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Home , Implied volatility, Jump diffusion, Stochastic volatility, Valuation of options

Pricing for a Stochastic- Jump-Diffusion Model with Log-Uniform Jump-Amplitudes∗

Guoqing Yan† and Floyd B. Hanson†

Abstract— An alternative option pricing model is proposed, are critical ingredients of the data-generating mechanism. in which the stock prices follow a diffusion model with square is needed to calibrate the longer ma- root stochastic volatility and a jump model with log-uniformly turities and jumps to reflect shorter maturity option pricing. distributed jump amplitudes in the stock price process. The stochastic-volatility follows a square-root and mean-reverting As discussed in Andersen, Benzoni and Lund [2], Bates [5] . Fourier transforms are applied to solve the and Bakshi, Cao and Chen [3], the most reasonable model problem for risk-neutral European option pricing under this of stock prices would include both stochastic-volatility and compound stochastic-volatility jump-diffusion (SVJD) process. jump-diffusion. Characteristic formulas and their inverses simplified by integra- In this paper, an alternative stochastic-volatility jump- tion along better equivalent contours are given. The numerical implementation of pricing formulas is accomplished by both fast diffusion model is proposed, which has square-root Fourier transforms (FFTs) and more highly accurate discrete and mean-reverting stochastic-volatility process and log- Fourier transforms (DFTs) for verifying results and for different uniformly distributed jump amplitudes in Section II. In output. Sect. III, a formal “closed form solution” (according to Key words: stochastic-volatility, jump-diffusion, risk neutral Heston [14]) for risk-neutral pricing of European options option pricing, uniform jump-amplitudes. is given by first converting the problem into characteris- tic functions (Fourier transforms) , then using the Fourier I.INTRODUCTION inversion formula for probability distribution functions to Despite the great success of the Black-Scholes model [6], find a more numerically robust form (but, not everyone [22], this log-normal pure diffusion model fails to reflect the would call it closed). In Sections IV and V, the solution three empirical phenomena: (1) the asymmetric leptokurtic to the problem is computed from the Fourier inverse using features, that is, the return distribution is skewed to the left, both fast Fourier transforms (FFTs) for speed and more and has a higher peak and two heavier tails than those of general discrete Fourier transforms (DFTs) for accuracy and the normal distribution; (2) the , that is, the validation. In Sect. VI, the conclusions are given. is not a constant as assumed in the Black- Scholes model; and (3) the large random fluctuations such II. STOCHASTIC-VOLATILITY JUMP-DIFFUSION MODEL as crashes and rallies. It is assumed that a risk-neutral probability measure M Therefore, many financial engineering studies have been exists, the asset price S(t), under this measure, follows a undertaken to modify and improve the Black-Scholes model jump-diffusion process, with zero-mean at the risk neutral to explain some or all of the above three empirical phe- rate r and conditional variance V (t) , nomena. Popular models include, for example, (i) the jump- diffusion models of Merton [23] and Kou [21]; (ii) the dS(t) = S(t) (r − λJ¯)dt + V (t)dWs(t) constant-elasticity-of-variance model of Cox and Ross [8]; +J(Q )dN(t)) , p (1) (iii) the stochastic-volatility models of Hull and White [17], Stein and Stein [25] and Heston [14]; (iv) the stochastic- It is only necessary to know that the risk-neutral measure volatility and stochastic-interest-rates models of Amin and exists [16]. Since Scott [24] finds that volatility Ng [1], Bakshi and Chen [4], and Scott [24]; (v) the has little impact on short-term option prices, the interest rate stochastic-volatility jump-diffusion models of Bates [5], and r will be assumed constant in this paper. Scott [24]. The instantaneous volatility follows a pure mean-reverting Jarrow and Rosenfeld [18] and Jorion [19] demonstrate and square-root diffusion process, given as that jumps are empirically important for several financial markets. Recently, some empirical research result indicate dV (t)= k (θ − V (t)) dt + σ V (t)dWv(t). (2) that both stochastic-volatility and discrete jump components The variables Ws(t) and Wv(t)pare standard Brownian ∗This work is supported in part by the National Science Foundation under motions for S(t) and V (t), respectively, with E[dWi(t)]=0, Grant DMS-02-07081. Any conclusions or recommendations expressed in Var[dWi(t)] = dt, for i = s or v, and correlation this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Submitted