Multiscale Stochastic Volatility Models

Transcription

1 Multiscale Stochastic Volatility Models Jean-Pierre Fouque University of California Santa Barbara 6th World Congress of the Bachelier Finance Society Toronto, June 25, 2010

2 Multiscale Stochastic Volatility for Equity, Interest-Rate and Credit Derivatives J.-P. Fouque, G. Papanicolaou, R. Sircar, K. Sølna Cambridge University Press. To appear (soon...)

3 Price Expansion P: price of a vanilla European option (to start with) P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 4 2 σσp 0 + D 1 = S S (Delta), D 2 = S 2 2 S 2 (Gamma) σ = σ (V ega) v i = v i (τ), payoff independent, τ = time-to-maturity P 0 is typically a constant volatility price closed-form formula Black-Scholes in Equity (Vasicek or CIR in Fixed Income, Black-Cox in Credit,...) Where do we get such an expansion? What do we expect from it?

4 Wish List P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 4 2 σσp 0 + Accuracy: the truncated expansion should be a good approximation (v i 0 fast enough) Stability: the coefficients v s should be stable in time short-time tight-fit vs. long-time rough fit Should be useful for hedging under physical measure (the v s are calibrated under risk-neutral) Should lead to practical consistent pricing of exotic derivatives Let s look at calibration first

5 Wish List P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 4 2 σσp 0 + Accuracy: the truncated expansion should be a good approximation (v i 0 fast enough) Stability: the coefficients v s should be stable in time short-time tight-fit vs. long-time rough fit Should be useful for hedging under physical measure (the v s are calibrated under risk-neutral) Should lead to practical consistent pricing of exotic derivatives Let s look at calibration first

6 Wish List P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 4 2 σσp 0 + Accuracy: the truncated expansion should be a good approximation (v i 0 fast enough) Stability: the coefficients v s should be stable in time short-time tight-fit vs. long-time rough fit Should lead to practical consistent pricing of path-dependent derivatives Should be useful for hedging under physical measure (the v s are calibrated under risk-neutral) Let s look at calibration first

7 Wish List P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 4 2 σσp 0 + Accuracy: the truncated expansion should be a good approximation (v i 0 fast enough) Stability: the coefficients v s should be stable in time short-time tight-fit vs. long-time rough fit Should lead to practical consistent pricing of path-dependent derivatives Should be useful for hedging under physical measure (the v s are calibrated under risk-neutral) Let s look at calibration first

8 Wish List P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 4 2 σσp 0 + Accuracy: the truncated expansion should be a good approximation (v i 0 fast enough) Stability: the coefficients v s should be stable in time short-time tight-fit vs. long-time rough fit Should lead to practical consistent pricing of path-dependent derivatives Should be useful for hedging under physical measure (the v s are calibrated under risk-neutral) Let s look at calibration first

9 Calibration on Implied Volatilities For vanilla European options we have: σ P 0 = τ σd 2 P 0 so that P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 στ σp 0 + v 3 στ D 1 σ P 0 + For Calls, P 0 = C BS and by direct computation { P = C BS + v 0 + v ( 2 στ + v 1 + v ) ( 3 1 d )} 1 στ σ σ C BS + τ where d 1 = LM+(r+1 2 σ2 )τ σ, and LM log(k/s) τ Expanding the implied volatility I = σ + I 1 + P C BS ( σ + I 1 + ) = C BS + I 1 σ C BS + = I 1 = v 0 + v ( 2 στ + v 1 + v ) ( 3 1 d ) 1 στ σ + τ Affine in LMMR: I = b + a LM τ + (quartic in LM) + where the term structure of the v s (τ dependence) is important.

