Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models

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1 Large Deviations and Stochastic Volatility with Jumps: TU Berlin with A. Jaquier and A. Mijatović (Imperial College London) SIAM conference on Financial Mathematics, Minneapolis, MN July 10, 2012

2 Implied Volatility Implied Volatility For strike K 0 and time-to-maturity T > 0 implied volatility is the quantity σ imp (K, T ) 0 that solves C Black-Scholes (K, T ; σ imp (K, T )) = e rt E Q [(S T K) + ] Goal: Understand qualitatively how the stochastic model for S determines σ imp (K, T ) and thus the shape of the implied volatility surface. Tool: Asymptotic Analysis: K = e xt, T, using a Large Deviation Principle (LDP).

3 Implied Volatility Implied Volatility For strike K 0 and time-to-maturity T > 0 implied volatility is the quantity σ imp (K, T ) 0 that solves C Black-Scholes (K, T ; σ imp (K, T )) = e rt E Q [(S T K) + ] Goal: Understand qualitatively how the stochastic model for S determines σ imp (K, T ) and thus the shape of the implied volatility surface. Tool: Asymptotic Analysis: K = e xt, T, using a Large Deviation Principle (LDP).

4 Implied Volatility Implied Volatility For strike K 0 and time-to-maturity T > 0 implied volatility is the quantity σ imp (K, T ) 0 that solves C Black-Scholes (K, T ; σ imp (K, T )) = e rt E Q [(S T K) + ] Goal: Understand qualitatively how the stochastic model for S determines σ imp (K, T ) and thus the shape of the implied volatility surface. Tool: Asymptotic Analysis: K = e xt, T, using a Large Deviation Principle (LDP).

5 Implied Volatility Surface

6 1 Affine Stochastic Volatility Models 2 Large deviations and option prices 3 Examples

7 Affine Stochastic Volatility Models (1) X t... log-price-process V t... a latent factor (or factors), such as stochastic variance or stochastic arrival rate of jumps. S t := exp(x t )... price-process. We assume it is a true martingale under the pricing measure Q. For simplicity we assume zero interest rate r = 0. Definition We call (X, V ) an affine stochastic volatility model, if (X, V ) is a stochastically continuous, conservative and time-homogeneous Markov process, such that [ ] E Q e uxt+wvt X0 = x, V 0 = v = e ux exp (φ(t, u, w) + vψ(t, u, w)) for all (u, w) C where the expectation is finite.

8 Affine Stochastic Volatility Models (1) X t... log-price-process V t... a latent factor (or factors), such as stochastic variance or stochastic arrival rate of jumps. S t := exp(x t )... price-process. We assume it is a true martingale under the pricing measure Q. For simplicity we assume zero interest rate r = 0. Definition We call (X, V ) an affine stochastic volatility model, if (X, V ) is a stochastically continuous, conservative and time-homogeneous Markov process, such that [ ] E Q e uxt+wvt X0 = x, V 0 = v = e ux exp (φ(t, u, w) + vψ(t, u, w)) for all (u, w) C where the expectation is finite.

9 Affine Stochastic Volatility Models (2) We can prove that φ and ψ are differentiable in t, and thus that (X t, V t ) t 0 is a regular affine process in the sense of Duffie et al. [2003]. Implies in particular that (X t, V t ) t 0 is a semi-martingale with absolutely continuous characteristics. The class of ASVMs such defined, includes many important stochastic volatility models: the Heston model with and without added jumps, the models of Bates [1996, 2000] and the Barndorff-Nielsen-Shephard (BNS) model. Exponential-Lévy models and the Black-Scholes model can be treated as degenerate ASVMs.

10 Affine Stochastic Volatility Models (2) We can prove that φ and ψ are differentiable in t, and thus that (X t, V t ) t 0 is a regular affine process in the sense of Duffie et al. [2003]. Implies in particular that (X t, V t ) t 0 is a semi-martingale with absolutely continuous characteristics. The class of ASVMs such defined, includes many important stochastic volatility models: the Heston model with and without added jumps, the models of Bates [1996, 2000] and the Barndorff-Nielsen-Shephard (BNS) model. Exponential-Lévy models and the Black-Scholes model can be treated as degenerate ASVMs.

11 Affine Stochastic Volatility Models (2) We can prove that φ and ψ are differentiable in t, and thus that (X t, V t ) t 0 is a regular affine process in the sense of Duffie et al. [2003]. Implies in particular that (X t, V t ) t 0 is a semi-martingale with absolutely continuous characteristics. The class of ASVMs such defined, includes many important stochastic volatility models: the Heston model with and without added jumps, the models of Bates [1996, 2000] and the Barndorff-Nielsen-Shephard (BNS) model. Exponential-Lévy models and the Black-Scholes model can be treated as degenerate ASVMs.

