Option Pricing under Risk-Minimization Criterion in an Incomplete Market with the Finite Difference Method

We study option pricing with risk-minimization criterion in an incomplete market where the dynamics of the risky underlying asset is governed by a jump diffusion equation with stochastic volatility. We obtain the Radon-Nikodym derivative for the minimal martingalemeasure and a partial integro-differential equation (PIDE) of European option.Thefinite differencemethod is employed to compute the European option valuation of PIDE.


Introduction
Option pricing problem is one of the predominant concerns in the financial market.Since the advent of the justly celebrated Black-Scholes option pricing formula in [1], there has been an increasing amount of the literature describing the theory and its practice.In the Black-Scholes model, the appreciation rate and the volatility rate are assumed to be constants.However, more and more empirical evidence has revealed that these assumptions are not consistent with reality.Hence, many option valuation models which relax some of the restrictive assumptions in the Black-Scholes framework have been proposed and tested, such as the stochastic volatility models in [2][3][4][5][6][7][8], the jump diffusion or the Lévy process models in [9][10][11][12].The model to be studied in this paper takes into account not only stochastic volatility based on Heston's model in [8], but also the jump diffusion case.Therefore, several new problems and difficulties are exposed in our model.
Different from the Black-Scholes framework which uses jump diffusion to describe the price dynamics of underlying asset, the market of our model is incomplete; that is, it is not possible to replicate the payoff of every contingent claim by a portfolio, and there are several equivalent martingale measures.How to choose a consistent pricing measure from the set of equivalent martingale measures becomes an important problem.This means that we need to find some criterions to determine one from the set of equivalent martingale measures in some economically or mathematically motivated fashion.Föllmer and Leukert (2000), Kallsen (1999), Cvitanić et al. (2001), and Bielecki and Jeanblanc (2008) in [13][14][15][16] identified a unique equivalent martingale measure by utility maximization.Then, the option valuation under the minimal martingale measure was further developed by several researchers.Schweizer (1991) and Föllmer and Schweizer (1991) in [17,18] found that under the minimal martingale measure, a unique risk-minimizing (or optimal) strategy hedging of contingent claims in incomplete market exists.In our paper, we name the criterion under the minimal martingale measure as a risk-minimization criterion.Thus, in the incomplete market, option pricing is approximately possible with risk-minimization criterion.As presented in this paper, our work is based on the task of Föllmer and Schweizer in [17,18], and the purpose of this paper is to find the minimal martingale measure and the measure switch of asset prices processes with stochastic volatility and jump diffusion.By employing the minimal martingale measure, we obtain the Radon-Nikodym derivative and a partial integrodifferential equation (PIDE) for the European option.
However, it is difficult to get the exact solution of the PIDE in our model.Several numerical methods have been proposed to solve the PIDE approximately.The methods include numerical integration by Chiarella and Ziogas (2009) in [19], finite elements by Matache et al. (2005) in [20], the method of lines by Meyer (1998) in [21], and the finite difference methods (FDM) including those by Carr and Hirsa (2003) in [22], d 'Halluin et al. (2004) in [23], and Briani et al. (2004) in [24].This paper employs the FDM to compute the valuation of European option, and American option.Then further studies on using other methods to compute the solution of PIDE and derive the pricing formula of another option will be included in our future study.
The rest of the paper is organized as follows.In Section 2, we present the model for the underlying market and Doob-Meyer decomposition of the risky asset.In Section 3, we investigate an explicit representation of the density process of the minimal martingale measure.In Section 4, we derive a PIDE for the European option.The European option pricing with FDM is then studied in Section 5. Numerical results are shown in Section 6, and conclusions are given in Section 7.

The Model
In this paper, we consider the financial market with the following two basic assets: (i) a Bond whose price   at time  is given by (ii) a Stock whose price   at time  is given by Here,  ∈ [0, ],  ∈  0 ⊂ R/{0}, and  > 0; on the filtered complete space (Ω, F, {F  } ≥0 , ), there are    ,    which both are 1-dimensional Brownian motions with       =  and (, ) = (, ) − ]() which is the compensated jump measure of (⋅).(, ) is the jump measure, and ]() is the Lévy measure of the Lévy process (⋅).() is given by () = ∫  0 ∫  0 ( − 1)(, ),  ≥ 0. Additionally, ∫  0 ( − 1) 2 ]() < ∞.From above settings, the risk is at least two dimensional including the Brownian motion and Lévy process but only one risky asset in the market.It is impossible to exactly replicate the payoff of a given option by a dynamic portfolio strategy of the basic assets which is self-financing.We all know that if the dynamic portfolio strategy exists, then the initial cost of the portfolio must equal the price of the option, and the market is called complete; otherwise, an arbitrage opportunity exists, and the market is called incomplete.Therefore, the market of our model is incomplete since the perfect replication is impossible.
Compared with the Black-Scholes model, our model remedies some serious drawbacks of the Black-Scholes model, such as the constant volatility assumption and market completeness assumption.Bakshi et al. in [10] systematically analyzed the performance of the stochastic volatility model, the jump diffusion model, and the stochastic interest rate model, and concluded that a model with stochastic volatility and the jump diffusion was a better alternative to the Black-Scholes model, because the former not only performed far better but also was practically implementable.An excellent innovation of our model is combining stochastic volatility model and the jump diffusion model.Consequently, our model is better than the Black-Scholes model to portray realistic financial markets in theory and practice.Now, we want to find a unique optimal strategy hedging of contingent claims under risk-minimization criterion.Based on [18], it is equivalent to find the minimal martingale measure from the set of equivalent martingale measures and then obtain approximate prices of contingent claims.
With the Doob-Meyer decomposition, the discounted risky asset price process, Ŝ =  − ∫  0      , is a special semimartingale and can be written as with where   is the martingale part of Ŝ and   is the predictable process of finite variation.

