Martingales and Partial Differential Equations Price Valuation Models for European Put Options

Abstract

In financial mathematics, options are seen as financial transactions that gives the holder the right rather than obligations to buy or sell some specific quantity of an asset in the eminent future at a static price often called the strike price on or before the expiration date of the option contract. In this research work, we examined a typical model in finance, Constant Elasticity of Variance (CEV). Having derived its respective Stochastic Differential Equations (SDEs), we obtained the various Martingales and Partial Differential Equations (PDEs) option price valuation formulas. These were done by using the replicating and riskless portfolio methods. Also, we were able to establish the equivalence of the Martingales and the PDEs methods for European put option pricing for the two different SDE models considered in this research. We, then applied the Girsanov theorem, Martingale theorem as well as Feynman-Kac theorem. Results obtained, show that using our approach described in this research work, we can actually prove the equivalence of the two (Martingales and PDEs) methods by first, beginning with the Martingales option price valuation formula and finally arrive at the Black-Scholes parabolic PDEs in converse.

1. Introduction

In finance, derivatives are usually tools that are attached to a specific asset and through which definite risks in finance can be disposed or handled in financial markets. Various transactions in form of derivatives ought to be treated separately. The value of derivatives in finance is often obtained from the exact prices of the fundamental asset value. Since the future reference price of the underlying item is not known with certainty, the value of the financial derivative at maturity can only be predicted. These derivatives in finance, can be applied for various reasons such as arbitrage, hedging, etc. [1].

Options are one of the examples of derivatives in finance. They are one of the most vital financial instruments. Their recent applications in hedging and arbitrage have earned it widespread attention. There are basically four of its kinds; European option, Asian option, American Option and barrier option [2]. Many reasons may inspire an investor to trade in options rather than trading on the commodities or stocks directly. One of such key reasons is that options save transactions and aids an investor in avoiding risks and market restrictions and fluctuations. Trading options in conjunction with their stock portfolios aids investors in carefully adjusting the risk and return of their investments. Since its introduction in financial mathematics, by Black, Scholes and Merton, the interest in derivatives trading has greatly increased [3].

Determining the values of financial derivatives such as options has since constituted a major problem in the field of financial mathematics. There are two key approaches to derive valuation formulas for a specific financial derivative such as options. These are the Martingales and Partial Differential Equations (PDEs), approaches [4]. In the Martingale approach, the replicating portfolio which is obtained from underlying assets is expected to be self-financing. Since there is no arbitrage, the identified numeraire that converts the replicating portfolio into martingales is used. Hence, value of the derivative is the expectation of the value of the payoff at expiration.

In the case of the PDE approach, a riskless portfolio is formed with the elimination of the stochastic component such that the Stochastic Differential Equation turns to a PDE. A boundary condition is then specified and the resulting PDE is solved by applying the Feynman-Kac theorem. Let us state here that the two approaches as described above are equal [5], even though they both generate different option price valuation formulas; but showing a detailed proof for this is often a difficult task.

This chapter will be centered on the use of advanced mathematical tools (Girsanov and Feynman-Kac Theorems) to carry out a comparative analysis of the Martingale and PDE approaches in the pricing of options using Stochastic Differential Equation (SDE) model in finance as practical examples.

1.1 Definition of terms

In a market economy, we assume there are $N$ assets with values $S_1\left(t\right),\cdots ,S_N\left(t\right)$ at time $t,$ each of which follows a stochastic process.

1.1.1 Option value

Option values are often a function of the underlying asset and time, $u_t=f\left(S_t,t\right)$. The calculation of the price of an option (Premium) is our primary concern. For call option, the fundamental value is given by $\left(S_t-K,0\right)$ and for put option, it is given as $\left(K-S_t,0\right),$ $0\le t\le T.$ This is actually the value representing the profit an investor gains by exercising his right on the option. The call value is the difference between the underlying price and its fundamental value [6].

1.1.2 Options and replication

Payoff, $V_T$ at time $T$ of an option is always a function of the underlying asset. We look out for a self-financing trading strategy that replicates the same payoff such that ${\prod }_T=V_T$. This implies a replicating strategy and replicating portfolio [7].

1.1.3 Self-financing portfolio

A portfolio allocation ${\left({\xi }_t,{\eta }_t\right)}_{\mathit{tϵ}{\mathbb{R}}_{+}}$ with price (value) $V_t$ given by

$$V_t={\xi }_t{S}_t+{\eta }_t{A}_t,tϵ{\mathbb{R}}^{+}$$E1

is self-financing if it satisfies the relation

$$d{V}_t={\eta }_{t}d{A}_t+{\xi }_t{\mathit{dS}}_t$$E2

where ${\xi }_t$ is the stock shares in $S_t$ and ${\eta }_t$ is the deposit in the bank.

1.1.4 Risk-neutral measure

A probability measure ${\mathbb{P}}^{\ast }$ on a sample space, $\mathrm{\Omega }$ is called a risk-neutral measure if it satisfies

$${\mathbb{E}}^{\ast }\left[S_t\left|{\mathcal{F}}_u\right.\right]=e^{r\left(t-u\right)}{S}_u,0\le u\le t$$E3

Where ${\mathbb{E}}^{\ast }$ denotes the expectation under ${\mathbb{P}}^{\ast }$.

1.1.5 Numeraires

Numeraires are assets with positive price, $N_t>0$, for all given $t$. The comparative price $\widetilde{S_t}$ of an underlying asset is the ratio of the stock price $S_t$ to its numeraire price $N_t$, i.e. ${\tilde{S}}_t=\frac{S_t}{N_t}$ and $S$ is usually valued in terms of measured in units of $N$.

1.1.6 Equivalent martingale measure

Let $\mathbb{Q}$ and $\mathbb{P}$ be probability measures on a sample space $\mathrm{\Omega }$. A probability measure, $\mathbb{Q}$ is said to be an an Equivalent Martingale Measure (EMM) for a numeraire $N_t,$ if the following conditions are true:

1. $\mathbb{Q}˜\mathbb{P}$ and,

2. The relative price ${\tilde{S}}_t$ is a martingale under $\mathbb{Q}$. This implies

   $${\mathrm{E}}^{\mathbb{Q}}\left[\left.\frac{S_T}{N_T}\right|{\mathcal{F}}_t\right]=\frac{S_t}{N_t}\text{for}T>t.$$E4

For a complete market economy, given a numeraire $N,$ one can obviously find a unique EMM $\mathbb{N}$ such that asset prices discounted by $N$ are martingale under$\mathbb{N}$. On the contrary, given an EMM, $\mathbb{N}$ we can always find an exclusive numeraire $N$ so that asset prices discounted by $N$ are also, martingale under $\mathbb{N}.$

1.1.7 Martingale

An integrable process ${\left(X_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$ is said to be a martingale with respect to the filtration ${\left({\mathcal{F}}_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$ if

$$E\left[X_t\left|{\mathcal{F}}_s\right.\right]=X_s,0\le s\le t.$$E5

1.1.8 Fundamental theorem of arbitrage

This Theorem states that if the market is complete, then for each and every numeraire $N_t$ there exists a unique Equivalent Martingale Measure $\mathbb{N}$ such that the relative price of the assets using that numeraire, is a martingale. Therefore, $\frac{S_t}{N_t}$ is a martingale under$\mathbb{N}$. Hence

$$E^{\mathbb{N}}\left[\frac{S_T}{N_T}\left|{\mathcal{F}}_t\right.\right]=\frac{S_t}{N_t}$$E6

If we choose another numeraire $M_t,$ then $\frac{S_t}{M_t}$ is no longer a martingale under $\mathbb{N},$ but there exists another unique equivalent martingale measure, ${\mathbb{M}}_t$ so that $\frac{S_t}{M_t}$ is a martingale under$\mathbb{M}$. Hence,

$$E^{\mathbb{M}}\left[\frac{S_T}{M_T}\left|{\mathcal{F}}_t\right.\right]=\frac{S_t}{M_t}$$E7

Other financial terms relevant to the discussions in this research work are defined as follows:

Asset (e.g. Stock, interest rates, commodities, etc.): these are objects that provides a claim to future cash flow.

Strike price: this is defined as a fixed price for exercising rights on an option.

Portfolio: a portfolio (or stock portfolio) is an investment in several assets at the same time. The idea of holding portfolio is that it is expected that this reduces the risk in investment.

2. Methods

2.1 Assumptions, parameters and variables of option price models

2.1.1 Assumptions

The assumptions used in the solutions of the model are as follows:

1. All assets (stock) considered this research work are dividends free.

2. There exists a riskless investment portfolio with a guaranteed return on investment and no chance of default.

3. There exists no arbitrage in the market which implies a complete market economy.

2.1.2 Parameters and variables

The value of the option depends on the following parameters and variables:

$S\left(t\right),X\left(t\right),X_t\text{or}{S}_t:$ the underlying asset (stock price) at time $t$.

$\sigma :$ the volatility of the underlying asset (measure of the standard deviation of the returns of the asset).

$K:$ the exercise or strike price.

$T:$ time of expiry.

$\mathrm{dt}:$ step-size.

$r:$ interest rate.

$\mu :$ average rate of growth of the asset.

$\mathit{dS}\text{or}\mathit{dX}:$ change in stock price (always positive).

