribution of the underlying asset. However, if these prices were used to constrain the distribution that otherwise had maximum entropy (or minimum cross-entropy), a unique distribution was obtained. Their research showed that the maximum entropy distribution was able to fit a known density accurately, given simulated option prices at different strikes [

18].

Buchen and Kelly’s method has important meanings and attracted some authors to continue their research and compare it with other methods. One criticism often raised is that the method of finding the form of the density using Lagrange multipliers is not rigorous. Instead of the Lagrange multiplier method, Neri and Schneider [

19] set up a simple and robust algorithm for the maximum entropy distribution. This algorithm worked well in practice and led to the correct form, and they also gave a complete mathematical proof. Their proof used results of Csiszár [

62] that gave additional insights into “distances” between distributions and establish remarkable “geometric” results.

In the field of option pricing, Gulko [

15] has to be mentioned here. He formulated the maximum-entropy framework by introducing the EPT in 1997. In his opinion, the EPT offers the construction of unique risk-neutral probabilities in both complete and incomplete market economies. The EPT suggests that, in information efficient markets, plausible market beliefs are not equally likely and that the maximum-entropy or maximum missing information must prevail. He also applied the EPT to the canonical option valuation problems—stock option pricing [

16] and bond option pricing [

17]. He claimed that the maximum-entropy density

p(

r) is a joint normal density in the return space

$R={R}^{n}$ which is a dart board and repeat the dart-board argument on

R. He showed that a joint normal density solves the following program:

subject to

${{\displaystyle \int}}_{R}udr=1,\text{}{{\displaystyle \int}}_{R}ur{r}^{T}dr=V$,

u(

r) > 0 on

R.

The maximum is a probability density of the form

$\frac{exp\left({{\displaystyle \sum}}_{ij}{\lambda}_{ij}{r}_{i}{r}_{j}\right)}{{{\displaystyle \int}}_{R}exp\left({{\displaystyle \sum}}_{ij}{\lambda}_{ij}{r}_{i}{r}_{j}\right)dr}$, where

$\text{}{\lambda}_{ij}$ is the Lagrange multiplier associated with the moment constraint

v_{ij},

r = [

r_{i}]

$\in {R}^{n}$ is the column vector of random stock returns, where

r_{i} is a random return on asset

i for 1 ≤

i ≤

n,

p(

r) is the maximum-entropy joint probability density of stock returns, and

μ = [

μ_{i}]

≡ E(

r) with

μ_{i} ≡

E(

r_{i}) for 1 ≤

i ≤

n,

V = [

v_{j}] is a known covariance matrix of the returns with

v_{ij} = E[(

r_{i}−

μ_{i})(

r_{j}−

μ_{j})] for 1 ≤

i,

j ≤

n. Assume the return on any stock cannot be expressed as a linear combination of the returns on other stocks, so the covariance matrix

V is nonsingular. Therefore, in a stock market, the maximum-entropy density

p(

r) is a joint normal as follows:

where |

V| denotes the determinant of

V. Also, by analogy with the univariate case, the joint density

f(

S_{T}) of the random stock prices

${S}_{T}\in {R}_{+}^{n}$ is a multivariate lognormal.

With current stock price

S, current riskless bond price

P and strike price

K, the gamma density is:

Where

$u=\frac{\left(\frac{S}{P}\right)}{{\sigma}^{2}},v=\frac{{\left(\frac{S}{P}\right)}^{2}}{{\sigma}^{2}}$.

Then the cumulative gamma distribution function can be denoted by:

and the complementary cumulative gamma function can be denoted by:

Therefore, the gamma prices of European call and put options on dividend protected stocks are respectively as following:

The gamma formulae above require only four “observable” inputs: K, P, S, and σ^{2}. These formulae have the structure and simplicity of the Black-Scholes stock option model. They also feature many Black-Scholes-like properties. However, unlike the Black-Scholes model, the gamma model imposes no restrictions on the dynamics of the stock price or stock returns. Instead, the gamma model parameterizes the price dynamics of an imaginary constant-cash-flow security. The price process S(t) for the actual stock is immune to this parameterization. For example, the parameterization of x(t) does not prevent the volatility of the actual stock price from being random. Also, unlike the Black-Scholes model, the gamma formula is valid for arbitrary dynamics of short-term interest rates.

Similar to the stock option pricing, Gulko applied the EPT to the famous Vasicek-Jamshidian model (the model specifies that the instantaneous interest rate follows the stochastic differential equation:

where

W_{t} is a Wiener process under the risk neutral framework modeling the random market risk factor, in that it models the continuous inflow of randomness into the system. The standard deviation parameter,

σ, determines the volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters

b,

a and

σ, together with the initial condition

r_{0} , completely characterize the dynamics, and can be quickly characterized as follows, assuming

a to be non-negative) [

59] and made some useful improvements [

17]. Gulko’s new model was different from Vasicek-Jamshidian model in the following ways that show its advantages. First, unlike the Vasicek-Jamshidian model, the Call formula did not restrict movements of term structure of interest rates. Second, the Vasicek-Jamshidian model was valid only in a complete market, while the Call formula was valid in both complete and incomplete markets. Third, the Vasicek-Jamshidian model was suitable for pricing options on default-free bonds only, while the Call formula (43) was suitable for pricing options on both default-free and risky bonds.

However, Gulko’s method can only solve the problem of European bond option pricing. American bond options are generally believed difficult to price since the process requires numerical methods. In order to apply the entropy method to American bond option pricing, Zhou

et al. formulated a new entropy model on the basis of Gulko’s entropy pricing theory as well as Geske-Johnson’s method of analytical approximation of American options [

63]. In another thesis Zhou

et al. [

64] believed that the option pricing in incomplete markets should differ from that in complete markets. Therefore the classical Black-Scholes option pricing model may be unsuitable in incomplete markets. They developed an analytical formula to value caps, floors, collars and swaptions of interest rates with parallel to Black-Scholes option pricing model on the basis of the entropy pricing method. Zhou and Wang [

65] extended the research to the hedging parameters in incomplete markets. They followed the Black-Scholes model to define and formulate a series of hedging parameters, and performed numerical simulations based on the entropy model. They compared the results with those obtained under the framework of the Black-Scholes model. Their results showed that the sensitivity degrees of the entropy model were larger than those of the Black-Scholes model and therefore proved the entropy model is a new and better method for the risk management of derivatives in an incomplete market.

Recently, Trivellato illustrated some financial applications of the Tsallis and Kaniadakis deformed exponentials [

66,

67]. The Kaniadakis exponential [

68] was introduced to define a new family of martingale measures based on the standard entropy martingale measure [

27] and the well-known

p-martingale measures [

30]. It has proven to be suitable to explain a very large class of experimentally observed phenomena [

69,

70,

71]. The

φ-logarithm was used to introduce the notion of

φ-divergence between two probability measures, which extends the standard Kullback-Leibler divergence. The minimization of this deformed divergence was proposed as a general criterion to select a pricing measure in incomplete markets. He investigated the relationships between this relative entropy and the deformed Tsallis and Kaniadakis relative entropies, and illustrated their applications in finance, especially generalizing the well-known Black-Scholes model. The Kaniadakis entropy

${H}_{\alpha}^{K}\left(Q|P\right)$ can be described as follows:

where

$\alpha \in \left[-1,1\right]$.