Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility

Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility José Enrique Figueroa-López 1 1 Department of Statistics Purdue University Statistics, Jump Processes, and Malliavin Calculus: Recent Applications Barcelona GSE Summer Forum June 26, 214 Joint work with Sveinn Ólafsson from Purdue University

Outline 1 The Setup Tempered-Stable-Like Processes A stochastic volatility financial model with Lévy jumps 2 Problem Formulation Some relevant literature 3 High-order Expansions for (near) At-the-money Options Tempered-stable-like Lévy with and without Brownian component Stochastic volatility model with tempered-stable-like jumps 4 Conclusions

The Setup Tempered-Stable-Like Processes Lévy Process 1 Lévy process {X t } t X = Independent Increments: t < t 1 < < t n = X t1 X t,..., X tn X tn 1 are independent Stationary Increments s < t = X t X D s = X t s The trajectories of the process, t X t(ω), are right-continuous with left-limits 2 The distribution law of {X t } t is determined by the distribution of X 1 : If L(X 1 ) N (, 1), then X t = W t is a standard Brownian Motion; If L(X 1 ) Poisson(λ), then X t = N t is a Poisson process with intensity λ;

The Setup Tempered-Stable-Like Processes Tempered-stable-like processes 1 A Lévy process {X t } t whose distribution at t = 1 has a characteristic function of the form: E ( ( ) e iux ) ( 1 = exp ibu + e iux ) 1 iux1 { x 1} s(x)dx, R\{} ( where s(x) = C x x ) q(x) x α 1 for some constants C(1), C( 1) (, ), α (, 2), and a function q : R\{} [, ) such that q(x) x 1, sup q(x) <. x 2 b, α, and s are called the drift", the index of jump activity, and the Lévy density of the process.

The Setup Tempered-Stable-Like Processes Connection to Stable Processes 1 If q(x) 1, the resulting Lévy process is a Stable Lévy Process {Z t } t ; 2 For a suitable c R, the process Z t := Z t ct is self-similar: {h 1/α Zht } t D = { Zt } t (h > ). If α > 1, c = EZ 1 ; If α < 1, c = xs(x)dx is the drift of Z. x 1 3 Distributions are often too fat" for some applications (e.g., finance): E( Z t p ) =, for any p > α. 4 The function q(x) can mitigate the intensity of large jumps so that E( X t p ) < x p α 1 q(x)dx <. x 1 5 Notation: Throughout, Z is called the strictly stable processes ( associated ) with X. This is characterized by its Lévy density s(x) := C x α 1 and that its center" is. x x

The Setup Tempered-Stable-Like Processes Short and long time behavior of TSP 1 In short-time, {X t } t behaves like a stable process (cf. Rosenbaum and Tankov (211)): 1 < α < 2: {h 1/α X ht } D { Z t} t, (h ), for the strictly α-stable process Z associated with X; < α < 1: {h 1/α (X ht cht)} D { Z t} t, (h ), for the strictly α-stable process Z associated with X; 2 In long-time, {X t } t behaves like a Brownian Motion: {h 1/2 X ht } D {B t } t, (h ), where {B t } t is a suitable Brownian motion.

The Setup A stochastic volatility financial model with Lévy jumps Why stochastic volatility? 1 The Black-Scholes model (S t = S e Xt with X t = σw t + bt) offers tractable solutions, but is not in line with many stylized features observed in asset and option prices. 2 Exponential Lévy models (S t = S e Xt with a Lévy process X t ) generalize the Black-Scholes framework by allowing jumps in stock prices while preserving the independence and stationarity of returns. 3 They allow to generate implied volatility smiles similar to the ones observed in practice. 4 Exponential Lévy models still have their shortcomings: Independence of log-returns is not consistent with some fine stylized features of return s time series (e.g., volatility clustering and leverage) Exponential Lévy models are not capable of capturing the evolution of the implied volatility surface in time

The Setup A stochastic volatility financial model with Lévy jumps Introducing a stochastic volatility component 1 Consider a log-return process L t := log (S t /S ) such that dl t = dv t }{{} Continuous Component + dx t }{{} Jump Component = µ(y t )dt + σ(y t )dw t + dx t ; 2 Y = {Y t } t is a latent (hidden) risky factor with dynamics dy t = α(y t )dt + γ(y t )dw t, Y = y ; 3 X = {X t } t is a pure-jump tempered-stable-like process as before; 4 (W, W ) are correlated B.M. s independent of X; 5 If µ(y) and σ(y) σ(y ), L is a Lévy process with triplet (σ(y ), b, ν).