12 Affine Stochastic Volatility Models (2) We can prove that φ and ψ are differentiable in t, and thus that (X t, V t ) t 0 is a regular affine process in the sense of Duffie et al. [2003]. Implies in particular that (X t, V t ) t 0 is a semi-martingale with absolutely continuous characteristics. The class of ASVMs such defined, includes many important stochastic volatility models: the Heston model with and without added jumps, the models of Bates [1996, 2000] and the Barndorff-Nielsen-Shephard (BNS) model. Exponential-Lévy models and the Black-Scholes model can be treated as degenerate ASVMs.

13 Affine Stochastic Volatility Models (3) Define F (u, w) = t φ(t, u, w) t=0 R(u, w) = t ψ(t, u, w) t=0. The functions φ and ψ satisfy... Generalized Riccati Equations t φ(t, u, w) = F (u, ψ(t, u, w)), φ(0, u, w) = 0 t ψ(t, u, w) = R(u, ψ(t, u, w)), ψ(0, u, w) = w.

14 Affine Stochastic Volatility Models (4) F and R are functions of Lévy-Khintchine form We call F (u, w), R(u, w) the functional characteristics of the model. The martingale condition on exp (X t ) implies that F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0. We also define χ(u) = w R(u, w) w=0.

15 Affine Stochastic Volatility Models (4) F and R are functions of Lévy-Khintchine form We call F (u, w), R(u, w) the functional characteristics of the model. The martingale condition on exp (X t ) implies that F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0. We also define χ(u) = w R(u, w) w=0.

16 Affine Stochastic Volatility Models (4) F and R are functions of Lévy-Khintchine form We call F (u, w), R(u, w) the functional characteristics of the model. The martingale condition on exp (X t ) implies that F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0. We also define χ(u) = w R(u, w) w=0.

17 Affine Stochastic Volatility Models (4) F and R are functions of Lévy-Khintchine form We call F (u, w), R(u, w) the functional characteristics of the model. The martingale condition on exp (X t ) implies that F (0, 0) = R(0, 0) = F (1, 0) = R(1, 0) = 0. We also define χ(u) = w R(u, w) w=0.

18 Example: Heston Model Heston in SDE form dx t = V t 2 dt + V t dwt 1, dv t = λ(v t θ) dt + ζ V t dwt 2, where W 1, W 2 are BMs with correlation ρ ( 1, 1), and ζ, λ, θ > 0 Functional Characteristics of the Heston Model F (u, w) = λθw, R(u, w) = 1 2 (u2 u) + ζ2 2 w 2 λw + uwρζ. Moreover we have χ(u) = ρζu λ.

19 Example: Heston Model Heston in SDE form dx t = V t 2 dt + V t dwt 1, dv t = λ(v t θ) dt + ζ V t dwt 2, where W 1, W 2 are BMs with correlation ρ ( 1, 1), and ζ, λ, θ > 0 Functional Characteristics of the Heston Model F (u, w) = λθw, R(u, w) = 1 2 (u2 u) + ζ2 2 w 2 λw + uwρζ. Moreover we have χ(u) = ρζu λ.

20 Example: Barndorff-Nielsen-Shephard Model Barndorff-Nielsen-Shephard (BNS) Model in SDE form dx t = (δ 1 2 V t)dt + V t dw t + ρ dj λt, dv t = λv t dt + dj λt, where λ > 0, ρ < 0 and (J t ) t 0 is a Lévy subordinator with the Lévy measure ν. Functional Characteristics of BNS Model where κ(u) is the cgf of J. F (u, w) = λκ(w + ρu) uλκ(ρ), R(u, w) = 1 2 (u2 u) λw.

21 Example: Barndorff-Nielsen-Shephard Model Barndorff-Nielsen-Shephard (BNS) Model in SDE form dx t = (δ 1 2 V t)dt + V t dw t + ρ dj λt, dv t = λv t dt + dj λt, where λ > 0, ρ < 0 and (J t ) t 0 is a Lévy subordinator with the Lévy measure ν. Functional Characteristics of BNS Model where κ(u) is the cgf of J. F (u, w) = λκ(w + ρu) uλκ(ρ), R(u, w) = 1 2 (u2 u) λw.

22 1 Affine Stochastic Volatility Models 2 Large deviations and option prices 3 Examples

23 Large deviations theory Definition The family of random variables (Z t ) t 0 satisfies a large deviations principle (LDP) with the good rate function Λ if for every Borel measurable set B in R, 1 inf x B Λ (x) lim inf o t t log P (Z t B) lim sup t 1 t log P (Z t B) inf Λ (x). x B Continuous rate function If Λ is continuous on B, then a large deviation principle implies that ( ) P (Z t B) exp t inf x B Λ (x) for large t.