10 Calibration on Implied Volatilities For vanilla European options we have: σ P 0 = τ σd 2 P 0 so that P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 στ σp 0 + v 3 στ D 1 σ P 0 + For Calls, P 0 = C BS and by direct computation { P = C BS + v 0 + v ( 2 στ + v 1 + v ) ( 3 1 d )} 1 στ σ σ C BS + τ where d 1 = LM+(r+1 2 σ2 )τ σ, and LM log(k/s) τ Expanding the implied volatility I = σ + I 1 + P C BS ( σ + I 1 + ) = C BS + I 1 σ C BS + = I 1 = v 0 + v ( 2 στ + v 1 + v ) ( 3 1 d ) 1 στ σ + τ Affine in LMMR: I = b + a LM τ + (quartic in LM) + where the term structure of the v s (τ dependence) is important.

11 Calibration on Implied Volatilities For vanilla European options we have: σ P 0 = τ σd 2 P 0 so that P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 στ σp 0 + v 3 στ D 1 σ P 0 + For Calls, P 0 = C BS and by direct computation { P = C BS + v 0 + v ( 2 στ + v 1 + v ) ( 3 1 d )} 1 στ σ σ C BS + τ where d 1 = LM+(r+1 2 σ2 )τ σ, and LM log(k/s) τ Expanding the implied volatility I = σ + I 1 + P C BS ( σ + I 1 + ) = C BS + I 1 σ C BS + = I 1 = v 0 + v ( 2 στ + v 1 + v ) ( 3 1 d ) 1 στ σ + τ Affine in LMMR: I = b + a LM τ + (quartic in LM) + where the term structure of the v s (τ dependence) is important.

12 Goal: fit Calibration Examples I = b + a LM τ + (quartic in LM) + to the observed implied volatility surface. We typically fit the parameters a, b,... by regressing in LMMR maturity-by-maturity, then we fit their dependence in τ. We will see that our expansion leads to a, b which are affine in τ. Some examples

13 0.5 τ=43 days days days Implied Volatility LMMR LMMR LMMR τ=197 days days days Implied Volatility LMMR LMMR LMMR S&P 500 Implied Volatility data on June 5, 2003 and fits to the affine LMMR approximation for six different maturities.

14 m 0 + m 1 τ τ(yrs) b 0 + b 1 τ τ(yrs) S&P 500 Implied Volatility data on June 5, 2003 and fits to the two-scales asymptotic theory. The bottom (rep. top) figure shows the linear regression of b (resp. a) with respect to time to maturity τ.

15 a b Trading Day Number

16 Higher Order Expansion I 4 a j (τ) (LM) j + 1 τ Φ t, j=0

17 0.5 5 June, 2003: S&P 500 Options, 15 days to maturity June, 2003: S&P 500 Options, 71 days to maturity Implied Volatility Implied Volatility Log Moneyness + 1 Log Moneyness June, 2003: S&P 500 Options, 197 days to maturity June, 2003: S&P 500 Options, 379 days to maturity Implied Volatility Implied Volatility Log Moneyness Log Moneyness + 1 S&P 500 Implied Volatility data on June 5, 2003 and quartic fits to the asymptotic theory for four maturities.

18 a 4 2 a τ (yrs) τ(yrs) a a τ (yrs) S&P 500 Term-Structure Fit using second order approximation. Data from June 5, 2003.

19 Equity for instance. Under physical measure: Stochastic Volatility Models ds t S t = µdt + σ t dw (0) t σ t = f(y t, Z t, ) dy t = α(y t )dt + β(y t )dw (1) t dz t = c(z t )dt + g(z t )dw (2) t Volatility factors can be differentiated by their time scales

20 Multiscale Stochastic Volatility Models σ t = f(y t,z t ) Y t is fast mean-reverting (ergodic on a fast time scale): dy t = 1 ε α(y t)dt + 1 ε β(y t )dw (1) t, 0 < ε 1 Z t is slowly varying: dz t = δc(z t )dt + δ g(z t )dw (2) t, 0 < δ 1 Separation of time scales: ε << T << 1/δ 1 T T 0 σ 2 t dt = 1 T Local Effective Volatility: T 0 f 2 (Y t, Z t )dt f 2 (, z) Φ Y σ 2 (z) f 2 (,z) Φ Y

