Jump-Diffusion Models

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Jump-Diffusion Models where F is the risk-neutral jump distribution. The jump compensator is then given by Jump-diffusion (JD) models are particular cases of Z exponential Lévymodels in which the frequency of µJ = −λ ξ F (dξ) jumps is finite. They can be considered as pro- totypes for a large class of more complex models To simplify the presentation, we henceforth assume such as the stochastic volatility plus jumps model zero dividends so thatµ ˆ = r, the risk-free rate. of Bates [1]. Consider a market with a riskless asset (the Characteristic Function bond) and one risky asset (the stock) whose price Define the forward price F := S er T . If x := at time t is denoted by St. In a JD model, the SDE 0 t for the stock price is given as log St/F ) is a Lévyprocess, its characteristic func- iuxT tion φT (u) := E e has the Lévy-Khintchine dSt = µ St− dt + σ St− dZt + St− dJt (1) representation

n 2 2 2 where Zt is a Brownian motion and φT (u) = exp i u (µJ − σ /2) T − u σ /2 T Z N iuξ o Xt +T e − 1 ν(ξ) dξ . (3) Jt = Yi i=1 Typical assumptions for the distribution of jump is a compound Poisson process where the jump sizes sizes are: normal as in the original paper by Merton Y are independent and identically distributed with [6] and double-exponential as in Kou [4]. In the i Merton model, the L´evydensity ν(·) is given by distribution F and the number of jumps Nt is a Poisson process with jump intensity λ. The as- λ (ξ − α)2 ν(ξ) = √ exp − set price St thus follows geometric Brownian mo- 2 π δ 2 δ2 tion between jumps. Monte Carlo simulation of the process can be carried out by ﬁrst simulating where α is the mean of the log-jump size log J and δ the standard deviation of jumps. This leads to the number of jumps Nt, the jump times, and then simulating geometric Brownian motion on intervals the explicit characteristic function between jump times. 1 φ (u) = exp i u ω T − u2 σ2 T The SDE (1) has the exact solution: T 2 2 2 2 St = S0 exp µ t + σ Zt − σ t/2 + Jt (2) +λ T ei u α−u δ /2 − 1

Merton [6] considers the case where the jump sizes In the double-exponential case Yi are normally distributed. −α+ ξ −α− ξ ν(ξ) = λ p α+ e 1ξ≥0 + (1 − p) α− e 1ξ<0

Risk-neutral drift where α+ and α− are the expected positive and If the above model is used as a pricing model, the negative jump sizes respectively and p is the rela- drift µ in (1) is given by the risk-neutral driftµ ˆ tive probability of a positive jump. This gives the explicit characteristic function plus a jump compensator µJ : 1 φ (u) = exp i u ω T − u2 σ2 T µ =µ ˆ + µJ T 2 p 1 − p To identify µJ , taking expectations of equation (1) +λ T − and from the deﬁnition ofµ ˆ, α+ − i u α− + i u with E[dSt] =µ ˆ St dt = µ St dt 1 p 1 − p Z ω = − α2 − λ − +λ ξ F (dξ) St dt 2 α+ + 1 α− − 1

1 Pricing of European options with √ Given a characteristic function, European call op- log(S/K) + r T σ T d1 = √ + ; tions can be priced using Fourier methods as in σ T 2 Lewis [5]: √ log(S/K) + r T σ T d2 = √ − ; C(S, K, T ) σ T 2 √ 1 Z ∞ du = e−r T F − FK 2 1 Jump to Ruin π 0 u + 4 Yi In the case where e = 0 with probability 1, µJ = −iuk ×Re e φT (u − i/2) (4) λ and equation (6) simpliﬁes to