2.2 Preliminaries for model derivation

2.2.1 Girsanov theorem

The basic characteristics of the Girsanov theorem is that of the exponential Martingale

$$Z_t=\mathit{exp}\left(-\frac{1}{2}{\int }_0^t{\theta }_s^2\mathit{ds}+{\int }_0^t{\theta }_{s}d{W}_s\right).$$E8

Then, the Girsanov Theorem is stated as follows:

Theorem 2.1 (Girsanov Theorem) [8]: The process $\widetilde{W_t}=W_t+{\int }_0^t{\varphi }_s\mathit{ds}$ is Brownian motion under the measure $\mathbb{Q}.$

Theorem 2.2: Let ${\left({\mathrm{\varphi }}_t\right)}_{\mathit{tϵ}\left[0,T\right]}$ be a stochastic process satisfying the Novikov integrability condition

$$E\left[\mathit{exp}\left(\frac{1}{2}{\int }_0^T{\left|\left.\varphi \right|\right.}^2\mathit{dt}\right)\right]<\infty$$

and let $\mathbb{Q}$ denote the probability measure defined by

$$\frac{d\mathbb{Q}}{d\mathbb{P}}=\mathit{exp}\left(-{\int }_0^T{\varphi }_{s}d{W}_s-\frac{1}{2}{\int }_0^T{\varphi }_s^2\mathit{ds}\right)$$

then

$${\tilde{W}}_t=W_t+{\int }_0^t{\varphi }_s\mathit{ds},tϵ\left[0,T\right],$$

Is certainly a Brownian motion under the measure, $\mathbb{Q}.$

Theorem 2.3 (Martingale Representation Theorem) [8]: Assuming $M_t$ is an ${\mathcal{F}}_t-$martingale where ${\left\{{\mathcal{F}}_t\right\}}_{t\ge 0}$ is the filteration generated by the $n$-dimensional Brownian motion, $W_t=\left(W_t^{\left(1\right)},\dots ,W_t^{\left(n\right)}\right).$ If $E\left[M_t^2\right]<\infty$ for all $t$ then there exists a unique $n$-dimensional adapted stochastic process, ${\varphi }_t$ such that

$$M_t=M_0+{\int }_0^t{\varphi }_s^{T}d{W}_t\text{for}\mathrm{all}t\ge 0$$

where ${\varphi }_s^T$denotes the transpose of the vector, ${\varphi }_s.$

Lemma 2.1 [9]: Let ${\left({\xi }_t,{\eta }_t\right)}_{\mathit{tϵ}{\mathbb{R}}_{+}}$ be a portfolio strategy with value

$$V_t={\eta }_t{A}_t+{\xi }_t{S}_t,tϵ{\mathbb{R}}^{+}$$

The following statements are equivalent:

(i) the portfolio strategy ${\left({\xi }_t,{\eta }_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$ is self-financing,

(ii)

$$\widetilde{V_t}={\tilde{V}}_0+{\int }_0^t{\xi }_{u}d{X}_u,tϵ$$E9

2.2.2 Ito formula

Let us consider the general Ito’s formula which is applicable to Ito processes of the type [8]

$$X_t=X_0+{\int }_0^t{\mu }_s\mathit{ds}+{\int }_0^t{\sigma }_{s}d{W}_s,tϵ{\mathbb{R}}^{+}$$E10

or in differential notation

$$d{X}_t={\mu }_t\mathit{dt}+{\sigma }_{t}d{W}_t$$

where ${\left({\mu }_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$ and ${\left({\sigma }_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$ are square-integrable adapted processes.

Lemma 2.2: For any Ito process ${\left(X_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$ of the form Eq. (9) and any $\mathit{fϵ}{C}^{1,2}\left({\mathbb{R}}^{+}\times \mathbb{R}\right)$ and $Z_t=f\left(t,X_t\right)$ we have,

$$Z_t=f\left(0,X_0\right)+{\int }_0^t{\mu }_s\frac{\partial f}{\partial x}\left(s,X_s\right)\mathit{ds}+{\int }_0^t{\sigma }_s\frac{\partial f}{\partial x}\left(s,X_s\right)d{B}_s+{\int }_0^t\frac{\partial f}{\partial t}\left(s,X_s\right)\mathit{ds}+\frac{1}{2}{\int }_0^t{\left|{\sigma }_s\right|}^2\frac{{\partial }^{2}f}{\partial x^2}\left(s,X_s\right)\mathit{ds}$$

or in differential form

$$\begin{array}{ll}d{Z}_t& =\frac{\partial f}{\partial t}\left(t,X_t\right)\mathit{dt}+\frac{\partial f}{\partial x}\left(t,X_t\right)d{X}_t+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}\left(t,X_t\right){\left(d{X}_t\right)}^2\\ & =\left(\frac{\partial f}{\partial t}\left(t,X_t\right)+\frac{\partial f}{\partial x}\left(t,X_t\right){\mu }_t+\frac{1}{2}\frac{{\partial }^{2}f}{\partial x^2}\left(t,X_t\right){\sigma }_t^2\right)\mathit{dt}+\frac{\partial f}{\partial x}\left(t,X_t\right){\sigma }_{t}d{W}_t.\end{array}$$E11

2.2.3 The Black-Scholes PDE

The Black-Scholes Partial Differential Equation (PDE) for the price of a self-financing portfolio is given as:

$$\mathit{rg}\left(t,x\right)=\frac{\partial g}{\partial t}\left(t,x\right)+\mathit{rx}\frac{\partial g}{\partial x}\left(t,x\right)+\frac{1}{2}{\sigma }^2{x}^2\frac{{\partial }^{2}g}{d{x}^2}\left(t,x\right),x>0,tϵ\left[0,T\right]$$

Proposition 2.1 [9]: Let ${\left({\xi }_t,{\eta }_t\right)}_{\mathit{tϵ}{\mathbb{R}}^{+}}$be a portfolio strategy such that.

1. ${\left({\xi }_t,{\eta }_t\right)}_{\mathit{tϵ}{\mathbb{R}}_{+}}$ is self-financing,

2. the value $V_t={\eta }_t{A}_t+{\xi }_t{S}_t,tϵ{\mathbb{R}}^{+}$, takes the form

   $$V_t=g\left(t,S_t\right)$$

for some $\mathit{gϵ}{C}^{1,2}\left(\left(0,\infty \right)\times \left(0,\infty \right)\right)$.

Then the function $g\left(t,x\right)$ satisfies the Black-Scholes PDE

$$\mathit{rg}\left(t,x\right)=\frac{\partial g}{\partial t}\left(t,x\right)+\mathit{rx}\frac{\partial g}{\partial x}\left(t,x\right)+\frac{1}{2}{\sigma }^2{x}^2\frac{{\partial }^{2}g}{d{x}^2}\left(t,x\right),x>0,tϵ\left[0,T\right]$$E12

and ${\xi }_t$ is given by

$${\xi }_t=\frac{\partial g}{\partial x}\left(t,S_t\right),tϵ{\mathbb{R}}^{+}.$$

2.2.4 The Black-Scholes European option price models

In a Black-Scholes European option price models, the concept of arbitrage allows one to establish relationships between prices and hence to determine them. A primary step which determines the theory of valuation of options is the issue of arbitrage. In the real sense of finance, there no prospects of making riskless profit but in most cases in finance, we assume the possibilities of riskless investment which assures return on investment with no choice of uncertainties. Such assumptions can be envisaged in situations of cash deposits in banks or in cases of government bond.

2.2.5 European call and put options

Financial derivatives such as options are financial gears that provides the option holder the right, but not compulsion, for a financial transaction of an underlying asset at a specific time and price [10]. The payoff function of the European Call Option with strike price$K$, is given by $F\left(x\right)={\left(x-K\right)}^{+}$ and the Black -Scholes PDE reads

$$\left\{\begin{array}{c}r{g}_c\left(t,x\right)=\frac{{\partial g}_c}{\partial t}\left(t,x\right)+\mathit{rx}\frac{{\partial g}_c}{\partial x}\left(t,x\right)+\frac{1}{2}{\sigma }^2{x}^2\frac{{\partial }^2{g}_c}{\partial x^2}\left(t,x\right)\\ g_c\left(T,x\right)={\left(x-K\right)}^{+}\end{array}\right.$$E13

As earlier stated, call options provide the buyers the right to buy assets at a fixed price, $K$ at the maturation time of the option. At a maturation times, if the asset $\left(x\right)>$ $K;$ the buyer gains instantaneous profit,$x-K$. However, if $\left(x\right)<$ $K;$ the buyer refuses to execute the option.

Similarly, for European put options, the Black-Scholes PDE is given as

$$\left\{\begin{array}{c}r{g}_p\left(t,x\right)=\frac{{\partial g}_p}{\partial t}\left(t,x\right)+\mathit{rx}\frac{{\partial g}_p}{\partial x}\left(t,x\right)+\frac{1}{2}{\sigma }^2{x}^2\frac{{\partial }^2{g}_p}{\partial x^2}\left(t,x\right)\\ g_p\left(T,x\right)={\left(K-x\right)}^{+}\end{array}\right.$$E14

For the put option, the option buyer has the right to sell the asset at a agreed price at the maturation date. At option expiry tine, if $\left(x\right)<$ $\left(K\right);$ then the buyer gains in a profit, $K-x.$ But if $\left(x\right)>\left(K\right),$ he rescinds his decision to execute his right on the option.