Problem Formulation The General Problems 1 Let us first take the drifts b and µ such that S t := e Lt is a martingale; in particular, q must be such that x 1 ex q(x)x α 1 dx < and E (S t ) = E ( e ) Lt = 1; 2 Consider the functional: [ (e L Π t := E t 1 ) ] + = E [ max ( e Lt 1, )]. 3 By Dominated Convergence Theorem, Π t when t. 4 The General Problems: We want to characterize the rate of convergence of Π t as t ; Characterize the effect of the different model s parameters in the short-time asymptotic behavior of Π t.

Problem Formulation Motivation 1 In finance, S E [ ( e Lt 1 ) + ] represents the price of an At-the-money (ATM) European Call Option with time-to-expiration t written on a stock whose price process is modeled by t S t := S e Lt. 2 Our results shed light on the behavior of ATM option prices close to expiration under a stochastic volatility model with Lévy type jumps. 3 In general, the call option price with time-to-expiry t and strike K = S e κ is [ E (S t K ) +] = S E [ ( e Xt e κ) + ]. 4 In mathematics, ϕ K (S) = (S K ) + are natural building blocks of convex functions f : R + R + : f (S) = f () + f +()S + (S K ) + µ(dk ).

Problem Formulation Some relevant literature Some relevant literature Two distinct regimes: Not ATM and ATM. Not ATM (K S κ ) 1 Tankov (211): Leading order term for general Lévy process: ( E e X t e κ) + (e = (1 e κ ) + + t x e κ)+ s(x)dx + o(t). 2 F-L & Forde (212): High-order term for quite general Lévy process; ( E e X t e κ) + (e = (1 e κ ) + + t x e κ)+ s(x)dx + t 2 2 d 2(κ) + o(t 2 ), where d 2 (κ) has an explicit form in terms of s.

Problem Formulation Some relevant literature Some relevant literature. Cont... ATM (K = S κ = ) 1 Tankov (211), Roper (211): Leading order term for a general bounded variation Lévy process (α < 1): ( + { } E e X t (e 1) = t max x 1 )+ (1 s(x)dx, e x )+ s(x)dx + o(t). 2 Tankov (211): Leading term for a pure-jump Lévy process with stable-like small-jump behavior with α > 1: ( ) + E e X t 1 = t 1/α E ( Z + ) ( + o t 1/α), (t ), where Z is the strictly α-stable process associated with X; 3 If C(1) = C( 1) =: C, 1 d 1 := E( Z + 1 ) = 1 π Γ (1 1/α) (2CΓ( α) cos (πα/2) )1/α. 4 Tankov (211): If L = σw + X with non-zero σ, ( ) + E e σw t +X t 1 = t 1/2 σ + o(t 1/2 ), (t ). 2π

Tempered-stable-like Lévy with and without Brownian component Second-Order Expansions Important question: What is the accuracy of the previous asymptotics? F-L, Gong, and Houdré (212): For a tempered-stable-like Lévy process X with α (1, 2), Π t = E ( e Xt 1 ) + = d1 t 1 α + d2 t + o(t), (t ), where d 1 = E( Z + 1 ) and d 2 = C(1)P( Z 1 < ) C( 1)P( Z 1 ) (e x q(x) q(x) x) x α 1 dx (e x q(x) q(x) x) x α 1 dx.

Tempered-stable-like Lévy with and without Brownian component Second-Order Expansions. Continue... F-L, Gong, and Houdré (212): For a tempered-stable-like process X with α (1, 2) and a nonzero independent Brownian component σw t, where Π t = E ( e Xt +σwt 1 ) + = d1 t 1 2 + d2 t 3 α 2 + o(t 3 α 2 ), (t ), d 1 := σe ( W + ) σ 1 = 2π C(1) + C( 1) d 2 := σ 1 α E ( W 1 1 α) 2α(α 1) ( = 21 α Γ 1 α ) (C(1) + C( 1)) σ 1 α. π 2 2α(α 1)

Tempered-stable-like Lévy with and without Brownian component CGMY or Kobol Model 1 Lévy density: s(x) = C x Y 1 ( e x G 1x< + e x M 1x> ), G, M, C >, α (, 2); 2 Pure-jump Lévy model (σ = ): d 1 = E( Z + 1 ) = 1 ( π Γ 1 1 ) ( ( ) ) 2CΓ( Y ) πy 1/Y Y cos 2 d 2 = CΓ( Y ) 2 3 Mixed Lévy model (σ ): d 1 = σ 2π, ( (M 1) Y M Y (G + 1) Y + G Y ), ( d 2 := 21 Y Γ 1 Y ) Cσ 1 Y π 2 Y (Y 1).