to Proceedings of American Corr[dW (t), dW (t)] = ρ. (3) Control Conference, pp. 1-7, 13 September 2006. s v † Department of Mathematics, , and Computer Science Uni- versity of Illinois at Chicago, 851 S Morgan St.; m/c 249, Chicago, IL In (1), J(Q) is the Poisson jump-amplitude, Q is an 60607–7045, USA; E–Mail: [email protected] or [email protected] underlying Poisson amplitude mark process selected [10], [13], [12], [26], [9] so that III. EUROPEAN PRICE We let C denote the price at time t of a European style Q = ln(J(Q)+1), (4) call option on S(t) with K and time T . Using the fact that the terminal payoff of a European for convenience, N(t) is the standard Poisson jump counting call option on the underlying stock S with strike price K process with jump intensity λ, and is max(S(T ) − K, 0) and assuming the short-term interest rate r is constant over the lifetime of the option, the price E[dN(t)] = λdt = Var[dN(t)]. of the European call at time t equals discounted, conditional expected payoff Also, the symbolic jump term J(Q)dN(t) in (1) denotes the Poisson sum, C(S(t),V (t),t; K,T ) −r(T −t) = e EM[max[ST − K, 0]|S(t),V (t)] dN(t) −r(T −t) ∞ = e ST p (ST |S(t),V (t))dST J(Q)dN(t)= J(Qi), (5) K M ∞ (9) i=1 −K pM`R(ST |S(t),V (t))dST X K R ´ where Qi is the ith jump-amplitude random variable taken = S(t)P1(S(t),V (t),t; K,T ) −r(T −t) from a set of independent, identically distributed (IID) ran- −Ke P2(S(t),V (t),t; K,T ), dom variables. where EM is the expectation with respect to the risk-neutral Let the density of the jump-amplitude mark Q be uni- probability measure, formly distributed: pM(ST |S(t), V (t)) 1 1, a ≤ q ≤ b φQ(q)= , (6) is the corresponding conditional density given (S(t), V (t)), b − a 0, else   P1(S(t),V (t),t; K,T ) (10) where a < 0 < b. The mark Q has moments, such that the −r(T −t) ∞ mean is = e K ST pM(ST |S(t),V (t))dST /S(t) ∞ R µQ ≡ EQ[Q]=0.5(b + a) = K ST pM(ST |S(t),V (t))dST /EM[S(T )|S(t),V (t)], R and variance is by the risk-neutral property, is the risk-neutral probability that S(T ) > K (since the integrand is nonnegative and the 2 2 σQ ≡ VarQ[Q] = (b − a) /12. integral over [0, ∞) is one) and risk-neutral in-the-money (ITM) probability is The jump-amplitude J itself has mean P2(S(t),V (t),t; K,T ) = Prob[S(T ) > K|S(t),V (t)] (11) ¯ J ≡ E[J(Q)] = (exp(b) − exp(a))/(b − a) − 1. (7) is the complementary risk neutral distribution function. The Choosing the log-uniform distribution has several reasons: European option evaluation problem is to evaluate P1 and first, since the exponentially small tails of the log-normal or P2 under the distribution assumptions embedded in the log-double-exponential distribution are contrary to the flat risk neutral probability measure. The difficulty is that the and thick tails of the long time financial market log-return cumulative distribution function for most distributions is data. Next, around the near-zero peak of the log-double- infeasible [5]. exponential and the log-normal, the jumps are small, so are We follow the similar treatments in Bates [5], Hes- ton [14] and Bakshi et al. [3]. We apply the Dynkin theorem not qualitatively different or separately detectable from the continuous diffusion fluctuations. Moreover, an infinite jump (see [9, Chapter 8] for instance) to derive the partial integro- differential equation (PIDE) satisfied by the value of an domain is unrealistic , since the jumps should be bounded option. The Dynkin theorem establishes a relationship be- in real world financial markets and infinite domain leads to unrealistic restrictions in portfolio optimization [11]. tween stochastic differential equations and partial differential equations. For the jump-diffusion in the time-inhomogeneous The square-root stochastic-volatility process (2) has two case and in one-dimension, given the stochastic differential major advantages. First, the model can allow for systematic equation for , . The second is that the process generates X(t) an analytically tractable method of pricing options without dX(t) = f(X(t),t)dt + g(X(t),t)dW (t) sacrificing accuracy of requiring undesirable restrictions on +h(X(t),t,Q)dP (t; X(t),t) parameter values [5]. By the Itˆo’s chain rule and under a risk-neutral probability the Dynkin theorem states that the expectation, where T is measure M, the log-return process ln(S(t)) satisfies the SDE the terminal time, u(x, t)= E[U(X(T ))|X(t)= x] d ln(S(t)) = (r − λJ¯ − V (t)/2)dt + V (t)dWs(t) is the solution to the following PIDE: +QdN(t) , p (8) ∂u ∂u 1 ∂2u 0 = + f + g2 where r is the risk-free interest rate, such that ∂t ∂x 2 ∂x2 EM[S(T )|S(t0)] = S(t0) exp(r(T − t0)) for some ¯ +λ (u(x + h(x,t,q),t) − u(x,t))φQ(q)dq, initial time t0 with risk-neutral drift µM = r − λJ. ZQ