2.3 Description of martingale and PDEs options price valuation approach

2.3.1 Martingale approach

In this approach, options are not part of the traded assets $S_1\left(t\right),\dots ,S_N\left(t\right),$ so cannot be priced directly. However, using the arguments in Section 1.1.3, we can form a replicating portfolio $\prod \left(t\right)={\sum }_{i=1}^N{a}_i\left(t\right)S_i\left(t\right)$ that reproduces the option price at maturation time, such that $V_t={\mathrm{\Pi }}_t$ at $t>0$ as well as $V_T={\mathrm{\Pi }}_T$. However, theorem of Arbitrage states with a numeraire $N_t,$ each comparative price of the asset is a Martingale relative to the measure$\mathbb{N}$, as well as $V_t/N_t$ as they are blend of Martingale. This implies that $V_t/N_t$ becomes

$$E^{\mathbb{N}}\left[\left.\frac{V_T}{N_T}\right|{\mathcal{F}}_t\right]=\frac{V_t}{N_t}$$E15

from which the time, $t$ price of the option, $V_t$, is

$$V_t=N_t{E}^{\mathbb{N}}\left[\left.\frac{V_T}{N_T}\right|{\mathcal{F}}_t\right]$$E16

In the Black-Scholes economy, we have two assets, a stock $S_t$ that follows the Stochastic Differential Equation (SDE)

$$\mathit{dS}=\mathit{\text{rSdt}}+\mathit{\text{σSdW}}$$E17

and a fixed bond $B$ such that Eq. (17) is re-written as

$$d{S}_t=\mathit{rdt}+\sigma S_{t}d{W}_t$$E18

and

$$d{B}_t=r{B}_t\mathit{dt}.$$

Applying Girsanov’s theorem, the process for $d{S}_t$ becomes

$$d{S}_t=r{S}_t\mathit{dt}+\sigma S_{t}d{W}_t^{\mathbb{B}}$$E19

where $d{W}_t^{\mathbb{B}}=d{W}_t+\frac{\mu -r}{\sigma }\mathit{dt}.$

It is straightforward to solve for $B_t=exp\left({\int }_0^t\mathit{rdu}\right)=e^{\mathit{rt}}$. We apply Ito’s Lemma (Lemma 2.2) to the function $\mathit{ln}{B}_t$. Then $ln{B}_t$ follows the SDE

$$dln{B}_t=r_t\mathit{dt}.$$

Integrating from 0 to $t$ we have,

$$ln{B}_t-ln{B}_0={\int }_0^t{r}_u\mathit{du}$$

Taking exponential of both sides, so the solution to the SDE is $B_t=exp\left({\int }_0^t\mathit{rdu}\right)$ since the time zero value of the bond is $B_0=1.$ When interest rates are constant then $r_t=r$ and $B_t=e^{\mathit{rt}}$. Hence, we apply $B_t$ to be the numeraire such that ${\tilde{S}}_t=\frac{S_t}{B_t}$ becomes a Martingale under the measure, $\mathbb{B}$. The payoff for European Put is given as$V_T={\left(K-S_T\right)}^{+}$, hence, by Eq. (16), the Put price at time-$t$ price is

$$V_t=B_t{E}^{\mathbb{B}}\left[\left.\frac{{\left(K-S_T\right)}^{+}}{B_T}\right|{\mathcal{F}}_t\right]=e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[{\left(K-S_T\right)}^{+}\left|{\mathcal{F}}_t\right.\right]$$E20

We can either evaluate this expectation directly or obtain the solution for the European put option price.

Alternatively, when we chose a different numeraire. $B_t,$ is subjective, and we can use $S_t$. We have seen earlier that $\frac{S_t}{B_t}$ is already a martingale under $\mathbb{B}$. Therefore, $\frac{B_t}{S_t}$ would be Martingale, under $\mathbb{S}.$ In Eq. (20) the value $V_t$ of the Put is derived from

$$\frac{V_t}{B_t}=E^{\mathbb{B}}\left[\left.\frac{V_T}{B_T}\right|{\mathcal{F}}_t\right]$$

Equivalently, using $S_t$ as the numeraire, the same value $V_t$ can be derived from

$$\frac{V_t}{S_t}=E^{\mathbb{S}}\left[\left.\frac{V_T}{S_T}\right|{\mathcal{F}}_t\right]$$

The European Put has payoff $V_T={\left(K-S_T\right)}^{+}$, so the time-$t$ price of the Put is

$$V_t=S_t{E}^{\mathbb{S}}\left[\left.{\left(\frac{K}{S_T}-1\right)}^{+}\right|{\mathcal{F}}_t\right]=S_t{E}^{\mathbb{S}}\left[\left.\frac{{\left(K-S_T\right)}^{+}}{S_T}\right|{\mathcal{F}}_t\right]$$E21

Even though the expression in Eqs. (20) and (21) are different, they both produce the same solution because the change in numeraire does not affect the option values once an Equivalent Martingales Measure (EMM) has been defined. So, $S_t=B_t=exp\left({\int }_0^t\mathit{rdu}\right)=e^{\mathit{rt}}$ which makes equation Eqs. (20) and (21) produce the same solution.

2.3.2 PDE approach

Financial portfolios are pairs of adapted stochastic processes such that at date, t they can only depend on the path of a Brownian motion at time, $t$ [11]. In this approach, Stochastic Differential Equations (SDEs) are converted into a Partial Differential Equation (PDE) with a set boundary condition. The stochastic components of the SDEs are removed and the PDEs becomes easier to solve. A set of initial values or boundary conditions are often provided to have a unique solution. Whereas, the PDE does not give exact solutions, it can then be solved numerically [12].

It is important also to state here that as we shall see later, the Feynman-Kac Theorem (Theorem 2.4 or 2.5), is an essential tool for the pricing of options using the PDE approach [13]. We shall consider an existing model in mathematical finance and then proceed to solve them analytically by applying the two approaches, Martingale and PDEs methods to obtain options price formulas. The SDE model of interest in this chapter is that of Constant Elasticity of Variance (CEV). In this model (CEV), the volatility is assumed to be constant.

2.4 Derivation of options price valuation formula

2.4.1 The Constant Elasticity of Variance (CEV) SDE model

Here, we examine European put option pricing in the CEV SDE model Cox (1975). In this case, we have two assets; the cash deposit in the bank account $B\text{is given}\mathrm{by}{B}_t=e^{\mathit{rt}}$ and the stock $X$ follow the SDE

$$d{X}_t=r{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{W}_t,X_0>0$$E22

with constants $\sigma >0\text{and}\alpha >0$ where $W_t$ is a standard Brownian motion, $t\in \left[0,\infty \right],$ [14]. We proceed to compute the Martingale and PDE options price valuation formula for $u\left(t,X_t\right)$.

In order to compute the Martingale option price valuation formula for the SDE model in Eq. (22), we assume $\varphi ={\left({\eta }_t,{\xi }_t\right)}_{tϵ\left[0,T\right]}$ be portfolio strategy with price,

$$V_t\left(\varphi \right)={\eta }_t{B}_t+{\xi }_t$$E23

and that it satisfies the self-financing condition

$$d{V}_t\left(\varphi \right)={\eta }_{t}d{B}_t+{\xi }_{t}d{X}_t=r{e}^{\mathit{rt}}{\eta }_t\mathit{dt}+{\xi }_{t}d$$E24

$\text{or}$ equivalently

$$V_t\left(\varphi \right)=V_0\left(\varphi \right)+{\int }_0^t{\eta }_{t}d{B}_s+{\int }_0^t{\xi }_{t}d{X}_s$$E25

We proceed by applying notion of change in numeraire. Certainly, the asset prices are strictly positive, that is $B_t>0$, then the market can be normalized by considering

$${\tilde{B}}_t=B_t^{-1}{B}_t=1$$

and

$${\tilde{X}}_t=B_t^{-1}{X}_t=e^{-\mathit{rt}}{X}_t$$

Thus, market normalization implies taking the price $B_t$ of the riskless investment as the numeraire (unit of price) and then calculating other prices in terms of this numeraire.

