We study the pricing of American options in an incomplete market in which the dynamics of the underlying risky asset is driven by a jump diffusion process with stochastic volatility. By employing a risk-minimization criterion, we obtain the Radon-Nikodym derivative for the minimal martingale measure and consequently a linear complementarity problem (LCP) for American option price. An iterative method is then established to solve the LCP problem for American put option price. Our numerical results show that the model and numerical scheme are robust in capturing the feature of incomplete finance market, particularly the influence of market volatility on the price of American options.
1. Introduction
Since the pioneering work of Black-Scholes [1] and Merton [2] on the pricing of options, option pricing has become one of the predominant concerns in financial market. By relaxing some of the restrictive assumptions, such as the assumption of constant volatility and the geometric Brownian motion for the price of the underlying asset, many option pricing models have been proposed. Today, an increasing amount of literature has focused on stochastic volatility models (see, e.g., [3]), jump diffusion or Lévy process models [4, 5], which fit some features of the observed empirical market data. In this paper, in constructing our price model, we will take account of both the feature of stochastic volatility as in Heston's model and the feature of jump diffusion as in Bates [6]. The combination of the stochastic volatility and jump diffusion in the same model enables the model to capture the nature of the real financial market and yields a challenging and difficult mathematical problem for analysis.
Because of infrequent trading or by the presence of additional sources of uncertainty, the completeness of the market often breaks down. Thus, replicating the payoff of a contingent claim by a portfolio is possible, and various equivalent martingale measures (EMM) have been adopted. The question is that, among the equivalent martingale measures, one needs to establish certain criterions to determine the appropriate measure to be adopted. A unique equivalent martingale measure by using utility maximization has been identified by many authors including Kallsen [7], Bielecki and Jeanblanc [8], and Cvitanić et al. [9]. It was also found that, under the minimal martingale measure, there exists a unique risk-minimizing strategy hedging of contingent claims in complete market [10]. The criterion under the minimal martingale measure is thus referred to as risk-minimization criterion (see [11]). According to this, it is possible to price option with risk-minimization criterion. In this paper, we use the minimal martingale measure and consider the switch of assets prices with jump diffusion and stochastic volatility and obtain the Radon-Nikodym derivative and a linear complementarity problem for the pricing of American option.
Unlike European options, American options, which can be exercised any time before the expiring date, have to be priced usually numerically. Comparing to European options which lead to a partial integrodifferential equation, American options lead to inequalities in the form of a linear complementarity problem (LCP). In the American option pricing context, several methods have been proposed to approximate the linear complementarity problems resulting from American option pricing. These include the finite difference method, the operator splitting method, the multigrid method, and the penalty method [12]. Moreover, Tavella and Randall [13] described a stationary iterative method for pricing European options, and then Salmi and Toivanen [14] proposed a generalization of this iterative method to price American option. But they did not consider any multifactor jump diffusion models such as stochastic volatility models with jumps. Our paper aims to further develop previous numerical method to deal with multifactor jump diffusion models via the employment of the risk-minimization criterion to obtain the Radon-Nikodym derivative for the minimal martingale measure.
The rest of the paper is organized as follows. In Section 2, we discuss the pricing model with stochastic volatility and jump diffusion under the minimal martingale measure and then derive the linear complementarity problem under the risk-minimization criterion. Section 3 presents the discrete space and time approximation for the LCP problem. The method of projected successive overrelaxation (projected SOR) is used to yield our results. Numerical results and discussion are reported in Section 4.
2. The Model under Minimal Martingale Measure
Taking into account the jumps in the evolution process of the underlying asset, we introduce the following stochastic volatility (1) based on the work in Heston [3]:
(1)dXtXt-=μdt+VtdWt1+∫R(y-1)N~(dt,dy),dVt=α(φ-Vt)dt+σVtdWt2,
where Wt=(Wt1,Wt2) is defined on the filtered complete space (Ω,ℱ,{ℱt}t≥0,P) with 2-dimensional standard Brownian motion with correlation coefficient dWt1dWt2=ρdt,ρ∈[-1,1]. The asset price has drift μ and N-(dt,dy)=N(dt,dy)-v(dy)dt is the compensated jump measure, N(dt,dy) denotes the jump measure, and v(dy) is the Lévy measure, σ governs the volatility of the variance process, and α and φ are the mean reversion coefficients to ensure that the variance will drift back to φ at the rate of α.