2 subject to the final condition u(x, T ) = U(x), given the A. Characteristic Function Formulation for Solution terminal distribution U(x) (see [9, Chapter 8]for instance). The corresponding characteristic functions for We make a change of variable from S(t) to Pj (ℓ,v,t; κ,T ), with respect to the κ variable, for , defined by L(t) = ln(S(t)) j =1:2 b ∞ iyκ with SDE given in (8) and inverse S(t) = exp(L(t)). Eq. (9) fj (ℓ,v,t; y,T ) ≡− e dPj (ℓ,v,t; κ,T ), (17) is rewritten as Z−∞ C(S(t),V (t),t; K,T )er(T −t) with a minus sign to account for the negativityb of the measure dP , will also satisfy similar PIDEs: = EM[max(ST − K, 0)|S(t),V (t)], j

r(T −t) ∂fj now C(S(t), V (t),t; K,T )e is the conditional expec- b + Aj [fj ](ℓ,v,t; κ,T )=0, (18) ∂t tation of the composite process, where M is an appropri- ate theoretical risk-neutralizing measure. Applying the two- for j = 1:2, again suppressing PIDE parameters (y,T ), with dimensional Dynkin theorem for the price dynamics (1) and the respective boundary conditions: (2), we obtain that the value of a European-style option, as iyℓ fj (ℓ,v,T ; y,T )=+e , (19) a function of the stock log-return L(t) denoted by since from (17) and (14-16) C(L(t), V (t),t; κ,T ) ≡ C(S(t), V (t),t; K,T ), dPj (ℓ,v,T ; κ, T )= d1ℓ>κ = dH(ℓ − κ)= −δ(κ − ℓ)dκ. i.e., b To solveb for the characteristic function explicitly, letting τ = C(ℓ,v,t; κ, T )=EM[max(exp(L(T ))−K, 0)|L(t)=ℓ,V (t)=v] T − t be the time-to-go we conjecture that the function is respectively given by: andb κ ≡ ln(K), satisfies the following PIDE: fj (ℓ,v,t; y,t + τ)=exp(gj (τ)+hj (τ)v+iyℓ+βj(τ)) , (20) ∂C ∂C 1 ∂C 0 = +A[C](ℓ,v,t; κ, T )≡ + r−λJ¯− v for j = 1:2, where with β (τ) = rτδ and the boundary ∂t ∂t „ 2 « ∂ℓ j j,2 b b b conditions: b 2 2 2 ∂C 1 ∂ C ∂ C 1 2 ∂ C +k(θ−v) + v +ρσv + σ v (12) gj (0)=0= hj (0). ∂v 2 ∂ℓ2 ∂ℓ∂v 2 ∂v2 ∞b b b b This conjecture exploits the linearity of the coefficient in the −rC+λ C(ℓ + q,v,t)−C(ℓ,v,t) φQ(q)dq, Z−∞“ ” PIDEs (18). b b b By substituting (20) into (18) and cancelling the common From (9), in the current state variables, factor