Hence, the discounted portfolio becomes

$${\tilde{V}}_t\left(\varphi \right)=B_t^{-1}{V}_t\left(\varphi \right)=e^{-\mathit{rt}}\left({\eta }_t{B}_t+{\xi }_t{X}_t\right)={\eta }_t+{\xi }_t{\tilde{X}}_t$$

and applying integration by parts, we have

$$d{\tilde{V}}_t\left(\varphi \right)=B_t^{-1}d{V}_t\left(\varphi \right)-r{e}^{-\mathit{rt}}{V}_t\left(\varphi \right)\mathit{dt}+\left(d{B}_t^{-1}\right)\left(d{V}_t\left(\varphi \right)\right).$$

Since $\left(d{B}_t^{-1}\right)\left(d{V}_t\left(\varphi \right)\right)=0.$ Hence, we have that,

$$d{\tilde{V}}_t\left(\varphi \right)=B_t^{-1}d{V}_t\left(\varphi \right)-r{e}^{-\mathit{rt}}{V}_t\left(\varphi \right)\mathit{dt}$$E26

from Eq. (24), it is known that

$$d{V}_t\left(\varphi \right)={\eta }_{t}d{B}_t+{\xi }_{t}d{X}_t,$$

i.e., $\varphi$ is self-financing, then

$$\begin{array}{ll}d{\tilde{V}}_t\left(\varphi \right)& =e^{-\mathit{rt}}\left\{r{\eta }_t{e}^{\mathit{rt}}\mathit{dt}+{\xi }_{t}d{X}_t\right\}-r\left({\eta }_t+{\xi }_t{\tilde{X}}_t\right)\mathit{dt}\\ & =r{e}^{-\mathit{rt}}{e}^{\mathit{rt}}{\eta }_t\mathit{dt}+e^{-\mathit{rt}}{\xi }_{t}d{X}_t-r{\eta }_t\mathit{dt}-r{\xi }_t{e}^{-\mathit{rt}}{X}_t\mathit{dt}\\ & =r{\eta }_t\mathit{dt}+e^{-\mathit{rt}}{\xi }_{t}d{X}_t-r{\eta }_t\mathit{dt}-r{\xi }_t{e}^{-\mathit{rt}}{X}_t\mathit{dt}\\ d{\tilde{V}}_t\left(\varphi \right)& ={\xi }_t{e}^{-\mathit{rt}}d{X}_t-r{\xi }_t{e}^{-\mathit{rt}}{X}_t\mathit{dt}={\xi }_t\left\{e^{-\mathit{rt}}d{X}_t+d\left(e^{-\mathit{rt}}\right)X_t\right\}\\ d{\tilde{V}}_t\left(\varphi \right)& ={\xi }_{t}d{\tilde{X}}_t\end{array}$$

which yields that ${\tilde{V}}_t\left(\varphi \right)$ is self-financing because $d{\tilde{B}}_t=d\left(1\right)=0.$ Using Eq. (26) and assuming that $d{\tilde{V}}_t\left(\varphi \right)={\xi }_{t}d{\tilde{X}}_t$ (this implies that $\varphi$ is a self-financing portfolio). Financially speaking, this implies that changes in portfolio values depends only on changes in the prices or values of assets and not on units in which we measure the asset prices.

In integral form, we recall that a discounted market, is written as

$${\tilde{V}}_t\left(\varphi \right)={\tilde{V}}_0\left(\varphi \right)+{\int }_0^t{\xi }_{s}d{\tilde{X}}_s,tϵ{\mathbb{R}}^{+}$$E27

Next, we need to show that Eq. (27) is a Martingale under$\mathbb{B}$. Then, by Girsanov’s Theorem, we can define a probability measure $\mathbb{B}$ and the process

$${\tilde{W}}_t=\frac{\mu -r}{\sigma }t+W_t,$$

is a Brownian motion under $\mathbb{B}$.

Now from Eq. (22) if we now compute $d{\tilde{X}}_t$, we get

$$\begin{array}{ll}d{\tilde{X}}_t& =d\left(e^{-\mathit{rt}}{X}_t\right)=-r{e}^{-\mathit{rt}}{X}_t\mathit{dt}+e^{-\mathit{rt}}d{X}_t\\ & =-r{\tilde{X}}_t\mathit{dt}+e^{-\mathit{rt}}\left[r{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{\tilde{W}}_t\right]\\ & =-r{\tilde{X}}_t\mathit{dt}+r{e}^{-\mathit{rt}}{X}_t\mathit{dt}+\sigma X_t^{\alpha }{e}^{-\mathit{rt}}d{\tilde{W}}_t\\ & =-r{\tilde{X}}_t\mathit{dt}+r{\tilde{X}}_t\mathit{dt}+\sigma X_t^{\alpha }{e}^{-\mathit{rt}}d{\tilde{W}}_t\\ d{\tilde{X}}_t& =\sigma {\tilde{X}}_t^{\alpha }d{\tilde{W}}_t\end{array}$$

or unambiguously, let

$$ln\left(\frac{{\tilde{X}}_t}{{\tilde{X}}_0}\right)=\sigma {\tilde{X}}_t^{\alpha }d{\tilde{W}}_t$$

We proceed to look for the solution of the preceding equation using Ito’s formula.

let,

$$Y\left(t\right)=ln{\tilde{X}}_t^{\alpha },f_t=\frac{\partial \left(ln{\tilde{X}}_t^{\alpha }\right)}{\partial t}=0,f_x=\frac{\partial \left(ln{\tilde{X}}_t^{\alpha }\right)}{\partial {\tilde{X}}_t^{\alpha }}=\frac{1}{{\tilde{X}}_t^{\alpha }},f_{\mathit{xx}}=-\frac{1}{{\tilde{X}}_t^{2\alpha }}$$

Noting that, $u\left(t\right)=\mu X_t,v\left(t\right)=\sigma {\tilde{X}}_t^{\alpha }$ and so we have

$$\begin{array}{ll}d\left(ln{\tilde{X}}_t^{\alpha }\right)& =\left[0+0\times \frac{1}{{\tilde{X}}_t^{\alpha }}+\frac{1}{2}{\left(\sigma {\tilde{X}}_t^{\alpha }\right)}^2\left(-\frac{1}{{\tilde{X}}_t^{2\alpha }}\right)\right]\mathit{dt}+\frac{1}{{\tilde{X}}_t^{\alpha }}\sigma {\tilde{X}}_t^{\alpha }d{\tilde{W}}_t\\ & \Rightarrow d\left(ln{\tilde{X}}_t^{\alpha }\right)=\frac{1}{2}\left(\frac{-{\sigma }^2{\tilde{X}}_t^{2\alpha }}{{\tilde{X}}_t^{2\alpha }}\right)\mathit{dt}+\frac{\sigma {\tilde{X}}_t^{\alpha }d{\tilde{W}}_t}{{\tilde{X}}_t^{\alpha }}\\ d\left(ln{\tilde{X}}_t^{\alpha }\right)& =-\frac{1}{2}{\sigma }^2\mathit{dt}+\mathit{\sigma d}{\tilde{W}}_t.\end{array}$$

Integrating both sides of the above equation and simplifying, we get

$$\begin{array}{l}ln{\tilde{X}}_t^{\alpha }-ln{\tilde{X}}_0^{\alpha }={\int }_0^t-\frac{1}{2}{\sigma }^2\mathit{dt}+{\int }_0^t\mathit{\sigma d}{\tilde{W}}_t\\ ln{\tilde{X}}_t^{\alpha }=ln{\tilde{X}}_0^{\alpha }-\frac{1}{2}{\sigma }^{2}t+\sigma {\tilde{W}}_t\end{array}$$

since $W_0=0.$

Taking exponential of both sides and simplifying, we get

$$\begin{array}{l}{e}^{ln{\tilde{X}}_t^{\alpha }}=e^{\left[ln{\tilde{X}}_0^{\alpha }-\frac{{\sigma }^2}{2}t+\sigma {\tilde{W}}_t\right]}\\ {\tilde{X}}_t^{\alpha }=e^{ln{\tilde{X}}_0^{\alpha }}.e^{\left[-\frac{{\sigma }^2}{2}t+\sigma {\tilde{W}}_t\right]}\\ \therefore {\tilde{X}}_t^{\alpha }={\tilde{X}}_0^{\alpha }\mathit{exp}\left[\sigma {\tilde{W}}_t-\frac{{\sigma }^2}{2}t\right]\end{array}$$E28

We now go ahead to show that the above solution in Eq. (28) satisfies the properties of a Martingale under the measure, $\mathbb{B}$. First, we find a measure $\mathbb{B}$ which uses $B_t$ as a numeraire in order to turn it to a Martingale. We now have

$$d{X}_t=r_t{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{W}_t^{\mathbb{B}}$$

where, $W_t^{\mathbb{B}}=W_t+\frac{\mu -r_t}{\sigma }t,$i.e. this is derived from the application of Girsanov theorem.

By using $B_t$ as the unit price (numeraire), the discounted aaset price ${\tilde{X}}_t=\frac{X_t}{B_t}$ and ${\tilde{X}}_t$ are martingales. Hence, by the application of Ito’s Lemma to ${\tilde{X}}_t$, we obtain

$$d{\tilde{X}}_t=\frac{\partial \tilde{X}}{\partial B}d{B}_t+\frac{\partial \tilde{X}}{\partial X}d{X}_t$$E29

Since all terms involving the second order derivatives are zero. i.e. $\sigma$ is zero when we compare the discounted stock price ${\tilde{X}}_t=\frac{X_t}{B_t}$ with Eq. (23). Expanding Eq. (29), we have

$$\begin{array}{ll}d{\tilde{X}}_t& =-\frac{X_t}{B_t^2}d{B}_t+\frac{1}{B_t}d{X}_t\\ & =-\frac{X_t}{B_t^2}\left(r_t{B}_t\mathit{dt}\right)+\frac{1}{B_t}\left(r_t{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{W}_t^{\mathbb{B}}\right)\\ & =-\frac{X_t{r}_t\mathit{dt}}{B_t}+\frac{1}{B_t}{r}_t{X}_t\mathit{dt}+\frac{\sigma X_t^{\alpha }d{W}_t^{\mathbb{B}}}{B_t}\\ & \Rightarrow d{\tilde{X}}_t=\sigma {\tilde{X}}_t^{\alpha }d{W}_t^{\mathbb{B}}\end{array}$$