According to the discussions in the reference by Föllmer and Schweizer [10], the optimal hedging strategy may be established with the minimal martingale measure and it has been proved that the minimal martingale measure can be determined uniquely. Using Theorem 3.1 in Heston [3], the Radon-Nikodym derivative under the minimal martingale measure q is
(2)Zt=exp(-∫0tθuVudWu1-12∫0tθu2Vudu+∫0t∫Rln(1-θu(y-1))N(du,dy)+∫0t∫Rθu(y-1)v(dy)du),
where
(3)θu=μu-ruVu+∫R(y-1)2v(dy)
and rt is the risk-free rate of interest in the discounted risky asset price process X^t=e-∫0truduXt. The derivation of (2) is given in Appendix A. Correspondingly, the Brownian motion and the compensated Poisson process under the minimal martingale measure are derived as follows:
(4)W~t1=Wt1+∫0tθuVudu,W~t2=Wt2+ρ∫0tθuVudu,v~(dy)dt=(1-θu(y-1))v(dy)dt,N~(dt,dy)=N(dt,dy)-v~(dy)dt.
In our model, there are two dimensions of risks from the Lévy process and the Brownian motion. Thus the financial market is not complete and such a claim has an intrinsic risk. In this context, we construct the pricing model under the risk-minimization criterion by employing the minimal martingale measure method, and therefore model (1) can be written as follows:
(5)dXtXt-=μdt+VtdW~t1+∫R(y-1)N~(dt,dy)-θtVtdt-∫Rθt(y-1)2v(dy)dt,dVt=α(φ-Vt)dt+σVtdW~t2-ρσθtVtdt.
The value of the American put option P(t,Xt,Vt) at time t, under strike price K and maturity date T, is
(6)P(t,Xt,Vt)=maxτEq[e-∫tτrudu(K-Xτ)+∣Ft],
where τ, representing the stopping times in [0,T], is to be determined to maximize the discounted option payoff.
Under the minimal martingale measure q, we obtain the following partial integrodifferential equation (see Appendix B for the detailed derivation):
(7)∂P∂t=(r+K0-θtK1)P(t,X,V)-∂P∂X(r-K1+θtK2)X-∂P∂V(α(φ-V)-ρσθtV)-12∂2P∂X2VX2-∂2P∂X∂VXVσρ-12∂2P∂V2Vσ2-∫RP(t,Xy,V)v(dy)+θt∫RP(t,Xy,V)(y-1)v(dy),
where
(8)K0=∫Rv(dy),K1=∫R(y-1)v(dy),K2=∫R(y-1)2v(dy).
Under Merton’s jump diffusion model, we assume v(dy)=λf(y)dy, where λ is the jump intensity and f(y) is distribution of the jump size y, and assume that the stochastic volatility V is the log-normal distribution with the density
(9)f(y)=1yδ2πexp(-(logy-γ)22δ2).
Let λ=1; thus the notations under the log-normal density functions above are K0=1,K1=eδ/2+γ-1, and K2=(1/1-2δ2)eγ2/(1-2δ2)-2eδ/2+γ+1.
Now we take the time transform τ=T-t and introduce an operator L by
(10)LP~=-(r+K0-θK1)P~(τ,X,V)+∂P~∂X(r-K1+θK2)X+∂P~∂V(α(φ-V)+ρσθV)+12∂2P~∂X2VX2+∂2P~∂X∂VXVσρ+12∂2P~∂V2Vσ2+∫RP~(τ,Xy,V)v(dy)-θ∫RP~(τ,Xy,V)(y-1)v(dy).
Then, the value of the American put option satisfies the following linear complementarity problem:
(11)LCP:∂P~∂τ-LP~τ≥0,P~τ≥(K-Xτ)+,(∂P~∂τ-LP~τ)[P~τ-(K-Xτ)+]=0
with the boundary conditions
(12)P~(τ,0,V)=K,∂P~(τ,X-,V)∂X=0,P~(τ,X,0)=(K-X)+,∂P~(τ,X,V-)∂V=0
and the initial condition
(13)P~(0,X,V)=(K-X)+.
