Heston vs Heston. Antoine Jacquier. Department of Mathematics, Imperial College London. ICASQF, Cartagena, Colombia, June 2016

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1 Department of Mathematics, Imperial College London ICASQF, Cartagena, Colombia, June Joint work with Fangwei Shi June 18, 2016

2 Implied volatility About models Calibration Implied volatility Asset price process: (S t = e Xt ) t 0, with X 0 = 0 No dividend, no interest rate Black-Scholes-Merton (BSM) framework: C BS (τ, k, σ) := E 0 (e X τ e k) = N (d +) e k N (d ), + d ± := k σ τ ± 1 2 σ τ Spot implied volatility σ τ (k): the unique (non-negative) solution to C observed (τ, k) = C BS (τ, k, σ τ (k)) Implied volatility: unit-free measure of option prices

3 Implied volatility About models Calibration Which model to calibrate the volatility surface? Stochastic volatility models (Heston, 3/2, Stein-Stein): easy to simulate (even for path-dependent options), realistic overall implied volatility surface Lévy / jump / affine models: provide a better short-dated fit to the implied volatility surface (steeper skew for out-of-the-money Puts), but jumps notoriously difficult to hedge Local volatility models: not easy to calibrate to the observed surface Local-stochastic volatility models: very appealing, but technically challenging

4 Implied volatility About models Calibration Which model to calibrate the volatility surface? Stochastic volatility models (Heston, 3/2, Stein-Stein): easy to simulate (even for path-dependent options), realistic overall implied volatility surface Lévy / jump / affine models: provide a better short-dated fit to the implied volatility surface (steeper skew for out-of-the-money Puts), but jumps notoriously difficult to hedge Local volatility models: not easy to calibrate to the observed surface Local-stochastic volatility models: very appealing, but technically challenging However the implied volatility is not available in closed form for most models Its asymptotic behaviour (small/large k, τ) however provides us with closed-form approximations, information about the impact of parameters

5 Implied volatility About models Calibration Implied volatility (σ τ (k)) asymptotics as k, τ 0 or τ : Berestycki-Busca-Florent (2004): small-τ using PDE methods for diffusions Henry-Labordère (2009): small-τ asymptotics using differential geometry Forde et al(2012), Jacquier et al(2012): small/ large τ using large deviations Lee (2003), Benaim-Friz (2009), Gulisashvili ( ), Caravenna-Corbetta (2016), De Marco-Jacquier-Hillairet (2013): k Laurence-Gatheral-Hsu-Ouyang-Wang (2012): small-τ in local volatility models Fouque et al( ): perturbation techniques for slow and fast mean-reverting stochastic volatility models Mijatović-Tankov (2012): small-τ for jump models Related works: Kim, Kunitomo, Osajima, Takahashi (1999-) : asymptotic expansions based on Kusuoka-Yoshida-Watanabe method (expansion around a Gaussian) Deuschel-Friz-Jacquier-Violante (CPAM 2014), De Marco-Friz (2014): small-noise expansions using Laplace method on Wiener space (Ben Arous-Bismut approach) Lorig-Pagliarani-Pascucci (2014- ): expansions around the money (k = 0) Note: from expansions of densities to implied volatility asymptotics is automatic (Gao-Lee (2013))

6 Implied volatility About models Calibration Take-away summary Classical stochastic volatility models generate a constant short-maturity ATM skew and a large-maturity one proportional to τ 1 ; However, short-term data suggests a time decay of the ATM skew proportional to τ α, α (0, 1/2) One solution: adding volatility factors (Gatheral s Double Mean-Reverting, Bergomi-Guyon), each factor acting on a specific time horizon But risk of over-parameterisation of the model In the Lévy case (Tankov, 2010), the situation is different, as τ 0: in the pure jump case with ( 1,1) x ν(dx) <, then σ2 τ (0) cτ; in the (α) stable case, σ 2 τ (0) cτ 1 2/α for α (1, 2); for out-of-the-money options, σ 2 τ (k) k 2 2τ log(τ) But short term at-the-money skew: constant