The solution to the SDE is

$${\tilde{X}}_t^{\alpha }={\tilde{X}}_0^{\alpha }\mathit{exp}\left[-\frac{{\sigma }^2}{2}t+\sigma W_t^{\mathbb{B}}\right].$$

To show that ${\tilde{X}}_t^{\alpha }$ is a Martingale under $\mathbb{B}$, we consider the expectation under $\mathbb{B}$ for $s<t$, hence we have,

$$E^{\mathbb{B}}\left[{\stackrel{\sim }{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\stackrel{\sim }{X}}_0^{\alpha }exp\left(-\frac{1}{2}{\sigma }^{2}t\right).E^{\mathbb{B}}\left[exp\left(\sigma W_t^{\mathbb{B}}\right)\left|{\mathcal{F}}_s\right.\right]$$

$$={\stackrel{\sim }{X}}_0^{\alpha }exp\left(-\frac{1}{2}{\sigma }^{2}t+\sigma W_s^{\mathbb{B}}\right).E^{\mathbb{B}}\left[exp\left(\sigma \left(W_t^{\mathbb{B}}-W_s^{\mathbb{B}}\right)\right)\left|{\mathcal{F}}_s\right.\right]$$

at time $s$ we have that $W_t^{\mathbb{B}}-W_s^{\mathbb{B}}$ is normally distributed with zero mean and variance $t$ as $N\left(0,t\right)$ which is identical in distribution to $W_{t-s}^{\mathbb{B}}$ at time zero. Hence, we can write

$$E^{\mathbb{B}}\left[{\tilde{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\tilde{X}}_0^{\alpha }exp\left(-\frac{1}{2}{\sigma }^{2}t+\sigma W_s^{\mathbb{B}}\right).E^{\mathbb{B}}\left[exp(\sigma \left(W_{t-s}^{\mathbb{B}}\right)\left|{\mathcal{F}}_0\right.\right]$$

Statistically, the moment generating function (mgf) of a random variable $X$ with normal distribution $N\left(\mu ,{\sigma }^2\right)$ is stated as

$$E\left[e^{\mathit{\varphi x}}\right]=exp\left(\mathit{\mu \varphi }+\frac{1}{2}{\varphi }^2{\sigma }^2\right).$$

Under $\mathbb{B}$ we have that $W_{t-s}^{\mathbb{B}}$ is $\mathbb{B}$- Brownian motion and distributed as $N\left(0,t-s\right)$. Therefore, the mgf of $W_{t-s}^{\mathbb{B}}$ is

$$E^{\mathbb{B}}\left[{\tilde{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\tilde{X}}_0^{\alpha }exp\left(-\frac{1}{2}{\sigma }^{2}t+\sigma W_s^{\mathbb{B}}\right).exp\left(\frac{1}{2}{\sigma }^2\left(t-s\right)\right)$$

where $\sigma =\varphi$ and we can then write

$$\begin{array}{l}{E}^{\mathbb{B}}\left[{\tilde{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\tilde{X}}_0^{\alpha }exp\left(-\frac{1}{2}{\sigma }^{2}t+\sigma W_s^{\mathbb{B}}\right)\mathit{exp}\left(\frac{1}{2}{\sigma }^2\left(t-s\right)\right)\\ E^{\mathbb{B}}\left[{\tilde{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\tilde{X}}_0^{\alpha }exp\left(-\frac{1}{2}{\sigma }^{2}s+\sigma W_s^{\mathbb{B}}\right)\\ \therefore E^{\mathbb{B}}\left[{\tilde{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\tilde{X}}_s^{\alpha }\end{array}$$

We thus have that

$$E^{\mathbb{B}}\left[{\tilde{X}}_t^{\alpha }\left|{\mathcal{F}}_s\right.\right]={\tilde{X}}_s^{\alpha }$$

which shows that ${\tilde{X}}_t^{\alpha }$ is a $\mathbb{B}$ Martingale.

Hence, we have that

$${\tilde{V}}_t\left(\varphi \right)={\tilde{V}}_0\left(\varphi \right)+{\int }_0^t{\xi }_{s}d{\tilde{X}}_s={\tilde{V}}_0\left(\varphi \right)+{\int }_0^t{\xi }_s\sigma {\tilde{X}}_t^{\alpha }d{\tilde{W}}_t$$

and ${\tilde{V}}_t\left(\varphi \right)$ is a stochastic integral with regard to the Brownian motion under the measure,$\mathbb{B}$. Applying the integrability condition in Theorem 2.2, yields

$$E^{\mathbb{B}}\left[{\int }_0^T\left|{\left.{\xi }_t\sigma {\tilde{X}}_t^{\alpha }\right|}^2\right.\mathit{dt}\right]<\infty$$

Hence, we have shown that ${\tilde{V}}_t\left(\varphi \right)$ is a martingale under $\mathbb{B}.$

Now since,

$${\tilde{V}}_t\left(\varphi \right)={\tilde{V}}_0\left(\varphi \right)+{\int }_0^t{\xi }_{s}d{\tilde{X}}_{s}tϵ{\mathbb{R}}^{+}$$

is a Martingale under $\mathbb{B},$ it follows from the Martingale properties of ${\tilde{V}}_t\left(\varphi \right)\text{under}\mathbb{B}$ that,

$$\begin{array}{ll}{\tilde{V}}_t\left(\varphi \right)& =E^{\mathbb{B}}\left[{\tilde{V}}_T\left|{\mathcal{F}}_t\right.\right]\\ & =e^{-\mathit{rT}}{E}^{\mathbb{B}}\left[V_T\left|{\mathcal{F}}_t\right.\right]\\ & =e^{-\mathit{rT}}{E}^{\mathbb{B}}\left[C\left|{\mathcal{F}}_t\right.\right]\end{array}$$E30

where $C$ is a contingent claim, $u\left(T,X_T\right).$

Note that $\varphi ={\left({\eta }_t,{\xi }_t\right)}_{tϵ\left[0,T\right]}$ hedges the claim $C,$ i.e. we have $V_T=C⟹V_T=u\left(T,X_T\right).$

Hence Eq. (30) becomes

$$V_t=e^{-\mathit{rt}}{\tilde{V}}_t=e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[u\left(T,X_T\right)\left|{\mathcal{F}}_t\right.\right]$$

Since the process ${\left(X_t\right)}_{tϵ{\mathbb{R}}_{+}}$ has the Markov property, the value

$$V_t=e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[\varphi \left(X_T\right)\left|{\mathcal{F}}_t\right.\right]$$E31

where $\varphi \left(X_T\right)=u\left(T,X_T\right)$ of the portfolio at $tϵ\left[0,T\right]$ can be written from Eq. (31) as a function $u\left(t,X_t\right)$ of $t\text{and}{X}_t.$ Given the payoff function,

$$u\left(T,X_T\right)={\left(K-X_T\right)}^{+},$$

Eq. (31) Becomes

$$u\left(t,X_t\right)=e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[{\left(K-X_T\right)}^{+}\left|{\mathcal{F}}_t\right.\right]$$

$$u\left(t,X_t\right)=E^{\mathbb{B}}\left[e^{-r\left(T-t\right)}{\left(K-X_T\right)}^{+}\right]$$

Alternatively, we have

$${\tilde{V}}_t\left(\varphi \right)={\tilde{V}}_0\left(\varphi \right)+{\int }_0^t{\xi }_{s}d{\tilde{X}}_s,tϵ{\mathbb{R}}^{+}$$

We the recall the basic theorem of Asset Pricing which stipulates that a market is arbitrage free if there is in existence, at least one Equivalent Martingale Measure (EMM). If we correlate all these facts with Theorem 2.3, then we have all instruments for asset pricing.

Hence, if $\varphi$ is a hedging strategy for $C$, we have that

$$C=V_0\left(\varphi \right)+{\int }_0^T{\eta }_{t}d{B}_t+{\int }_0^T{\xi }_{t}d{X}_t$$

and the discounted portfolio will be a replicating portfolio for the claim

$$\begin{array}{ll}\tilde{C}& =e^{-\mathit{rt}}C,i.e.\\ C& ={\tilde{V}}_T\left(\varphi \right)=V_0\left(\varphi \right)+{\int }_0^T{\xi }_{t}d{\tilde{X}}_t\end{array}$$

it follows from the martingale properties of $\tilde{V}\left(\varphi \right)$ under $\mathbb{B}$ that

$$E^{\mathbb{B}}\left[e^{-\mathit{rT}}C\left|{\mathcal{F}}_t\right.\right]=E^{\mathbb{B}}\left[{\tilde{V}}_T\left(\varphi \right)\left|{\mathcal{F}}_t\right.\right]={\tilde{V}}_t\left(\varphi \right)=e^{-\mathit{rt}}{V}_t\left(\varphi \right)$$

which yields

$$V_t\left(\varphi \right)=E^{\mathbb{B}}\left[e^{-r\left(T-t\right)}C\left|{\mathcal{F}}_t\right.\right].$$

Now, if

$$C=\varphi \left(X_T\right)=u\left(T,X_T\right),$$

then the arbitrage free price is given by

$${\pi }_t\left(C\right)=E^{\mathbb{B}}\left[e^{-r\left(T-t\right)}\varphi \left(X_T\right)\left|{\mathcal{F}}_t\right.\right]$$