3. Numerical Discretization
We use the uniform grid Δτ=Γ/n, Δx=X-/m, and Δv=V-/l on the domain [0,Γ]×[0,X-]×[0,V-]. Denoting P~i,j=P~(τ,iΔx,jΔv) and using the central difference scheme, we have
(14)∂P~∂x(τ,xi,vj)=P~i+1,j(τ)-P~i-1,j(τ)2Δx,∂2P~∂x2(τ,xi,vj)=P~i+1,j(τ)-2P~i,j(τ)+P~i-1,j(τ)(Δx)2,∂P~∂v(τ,xi,vj)=P~i,j+1(τ)-P~i,j-1(τ)2Δv,∂2P~∂v2(τ,xi,vj)=P~i,j+1(τ)-2P~i,j(τ)+P~i,j-1(τ)(Δv)2,∂2P~∂x∂v(τ,xi,vj)=P~i+1,j+1(τ)+P~i-1,j-1(τ)-P~i+1,j-1(τ)-P~i-1,j+1(τ)4ΔxΔv.
Through a long derivation, we obtain from (10) the following set of discrete equations and inequalities in matrix form:
(15)∂P~∂τ-CP~≥0,P~≥g,(∂P~∂τ-CP~)T(P~-g)=0,
where P~=(P~00P~10P~20⋯P~m0P~01P~11P~21⋯P~m1⋯P~0lP~1lP~2l⋯P~ml)T,C=D-R, in which D is a block tridiagonal matrix due to the first six terms in (10), and R is a full matrix contributed from the two integral terms of (10). The matrix D can be expressed in the form
(16)D=(B1D1E10…0000B2D2E2…000⋮⋮⋮⋮⋱⋮⋮⋮0000…Bl-1Dl-1El-10000…0B¯lDl),Bj=(a1,jb1,j-a1,j0…0000a2,jb2,j-a2,j…000⋮⋮⋮⋮⋱⋮⋮⋮0000…am-1,jbm-1,j-am-1,j0000…00bm,j),B¯l=(2a1,l0…002a2,l…0⋮⋮⋱⋮00…2am,l),Dj=(c1,jd1,j-c1,j0…0000c2,jd2,j-c2,j…000⋮⋮⋮⋮⋱⋮⋮⋮0000…cm-1,jdm-1,j-cm-1,j0000…00dm,j),Ej=(a1,j-b1,ja1,j0…0000a2,j-b2,ja2,j…000⋮⋮⋮⋮⋱⋮⋮⋮0000…am-1,j-bm-1,jam-1,j0000…02am,j-bm,j)
in which aij=-ijσρ/4, bij=(α(φ-jΔv)-ρσθtjΔv-jσ2)/2Δv, cij=(i(r-k1+θtk2)-jΔvi2)/2, and dij=r+k0-θtk1+jΔvi2+(jσ2/Δv).
To calculate the matrix R due to the two integral terms in (10), let z=Xy and rewrite the integrals by
(17)I1(τ,X,V)=∫0∞P~(τ,Xy,V)f(y)dy=∫0∞P~(τ,z,V)f(z/X)Xdz,I2(τ,X,V)=∫0∞P~(τ,Xy,V)(y-1)f(y)dy=∫0∞P~(τ,z,V)(z/X-1)f(z/X)Xdz.
The above integrals can then be discretized by using linear interpolation, namely,
(18)I1(i,j)τ=∫0∞P~(τ,z,jΔv)f(z/iΔx)(iΔx)dz≈∑k=0m-1Ai,jτ,k,
where
(19)Ai,jτ,k=∫kΔx(k+1)ΔxP~(τ,z,jΔv)f(z/iΔx)(iΔx)dz=∫kΔx(k+1)Δx[(k+1)Δx-zΔxP~(τ,kΔx,jΔv)+z-kΔxΔxP~(τ,(k+1)Δx,jΔv)]×f(z/iΔx)(iΔx)dz=12[kP~(τ,(k+1)Δx,jΔv)-(k+1)P~(τ,kΔx,jΔv)]×[erf(μ-ln((k+1)/i)2σ)-erf(μ-ln(k/i)2σ)]+12iexp(σ22+μ)×[P~(τ,kΔx,jΔv)-P~(τ,(k+1)Δx,jΔv)]×[erf(μ-ln((k+1)/i)+σ22σ)-erf(μ-ln(k/i)+σ22σ)].