21 Multiscale Stochastic Volatility Models σ t = f(y t,z t ) Y t is fast mean-reverting (ergodic on a fast time scale): dy t = 1 ε α(y t)dt + 1 ε β(y t )dw (1) t, 0 < ε 1 Z t is slowly varying: dz t = δc(z t )dt + δ g(z t )dw (2) t, 0 < δ 1 Separation of time scales: ε << T << 1/δ (assuming f continuous in z): 1 T T 0 σ 2 t dt = 1 T Local Effective Volatility: T 0 f 2 (Y t, Z t )dt f 2 (, z) Φ Y σ 2 (z) f 2 (,z) Φ Y P 0 = P BS ( σ(z))

22 Multiscale Stochastic Volatility Models σ t = f(y t,z t ) Y t is fast mean-reverting (ergodic on a fast time scale): dy t = 1 ε α(y t)dt + 1 ε β(y t )dw (1) t, 0 < ε 1 Z t is slowly varying: dz t = δc(z t )dt + δ g(z t )dw (2) t, 0 < δ 1 Separation of time scales: ε << T << 1/δ (assuming f continuous in z): 1 T T 0 σ 2 t dt = 1 T Local Effective Volatility: T 0 f 2 (Y t, Z t )dt f 2 (, z) Φ Y σ 2 (z) f 2 (,z) Φ Y P 0 = P BS ( σ(z))

23 Market Prices of Volatility Risk Under the risk neutral measure IP chosen by the market: ds t = rs t dt + f(y t, Z t )S t dw (0) t ( 1 dy t = ε α(y t) 1 ) β(y t )Λ(Y t, Z t ) ε ( dz t = δ c(z t ) ) δ g(z t )Γ(Y t, Z t ) dt + 1 ε β(y t )dw (1) t dt + δ g(z t )dw (2) t d < W (0), W (1) > t = ρ 1 dt d < W (0), W (2) > t = ρ 2 dt Λ and Γ: market prices of volatility risk

24 Pricing Equation { } P ε,δ (t, x, y, z) = IE e r(t t) h(s T ) S t = x, Y t = y, Z t = z Feynman Kac: ( 1 ε L Y + 1 ε L ρ1,λ + L + δl ρ2,γ + δl Z + P ε,δ (T, x, y, z) = h(x) ) δ ε L ρ 12 P ε,δ = 0 with L = L BS (f(y, z)) = t + 1 ( 2 f2 (y, z)x 2 2 x + r x ) 2 x

25 Regular-Singular Perturbations P ε,δ = i,j ε i/2 δ j/2 P i,j = P 0 + ε P 1,0 + δ P 0,1 + L BS ( σ(z))p 0 = 0, P 0 (T, x) = h(x) = P 0 = P BS ( σ(z)) P 0 is independent of y and z is a parameter. bfl BS ( σ(z)) ( εp 1,0 ) + V ε 2 D 2 P BS + V ε 3 D 1 D 2 P BS = 0 ( ) bfl BS ( σ(z)) δp0,1 + 2 ( V δ 0 σ P BS + V δ 1 D 1 σ P BS ) = 0 P 1,0 (T,x) = P 0,1 (T,x) = 0 V δ 0 and V ε 2 are volatility level adjustments due to Γ and Λ resp. V δ 1 and V ε 3 are skew parameters proportional to ρ 2 and ρ 1 resp.