7 Implied volatility About models Calibration Rough volatility models In 2014, Gatheral-Jaisson-Rosenbaum (+ Bayer-Gatheral-Friz) proposed a fractional volatility model: where W H is a fractional Brownian motion ds t = S t σ t dz t, dσ t = ηdwt H, (1) Time series of the Oxford-Man SPX realised variance as well as implied volatility smiles of the SPX suggest that H (0, 1/2): short-memory volatility Is not statistically rejected by Ait-Sahalia-Jacod s test (2009) for Itô diffusions Main drawback: loss of Markovianity (H 1/2) rules out PDE techniques, and Monte Carlo is computationally intensive An intuitive remark: Mandelbrot-van Ness representation for fbm: 0 t Wt H = K 1 (s, t)dw s + 0 K 2 (s, t)dw s At time zero, the volatility process in (1) has already accumulated some randomness

8 Implied volatility About models Calibration (Classical) Stochastic volatility models: calibration? Consider the Heston model for the log-stock price process X := log(s): dx t = 1 2 V tdt + V t db t, X 0 = 0, dv t = κ(θ V t )dt + ξ V t dw t, V 0 = v 0 > 0, d B, W t = ρdt, All parameters can be calibrated by (local/global) minimisation given an observed BS 1 (C observed (K, T ) F t0 ) discrete implied volatility surface at time t 0 = 0 θ: long-term average; κ: mean-reversion speed; ξ: volatility of volatility; ρ: leverage effect; v 0 : initial variance

9 Implied volatility About models Calibration (Classical) Stochastic volatility models: calibration? Consider the Heston model for the log-stock price process X := log(s): dx t = 1 2 V tdt + V t db t, X 0 = 0, dv t = κ(θ V t )dt + ξ V t dw t, V 0 = v 0 > 0, d B, W t = ρdt, All parameters can be calibrated by (local/global) minimisation given an observed BS 1 (C observed (K, T ) F t0 ) discrete implied volatility surface at time t 0 = 0 θ: long-term average; κ: mean-reversion speed; ξ: volatility of volatility; ρ: leverage effect; v 0 : initial variance Initial???????

10 Implied volatility About models Calibration (Classical) Stochastic volatility models: calibration? Consider the Heston model for the log-stock price process X := log(s): dx t = 1 2 V tdt + V t db t, X 0 = 0, dv t = κ(θ V t )dt + ξ V t dw t, V 0 = v 0 > 0, d B, W t = ρdt, All parameters can be calibrated by (local/global) minimisation given an observed BS 1 (C observed (K, T ) F t0 ) discrete implied volatility surface at time t 0 = 0 θ: long-term average; κ: mean-reversion speed; ξ: volatility of volatility; ρ: leverage effect; v 0 : initial variance Initial??????? What is v 0? For the trader: small-maturity (one week) ATM implied volatility; Really observable? Or just another parameter? What does t 0 actually mean? What about yesterday? Yesterday is under P The option price is under Q

11 General Properties Asymptotics The Heston H(V) model Under the risk-neutral measure Q: dx t = 1 2 V tdt + V t db t, X 0 = 0, dv t = κ(θ V t)dt + ξ V tdw t, V 0 V, d B, W t = ρdt, (2) with κ, θ, ξ > 0 and ρ ( 1, 1) Assumption V F 0 ; = V is a continuous random variable supported on (v, v + ) [0, ]; M V (u) := E ( e uv ), for all u D V := {u R : E ( e uv ) < } (, 0]; D V contains at least an open neighbourhood of the origin: m := sup {u R : M V (u) < } belongs to (0, ] When V is a Dirac distribution (v = v +), the system (2) corresponds to the classical Heston model Clearly S = e X is a Q-martingale

12 General Properties Asymptotics Black-Scholes: dx t = Σ2 2 Asymptotic behaviour: a review dt + ΣdWt with ΛBS t (u) := log E ( e ) uxt = u(u 1)Σ2 t 2 ( u ) Λ BS (u) := lim tλ BS t = u2 Σ 2 t 0 t 2, for all u D BS = R In the language of Large deviations, X LDP(t, Λ BS ): { P(X t B) exp 1 } t inf x B Λ BS (x), where Λ BS (x) := sup {ux Λ(u)} = x2 u D BS 2Σ 2