Let e1ik=(1/2)(erf((μ-ln((k+1)/i))/2σ)-erf((μ-ln(k/i))/2σ)), and e2ik=(1/2)iexp(σ2/2+μ)(erf((μ-ln((k+1)/i)+σ2)/2σ)-erf((μ-ln(k/i)+σ2)/2σ)). Then (19) becomes
(20)Ai,jτ,k=(e2ik-(k+1)e1ik)P~(τ,kΔx,jΔv)+(ke1ik-e2ik)P~(τ,(k+1)Δx,jΔv),I1(i,j)τ≈∑k=0m-1Ai,jτ,k=e2,i0-2e1,i0+∑k=1m-1[(k-1)e1,ik-1-e2,ik-1+e2,ik-(k+1)e1,ik]+(m-1)e1,im-1-e2,im-1,
where
(21)A=(A00…00A0…0⋮⋮⋱⋮00…A0),A0=(a0,0a0,1a0,2…a0,m-1a0,ma1,0a1,1a1,2…a1,m-1a1,ma2,0a2,1a2,2…a2,m-1a2,m⋮⋮⋮⋮⋮⋮am,0am,1am,2…am,m-1am,m)
in which ai,0=e2,i0-e1,i0, ai,k=(k-1)e1,ik-1-e2,ik-1+e2,ik-(k+1)e1,ik, k=1,2,…,m-1, and ai,m=(m-1)e1,im-1-e2,im-1,i=0,1,…,m. Hence I1τ=AP~.
In a similar way, we obtain I2τ=BP~, where
(22)B=(B00…00B0…0⋮⋮⋱⋮00…B0),B0=(b00b01b02…b0,m-1b1,mb10b11b12…b1,m-1b1,mb20b21b22…b2,m-1b2,m⋮⋮⋮⋮⋮⋮bm0bm1bm2…bm,m-1bm,m)
in which bi0=e4i0-2e3i0-e2i0-2e1i0, bi,k=(k-1)e3,ik-1-e4,ik-1+e4,ik-(k+1)e3,ik-(k-1)e1,ik-1+e2,ik-1-e2,ik+(k+1)e1,ik,k=1,2,…,m-1, bim=(m-1)e3,im-1-e4,im-1-(m-1)e1,im-1+e2,im-1,i=0,1,…,m, e3ik=(1/2)exp(σ2/2+μ)×[erf((μ-ln((k+1)/i)+σ2)/2σ)-erf((μ-ln(k/i)+σ2)/2σ)], and e4ik=(1/2i1-2σ2)exp(μ2/(1-2σ2))×[erf((1-2σ2(μ-ln((k+1)/i)))/2σ-2μσ/1-2σ2)-erf((1-2σ2(μ-ln(k/i)))/2σ-2μσ/1-2σ2)].
Hence, the integral part of the operator L is R=(A-θτB).
Now for American options we obtain a semidiscrete LCP. The Crank-Nicolson method [15] is then used for time discretization. Let Δτ=Γ/n; we obtain
(23)(I-12ΔτC(n))P~(n)≥(I+12ΔτC(n+1))P~(n+1),(I-12ΔτC(n))P~(n)≥iiiiiiiP~(n+1)+P~(n)≥2g,((I-12ΔτC(n))P~(n)-(I+12ΔτC(n+1))P~(n+1))Tiiiiiiiii×(P~(n+1)+P~(n)-2g)=0
which can be denoted by
(24)LCP(M,P~(n+1),NP~(n),g):NP~(n)≥MP~(n+1),P~(n+1)+P~(n)≥2g,(NP~(n)-MP~(n+1))T×(P~(n+1)+P~(n)-2g)=0,
where M=I+(1/2)ΔτC(n+1) and N=I-(1/2)ΔτC(n).
The projected SOR algorithm PSOR(M,P~(n+1),NP~(n),g) can then be used to solve the LCP problem. The algorithm is as follows.