26 Regular-Singular Perturbations P ε,δ = i,j ε i/2 δ j/2 P i,j = P 0 + ε P 1,0 + δ P 0,1 + L BS ( σ(z))p 0 = 0, P 0 (T, x) = h(x) = P 0 = P BS ( σ(z)) P 0 is independent of y and z is a parameter. L BS ( σ(z)) ( ) εp 1,0 + V ε 2 D 2 P BS + V3D ε 1 D 2 P BS = 0 ( ) L BS ( σ(z)) δp0,1 + 2 ( ) V0 δ σ P BS + V1D δ 1 σ P BS = 0 P 1,0 (T,x) = P 0,1 (T,x) = 0 V δ 0 and V ε 2 are volatility level adjustments due to Γ and Λ resp. V δ 1 and V ε 3 are skew parameters proportional to ρ 2 and ρ 1 resp. Important: these Black-Scholes equations will hold for exotic options with additional boundary conditions, but with the same group parameters V s

27 Regular-Singular Perturbations P ε,δ = i,j ε i/2 δ j/2 P i,j = P 0 + ε P 1,0 + δ P 0,1 + L BS ( σ(z))p 0 = 0, P 0 (T, x) = h(x) = P 0 = P BS ( σ(z)) P 0 is independent of y and z is a parameter. L BS ( σ(z)) ( ) εp 1,0 + V ε 2 D 2 P BS + V3D ε 1 D 2 P BS = 0 ( ) L BS ( σ(z)) δp0,1 + 2 ( ) V0 δ σ P BS + V1D δ 1 σ P BS = 0 P 1,0 (T,x) = P 0,1 (T,x) = 0 V δ 0 and V ε 2 are volatility level adjustments due to Γ and Λ resp. V δ 1 and V ε 3 are skew parameters proportional to ρ 2 and ρ 1 resp. Important: these Black-Scholes equations will hold for exotic options with additional boundary conditions, but with the same group parameters V s

28 Explicit formulas for Vanilla European Options Notation: T t = τ εp1,0 = τ (V ε 2D 2 P BS + V ε 3D 1 D 2 P BS ) easily checked by using L BS D i = D i L BS bf δp 0,1 = τ ( V δ 0 σ P BS + V δ 1 D 1 σ P BS ) easily checked by using P BS = τ σd 2 P BS and then L BS D i = D i L BS. Back to our expansion P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 0 = τv δ 0, v 1 = τv δ 1 v 2 = τv ε 2, v 3 = τv ε 3 In terms of calibration to implied volatilities:

29 Explicit formulas for Vanilla European Options Notation: T t = τ εp1,0 = τ (V ε 2D 2 P BS + V ε 3D 1 D 2 P BS ) easily checked by using L BS D i = D i L BS δp0,1 = τ ( V δ 0 σ P BS + V δ 1D 1 σ P BS ) easily checked by using first P BS = τ σd 2 P BS and then L BS D i = D i L BS Back to our expansion P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 0 = τv δ 0, v 1 = τv δ 1 v 2 = τv ε 2, v 3 = τv ε 3 In terms of calibration to implied volatilities:

30 Explicit formulas for Vanilla European Options Notation: T t = τ εp1,0 = τ (V ε 2D 2 P BS + V ε 3D 1 D 2 P BS ) easily checked by using L BS D i = D i L BS δp0,1 = τ ( V δ 0 σ P BS + V δ 1D 1 σ P BS ) easily checked by using P BS = τ σd 2 P BS and then L BS D i = D i L BS. Back to our expansion P = P 0 + v 0 σ P 0 + v 1 D 1 σ P 0 + v 2 D 2 P 0 + v 3 D 1 D 2 P 0 + v 0 = τv δ 0, v 1 = τv δ 1 v 2 = τv ε 2, v 3 = τv ε 3 In terms of calibration to implied volatilities

31 Implied Volatility Calibration Formulas σ + V 2 σ + V ( 3 2r (1 2 σ σ ) + τ V V 1 2 (1 2r σ 2 ) ) }{{} intercept b + ( V3 σ 3 + τ V 1 σ 2 ) }{{} slope a LMMR Either one estimates σ from historical data (preferred for hedging where V 0 and V 2 do not appear), and then fitting maturity-by-maturity and regressing in τ, one gets: 1. V 1 and V 3 from the slope a 2. V 0 and V 2 from the intercept b or one uses the adjusted effective volatility σ σ 2 + 2V 2 calibrated from option data, along with V 0, V 1, and V 3 σ + V ( 3 2r (1 2σ σ ) + τ V V ) 1 2r (1 2 σ ) + 2 ( V3 σ 3 + τ V 1 σ 2 ) LMMR

