Option Pricing for a Stochastic-Volatility Jump-Diffusion Model with Log-Uniform Jump-Amplitudes Floyd B. Hanson and Guoqing Yan Department of Mathematics, Statistics, and Computer Science University of Illinois at Chicago ThB06: Computational Methods in Control Analysis, 15 June 2006 in Proceedings of 2006 American Control Conference, Minneapolis, pp. 2989-2994, 14 June 2006. This material is based upon work supported by the National Science Foundation under Grant No. 0207081 in Computational Mathematics. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. F. B. Hanson and G. Yan 1 UIC and FNMA
Outline 1. Introduction. 2. Stochastic-Volatility Jump-Diffusion Model. 3. European Option Prices. 4. Computing Fourier integrals and Inverses. 5. Numerical Results for Call and Put Options. 6. Conclusions. F. B. Hanson and G. Yan 2 UIC and FNMA
1. Introduction Classical Black-Scholes (1973) model fails to reflect the three empirical phenomena: Non-normal features: return distribution skewed negative and leptokurtic, with higher peak and heavier tails; Volatility smile: implied volatility not constant as in B-S model; Large, sudden movements in prices: crashes and rallies. Recently empirical research (Andersen et al.(2002), Bates (1996) and Bakshi et al.(1997)) imply that most reasonable model of stock prices includes both stochastic volatility and jump diffusions. Stochastic volatility is needed to calibrate the longer maturities and jumps are needed to reflect shorter maturity option pricing. Log-uniform jump amplitude distribution is more realistic and accurate to describe high-frequency data; square-root stochastic volatility process allows for systematic volatility risk and generates an analytically tractable method of pricing options. F. B. Hanson and G. Yan 3 UIC and FNMA
2. Stochastic-Volatility Jump-Diffusion Model Assume asset price S(t), under a risk-neutral probability measure M, follows a jump-diffusion process and conditional variance V (t) follows a square-root mean-reverting diffusion process: ( ds(t) = S(t) (r λ J)dt + ) V (t)dw s (t) dn(t) + k=1 S(T k )J(Q k), dv (t) = k (θ V (t)) dt + σ V (t)dw v (t). (2) r = constant risk-free interest rate; W s (t) and W v (t) are standard Brownian motions with correlation: Corr[dW s (t), dw v (t)] = ρ; J(Q) = Poisson jump-amplitude, Q = underlying Poisson amplitude mark process selected so that Q = ln(j(q) + 1); N(t) = compound Poisson process with jump intensity λ. (1) F. B. Hanson and G. Yan 4 UIC and FNMA
Density of jump-amplitude mark Q is uniformly distributed: φ Q (q) = 1 1, a q b b a 0, else, a < 0 < b Mark Mean: µ j E Q [Q] = 0.5(b + a); Mark Variance: σ 2 j Var Q[Q] = (b a) 2 /12; Jump-Amplitude Mean: J E[J(Q)] E[exp(Q) 1] =(exp(b) exp(a))/(b a) 1. Realism: Jump amplitudes are finite; NYSE uses circuit breakers that limit very large jumps; in optimal portfolio problem finite distributions allow realistic borrowing and short-selling. By Itô s chain rule, log-return process ln(s(t)) satisfies SDE: dln(s(t)) = (r λ J V (t)/2)dt + V (t)dw s (t) dn(t) + k=1 Q k. (3) F. B. Hanson and G. Yan 5 UIC and FNMA
3. European Option Prices 3.1 Probability Distribution Function: Price of European call option under risk-neutral probability measure: C(S(t), V (t), t; K, T) = e r(t t) E M [max[s T K, 0] S(t), V (t)] = S(t)P 1 (S(t), V (t), t; K, T) Ke r(t t) P 2 (S(t), V (t), t; K, T); C(S(t), V (t), t; K, T)e r(t t) = E M [max(s T K, 0) S(t), V (t)], the conditional expectation of the composite process; Change of variables: L(t) = ln(s(t)) and κ = ln(k), so Ĉ(L(t), V (t), t; κ, T) C(S(t), V (t), t; K, T) in terms of processes or for PDEs Ĉ(l, v, t; κ, T) C(exp(l), v, t;exp(κ), T); Applying the two-dimensional Dynkin s theorem, (4) F. B. Hanson and G. Yan 6 UIC and FNMA