−λ T λ T where the log-strike k := log (K/F ). C(S, K, T ) = e CBS(S e , K, r, σ, T ) (7) which is the Black-Scholes formula with a shifted Valuation equation interest rate r → r + λ. This special case of the Assuming the process (1), and a constant risk-free JD model where the stock price jumps to zero (or rate r, and further supposing that the market is ruin) whenever there is a jump, is the simplest pos- complete, the value V (S, t) of a European-style op- sible model of default. Equation (7) for the option tion satisfies: price in the jump-to-ruin model may also be derived from a Black-Scholes style replication argument us- ∂V 1 ∂2V ∂V Z ing stock and bonds of the issuer of the stock; upon + σ2 S2 + r S − r V + λ F (dξ) ∂t 2 ∂S2 ∂S default of the issuer, both stock and bonds jump to n ∂V o zero. The cost of funding stock with bonds of the V (S eξ, t) − V (S, t) − eξ − 1 S = 0 ∂S issuer is r + λ in this picture, which explains the (5)simple form (7) of the solution. where for ease of notation, V denotes V (S, t). Local volatility Equation (5) is a partial integro-differential equation (PIDE) which can be solved using finite differ- There is a particulary simple expression for Dupire ence methods [2]. local volatility in the jump-to-ruin model. It is given by Gatheral [3] A Valuation Formula for European Options √ 2 2 N(d2) σloc(K,T ; S) = σ + 2 λ σ T 0 Merton [6] derived an exact solution of the valua- N (d2) tion equation (5) for a European-style call option with √ with strike K and time-to-expiration T which has log(S/K) + λ T σ T the form of an infinite sum of Black-Scholes-like d2 = √ − ; σ T 2 terms: As K → ∞, d2 → −∞ and the correction term ∞ X e−λ T (λ T )n Z vanishes, and as K → 0, the correction term ex- C(S, K, T ) = F (dξ) n! n plodes. In addition, as the hazard rate λ increases, n=0 so does d2, increasing local volatility for low strikes ξ µJ T × CBS(S e e , K, r, σ, T ) K relative to high strikes. (6) where Fn is the distribution of the sum of n independent jumps and CBS(·) denotes the Black- Scholes solution, which is given as

−r T CBS(S, K, r, σ, T ) = SN(d1) − K e N(d2)

2 Normally distributed jumps [5] Lewis, A.L. (2000) Option Valuation under Stochastic Volatility with Mathematica Code, Fi- Merton [6] also shows that if jumps are normally nance Press, Newport Beach, CA. distributed with Y ∼ N(α, δ), equation (6) again i [6] Merton, R.C. (1976) Option pricing when un- simpliﬁes considerably to give derlying stock returns are discontinuous. J. Finan- C(S, K, T ) cial Economics 3, 125–144. ∞ 0 X e−λ T (λ0 T )n = C (S, K, r , σ ,T ) n! BS n n n=0 (8) with

2 λ0 = λ eα+δ /2 2 2 2 σn T = σ T + n δ 2 rn T = (r + µJ ) T + n α + δ /2

Each term CBS(S, K, rn, σn,T ) in equation (8) is the value of the option conditional on there being exactly n jumps during its life.

Implied volatility smile If there were no jumps in this model, the implied volatility smile would be ﬂat. Jumps in the JD stock price process induce an implied volatility smile whose short time limit (see e.g., [3]) is given as ∂ K σ2 (K,T ) → −2 µ as T → 0. ∂K BS J

The greater µJ , because jumps are either more fre- quent or more negatively skewed, the more negative is the implied volatility skew.

Jim Gatheral. References [1] Bates, D. (1996) Jumps and stochastic volatility: the exchange rate processes implicit in Deutschemark options. Rev. Fin. Studies 9, 69– 107. [2] Cont, R. and Voltchkova, E. (2005). A finite difference scheme for option pricing in jump- diffusion and exponential Lévy models, SIAM Journal on Numerical Analysis 43(4), 1596-1626. [3] Gatheral, J. (2006) The Volatility Surface, John Wiley & Sons, Hoboken, Chapter 5. [4] Kou, S. (2002) A jump-diffusion model for option pricing. Management Science 48, 1086–1101.