For n=1, dim(M),
(25)rn=NP~n-∑jMi,jP~j,P~n=max{P~j+ωrnMi,j,gn},
where the relaxation parameter ω=2/(1+1-φ2), wherein φ=maxi{(1/Mi,i)∑j≠i|Mi,j|}.
4. Numerical Results and Discussion
In this section, we present a set of results to demonstrate the feature of the model and the robustness of the model for the determination of American put option price and then investigate the influence of market volatility. The basic parameter values used in the computation are given in Table 1. The numerical algorithm is implemented in computer by using the MATLAB 2012 software package. Figure 1 shows the payoff as the function of stock price. The numerical solutions obtained by solving the LCP problem (24) are showed in Figure 2. We also compute the value of American option price under various values of volatility. The results are plotted in Figure 3.
The basic parameter values used in numerical investigation.
μ
0.23
κ
3.46
φ
0.08942
δ
0.0001
r
0.039
σ
0.14
λ
0.77
γ
−0.086
ρ
−0.82
T
1
Smax
200
vmax
1
K
100
N
3000
I
30
J
30
The payoff as a function of stock price (stock price = stock step * 20/3; volatility = volatility step * 1/30).
The American option price.
American option price under different volatilities.
From the numerical results, we can see that volatility has no influence on the payoff of American option, but it plays an important role in American option pricing. From Figures 2 and 3, we can see that, when the stock price is relatively low, the influence of volatility is not significant, while, as the stock price increases, the influence of volatility becomes more and more significant. It can also be found that, at the same level of stock price, the larger the volatility is, the more the influence it plays on American option price is, which is in agreement with the market reality. Thus, the results show that the model and numerical scheme are robust in capturing the feature of incomplete finance market, particularly the influence of market volatility on the price of American options.
AppendicesA. Derivation of the Radon-Nikodym Derivative under the Minimal Martingale Measure <inline-formula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M119">
<mml:mrow>
<mml:mi>q</mml:mi></mml:mrow>
</mml:math></inline-formula>
This appendix derives the Radon-Nikodym derivative under the minimal martingale measure q and then consequently obtains (5) for American option pricing model.
According to the Doob-Meyer decomposition, the discounted risky asset price process, X^t=e-∫0trsdsXt, is a special semimartingale and can be written as(A.1)X~t=X~0+Mt+At
with
(A.2)Mt=∫0tX~uVudWu1+∫0t∫RX~u-(y-1)N~(du,dy),(A.3)At=∫0tX~u(μu-ru)du,
where Mt is the martingale part of X~t and At is the predictable process of finite variation.
The minimal martingale measure q is unique [10] if there exists a predictable process βt that satisfies
(A.4)Zt=dqdp=1+∫0tβudMu.
By using (A.4), we get the theorem below for calculating the Radon-Nikodym derivative.
Theorem 1.
The Radon-Nikodym derivative under the minimal martingale measure q is
(A.5)Zt=exp(-∫0tθuVudWu1-12∫0tθu2Vudu+∫0t∫Rln(1-θu(y-1))N(du,dy)+∫0t∫Rθu(y-1)v(dy)du).
Proof.
According to the Girsanov transformation theory, the predictable process of bounded variation can be computed in terms of Zt,
(A.6)-dAt=d〈M,Z〉tZt-,
where 〈M,Z〉t is the quadratic variation process between M and Z. Under the minimal martingale measure q, the predicable process of bounded variation in the Doob-Meyer decomposition of M is given by
(A.7)d〈M,Z〉tZt-=βtd〈M〉tZt-=-dAt.
Using (A.4) and (A.7), we have
(A.8)Zt=1-∫0tZu-dAud〈M〉udMu.
From (A.2), we get
(A.9)〈M〉t=〈∫0tX~uVudWu1+∫0t∫RX~u-yN~(du,dy)〉=∫0tX~u2(Vu)2du+∫0t∫RX~u-2(y-1)2v(dy)du=∫0tX~u2(Vu+∫R(y-1)2v(dy))du.
Let dYu=-(dAu/d〈M〉u)dMu; then (A.8) can be written as Zt=1+∫0tZu-dYu. Hence,
(A.10)dYu=X~u(μu-ru)duX~u2(Vu+∫R(y-1)2v(dy))du×(X~uVudWu1+∫RX~u-(y-1)N~(du,dy))=(μu-ru)(VudWu1+∫R(y-1)N~(du,dy))Vu+∫R(y-1)2v(dy).