24 Large deviations theory Definition The family of random variables (Z t ) t 0 satisfies a large deviations principle (LDP) with the good rate function Λ if for every Borel measurable set B in R, 1 inf x B Λ (x) lim inf o t t log P (Z t B) lim sup t 1 t log P (Z t B) inf Λ (x). x B Continuous rate function If Λ is continuous on B, then a large deviation principle implies that ( ) P (Z t B) exp t inf x B Λ (x) for large t.

25 The Gärtner-Ellis theorem Assumption A.1: For all u R, define ) (e utzt Λ z (u) := lim t t 1 log E = lim t t 1 Λ z t (ut) as an extended real number. Denote D Λ z := {u R : Λ z (u) < } and assume that (i) the origin belongs to DΛ ; z (ii) Λ z is essentially smooth, i.e. Λ z is differentiable throughout DΛ o and is steep at the boundaries. z Theorem (Gärtner-Ellis) Under Assumption A.1, the family of random variables (Z t ) t 0 satisfies the LDP with rate function (Λ z ), defined as the Fenchel-Legendre transform of Λ z, (Λ z ) (x) := sup{ux Λ z (u)}, for all x R. u R

26 The Gärtner-Ellis theorem Assumption A.1: For all u R, define ) (e utzt Λ z (u) := lim t t 1 log E = lim t t 1 Λ z t (ut) as an extended real number. Denote D Λ z := {u R : Λ z (u) < } and assume that (i) the origin belongs to DΛ ; z (ii) Λ z is essentially smooth, i.e. Λ z is differentiable throughout DΛ o and is steep at the boundaries. z Theorem (Gärtner-Ellis) Under Assumption A.1, the family of random variables (Z t ) t 0 satisfies the LDP with rate function (Λ z ), defined as the Fenchel-Legendre transform of Λ z, (Λ z ) (x) := sup{ux Λ z (u)}, for all x R. u R

27 From LDP to option prices Theorem (Option price asymptotics) Let x be a fixed real number. If (X t /t) t 1 satisfies a LDP under Q with good rate function Λ, the asymptotic behaviour of a put option with strike e xt reads [ (e lim t t 1 log E xt e ) ] { x Λ Xt = (x) if x Λ (0), + x if x > Λ (0). Analogous results can be obtained for call options using a measure change to the share measure. By comparing to the Black-Scholes model, the results can be transferred to implied volatility asymptotics.

28 LDP for affine models Definition: We say that the function R explodes at the boundary if lim n R (u n, w n ) = for any sequence {(u n, w n )} n N DR o converging to a boundary point of DR o. Theorem Let (X, V ) be an ASVM with χ(0) < 0 and χ(1) < 0 and assume that F is not identically null. If R explodes at the boundary, F is steep and {(0, 0), (1, 0)} D o F, then a LDP holds for X t/t as t. Lemma Under the same assumptions, if either of the following conditions holds: (i) m and µ have exponential moments of all orders; (ii) (X, V ) is a diffusion; then a LDP holds for X t /t as t.

29 LDP for affine models Definition: We say that the function R explodes at the boundary if lim n R (u n, w n ) = for any sequence {(u n, w n )} n N DR o converging to a boundary point of DR o. Theorem Let (X, V ) be an ASVM with χ(0) < 0 and χ(1) < 0 and assume that F is not identically null. If R explodes at the boundary, F is steep and {(0, 0), (1, 0)} D o F, then a LDP holds for X t/t as t. Lemma Under the same assumptions, if either of the following conditions holds: (i) m and µ have exponential moments of all orders; (ii) (X, V ) is a diffusion; then a LDP holds for X t /t as t.

30 Implied Volatility in ASVMs Theorem (Implied Volatility Asymptotics for ASVMs) Let (X, V ) be an affine stochastic volatility model with functional characteristics F (u, w) and R(u, w) satisfying the assumptions from above. Let Λ(u) = F (u, w(u)) where w(u) is the solution of Then where σ (x) = 2 R(u, w(u)) = 0. lim σ imp(t, e xt ) = σ (x) t [ sgn(λ (1) x) Λ (x) x + sgn(x Λ (0)) ] Λ (x), and Λ (x) = sup u R (xu Λ(u)).