Minimal Martingale Measure
We introduce the nations of minimal martingale measure in this section.Föllmer and Schweizer (1991) [18] noticed that the optimal hedging strategy can be computed in terms of the minimal martingale measure.Furthermore, it is uniquely determined.Hence, under the minimal martingale measure, the Radon-Nikodym derivative can be found and computed.
Before that, we define the minimal martingale measure.
Definition 1 (see [18]).A local martingale measure P, equivalent to the original measure , is called minimal if P =  on F  and if any square-integrable -martingale  which is  orthogonal to  remains a local martingale under P.
(ii) P exists if and only if there exists a predictable process   that satisfies Using Theorem 2, we obtain the following theorem for computing the Radon-Nikodym derivative.

Theorem 3. The Radon-Nikodym derivative under the minimal martingale measure P is
Proof.The theory of the Girsanov transformation shows that the predictable process of bounded variation can also be computed in terms of   : Throughout this paper, we make use of the notations that ⟨, ⟩  defined the quadratic variation process between  and  and denote ⟨⟩  = ⟨, ⟩  .Under P, the predictable process of bounded variation in the Doob-Meyer decomposition of  is given by Using ( 6) and ( 9), we have Denote   = −(  /⟨⟩  )  , then (10) can be written as From ( 4), we get Hence, From ( 11), we know that   is the Doléans-Dade exponential.Thus, we obtain where Solving ( 14), we obtain   in Theorem 3.
Remark 4. The Brown motions under the minimal martingale measure P are and the compensatory of (, ) is Remark 5. Equation ( 2) under the minimal martingale measure P is written as To guarantee that Ŝ is a martingale under the minimal martingale measure P, the following corollary is necessary.

Corollary 6. Under the minimal martingale measure P, Ŝ𝑡 is a martingale if and only if
Proof.Substituting   =  ∫  0    Ŝ into (18), since Ŝ is a martingale, the drift term must be identical to zero.Then, we can get (19).

Partial Integro-Differential Equation for European Call Option
Under the minimal martingale measure P, the price of the European call option (,   ,   ) at time  with strike price  and maturity date  is given by and (,   ,   ) = (  − ) + .By the fact that the discounted price of the European call option is a martingale under P, we can obtain the following theorem.
Proof.The total derivative of the discounted option price is Mathematical Problems in Engineering 5 We make the drift term zero, since the discounted price of the European put option is a martingale.Then, we obtain in Theorem 7.

Discretization.
To solve (38), we use a finite difference scheme with the following nation and approximations: Now, we evaluate the integral term.First, by performing a change of variable  :=  ⋅ ,  = / and  = /, the Ι(,   ,   ) and ΙΙ(,   ,   ) can be discretized by using the linear interpolation.Thus, Now, by using linear interpolation, we get approximation the following: Denote the following: Then, we have  , , =  ( where (39)

Numerical Results
In this section, we adopted the basic set of appropriate parameter values of the model shown in Table 1 as our basic set of parameter values to solve the finite difference equation (38).
Figure 1 shows the payoff with the price and the volatility.The solution of (38) is shown in Figure 2. All numerical experiments have been implemented by MATLAB R2011b software on a 2.0-GHz Intel Core PC.Now, we suppose that the underlying stock price process follows (2) and there is a European call option with  0 = 100, √ 0 = 0.35, and, other parameters in Table 1.Table 2 shows the European option prices solved by FDM with different  which stands for the different accuracy.

Conclusion
With risk-minimization criterion, we employ the minimal martingale measure to solve the pricing problem in an incomplete market.Then, we obtain the Radon-Nikodym derivative under the minimal martingale measure and a partial integro-differential equation (PIDE) for the European option.Since it is difficult to get the exact solution of PIDE in our model, a FDM scheme is proposed to compute the solution approximately.Finally, we complete a European call option price, and it is shown that our method is stable and locally accurate.

Figure 1 :
Figure 1: The payoff with the price and the volatility.

Table 1 :
Basic set of appropriate parameter values of the model.

Table 2 :
European option prices.