of fj , where ℓ κ−r(T−t) C(ℓ,v,t; κ, T )=e P1(ℓ,v,t; κ, T )−e P2(ℓ,v,t; κ, T ), ′ ′ 1 0 = −gj (τ)−vhj (τ)−rδj,2 + r−λJ¯ ± v iy whereb κ = ln(K),b inserting this into (12)b and separating „ 2 « 1 2 1 2 2 assumed independent terms P1 and P2, produces two PIDEs +(k(θ − v)+ρσvδj, )hj − vy +ρσviyhj + σ vh 1 2 2 j for the risk neutralized probabilities Pi(ℓ,v,t; κ,T ) for i = ∞ 1, 2: (iy+δj,1)q b b −λJδ¯ j,1 +λ (e −1)φQ(q)dq, (21) Z−∞ ∂P 0 = 1 + A [P ](ℓ,v,t; κ, T ) ∂t 1 1 where ±1=+δj,1 − δj,2, and by separating the order v b ∂P b ∂P ∂P and order one terms to reduce to two ordinary differential ≡ 1 +A[P ](ℓ,v,t; κ, T )+v 1 +ρσv 1 (13) ∂t 1 ∂ℓ ∂v equations(ODEs), b ∞ b b b q 1 + r − λJ¯ P1 +λ (e − 1)P1(ℓ + q,v,t)φQ(q)dq; ′ 2 2 Z hj (τ) = σ hj (τ) + (ρσ(δj,1 + iy) − k)hj (τ) ` ´ −∞ 2 b b 1 1 where the (κ,T ) dependence has been suppressed, subject ± iy − y2 (22) to the boundary condition at the expiration time t = T : 2 2 and P1(ℓ,v,T ; κ,T )=1ℓ>κ, (14) ′ ¯ ¯ gj (τ) = kθhj (τ) + (r − λJ)iy − λJδj,1 − rδj,2 where 1 is the indicator function for the set ℓ>κ, and ∞ ℓ>κ b +λ (e(iy+δj,1)q − 1)φ (q)dq. (23) ∂P Q 0 = 2 + A [P ](ℓ,v,t; κ, T ) (15) −∞ ∂t 2 2 Z b To solve (22), we factor the RHS (right-hand-side) using ∂P2 b ≡ + A[P2](ℓ,v,t; κ, T )+ rP2; ∂t 2 2 b ηj = ρσ(iy + δj,1) − k & ∆j = ηj − σ iy(iy ± 1), subject to the boundary conditionb at the expirationb time t = q so we have T : 1 η + ∆ η − ∆ h′ (τ)= σ2 h + j j h + j j . P2(ℓ,v,T ; κ,T )=1ℓ>κ. (16) j 2 j σ2 j σ2     b 3 By separating of variables using partial fractions, C. by Put-Call Parity The price of a European put on the same stock can 1 1 1 be determined from the put-call parity. As both Hull [16] − − dhj = dτ, ηj ∆j ηj +∆j and Higham [15] that the put-call parity is based primarily ∆j hj + 2 hj + 2 ! σ σ on the properties of the maximum function, in absence of and integrating both sides, we obtain separation, friction terms like dividends, and hence is independent of any particular process, so that ηj −∆j hj + σ2 −rT ln = ∆j τ + C. C(S(0),V (0),t; K,T )+ Ke = P (S(0),V (0),t; K,T ) ηj +∆j hj + σ2 ! +S(0),