13 General Properties Asymptotics Black-Scholes: dx t = Σ2 2 Asymptotic behaviour: a review dt + ΣdWt with ΛBS t (u) := log E ( e ) uxt = u(u 1)Σ2 t 2 ( u ) Λ BS (u) := lim tλ BS t = u2 Σ 2 t 0 t 2, for all u D BS = R In the language of Large deviations, X LDP(t, Λ BS ): { P(X t B) exp 1 } t inf x B Λ BS (x), where Λ BS (x) := sup {ux Λ(u)} = x2 u D BS 2Σ 2 Classical Heston: ( Λ H(v0) (u) := lim tλ H(v 0) u ) t, for all u D H R, t 0 t ( ) { and likewise, X LDP t, Λ : P(X H(v 0 ) t B) exp 1 } t inf x B Λ H(v 0 ) (x)

14 General Properties Asymptotics Black-Scholes: dx t = Σ2 2 Asymptotic behaviour: a review dt + ΣdWt with ΛBS t (u) := log E ( e ) uxt = u(u 1)Σ2 t 2 ( u ) Λ BS (u) := lim tλ BS t = u2 Σ 2 t 0 t 2, for all u D BS = R In the language of Large deviations, X LDP(t, Λ BS ): { P(X t B) exp 1 } t inf x B Λ BS (x), where Λ BS (x) := sup {ux Λ(u)} = x2 u D BS 2Σ 2 Classical Heston: ( Λ H(v0) (u) := lim tλ H(v 0) u ) t, for all u D H R, t 0 t ( ) { and likewise, X LDP t, Λ : P(X H(v 0 ) t B) exp 1 } t inf x B Λ H(v 0 ) (x) Equating both probabilities above yields lim σ 2 x 2 t (x) = t 0 2Λ, for all x 0 (x) H(v 0 )

15 General Properties Asymptotics Heston, simple case: supp(v) = (v, v + ), v + < Λ H t ( (u) := E e ux t H ) [ ( = E E e ux t H ) ] ( ) V = E e ϕ(t,u)+ψ(t,u)v = e ϕ(t,u) M V (ψ(t, u)) so that and therefore and Λ H (u) := lim t 0 tλ H t ( u t ) = Λ H(v+) (u), for all u D H, { P(X t B) exp 1 } t inf x B Λ H(v (x) +), lim σt 2 (x) = t 0 x 2 2Λ, for all x 0 (x) H(v + )

16 General Properties Asymptotics Heston, simple case: supp(v) = (v, v + ), v + < Λ H t ( (u) := E e ux t H ) [ ( = E E e ux t H ) ] ( ) V = E e ϕ(t,u)+ψ(t,u)v = e ϕ(t,u) M V (ψ(t, u)) so that and therefore and Λ H (u) := lim t 0 tλ H t ( u t ) = Λ H(v+) (u), for all u D H, { P(X t B) exp 1 } t inf x B Λ H(v (x) +), lim σt 2 (x) = t 0 x 2 2Λ, for all x 0 (x) H(v + ) Remarks: This could correspond to a trader s view: she does not exactly observe the small-time ATM implied volatility, but only a proxy, and V could be uniformly distributed in (v, v + ) = (v0 atm ε, v0 atm + ε) Related to JP Fouque s recent result on uncertain v 0 Intuition: only the right tail of the distribution of V matters (for the asymptotics)

17 General Properties Asymptotics Heston, general case, Part I: v + = Theorem (J-Shi (2016)) Let h(t) t γ, with γ (0, 1] As t tends to zero, the following pointwise limits hold: lim t 0 h(t)λ H t ( u ) h(t) = 0, u R, if γ (0, 1/2), for any V, 0, u R, if γ [1/2, 1), v + <, Λ H(v+ )(u), u D H, if γ = 1, v + <, 0, u D m, if γ = 1/2, v + =, m <, and is infinite elsewhere, where D m := ( 2m, 2m) Whenever γ > 1 (for any V), or m < and γ > 1/2, the pointwise limit is infinite everywhere except at the origin

18 General Properties Asymptotics Heston, general case, Part II: v + =, m = Assumption A: v + = and V admits a smooth density f with log f (v) l 1 v l 2 as v tends to infinity, for some (l 1, l 2 ) R + (1, )