31 Implied Volatility in ASVMs (2) Corollary Under the assumptions from above 0 (Λ (0), Λ (1)) and for all x [Λ (0), Λ (1)] it holds that lim σ imp(t, e xt ) = [ 2 Λ (x) x + Λ (x)]. t Corollary Let (X, V ) be a non-degenerate affine stochastic volatility process that satisfies the assumptions from above. Then there exists a Lévy process Y, such that the limiting smiles of the models e X and e Y are identical. In the Heston model, the corresponding Lévy model is the Normal-Inverse-Gaussian (NIG) model.

32 Implied Volatility in ASVMs (2) Corollary Under the assumptions from above 0 (Λ (0), Λ (1)) and for all x [Λ (0), Λ (1)] it holds that lim σ imp(t, e xt ) = [ 2 Λ (x) x + Λ (x)]. t Corollary Let (X, V ) be a non-degenerate affine stochastic volatility process that satisfies the assumptions from above. Then there exists a Lévy process Y, such that the limiting smiles of the models e X and e Y are identical. In the Heston model, the corresponding Lévy model is the Normal-Inverse-Gaussian (NIG) model.

33 Implied Volatility in ASVMs (2) Corollary Under the assumptions from above 0 (Λ (0), Λ (1)) and for all x [Λ (0), Λ (1)] it holds that lim σ imp(t, e xt ) = [ 2 Λ (x) x + Λ (x)]. t Corollary Let (X, V ) be a non-degenerate affine stochastic volatility process that satisfies the assumptions from above. Then there exists a Lévy process Y, such that the limiting smiles of the models e X and e Y are identical. In the Heston model, the corresponding Lévy model is the Normal-Inverse-Gaussian (NIG) model.

34 1 Affine Stochastic Volatility Models 2 Large deviations and option prices 3 Examples

35 Example: Heston model & BNS model In the Heston model the rate function is Legendre transform of Λ(u) = λθ ( ζ 2 χ(u) + ) (u) where (u) = χ(u) 2 ζ 2 (u 2 u). The limiting volatility σ (x) can be explictly computed and coincides - after reparameterization - with the SVI parameterization of Jim Gatheral: ( 1 + ω 2 ρx + σ 2 Heston (x) = ω 1 2 (ω 2 x + ρ) ρ 2 ). In the BNS-Model we obtain ( u 2 Λ(u) = λκ 2λ + u ( ρ 1 )) uλκ(ρ). 2λ

36 Example: Heston model & BNS model In the Heston model the rate function is Legendre transform of Λ(u) = λθ ( ζ 2 χ(u) + ) (u) where (u) = χ(u) 2 ζ 2 (u 2 u). The limiting volatility σ (x) can be explictly computed and coincides - after reparameterization - with the SVI parameterization of Jim Gatheral: ( 1 + ω 2 ρx + σ 2 Heston (x) = ω 1 2 (ω 2 x + ρ) ρ 2 ). In the BNS-Model we obtain ( u 2 Λ(u) = λκ 2λ + u ( ρ 1 )) uλκ(ρ). 2λ

37 Example: Heston model & BNS model In the Heston model the rate function is Legendre transform of Λ(u) = λθ ( ζ 2 χ(u) + ) (u) where (u) = χ(u) 2 ζ 2 (u 2 u). The limiting volatility σ (x) can be explictly computed and coincides - after reparameterization - with the SVI parameterization of Jim Gatheral: ( 1 + ω 2 ρx + σ 2 Heston (x) = ω 1 2 (ω 2 x + ρ) ρ 2 ). In the BNS-Model we obtain ( u 2 Λ(u) = λκ 2λ + u ( ρ 1 )) uλκ(ρ). 2λ

38 Example: Heston model & BNS model In the Heston model the rate function is Legendre transform of Λ(u) = λθ ( ζ 2 χ(u) + ) (u) where (u) = χ(u) 2 ζ 2 (u 2 u). The limiting volatility σ (x) can be explictly computed and coincides - after reparameterization - with the SVI parameterization of Jim Gatheral: ( 1 + ω 2 ρx + σ 2 Heston (x) = ω 1 2 (ω 2 x + ρ) ρ 2 ). In the BNS-Model we obtain ( u 2 Λ(u) = λκ 2λ + u ( ρ 1 )) uλκ(ρ). 2λ

39 Numerical Illustration: BNS Model Γ-BNS model with a = , b = , v 0 = , γ = , (Schoutens) Solid line: asymptotic smile. Dotted and dashed: 5, 10 and 20 years generated smile.

40 Thank you for your attention!

41 David S. Bates. Jump and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. The Review of Financial Studies, 9: , David S. Bates. Post- 87 crash fears in the S&P 500 futures option market. Journal of Econometrics, 94: , D. Duffie, D. Filipovic, and W. Schachermayer. Affine processes and applications in finance. The Annals of Applied Probability, 13(3): , 2003.

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