32 Implied Volatility Calibration Formulas σ + V 2 σ + V ( 3 2r (1 2 σ σ ) + τ V V 1 2 (1 2r σ 2 ) ) }{{} intercept b + ( V3 σ 3 + τ V 1 σ 2 ) }{{} slope a LMMR Either one estimates σ from historical data (preferred for hedging where V 0 and V 2 do not appear), and then fitting maturity-by-maturity and regressing in τ, one gets: 1. V 1 and V 3 from the slope a 2. V 0 and V 2 from the intercept b or one uses the adjusted effective volatility σ σ 2 + 2V 2 calibrated from option data, along with V 0, V 1, and V 3 (preferred for pricing): σ + V ( 3 2r (1 2σ σ ) + τ V V ) ( 1 2r (1 2 σ ) V3 + 2 σ + τ V ) 1 LMMR 3 σ 2

33 Back to the Wish List: Accuracy If the payoff function h is smooth: P ε,δ = = (P 0 + εp 1,0 + εp 2,0 + ε 3/2 P 3,0 ) (P 0 + εp 1,0 + δp 0,1 ) then the residual R ε,δ satisfies and therefore R ε,δ = O(ε + δ). + δ ( P 0,1 + εp 1,1 + εp 2,1 ) + R ε,δ + O(ε + δ) + R ε,δ L ε,δ R ε,δ = O(ε + δ) R ε,δ (T) = O(ε + δ) If h is non-smooth (call option in particular), then use a careful regularization.

34 Path-Dependent Derivatives (Barrier, Asian,...) Calibrate σ, V 0, V 1 and V 3 on the implied volatility surface Solve the corresponding problem with constant volatility σ = P 0 = P BS (σ ) Use V 0, V 1 and V 3 to compute the source 2 (V 0 σ P BS + V 1 D 1 σ P BS) + V 3 D 1 D 2 P BS Get the correction by solving the SAME PROBLEM with zero boundary conditions and the source.

35 American Options Calibrate σ, V 0, V 1 and V 3 on the implied volatility surface Solve the corresponding problem with constant volatility σ = P and the free boundary x (t) Use V 0, V 1 and V 3 to compute the source 2 (V 0 σ P BS + V 1 D 1 σ P BS) + V 3 D 1 D 2 P BS Get the correction by solving the corresponding problem with fixed boundary x (t), zero boundary conditions and the source.

36 Cost of the Black-Scholes Hedging Strategy Infinitesimal cost: P BS (T, S T ) = h(s T ) P BS (t, S t ) = a t S t + b t e rt, a t = x P BS dp BS (t, S t ) (a t ds t + rb t e rt dt) }{{} self-financing part = 1 2 ( f 2 (Y t, Z t ) σ 2) D 2 P BS (t, S t )dt Cumulative financing cost: E BS (t) = 1 2 t 0 e rs ( f 2 (Y s,z s ) σ 2) D 2 P BS (s,s s )ds Choice of σ?

37 Choice of σ? Since Y t is fast mean-reverting (ε << 1), integrals like t ( f 2 (Y s,z s ) σ 2) Ψ s ds will be small with ε if 0 Therefore two choices: σ 2 = σ 2 (z) = f 2 (,z) Φ(Y) σ 2 = σ 2 (Z t ) and P BS = P BS (t, S t ; σ(z t )), in which case σ(z t ) needs to be estimated continuously (and dp BS revisited) σ 2 = σ 2 (Z 0 ) and P BS = P BS (t, S t ; σ(z 0 )) with f 2 (Y s,z s ) σ 2 = ( f 2 (Y s,z s ) σ 2 (Z t ) ) + ( σ 2 (Z t ) σ 2 (Z 0 ) ) in which case parameters are frozen at time zero, an additional cost of order δ comes from the second term (offset in practice by re-calibration at δ-frequency).