19 General Properties Asymptotics Heston, general case, Part II: v + =, m = Assumption A: v + = and V admits a smooth density f with log f (v) l 1 v l 2 as v tends to infinity, for some (l 1, l 2 ) R + (1, ) Theorem (J-Shi (2016)): Thin-tail case The only non-degenerate speed factor is γ 0 = l 2 /(1+l 2 ) (1/2, 1), and, for any u R, Λ γ0 (u) := lim t 0 t γ 0 Λ H t ( u t γ 0 ) ( = 1 1 ) ( ) 1 1 l 2 l 1 l 2 l 2 1 ( u 2 2 ) l 2 l 2 1 Corollary Λ γ 0 (x) := sup {ux Λ γ0 (u)} = cx 2γ l 2 0, with c := (2l 2 ) u R 1 1+l 1+l 2 l 2 1 (1 + l 2 ), Under Assumption A, X LDP ( t γ 0, Λ ) γ 0 and lim t 1+l 1 2 σt 2 (x) = x2(1 γ 0) t 0 2c

20 General Properties Asymptotics A slight detour via moderate deviations Theorem (J-Shi (2016)): Thin-tail case Let Xt α := t α X t Under Assumption A, let γ 1 := (l 2 1) 1 l 2 (i) For any γ (0, γ 1 ), set α = (1 γ/γ 0 )/2; then (X α t ) t 0 satisfies a LDP with speed t γ and good rate function Λ γ; (ii) if γ = γ 1, set α = 1 γ 1, then (X α t ) t 0 LDP(t γ, L ) with L (x) := sup u D H { ux ( 1 1 ) ( ) 1 } 1 l 2 1 ΛH(1) (u) γ 1, for all x R l 2 l 1 l 2 Remark: P (X α t When γ (γ 0, γ 1 ), then α < 0 and (i) yields estimates of the form { x) = P (X t xt α ) exp Λ γ (x) } t γ see also Friz-Gerhold-Pinter (2016) small time and large strike regime,

21 General Properties Asymptotics The fat-tail case Assumption B: There exists (γ 0, γ 1, γ 2, ω) R + R R N +, such that the following asymptotics holds for the log-mgf of V as u tends to m from below: log M V (u) = γ 1 log(m u) + γ 2 + o(1), ω = 1, γ 1 < 0, γ 0 [1 + γ 1 (m u) log(m u) + γ 2 (m u) + o (m u)] (m u) ω 1, ω 2, Theorem (J-Shi (2016)): Fat-tail case Under Assumption B, the implied volatility behaves as follows as t tends to zero: σt 2 (k) = ( ) k 1 2 2mt + o t 1/2, for ω = 1, k 2 2mt + c ( ) 1(k) 1 4mt 1/4 + o t 1/4, for ω = 2 Example: Ergodic distribution of the CIR process (Gamma distribution) (3) (4)

22 M V (u) = v + = + ; m = 1/2; Non-central χ 2 (q, λ) ( ) 1 λu exp, for all u D (1 2u) q/2 V = 1 2u the only suitable scale function is h(t) t (, 1 2 ),

23 M V (u) = v + = + ; m = 1/2; Non-central χ 2 (q, λ) ( ) 1 λu exp, for all u D (1 2u) q/2 V = 1 2u the only suitable scale function is h(t) t (, 1 2 ), Remark [Link with Forward-start options / smile in Heston]: Let X H(v 0 ), and fix a forward-starting date t > 0 [ ( ) ] [ St+τ ( ) ] [ ( ) ] E e x F 0 = E e X t+τ X t e x S F 0 =: E e X (t) τ e x F 0, t where (X τ (t) ) τ 0 H(V), with ( V β t χ 2 q = 4κθ ξ 2, λ = v 0e κt β t ), and β t := ξ2 4κ (1 e κt )

24 Folded Gaussian; V N (0, 1) The density of V reads f (v) = 2 ( π exp 12 ) v 2, for all v D V = R +, which satisfies Assumption A Simple computations yield ( z 2 ) M V (z) = 2 exp N (z), for any z R 2 Direct computations show that the only not-trivial scaling is h(t) t 2/3, and lim t 1/3 σt 2 (x) = (2x)2/3, for all x 0 t 0 3

25 Folded Gaussian; V N (0, 1)

26 Uniform distribution

27 Buena Suerte Para las Semifinales