Solving for hj satisfying the boundary conditions yields the or in other words, European put option price is solutions P (S(0),V (0),t; K,T ) = C(S(0),V (0),t; K,T ) 2 2 ∆j τ (η − ∆ )(e − 1) −rT j j (24) +Ke − S(0), (27) hj (τ)= 2 ∆ τ ; σ (ηj + ∆j − (ηj − ∆j )e j ) easily calculated once the call option price is known. and OMPUTING NVERSE OURIER NTEGRALS gj (τ) = ((r − λJ¯)iy − λJδ¯ j,1 − rδj,2)τ IV. C I F I ∞ (iy+δj,1)q +λτ (e − 1)φQ(q)dq (25) The inverse Fourier integral (26) can be computed by Z−∞ means of standard procedures of numerical integration with −∆j τ kθ (∆j + ηj )(1 − e ) some precautions. Two methods are compared: the discrete − 2 2 ln 1 − σ „ „ 2∆j « Fourier transform (DFT) with Gaussian Quadrature sub- integral refinement for accuracy and the other is the fast +(∆j + ηj )τ , « Fourier transform (FFT) for speed of computation. Applying the uniform distribution of jump-amplitude mark A. Discrete Fourier Transform (DFT) Approximations Q in our general formulas, leads to the following integral in (25): Since the integrand of (26) has a bounded limit as y → 0+, is otherwise smooth and decays very fast, it is rewritten in ∞ b iy δ q 1 iy δ q the general approximate form for DFT, (e( + j,1) −1)φ (q)dq = (e( + j,1) −1)dq Q b − a Z−∞ Za ∞ e(iy+δj,1)b −e(iy+δj,1)a I[F ](κ) ≡ F (y; κ)dy = −1, Z0 (b − a)(iy + δj,1) N N jh demonstrating the simplicity and utility of the log-uniform ≃ Ij (κ)= F (y; κ)dy, (28) − j=1 j=1 (j 1)h jump amplitude distribution. X XZ for sufficiently large N and such integrals are the basis of the discrete Fourier transform, where h is a fixed gross step size B. Inverse Fourier Transform Solution for Tail Probabilities depending on some integral cutoff Ry = max[y] ≃ N ∗ h. The sub-integrals on ((j − 1)h,jh) in (28) for j = 1:N are The tail probabilities P and P can be calculated by 1 2 computed by means of ten-point Gauss-Legendre formula for finding the inverse Fourier transforms of the characteristic refined accuracy need for oscillatory integrands and for the functions and are given (see Kendall et al. ([20]) by fact that it is an open quadrature formula that avoids any 1 1 +∞ non-smooth behavior as y → 0+. The number of steps N Pj (S(t), V (t),t; K,T )= + (26) is not static, but ultimately determined by a local stopping 2 π 0+ Z criterion: the integration loop is stopped if the ratio of the e−iy ln(K)f (ln(S(t)), V (t),t; y,T ) ·Re j dy, contribution of the last strip to the total integration becomes iy   smaller than 0.5e-7. By using formula (28) we also have for j =1:2, where the leading term of 1/2 is one to specify a suitable step size h. By trials, h =5 is a good half the residue at the y = 0 pole, Re[ ] denotes the choice that we can get sufficiently fast convergence and good real component of a complex number which arises from precision. We use the principal square root to evaluate the the principal value limits of the combined real parts of the square root of complex in our pricing formula. Also, we simplifying equivalent contour. This is what Heston [14] use principal branch for the value of the complex natural calls a ‘closed form solution”. The singularity at y = 0 is logarithm. only an apparent singularity in that the integrand of should be bounded as y → 0+ in most cases. Nevertheless, the B. Discrete and Fast Fourier Transform Comparisons infinite integrals involved by the Fourier transforms need to In addition to the general discrete Fourier transform be evaluated by some numerical integration method. (DFT), there is an extremely efficient way of computing the