using A as backward operator: 0 = b C t + A[ b C](l,v, t; κ, T) b C t + r λ J 1 2 v «b C l +k(θ v) C b v + 1 2 v 2 C b l + ρσv 2 C b 2 l v + 1 2 σ2 v 2 C b v r C b Z 2 +λ bc(l + q,v, t) C(l,v, b t) φ Q (q)dq, (5) and by substituting and separating variables, produce: PIDE for P 1, with boundary condition P 1 (l, v, T; κ, T) = 1 l>κ : 0 = P b 1 t + A 1[ P b 1 ](l, v, t; κ,t) P b 1 t + A[ P b 1 ](l, v, t; κ, T) + v P b 1 l +ρσv b P 1 v + `r λ J b P1 + λ (e q 1) b P 1 (l + q,v, t)φ Q (q)dq; (6) PIDE for P 2, with boundary condition P 2 (l, v, T; κ, T) = 1 l>κ : 0 = b P 2 t + A 2[ b P 2 ](l, v, t; κ,t) b P 2 t + A[ b P 2 ](l, v, t; κ, T) + r b P 2 ; (7) F. B. Hanson and G. Yan 7 UIC and FNMA
3.2 Characteristic Function: Corresponding characteristic functions defined by Fourier transforms, f j (l, v, t; y, T) Satisfying the same PIDEs as the P j (l, v, t; κ, T): e iyκ d P j (l, v, t; κ, T), (8) f j t + A j[f j ](l, v, t; κ, T) = 0, (9) where A j represents the corresponding full backward operators in (6) and (7) with boundary conditions, f j (l, v, T; y, T) = +e iyl, respectively for j = 1 : 2. Solution conjecture: f j (l, v, t; y, t + τ)=exp(g j (τ)+h j (τ)v+iyl+β j (τ)), (10) with β j (τ) = rτδ j,2 and BCs g j (0) = 0 = h j (0) for j = 1 : 2. F. B. Hanson and G. Yan 8 UIC and FNMA
3.3 Solution Details: For the Fourier transforms f j for j = 1 : 2, h j (τ) (η 2 j 2 j )(e jτ 1) σ 2 (η j + j (η j j )e jτ ) ; (11) g j (τ) = ((r λ J)iy λ Jδ j,1 rδ j,2 )τ where +λτ kθ σ 2 2 ln (e (iy+δ j,1)q 1)φ Q (q)dq (12) 1 ( j + η j )(1 e jτ ««) + ( j + η j )τ, 2 j η j = ρσ(iy + δ j,1 ) k & j = q η 2 j σ2 iy(iy ± 1); (e (iy+1)q 1)φ Q (q)dq = e(iy+1)b e (iy+1)a (b a)(iy + 1) 1. F. B. Hanson and G. Yan 9 UIC and FNMA
The tail probabilities P j for j = 1 : 2 are P j (S(t), V (t), t; K, T) = 1 2 + 1 [ ] + e iy ln(k) f j (ln(s(t)), V (t), t; y, T) Re π 0 + iy dy, (13) by complex integration on equivalent contours yielding a residue of 1/2 and a principal value integral in the limit to the left of the apparent singularity at y = 0 +, since the integrand is bounded in the singular limit. F. B. Hanson and G. Yan 10 UIC and FNMA
4. Computing Fourier Integrals and Inverses 4.1 Using 10-point Gauss-Legendre formula for DFTs: Re-write the Fourier integral as I(x) = 0 F(x)dx = lim N > NX j=1 Z jh (j 1)h F(x)dx. (14) Because of singularity at y = 0 and oscillatory behavior, discrete Fourier tranform (DFT) sub-integrals in (14) are computed by means of a highly accurate, ten-point Gauss-Legendre formula, which is also an open formula, not evaluating the function at the endpoints. N is not fixed but determined by a local stopping criterion: the integration loop is stopped if the ratio of the contribution of the last strip to the total integration becomes smaller than 0.5e-7. Step size h = 5: Good choice for fast convergence and good precision. F. B. Hanson and G. Yan 11 UIC and FNMA
4.2 Using Fast Fourier Transform (FFT): (After Carr and Madan (1999)) Initial call option price: C(S(t),V (t),t; K,T)= K e r(t t) (S(t) K)dP 2 (S(t),V (t),t; K, T); (15) Modified call option price to remove the singularity: C (mod) (S(t), V (t), t; κ, T) = e ακ C(S(t), V (t), t; K, T); (16) Corresponding Fourier transform of C (mod) (S(t), V (t), t; κ, T): Thus, Ψ(S(t),V (t),t; y, T) = C(S(t),V (t), t; K, T)= e ακ π e iyκ C (mod) (S(t), V (t),t; κ, T)dκ; (17) 0 e iyκ Ψ(S(t),V (t),t; y,t)dy; (18) F. B. Hanson and G. Yan 12 UIC and FNMA