Because Zt is the Doléans-Dade exponential, we obtain
(A.11)Zt=1+∫0tZu-dYu,Z0=1,dYu=-θu(VudWu1+∫R(y-1)N~(du,dy)),
where
(A.12)θu=μu-ruVu+∫R(y-1)2v(dy).
Solving equations (A.11), we obtain Zt as given in Theorem 2.
B. Derivation of the Partial Integrodifferential Equation (<xref ref-type="disp-formula" rid="EEq2.9">7</xref>)
Here we derive the partial integrodifferential equation (PIDE) (7), which is used in Section 2 to construct the linear complementarity problem (LCP) for the pricing of American put option.
Under the measure of minimal martingale q, the price of American put option P(t,Xt,Vt), having strike price K and maturity date T, is as follows:
(B.1)P(t,Xt,Vt)=maxτEq[e-∫tτrudu(K-Xτ)+∣Ft],
where τ, representing the stopping times in [0,T], is to be determined to maximize the discounted option payoff.
To derive the LCP problem for the American put option, we first need to derive the following theorem for fixed τ, which is for European put option.
Theorem 2.
The price of the European option satisfies the following PIDE:
(B.2)-rtP(t,Xt,Vt)+∂P∂t+∂P∂XrtXt-+∂P∂V×(α(φ-Vt)-ρσθtVt)+12∂2P∂X2VtXt2+∂2P∂X∂VXtVtσρ+12∂2P∂X2VtXt2+∫R(P(t,Xty,Vt)-P(t,Xt,Vt)-(y-1)∂P∂XXt-)×v(dy)=0.
Proof.
Using the Itô formula of semimartingales, we get
(B.3)d(e-∫tTruduP(t,Xt,Vt))=-rte-∫0truduP(t,Xt,Vt)dt+e-∫0trudu∂P∂tdt+e-∫0trudu∂P∂XdX+e-∫0trudu∂P∂VdV+12e-∫0trudu∂2P∂X2dXdX+e-∫0trudu∂2P∂X∂VdXdV+12e-∫0trudu∂2P∂V2dVdV+e-∫tTrudu∫R(P(t,Xty,Vt)-P(t,Xt,Vt))N(dt,dy).
By substituting (5) into the equation above, we obtain
(B.4)d(e-∫tTruduP(t,Xt,Vt))=e-∫0trudu(-rtP(t,Xt,Vt)dt+∂P∂tdt+∂P∂XXt-(μdt+VtdW~t1-θtVtdt-∫Rθt(y-1)2v(dy)dt-∫R(y-1)v~(dy)dt)+∂P∂V(α(φ-Vt)dt+σVtdW~t2-ρσθtVtdt)+12∂2P∂X2VXt-2dt+∂2P∂X∂VXVρσdt+12∂2P∂V2Vσ2dt+∫R(P(t,Xty,Vt)-P(t,Xt,Vt))N~(dt,dy)+∫R(P(t,Xty,Vt)-P(t,Xt,Vt))v~(dy)dt+∂P∂tdt).
By combining the like terms of dt, we get
(B.5)d(e-∫tTruduP(t,Xt,Vt))=e-∫0trudu(-rtP(t,Xt,Vt)+∂P∂t+∂P∂XrtXt-+∂P∂V(α(φ-Vt)ρσθt)+12∂2P∂X2VXt-2+∂2P∂X∂VXVρσ+12∂2P∂V2Vσ2+∫R(∂P∂XXt-P(t,Xty,Vt)-P(t,Xt,Vt)-(y-1)∂P∂XXt-)v~(dy))dt+e-∫0trudu(∂P∂XXt-VtdW~t1+∂P∂VσVtdW~t2+∫R(P(t,Xty,Vt)-P(t,Xt,Vt))N~(dt,dy)+∂P∂XrtXt-).
Because the discounted price of the European style option is a martingale, we set the drift term to zero, and consequently we obtain the equation as given in Theorem 2. By rearranging and simplifying (B.2), we obtain (7) in Section 2. Consequently, based on the operator L derived from this theorem, the LCP problem (11) is constructed in Section 2 for solving the American option model.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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