4 DFT by the fast Fourier transform (FFT), corresponding to integral into discrete Fourier transform: the inverse transform, N −ακk e −iκkyj N−1 CN (ℓ,v,t; κk,T ) = e − 2π π i N jk j=1 IN (sk)= e FN (yj )∆y, (29) X j=0 b FN (ℓ,v,t; yj ,T ; α)∆y, (33) X where sk = k ∗ ∆κ is the discrete stock price for k = 0 : for k =0=(N − 1), so FFT returns N values of κk where N −1, yj = j ∗∆y is the discrete Fourier variable for j = 0: now κk = −L + ℓ + k∆L and L = N∆L/2, to keep the N − 1. typically N is a power of 2, IN (sk) is the kth value at-the-money (ATM) strike price in the middle of the range. of the inverse I[F ](sk) from (28), FN (yj ) is the jth value of In order to apply the FFT, we need to let ∆L∆y = 2π/N. its integrand and ∆(y) is integral step size. The algorithm Hence there will be conflict that if we choose ∆y small reduces the number of matrix multiplication from O(N 2) to obtain a fine grid for the integration, then strike price to O(N log2 N). Unfortunately, the FFT cannot be directly spacings will be large. Following [7], Simpson’s rule is applied to evaluate the integral, since the integrand is singular incorporated into the summation to improve the accuracy, at the required evaluation point y =0 and precision is limited so N−1 to fixed step sizes ∆y. Our FFT approach to this problem is −ακk e −i 2π jk iy (L−ℓ) similar to that of Carr and Madan [7]. They developed some C (ℓ,v,t; κ ,T ) = e N e j N k π techniques around the problem to acquire a better evaluation j=0 X for the inverse Fourier transform to get the option price. b FN (ℓ,v,t; yj ,T ; α)∆y (34) In our model, f2(ℓ,v,t; κ,T ) is the characteristic function j of the tail probability with respect to κ. In order to remove ·[3 − (−1) − δj,0]/3, the non-smooth behavior at the pole y =0, the pole is shifted b to the imaginary axis by multiplying option price at time t where δj,k is the Kronecker delta and the factor in the by an exponential, following an idea of Carr and Madan last line is the Simpson’s term, preserving the fixed step (CarrMadan99). Since dP2 = −pMdK can be shown to be size needed for FFT. The summation in (34) is an exact application of the FFT. Appropriate and the Simpson ∞ α −r(T−t) ℓ s C(ℓ,v,t; κ, T )=−e b (e −e )dP2(ℓ,v,t; s,T ), (30) modified ∆y need to be chosen. We use α = 2.0 and Zκ ∆y =0.25. b b so that exponentially modified call price is b V. NUMERICAL RESULTS AND DISCUSSION b C(ℓ,v,t; κ,T ; α)= eακC(ℓ,v,t; κ,T ), (31) Both methods are implemented and validated against with each other. The two methods give the similar results within where thee exponential coefficientb α needs to be real double precision accuracy. The FFT method can compute and positive. The corresponding Fourier transform of the different levels strike price near at-the-money (ATM) C(ℓ, V (t),t; κ,T ; α) is defined by in about 5 seconds, which has advantage when one want ∞ a full view about the option price. However, because the e F(ℓ,v,t; y,T ; α)= eiyκC(ℓ,v,t; κ,T ; α)dκ. (32) regular spacings on log strike price are required, the method −∞ Z is not convenient to give out the results for one specific Hence, the call price can be expressede as the inverse Fourier strike price as can be implemented with the DFT. Using transform multiplied by the reciprocal of the exponential standard integration methods with the DFT has the advantage factor, of producing results for a given strike price in about 0.5 seconds. ∞ e−ακ The option prices from the stochastic-volatility jump- C(ℓ,v,t; κ,T ) = e−iyκF(ℓ,v,t; y,T ; α)dy 2π −∞ diffusion (SVJD) model are compared with those of Black- −ακ Z ∞ Scholes (BS) model. As expected, call and put option prices e −iyκ b = e F(ℓ,v,t; y,T ; α)dy. of SVJD model are higher than those of Black-Scholes π Z0 model with respect to the strike price. The reason is the The expression for F(ℓ,v,t; y,T ; α) is determined as fol- stochastic-volatility and jump increase the risk premium. lows: For longer the maturity time, the difference found to be ∞ ∞ are bigger. However, for same maturity time, the near-ATM F(ℓ,v,t; y,T ; α) = −e−r(T −t) eiyκ eακ option prices from two models have largest difference for Z−∞ Zκ strike price. s κ ·(e − e )dP2(ℓ,v,t; s,T )dsdκ See Figures 1-3 comparing the DFT results call option −r(T −t) prices for the SVJD model compared with the corresponding e f2(ℓ,v,t; y − (α + 1)i,T ) = b . Black-Scholes (pure diffusion) call option prices for the α2 + α − y2 + i(2α + 1)y maturity times T =0.1, 0.25 and 1.0 years, i.e., 36 days, In order to make (33) fit the application of the FFT, an one quarter and one year, respectively. For the corresponding approximation for C(ℓ,v,t; κ,T ) to transfer the Fourier DFT put option price results, see Figure 4-6 comparing

b 5 the results put option prices for the SVJD model compared Call Option Price for T = 1 with the corresponding Black-Scholes option prices for the SVJD BS maturity times T =0.1, 0.25 and 1.0 years, i.e., 36days, one 25 quarter and one year, respectively.

20 Call Option Price for T = 0.1

SVJD 25 BS 15

Call Option Price 10 20

5 15

80 90 100 110 120 130 10

Call Option Price Strike Price

5 Fig. 3. DFT call option prices for the SVJD model compared to the corresponding pure diffusion Black-Scholes values for parameter values r = 3%, S0 = $100 and T = 1 year for the options; while for the stochastic = 7% = 0 012 = 0 622 = 0 53 = 80 90 100 110 120 130 volatility they are σ , V . , ρ − . , θ . and k Strike Price 0.012; and for the uniform jump model they are a = −0.028, b = 0.026 and λ = 64.