Tempered-stable-like Lévy with and without Brownian component CGMY Model s(x) = C x α 1 ( e x /G 1 x< + e x /M 1 x> ), G, M, C > ATM Call Option Prices Pure Jump CGMY Model (C=.5,G=2,M=3.6,Y=1.5).35.3.25.2.15.1 IFT based Method MC based Method 1st order Approx. 2nd order Approx. ATM Call Option Prices General CGMY Model (σ=.4,c=.5,g=2,m=3.6,y=1.5.35.3.25.2.15.1 IFT based Method MC based Method 1st order Approx. 2nd order Approx..5.5.5 1 1.5 2 2.5 3 3.5 4 Time to maturity, T (in years) x 1 3.5 1 1.5 2 2.5 3 3.5 4 Time to maturity, T (in years) x 1 3 Figure: Comparisons of ATM call option prices computed by two methods (Inverse Fourier Transform and Monte-Carlo method) with the first- and second-order approximations.

Tempered-stable-like Lévy with and without Brownian component Additional Technical Conditions ( The Lévy density s(x) := C x x ) q(x) x α 1 is such that (i) α (1, 2), (ii) C(±1) (, ); (iii) q(x) x 1; (v) 1 q(x) lim =: ±q ( ± ) with q ( ± ) (, ); x ± x (vi) q(x) e x, for all x > ; (vii) q(x) 1, for all x < ; (viii) lim sup x ln q(x) x < ; Natural question: How necessary are these conditions? (iv) sup q(x) < ; x

Stochastic volatility model with tempered-stable-like jumps Specific Problems 1 Investigate the necessity of the technical conditions above. 2 The most liquid options are near" ATM: Π t (κ) := E [ ( e Lt e κ) + ], with κ ; Two approaches: Approximating Π t(κ) with Π t(); Consider another asymptotic regime of the form Π t(κ t) with κ t ; 3 Extend the result to handle a stochastic volatility component: Π t (κ t ) := E [ ( e Lt e κt ) + ], where dl t = dv t + dx t = µ(y t )dt + σ(y t )dw t + dx t, with dy t = α(y t )dt + γ(y t )dw t, Y = y;

Stochastic volatility model with tempered-stable-like jumps Pure-jump Case Theorem (F-L & Ólafsson, 213) ( Suppose the Lévy density s(x) := C x x ) q(x) x α 1 is such that (i) α (1, 2), (ii) C(±1) (, ); (iii) q(x) x 1; (v) 1 e x s(x)dx < ; (vi) x 1 x α 1 q(x) dx <. (iv) sup q(x) < ; x

Stochastic volatility model with tempered-stable-like jumps Pure-jump Case. Continuation... Theorem (F-L & Ólafsson, 213) Then, for κ t := θt + o(t), as t, for some θ R, Π t = E ( e Xt e κt ) + = d1 t 1 α + d2 t + o(t), (t ), where d 1 = E( Z + 1 ) and d 2 = C(1)P( Z 1 < ) C( 1)P( Z 1 ) Remark: The condition (vi) (e x q(x) q(x) x) x α 1 dx x 1 (e x q(x) q(x) x) x α 1 dx θp( Z 1 ). x α 1 q(x) dx <, is a necessary conditions for such a second-order expansion to exist.

Stochastic volatility model with tempered-stable-like jumps Nonzero Diffusion Component Theorem (F-L & Ólafsson, 213) Let dv t = µ(y t )dt + σ(y t )dw t, dy t = α(y t )dt + γ(y t )dw t, Y = y ; independent of X such that σ(y ) > and σ( ) is Lipschitz continuous at y. Then, for κ t := θt 3 α 2 + o(t 3 α 2 ), as t, for some θ R, Π t = E ( e Xt +Vt e ) ) κt + = d1 t 1 2 + d2 t 3 α 2 + o (t 3 α 2, (t ), where d 1 := σ(y ) 2π, d 2 := θ 2 + C(1) + C( 1) 2α(α 1) σ(y ) 1 α E ( W 1 1 α).