where Ψ(S(t),V (t), t; y, T) = e iyκ e ακ e r(t t) (S(t) K) κ dp 2 (S(t),V (t), t; κ, T)dκ = e r(t t) f 2 (y (α + 1)i) α 2 + α y 2 + i(2α + 1)y ; (19) Transfer the Fourier integral into discrete Fourier transform (DFT) and incorporate Simpson s rule (Carr and Madan (1999)) to increase accuracy of the FFT application for Fourier inverses: C(S(t), V (t), t; κ, T) = e ακ π NX j=1 e i 2π N jk e iy j(l ln(s)) Ψ(y j ) where α = 2.0 and dy = 0.25 are used. dy 3 [3 + ( 1)(j+1) δ j,1 ], (20) F. B. Hanson and G. Yan 13 UIC and FNMA
5. Numerical Results for Call and Put Options Two numerical algorithms give the same results within accuracy standard. The FFT method can compute different levels strike price near at-the-money (ATM) in 5 seconds. The standard integration method can give out the results for one specific strike price in about 0.5 seconds. The implementations are using MATLAB 6.5 and on the PC with 2.4GHz CPU. The option prices from the stochastic-volatility jump-diffusion (SVJD) model are compared with those of Black-Scholes model: Parameters: r = 3%, S 0 = $100; σ7%, V = 0.012, ρ = 0.622, θ = 0.53, k = 0.012; a = 0.028, b = 0.026, λ = 64. F. B. Hanson and G. Yan 14 UIC and FNMA
5.1 Call Option Prices: Call Option Price for T = 0.1 Call Option Price for T = 0.25 Call Option Price for T = 1 25 SVJD BS 25 SVJD BS 25 SVJD BS 20 20 20 Call Option Price 15 10 Call Option Price 15 10 Call Option Price 15 10 5 5 5 80 90 100 110 120 130 Strike Price 80 90 100 110 120 130 Strike Price 80 90 100 110 120 130 Strike Price (a) Call prices for T = 0.1. (b) Call prices for T = 0.25. (c) Call prices for T = 1.0. Figure 1: Call option prices for the SVJD model compared to the corresponding pure diffusion Black-Scholes values for T = 0.1, 0.25, 1.0. F. B. Hanson and G. Yan 15 UIC and FNMA
5.2 Put Option Prices: Put Option Price for T = 0.1 Put Option Price for T = 0.25 Put Option Price for T = 1 30 SVJD BS 30 SVJD BS 30 SVJD BS 25 25 25 Put Option Price 20 15 Put Option Price 20 15 Put Option Price 20 15 10 10 10 5 5 5 80 90 100 110 120 130 Strike Price 80 90 100 110 120 130 Strike Price 80 90 100 110 120 130 Strike Price (a) Put prices for T = 0.1. (b) Put prices for T = 0.25. (c) Put prices for T = 1.0. Figure 2: Put option prices for the SVJD model by put-call parity compared to the corresponding pure diffusion Black-Scholes values for T = 0.1, 0.25, 1.0. F. B. Hanson and G. Yan 16 UIC and FNMA
6. Conclusions Proposed an alternative stochastic-volatility, jump-diffusion (SVJD) model, stochastic volatility follows a square-root mean-reverting stochastic process and jump-amplitude has log-uniform distribution. Characteristic functions of the log-terminal stock price and the conditional risk neutral probability are analytically derived. The option prices are expressed in terms of characteristic functions in formally closed form. Two numerical computing algorithms using an accurate 10-point Gauss-Legendre discrete Fourier integral (DFT) formula and a fast FFT are compared. Same option prices are given by two methods for the SVJD model. Compared with those from Black-Scholes model, the SVJD model have higher option prices, especially for longer maturity and near at-the-money (ATM) strike price. F. B. Hanson and G. Yan 17 UIC and FNMA