Fig. 1. DFT call option prices for the SVJD model compared to the corresponding pure diffusion Black-Scholes values for parameter values r = Put Option Price for T = 0.1 3%, S0 = $100 and T = 0.1 years for the options; while for the stochastic SVJD volatility they are σ = 7%, V = 0.012, ρ = −0.622, θ = 0.53 and k = BS 0.012; and for the uniform jump model they are a = −0.028, b = 0.026 30 and λ = 64. 25

Call Option Price for T = 0.25 20

SVJD 25 BS 15 Put Option Price

10 20

5

15

80 90 100 110 120 130 Strike Price 10 Call Option Price

Fig. 4. DFT put option prices for the SVJD model compared to the 5 corresponding pure diffusion Black-Scholes values for parameter values r = 3%, S0 = $100 and T = 0.1 years for the options; while for the stochastic volatility they are σ = 7%, V = 0.012, ρ = −0.622, θ = 0.53 and k = 0.012; and for the uniform jump model they are a = −0.028, 80 90 100 110 120 130 Strike Price b = 0.026 and λ = 64.

Fig. 2. DFT call option prices for the SVJD model compared to the corresponding pure diffusion Black-Scholes values for parameter values r = characteristic functions in a formally closed form in terms of 3%, S0 = $100 and T = 1/4 years (i.e., one quarter) for the options; while a Fourier inverse transform on a reduced equivalent contour. for the stochastic volatility they are σ = 7%, V = 0.012, ρ = −0.622, Two numerical computing algorithms are implemented. The θ = 0.53 and k = 0.012; and for the uniform jump model they are a = −0.028, b = 0.026 and λ = 64. first is a general integral for the DFT accurately approxi- mated by 10-point Gauss-Legendre quadrature formula and second applies the FFT algorithm using an evenly-spaced VI. SUMMARY AND CONCLUSION Simpson’s rule enhancement technique due to Carr and An alternative stochastic-volatility jump-diffusion model Madan [7]. Also an exponential modification technique of is proposed with square root mean reverting for stochastic- theirs was used to move a pole at the origin in the inverse volatility combined with log-uniform jump amplitudes. Char- Fourier transform to a less numerically sensitive position on acteristic functions of the log of the terminal stock price the imaginary axis. Two methods give the similar results and and the conditional risk neutral option prices are derived have different advantages depending on the desired output. analytically. The option prices can be expressed in terms of The option prices from this alternative model are compared

6 Put Option Price for T = 0.25 [3] G. Bakshi, C. Cao and Z. Chen, “Empirical Performance of Alternative SVJD Option Pricing Models,” Journal of ,vol. 52, 1997, pp. 2003- BS 2049. 30 [4] G. Bakshi and Z. Chen, “An alternative model for Contingent Claims,”Journal of ,vol. 44, 1997, pp. 123-165. 25 [5] D. Bates, “Jump and Stochastic Volatility: Exchange Rate Processes Implict in Deutche Mark in Options”, Review of Financial Studies,