$X_t$ solves the following SDE

$$d{X}_t=r{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{\tilde{W}}_t$$

under $\mathbb{B}$. Hence, $X_t$ is a Markov process, and we have that

$$E^{\mathbb{B}}\left[e^{-r\left(T-t\right)}\varphi \left(X_T\right)\left|{\mathcal{F}}_t\right.\right]=E^{\mathbb{B}}{\left[e^{-r\left(T-t\right)}\varphi \left(X_T\right)\right]}_{\left|x=X_t\right.}$$E32

given the payoff function,

$$\varphi \left(X_T\right)=u\left(T,X_T\right)={\left(K-X_T\right)}^{+}.$$

Hence, Eq. (32) becomes

$$u\left(t,X_t\right)=E^{\mathbb{B}}\left[e^{-r\left(T-t\right)}{\left(K-X_T\right)}^{+}\right]$$

Next, we shall derive the PDE valuation formula for the above SDE model in Eq. (22). There are two assets in the model; the bank account $B$ is given by $B_t=e^{\mathit{rt}}$ and the stock X follows the SDE

$$d{X}_t=r{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{W}_t,X_0>0$$

with constants $\sigma >0$ and $\alpha >0,$where $W$ is a $Q-\text{Brownian}$ motion. Applying Ito’s lemma (Lemma 2.2), the value $u=u\left(X_t,t\right)$ of the Option written on $X$ follows the SDE

$$\mathit{du}=\left(\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}\right)\mathit{dt}+\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}$$E33

We transform the SDE for $u$ into PDE, and define some boundary conditions to allow a solution to the PDE. We then proceed to set up a risk-free self-financing and replicating portfolio $\prod$ consisting of $\eta$ units of the option and $\xi$ units of the asset, to have

$$d\prod =\mathit{\eta du}+\mathit{\xi dX}$$

Hence, the portfolio trails the SDE

$$\begin{array}{ll}d\prod & =\mathrm{\eta }\left[\left(\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}\right)\mathit{dt}+\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}\right]+\xi \left[\mathit{\text{rxdt}}+\sigma x^{\alpha }\mathit{dW}\right]\\ & =\eta \left(\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}\right)\mathit{dt}+\mathit{\eta \sigma }{x}^{\alpha }\frac{\partial u}{\partial x}\mathit{dW}+\mathit{\text{ξrxdt}}+\mathit{\xi \sigma }{x}^{\alpha }\mathit{dW}\\ d\prod & =\left[\eta \frac{\partial u}{\partial t}+\mathit{\eta rx}\frac{\partial u}{\partial x}+\frac{1}{2}\eta {\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}+\mathit{\xi rx}\right]\mathit{dt}+\left[\mathit{\xi \sigma }{x}^{\alpha }+\mathit{\eta \sigma }{x}^{\alpha }\frac{\partial u}{\partial x}\right]\mathit{dW}\end{array}$$E34

We take the choices of $\eta =1$ and $\xi =-\frac{\partial u}{\partial x}$ which eliminates the stochastic element from $d\prod$ and makes the portfolio risk-free. The portfolio being risk-free, it earns a riskless rate and the return on portfolio investment in small time increment $\mathit{dt}$ becomes

$$d\prod =r\prod \mathit{dt}$$

Substituting for $\xi$ in Eq. (34) and equating it with $r\prod \mathit{dt}$, produces a Black-Scholes PDE for $u\left(t,X_t\right)$.

$$\begin{array}{ll}d\prod & =\left[\eta \frac{\partial u}{\partial t}+\mathit{\eta rx}\frac{\partial u}{\partial x}+\frac{1}{2}\eta {\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}-\mathit{rx}\frac{\partial u}{\partial t}\right]\mathit{dt}+\left[-\sigma x^{\alpha }\frac{\partial u}{\partial x}+\mathit{\eta \sigma }{x}^{\alpha }\frac{\partial u}{\partial x}\right]\mathit{dW}=\left(\mathit{r\eta u}-\mathit{rx}\frac{\partial u}{\partial x}\right)\mathit{dt}\\ & ⟹\eta \frac{\partial u}{\partial t}+\mathit{\eta rx}\frac{\partial u}{\partial x}+\frac{1}{2}\eta {\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}-\mathit{rx}\frac{\partial u}{\partial x}=\mathit{r\eta u}-\mathit{rx}\frac{\partial u}{\partial x}\\ & ⟹\eta \frac{\partial u}{\partial t}+\mathit{\eta rx}\frac{\partial u}{\partial x}+\frac{1}{2}\eta {\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}-\mathit{r\eta u}=0\end{array}$$

but $\eta =1$, therefore

$$⟹\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}-\mathit{ru}=0.$$E35

We have been able to establish a risk-free portfolio with $X\text{and}u$ resulting to a PDE since $X\text{and}u$ are each determined by one source of ambiguity which is the Brownian motion $W$. This enhances the possibility of eliminating the stochastic element thereby resulting to a PDE.

However, we then progress to seek a solution to the PDE above by the application of the Feynman-Kac Theorem (Leon [15], Sarkka [16], Wiktorsson [17]) or numerically given a certain boundary condition. We note that the value $u_t=u\left(X_t,t\right)$ of European put option at time $t$ having a strike price $K$ with a constant interest rate, $r$ which follows the Black-Scholes PDE as stated in Eq. (35) with boundary condition$\left(X_T,T\right)={\left(K-X_T\right)}^{+}$.

2.5 Equivalence of the martingales and PDEs valuation formulas for option pricing models

The two stipulated methods for the derivation of option price valuation formulas as earlier discussed are equal. This can be ascertained through a rigorous proof using the examples earlier discussed in this work. The method employed by Heath and Schweizer [4] to show the equivalence of the two approaches is:

1. To prove that the valuation formula obtained through the Martingale method can be solved using Ito’s formula, which implies that under some stated verifiable necessary and sufficient conditions that it satisfies the corresponding valuation PDE formula, at least on the support of the underlying diffusion.

2. To establish that a solution exists for the derived PDE valuation formula.

However, in our research work, we looked at a possibility of establishing the equivalence of the two methods as earlier discussed. This is achieved by the application of Girsanov theorem and Feynman-Kac theorem to drive home our argument that the two approaches are equivalent. That is to show that the Martingale (No-arbitrage) method yields the Black-Scholes PDEs and vice versa.

In financial mathematics, the Feynman-Kac theorem provides the required relationship between parabolic Partial Differential Equations (PDEs) and stochastic processes. It represents solutions of a parabolic PDE forms of conditional expectations. Conversely, an important class of expectations of random processes can be computed by deterministic methods. The theorem can be in one dimension, and in multiple dimensions [12, 16].

Theorem 2.4: The Feynman-Kac Theorem in one dimension

Suppose that $x_t$ follows the stochastic process

$$d{x}_t=\mu \left(x_t,t\right)\mathit{dt}+\sigma \left(x_t,t\right)d{W}_t^{\mathbb{Q}}$$E36

where $W_t^{\mathbb{Q}}$ is a Weiner process under the measure $\mathbb{Q}$. If, $V\left(x_t,t\right)\text{is differentiable}$with respect to $x_t$ and $t$ and $V\left(x_t,t\right)$ follows the Partial Differential Equation (PDE) stated as

$$\frac{\partial V}{\partial t}+\mu \left(x_t,t\right)\frac{\partial V}{\partial x}+\frac{1}{2}\sigma {\left(x_t,t\right)}^2\frac{{\partial }^{2}V}{\partial x^2}-r\left(x_t,t\right)V\left(x_t,t\right)=0$$E37

and with boundary condition $V\left(X_T,T\right)$. The theorem asserts that $V\left(x_t,t\right)$ has the solution

$$V\left(x_t,t\right)=E^{\mathbb{Q}}\left[\left.e^{-{\int }_t^{T}r\left(x_u,u\right)\mathit{du}}V\left(X_T,T\right)\right|{\mathcal{F}}_t\right]$$E38

Note that the expectation is taken under the measure $\mathbb{Q}$ that makes the stochastic term in Eq. (36) Brownian motion. The generator of the process in Eq. (36) is defined as the operator

$$A=\mu \left(x_t,t\right)\frac{\partial }{\partial x}+\frac{1}{2}\sigma {\left(x_t,t\right)}^2\frac{\partial }{{\partial x}^2}$$E39

So, the PDE in $V\left(x_t,t\right)$ is sometimes written

$$\frac{\partial V}{\partial t}+\mathit{AV}\left(x_t,t\right)-r\left(x_t,t\right)V\left(x_t,t\right)=0$$E40

The Theorem can be used in two ways:

1. If $x_t$ adapts to the process in Eq. (36) and given a function $V\left(x_t,t\right)$ with initial boundary condition$V\left(X_T,T\right)$, then we get a solution for $V\left(x_t,t\right)$ as in Eq. (38).