Stochastic volatility model with tempered-stable-like jumps Outline of the proof (pure-jump case) Step1. Change of probability measure P P, where dp Ft = e Xt dp Ft : E ( e Xt e ) κt + = e x P (X t > κ t + x)dx (1) = t 1/α e κt = t 1/α e κt t 1/α κ t e t t 1/α κ t e t 1/αu P (t 1/α X t > u)du 1/αu P (t 1/α X t > u)du + t 1/α e κt e t 1/αu P (t 1/α X t > u)du (2) κ t P ( Z1 ) + t 1/α P ( Z 1 > u)du = κ t P ( Z1 ) + t 1/α E ( Z1 ) + (1) u = t 1/α (κ t + x), (2) t 1/α X t D Z1,

Stochastic volatility model with tempered-stable-like jumps Outline of the proof (pure-jump case) Step 2. To study the convergence in e t 1/αu P (t 1/α X t > u)du E ( Z1 ) +, consider another change P P such that, under P, X is a stable process: e t 1/αu P (t 1/α X t > u)du = where d P = e Ut dp Ft. Since Ẽ ( Z ) + 1 = Ft Ũ t = U t ẼU t, t 1/α 1 ( = 1 t 1 t e t 1/α u Ẽ ( e Ut 1 {t 1/α X t >u}) du P(t 1/α Z t u)du and letting e t 1/αu P (t 1/α X t > u)du Ẽ ( Z ) ) + 1 ( e v 1 ) P( Z + t (e v 1) P( Z + t + Ũt v)dv + Ũt v)dv + o(1)

Stochastic volatility model with tempered-stable-like jumps Outline of the proof (pure-jump case) Step 2. In terms of the compensated jump measure µ of Z (under P), t t Z t = x µ(ds, dx), Ũ t = ( ln q(x) x) µ(ds, dx), Thus, as t, R\{} t 1/α 1 ( = R\{} e t 1/αu P (t 1/α X t > u)du Ẽ ( Z ) ) + 1 ( e v 1 ) 1 t P( Z + t + Ũt v)dv (e v 1) 1 t P( Z + t + Ũt v)dv ( e v 1 ) ( ) x 1 {x + ln q(x) x v}c x α 1 dxdv x ( ) x (e v 1) 1 {x + ln q(x) x v}c x α 1 dxdv x

Stochastic volatility model with tempered-stable-like jumps Outline of the proof (pure-jump case) Step 3. The above approach requires the the conditions: 1 q(x) = O(x), lim sup x ln q(x) x <. To relax those, the idea is to approximate X by a process X (δ) such that X (δ) D X, as δ X X (δ) is of bounded variation It turns out that the condition x 1 x α 1 q(x) dx < is exactly what is needed to do that. E ) (δ) + X (e t e e κt E X t e E ( e Xt e κt ) + E X (δ) t ) (δ) + X (e t e e κt + E X t e X (δ) t.

Stochastic volatility model with tempered-stable-like jumps Outline of the proof (pure-jump case) Step 3. The above approach requires the the conditions: 1 q(x) = O(x), lim sup x ln q(x) x <. To relax those, the idea is to approximate X by a process X (δ) such that X (δ) D X, as δ X X (δ) is of bounded variation It turns out that the condition x 1 x α 1 q(x) dx < is exactly what is needed to do that. t 1/α 1 (t 1/α E ) ) (δ) + X (e t e κt d1 1 e t E X t e t 1/α 1 ( t 1/α E ( e Xt e κt ) + d1 ) X (δ) t ) ) t (t 1/α 1 1/α (δ) + X E (e t e κt d1 + 1 e t E X t e X (δ) t

Stochastic volatility model with tempered-stable-like jumps Outline of the proof (pure-jump case) Step 3. Since X X (δ) is of bounded variation, it turns out that 1 e lim sup t t E X t e X (δ) t K 1 q(x) x α dx x <δ for some K <. Making t, d (δ) 2 K 1 q(x) x α dx lim inf t x <δ lim sup t d (δ) 2 + K t 1/α 1 ( t 1/α E ( e Xt e κt ) + d1 ) ( t 1/α 1 t 1/α E ( e Xt e ) ) κt + d1 1 q(x) x α dx x <δ Finally, the results follows from making δ since d (δ) 2 d 2.

Conclusions Conclusions 1 Obtained the second-order short-time expansions for near" ATM European call option prices under stable-like small jumps and a possible nonzero independent diffusion component. 2 Characterized explicitly the effects of the different parameters into the behavior of ATM option prices and implied volatility near expiration.

Appendix Bibliography For Further Reading I Figueroa-López, J.E., Gong, R., & Houdré, C. High-order short-time expansions for ATM option prices under a tempered stable Lévy model. To appear in Mathematical Finance, 214-. Figueroa-López, J.E., & Ólafsson, S. Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility. Arxiv 214. Tankov, P. Pricing and hedging in exponential Lévy models: review of recent results, Paris-Princeton Lecture Notes in Mathematical Finance, Springer 21. Rosenbaum, R., and Tankov, P. Asymptotic results for time-changed Lévy processes sampled at hitting times. Stochastic processes and their applications, 121, 211.