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Put Option Price Transform,” J. Comp. Fin., vol. 2, 1999, pp. 61-73. 10 [8] J. C. Cox and S. A. Ross, “The for Alternative Stochastic Processes,” J. Fin. Economics, vol. 31, 1976, pp. 383-402. [9] F. B. Hanson, Applied Stochastic Processes and Control for Jump- 5 Diffusions: Modeling, Analysis and Computation, SIAM Books, Philadelphia, PA, to appear 2006; http://www2.math.uic. edu/˜hanson/math574/#Text . 80 90 100 110 120 130 Strike Price [10] F. B. Hanson and J. J. Westman, “Optimal Consumption and Portfolio Control for Jump-Diffusion Stock Process with Log-Normal Jumps,” Proc. 2002 Amer. Control Conf., 2002, pp. 4256-4261. Fig. 5. DFT put option prices for the SVJD model compared to the [11] F. B. Hanson and J. J. Westman, “Computational Methods for Port- corresponding pure diffusion Black-Scholes values for parameter values folio and Consumption Optimization in Log-Normal Diffusion, Log- r = 3%, S0 = $100 and T = 1/4 years for the options; while for the Uniform Jump Environments,” Proc. 15th International Symposium on stochastic volatility they are σ = 7%, V = 0.012, ρ = −0.622, θ = 0.53 Mathematical Theory of Networks and Systems, 9 CD pages, August and k = 0.012; and for the uniform jump model they are a = −0.028, 2002. b = 0.026 and λ = 64. [12] F. B. Hanson, J. J. Westman and Z. Zhu, “Multinomial Maximum Likelihood Estimation of Market Parameters for Stock Jump-Diffusion Put Option Price for T = 1 Models,” AMS Contemporary Mathematics, vol. 351, 2004, pp. 155- 169. 30 SVJD [13] F. B. Hanson and Z. Zhu, “Comparison of Market Parameters for BS Jump-Diffusion Distributions Using Multinomial Maximum Likeli- hood Estimation,” Proc. 43rd IEEE Conference on Decision and 25 Control, pp. 3919-3924, December 2004. [14] S. Heston, “A Closed-Form Solution For Option with Stochastic 20 Volatility with Applications to and Currency Options,” Review of Financial Study, vol. 6, 1993, pp. 337-343. [15] D. J. Higham, An Introduction to Financial Option Valuation: Math- 15 ematics, Stochastics and Computation, Cambridge University Press, Cambridge, UK, 2004. Put Option Price [16] J. C. Hull, Options, Futures, & Other Derivatives, 4th Edition, 10 Prentice-Hall, Englewood Cliffs, NJ, 2000. [17] J. Hull and A. White,“The Pricing of Options with Stochastic Volatil- 5 ities,” Journal of Finance, vol. 42, 1987, pp. 281-300. [18] R. A. Jarrow and E. R. Rosenfeld, “Jump Risks and the Intertemporal Capital Model,” J. Business, vol. 57 (3), 1984, pp. 337- 80 90 100 110 120 130 351. Strike Price [19] P. Jorion, “On Jump Processes in the Foreign Exchange and Stock Markets,” Rev. Fin. Studies, vol. 88 (4), 1989, pp. 427-445. [20] M. G. Kendall, A. Stuart and J. K. Ord, “Advanced Theory of Fig. 6. DFT put option prices for the SVJD model compared to the Statistics,” vol. 1, ”Distribution Theory,” Halsted Press, 1987. corresponding pure diffusion Black-Scholes values for parameter values [21] S. G. Kou,“A Model for Option Pricing,” Management = 3% = $100 = 1 r , S0 and T year for the options; while for the Science, vol. 48 (8),2002, pp. 1086-1101. = 7% = 0 012 = 0 622 = 0 53 stochastic volatility they are σ , V . , ρ − . , θ . [22] R. C. Merton, “Theory of Rational Option Pricing,” Bell J. Econ. = 0 012 = 0 028 and k . ; and for the uniform jump model they are a − . , Mgmt. Sci., vol. 4, 1973, pp. 141-183. = 0 026 = 64 b . and λ . [23] R. C. Merton, “Option Pricing when Underlying Stock Returns are Discontinuous,” J. Financial Economics, vol. 3, 1976, pp. 125-144. [24] L. Scott,“Pricing Stock Options in a Jump-Diffusion Model with with those from Black-Scholes model. The SVJD uniform Stochastic Volatility and Interest Rates: Applications of Fourier Inver- jump model has higher option prices, especially for longer sion Methods,” , vol. 7(4), 1997,pp. 413-424. [25] E. Stein and J. Stein, “Stock Price Distributions with Stochastic maturity and near at-the-money strike price. Volatility,” Review of Financial Studies,vol. 4, 1991, pp. 727-752. [26] Z. Zhu and F. B. Hanson, “Log-Double-Uniform Jump- REFERENCES Diffusion Model,” Dept. Math., Stat., and Comp. Sci., U. Illinois at Chicago, working paper, to be submitted, 2005, pp. [1] K. Amin and V. Ng, “Option Valuation with Systematic Stochastic 1-29. http://www.math.uic.edu/˜hanson/pub/ZZhu/ Volatility,” Journal of Finance, vol. 48, 1993, pp. 881-910. dbuniformfree2.pdf . [2] T.G. Andersen, L. Benzoni and J. Lund, “An Empirical Investigation of Continuous-Time Equity Return Models,” Journal of Finance, vol. 57, 2002, pp. 1239-1284.

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