2. If we know that the solution to $V\left(x_t,t\right)$ is given by Eq. (38) and that $x_t$ follows the process in Eq. (36), then we are assured that $V\left(x_t,t\right)$satisfies the PDE in Eq. (40)$.$

Theorem 2.5: Multidimensional version of the Feynman-Kac theorem

Suppose that $x_t$ follows the stochastic process in $n$ dimensions

$$d{x}_t=\mu \left(x_t,t\right)\mathit{dt}+\sigma \left(x_t,t\right)d{W}_t^{\mathbb{Q}}$$

where $x_t$ and $\mu \left(x_t,t\right)$ are each vector of dimension $n$, $W_t^{\mathbb{Q}}$ is a vector of dimension $m\text{of}\mathbb{Q}-\text{Brownian motion},\text{and}\sigma \left(x_t,t\right)$ is a matrix of size $n\times m$. In other words

$$d\left(\genfrac{}{}{0pt}{}{\begin{array}{c}{x}_1\left(t\right)\\ ⋮\end{array}}{x_n\left(t\right)}\right)=\left(\genfrac{}{}{0pt}{}{\begin{array}{c}{\mu }_1\left(x_t,t\right)\\ ⋮\end{array}}{{\mu }_n\left(x_t,t\right)}\right)\mathit{dt}+\left(\begin{array}{ccc}{\sigma }_{11}\left(x_t,t\right)& \cdots & {\sigma }_{1m}\left(x_t,t\right)\\ ⋮& \ddots & ⋮\\ {\sigma }_{n1}\left(x_t,t\right)& \cdots & {\sigma }_{\mathit{nm}}\left(x_t,t\right)\end{array}\right)\left(\genfrac{}{}{0pt}{}{\begin{array}{c}d{W}_1^{\mathbb{Q}}\left(t\right)\\ ⋮\end{array}}{d{W}_m^{\mathbb{Q}}\left(t\right)}\right)$$

The generator of the process is

$$A=\sum _{i=1}^n{\mu }_i\frac{\partial }{\partial x_i}+\frac{1}{2}\sum _{i=1}^n\sum _{j=1}^n{\left(\sigma {\sigma }^T\right)}_{\mathit{ij}}\frac{{\partial }^2}{\partial x_i\partial x_j}$$E41

where for notational convenience ${\mu }_i={\mu }_i\left(x_t,t\right),\sigma =\sigma \left(x_t,t\right),$ and ${\left(\sigma {\sigma }^T\right)}_{\mathit{ij}}$ is components $\left(i,j\right)$ in the matrix $\sigma {\sigma }^T$ of dimensions $\left(n\times n\right)$. Hence, the Partial Differential Equation (PDE) in $V\left(x_t,t\right)$ is given as

$$\frac{\partial V}{\partial t}+\mathit{AV}\left(x_t,t\right)-r\left(x_t,t\right)V\left(x_t,t\right)=0$$E42

and with boundary condition $V\left(X_T,T\right)\left|{\mathcal{F}}_t\right.$ has solution

$$V\left(x_t,t\right)=E^{\mathbb{Q}}\left[\left.e^{-{\int }_t^{T}r\left(x_u,u\right)\mathit{du}}V\left(X_T,T\right)\right|{\mathcal{F}}_t\right]$$E43

2.5.1 Equivalence of the martingale and PDEs option valuation formulas for constant elasticity of variance (CEV) SDE model

We now proceed to prove the equality of the Martingales and PDEs approaches for the given SDE model in finance.

We consider the CEV SDE model given in Eq. (22) as

$$d{X}_t=r{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{W}_t$$

and the fixed bond is given as

$$d{B}_t=r{B}_t\mathit{dt}$$

We apply the Girsanov’s Theorem so that the process for $d{X}_t$ for CEV model in Eq. (22) becomes

$$d{X}_t=r{X}_t\mathit{du}+\sigma X_t^{\alpha }d{W}_t^{\mathbb{B}}$$E44

The European put has payoff

$$u_T={\left(K-X_T\right)}^{+}$$

So in accordance with Eq. (21), the time, $t$ price of the put option is

$$u_t=u\left(t,X_t\right)=B_t{E}^{\mathbb{B}}\left[\frac{K-X_T}{B_T}\left|{\mathcal{F}}_{+}\right.\right]$$

were

$$\begin{array}{ll}{B}_t& =\mathit{exp}\left({\int }_0^t\mathit{rdu}\right)=e^{\mathit{rt}}\\ u_t& =e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[{\left(K-X_T\right)}^{+}\left|{\mathcal{F}}_t\right.\right]\end{array}$$E45

We ascertain that the discounted option price $\frac{u_t}{B_t}$ becomes a martingale only when the Black-Scholes parabolic PDE is fulfilled or solved. In a Black-Scholes model, the arbitrage-free price of an option $u_t$ is given as

$$u_t=N_t{E}^{\mathbb{N}}\left[\frac{u_T}{N_T}\left|{\mathcal{F}}_t\right.\right]$$E46

with numeraire $N_t=B_t$

$$u_t=u\left(X_t,t\right)=e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[u\left[X_T,T\right]\left|{\mathcal{F}}_t\right.\right]$$E47

where $\mathbb{B}$ is the measure under which the discounted stock price

$$\frac{X_t}{B_t}=e^{-\mathit{rt}}{X}_t$$

is a martingale. Under $\mathbb{B},$ the discounted option price,

$$Z\left(X_t,t\right)=e^{-\mathit{rt}}u\left(X_t,t\right),$$

is also a Martingale, which we write as

$$Z_t=e^{-\mathit{rt}}{u}_t$$

for convenience.

By Ito’s lemma, $Z\left(X_t,t\right)$ follows the process

$${\mathit{dZ}}_t=\frac{\partial Z}{\partial t}\mathit{dt}+\frac{\partial Z}{\partial X}d{X}_t+\frac{1}{2}\frac{{\partial }^{2}Z}{\partial X^2}{\left(d{X}_t\right)}^2$$E48

We need the Ito multiplication table given below

Now by differentiating$Z_t=e^{-\mathit{rt}}{u}_t$, we get

$$\frac{\partial Z}{\partial t}=-r{e}^{-\mathit{rt}}{u}_t+e^{-\mathit{rt}}\frac{\partial u}{\partial t}.$$E49

Substituting Eq. (49) into Eq. (48) and substituting for $d{X}_t$ for the CEV model in Eq. (22) yields

$$d{Z}_t=e^{-\mathit{rt}}\left[-r{u}_t+\frac{\partial u}{\partial t}\right]\mathit{dt}+\frac{\partial Z}{\partial X}\left[r{X}_t\mathit{dt}+\sigma X_t^{\alpha }d{W}_t\right]+\frac{1}{2}\frac{{\partial }^{2}Z}{\partial X^2}{\sigma }^2{X}_t^{2\alpha }\mathit{dt}$$E50

Note that

$$\frac{\partial Z}{\partial X}=e^{-\mathit{rt}}\frac{\partial u}{\partial X}\text{and}\frac{{\partial }^{2}Z}{\partial X^2}=e^{-\mathit{rt}}\frac{{\partial }^{2}u}{\partial X^2}.$$

Substituting for this in Eq. (50), we have

$$\begin{array}{ll}d{Z}_t& =e^{-\mathit{rt}}\left[-r{u}_t+\frac{\partial u}{\partial t}\right]\mathit{dt}+r{X}_t\frac{\partial Z}{\partial X}\mathit{dt}+\frac{\partial Z}{\partial X}\sigma X_t^{\alpha }d{W}_t+\frac{1}{2}\frac{{\partial }^{2}Z}{\partial X^2}{\sigma }^2{X}_t^{2\alpha }\mathit{dt}\\ d{Z}_t& =e^{-\mathit{rt}}\left[-r{u}_t+\frac{\partial u}{\partial t}\right]\mathit{dt}+e^{-\mathit{rt}}r{X}_t\frac{\partial u}{\partial X}\mathit{dt}+e^{-\mathit{rt}}\sigma X_t^{\alpha }\frac{\partial u}{\partial X}d{W}_t+\frac{1}{2}{e}^{-\mathit{rt}}{\sigma }^2{X}_t^{2\alpha }\frac{{\partial }^{2}u}{\partial X^2}\mathit{dt}\\ d{Z}_t& =e^{-\mathit{rt}}\left[-r{u}_t+\frac{\partial u}{\partial t}+r{X}_t\frac{\partial u}{\partial X}+\frac{1}{2}{\sigma }^2{X}_t^{2\alpha }\frac{{\partial }^{2}u}{\partial X^2}\right]\mathit{dt}+e^{-\mathit{rt}}\left[\sigma X_t^{\alpha }\frac{\partial u}{\partial X}\right]d{W}_t\end{array}$$E51

The conditional expectation under which the martingale option price is derived is taken under the measure, $\mathbb{B}$. Hence, the process $d{Z}_t$ must indicate this. Therefore, making a change of the measure to $\mathbb{B}$ in Eq. (51) to obtain

$$d{Z}_t=e^{-\mathit{rt}}\left[-r{u}_t+\frac{\partial u}{\partial t}+r{X}_t\frac{\partial u}{\partial X}+\frac{1}{2}{\sigma }^2{X}_t^{2\alpha }\frac{{\partial }^{2}u}{\partial X^2}\right]\mathit{dt}+e^{-\mathit{rt}}\left[\sigma X_t^{\alpha }\frac{\partial u}{\partial X}\right]\left[d{W}_t^{\mathbb{B}}-\left(\frac{\mu -r}{\sigma }\right)\mathit{dt}\right]$$E52

Since $d{W}_t^{\mathbb{B}}=d{W}_t+\left(\frac{\mu -r}{\sigma }\right)\mathit{dt}$ from the application of Girsanov’s Theorem that produced Eq. (44). Rearranging the terms in Eq. (52), we have

$$d{Z}_t=e^{-\mathit{rt}}\left[-r{u}_t+\frac{\partial u}{\partial t}+r{X}_t\frac{\partial u}{\partial X}+\frac{1}{2}{\sigma }^2{X}_t^{2\alpha }\frac{{\partial }^{2}u}{\partial X^2}\right]\mathit{dt}+e^{-\mathit{rt}}\left[\sigma X_t^{\alpha }\frac{\partial u}{\partial X}\right]d{W}_t^{\mathbb{B}}.$$E53

Hence, we have that in Eq. (51)

$$Z_t=e^{-\mathit{rt}}u\left[X_t,t\right],$$

the discounted time, $t$ option price, is a martingale under $\mathbb{B}$ only when the drift term in Eq. (53) is zero. That is when

$$\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}-r{u}_t=0.$$

We recognize this equation as the Black-Scholes PDE for the given SDE in Eq. (22). Obviously, it can be said that the price of the option can only be a Martingale if the Black-Scholes parabolic PDE is fulfilled. It suffices to state that the value of $u_t$ that solves the PDE given the boundary condition $u\left(T,X_T\right)={\left(K-X_T\right)}^{+}$ is the same with the $u_t$ in Eq. (45) and we can see that it is the same equation with the one derived in Eq. (35).

Moreso, we proceed to prove the equivalence using the Feynman-Kac theorem. Now given the derived PDE for the CEV model in Eq. (35), we proceed to solve the PDE by applying the Feynman-Kac theorem to ascertain if we can derive the Martingales formula.

$$\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}{\sigma }^2{x}^{2\alpha }\frac{{\partial }^{2}u}{\partial x^2}-\mathit{ru}=0$$

with the boundary condition

$$u\left(T,X_T\right)\mathit{=}{\left(K-X_T\right)}^{+}=\psi \left(x\right).$$

Let $X\left(t\right)$ solve the Stochastic Differential Equation

$$\mathit{dX}=\mathit{\text{rXdt}}+\sigma X^{\alpha }\mathit{dW}$$

with initial condition $x\left(t\right)=x.$

We define a process $x\left(t\right)$ on interval $\left[t,T\right]$ as follows:

$$\mathit{dX}=\mathit{\text{rxdt}}+\sigma x^{\alpha }\mathit{dW},x\left(t\right)=x$$

Applying Ito’s formula (Lemma 2.2) on $u\left(t,X_t\right)$, we have

$$\begin{array}{l}\mathit{du}=\frac{\partial u}{\partial t}\mathit{dt}+\frac{\partial u}{\partial x}\mathit{dX}+\frac{1}{2}\frac{{\partial }^{2}u}{{\partial x}^2}{\left(\mathit{dX}\right)}^2\\ {\left(\mathit{dX}\right)}^2=\left(\mathit{\text{rxdt}}+\sigma x^{\alpha }\mathit{dW}\right)\left(\mathit{\text{rxdt}}+\sigma x^{\alpha }\mathit{dW}\right)\\ {\left(\mathit{dX}\right)}^2={\left(\mathit{rx}\right)}^2{\left(\mathit{dt}\right)}^2+{\sigma }^2{x}^{2\alpha }{\left(\mathit{dW}\right)}^2+2\mathit{rx\sigma }{x}^{\alpha }\left(\mathit{dt}.\mathit{dW}\right)\end{array}$$

Using the Ito multiplication table, Table 1, we have

$\bullet$	$\mathit{dt}$	$d{W}_t$
$\mathit{dt}$	$0$	$0$
$d{W}_t$	$0$	$\mathit{dt}$

Table 1.

Ito’s multiplication table.

$${\left(\mathit{dX}\right)}^2={\sigma }^2{x}^{2\alpha }\mathit{dt}$$

Therefore

$$\begin{array}{ll}\mathit{du}& =\frac{\partial u}{\partial t}\mathit{dt}+\frac{\partial u}{\partial x}\left(\mathit{\text{rxdt}}+\sigma x^{\alpha }\mathit{dW}\right)+\frac{1}{2}\frac{{\partial }^{2}u}{{\partial x}^2}{\sigma }^2{x}^{2\alpha }\mathit{dt}\\ & =\frac{\partial u}{\partial t}\mathit{dt}+\mathit{rx}\frac{\partial u}{\partial x}\mathit{dt}+\sigma x^{\alpha }\frac{\partial u}{\partial x}\mathit{dW}+\frac{1}{2}\frac{{\partial }^{2}u}{{\partial x}^2}{\sigma }^2{x}^{2\alpha }\mathit{dt}\\ & =\frac{\partial u}{\partial t}\mathit{dt}+\mathit{rx}\frac{\partial u}{\partial x}\mathit{dt}+\frac{1}{2}\frac{{\partial }^{2}u}{{\partial x}^2}{\sigma }^2{x}^{2\alpha }\mathit{dt}+\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}\\ \mathit{du}& =\left[\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}\frac{{\partial }^{2}u}{{\partial x}^2}{\sigma }^2{x}^{2\alpha }\right]\mathit{dt}+\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}\end{array}$$E54

But, $\left[\frac{\partial u}{\partial t}+\mathit{rx}\frac{\partial u}{\partial x}+\frac{1}{2}\frac{{\partial }^{2}u}{{\partial x}^2}{\sigma }^2{x}^{2\alpha }\right]=\mathit{ru}$ as given in Eq. (35).

So, we now have

$$\mathit{du}=\mathit{\text{rudt}}+\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}$$

Integrating from $t\text{to}T$, we have

$${\int }_t^T\mathit{du}={\int }_t^T\mathit{\text{rudt}}+{\int }_t^T\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}$$

which gives

$$u\left(x\left(T\right),T\right)-u\left(x\left(t\right),t\right)={\int }_t^T\mathit{\text{rudt}}+{\int }_t^T\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}$$

Substituting the initial and terminal terms, we get

$$\psi \left(x\left(T\right)\right)-u\left(x,t\right)={\int }_t^T\mathit{\text{rudt}}+{\int }_t^T\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}$$

where $\psi \left(x\left(T\right)\right)={\left(K-X_T\right)}^{+}$

$$\psi \left(x\left(T\right)\right)-u\left(x,t\right)={\int }_t^T\mathit{\text{rudt}}+{\int }_t^T\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}.$$E55

Taking expectations of both sides, we have

$$\begin{array}{ll}E\left[\psi \left(x\left(T\right)\right)-u\left(x,t\right)\right]& =E\left[{\int }_t^T\mathit{\text{rudt}}\right]+E\left[{\int }_t^T\frac{\partial u}{\partial x}\sigma x^{\alpha }\mathit{dW}\right]\\ \mathrm{but},& E\left[{\int }_t^T\frac{\mathit{du}}{\mathit{dx}}\sigma x^{\alpha }\mathit{dB}\right]=0\end{array}$$E56

We now have

$$\begin{array}{l}E\left[u\left(x,t\right)\right]=E[\psi \left(x\left(T\right)\right]-E\left[{\int }_t^T\mathit{\text{rudt}}\right]\end{array}$$

$$\begin{array}{l}E\left[u\left(x,t\right)\right]=E\left[\psi \left(x\left(T\right)\right)-{\int }_t^T\mathit{\text{rudt}}\right]=E^{\mathbb{B}}\left[e^{-{\int }_t^T\mathit{\text{rsds}}}\psi \left(x\left(T\right)\right)|F_t\right]\end{array}$$

$$\begin{array}{l}u\left(x,t\right)=E^{\mathbb{B}}\left[e^{-r\left(T-t\right)}{\left(K-X_T\right)}^{+}|F_t\right]\end{array}$$

$$\begin{array}{l}\therefore u\left(x,t\right)=e^{-r\left(T-t\right)}{E}^{\mathbb{B}}\left[{\left(K-X_T\right)}^{+}\right]\end{array}$$

where $u\left(x,t\right)=u\left(X_t,t\right)$, which is the same option price valuation formula for the CEV model derived in Eq. (47).

3. Conclusion

In this chapter, we have examined the Martingale and PDEs approaches for the valuation of options price for two Stochastic Differential Equation models in finance using the replicating and riskless portfolio methods. The two SDE models studied are the Constant Elasticity of Variance (CEV) model and the Heston stochastic volatility model respectively. The model parameters are given in Section 2.1.2.

From this chapter, the following conclusions were drawn:

1. The Martingales and PDEs approach for the valuation of options price removes the stochastic components in Stochastic Differential Equation (SDE) models, thereby making the models riskless and hence easier for the resulting equations to be solved analytically or numerically.

2. That adequate understanding of the two approaches in valuation of financial Stochastic Differential Equation (SDE) models would aid an investor in the mitigation of the risk involved in financial derivative pricing and also in determining the quality of the choice (right) the holder of the option makes at the maturation date of the option.

3. Analytical results obtained through the application of Girsanov’s theorem for defining an Equivalent Martingale Measure and Feynman-Kac theorem respectively has shown that the two option price valuation formulas derived by both the Martingales and PDEs approaches are equivalent.