38 Corrected Hedging Strategy A careful analysis of the cost shows E 0 (t) = 1 2 t 0 e rs ( f 2 (Y s,z s ) σ 2 (Z t ) ) D 2 P BS (s,s s )ds = ε (B ε t + M ε t) + O(ε + δ), where M ε t is a martingale, and B ε t = ρ 1 2 t 0 e rs β(y s ) φ y f(y s,z s )D 1 D 2 P BS (s,s s )ds is a bounded variation bias which can be compensated by using the corrected hedging ratio a t given by x P BS + (T t)v 3 x D 1 D 2 P BS + (T t)v 1 x D 1 σ P BS The last term compensates for the bias generated by σ 2 (Z t ) σ 2 (Z 0 )

39 Examples of other: Models Regimes Applications

40 A Model with Volatility Time-Scale of Order One In the model σ t = f(y t,z t ), if one wants to: keep Y fast mean-reverting let Z be on a time scale comparable to maturity (or add one such factor) keep the computational tractability then, one needs to make sure that the SV model σ 2 (Z t ) is tractable. An interesting choice is the Heston model: A Fast Mean-Reverting Correction to Heston Stochastic Volatility Model with Matthew Lorig (PhD student, UCSB), where we develop this idea. An example of fit

41 A Model with Volatility Time-Scale of Order One In the model σ t = f(y t,z t ), if one wants to: keep Y fast mean-reverting let Z be on a time scale comparable to maturity (or add one such factor) keep the computational tractability then, one needs to make sure that the SV model σ 2 (Z t ) is tractable. An interesting choice is the Heston model: A Fast Mean-Reverting Correction to Heston Stochastic Volatility Model with Matthew Lorig (PhD student, UCSB), where we develop this idea. An example of fit

42 Days to Maturity = 65 Market Data Heston Fit Multiscale Fit Implied Volatility log(k/x) SPX Implied Volatilities from May 17, 2006

43 Fast Mean-Reverting SV and Short Maturities If the time scale of the fast mean-reverting factor Y is ε << 1, and if the maturity of interest is small but still large compared with ε, then, one can consider the regime ε << T ε << 1 It involves a non-trivial mixture of Large Deviation (short maturity) and Homogenization (fast mean reverting coefficient): Short maturity asymptotics for a fast mean reverting Heston stochastic volatility model with Jin Feng and Martin Forde (SIAM Journal on Financial Mathematics, Vol. 1, 2010). Interestingly, in this regime and for this model, we derive explicit formulas for the limiting implied volatility which looks like

44 Three parameters which control the implied volatility skew s level (θ), slope (ρ) and convexity (ν/κ).

45 A Cool Application to Forward-Looking Betas Discrete time CAPM model: R a R f = β a (R M R f ) + ǫ a Christoffersen, Jacobs, and Vainberg (2008, McGill University): β a = ( ) 1 ( ) 1 SKEWa 3 V AR a 2 SKEW M V AR M where VAR and SKEW are variance and risk-neutral skewness With Eli Kollman (PhD 2009, UCSB), we propose in Calibration of Stock Betas from Skews of Implied Volatilities (Applied Mathematical Finance, 2010): ˆβ a = ( V a,ǫ 3 V M,ǫ 3 ) 1/3 = ( a a,ǫ a M,ǫ ) 1/3 ( b a b M )

46 A Cool Application to Forward-Looking Betas Discrete time CAPM model: R a R f = β a (R M R f ) + ǫ a Christoffersen, Jacobs, and Vainberg (2008, McGill University): β a = ( ) 1 ( ) 1 SKEWa 3 V AR a 2 SKEW M V AR M where VAR and SKEW are variance and risk-neutral skewness With Eli Kollman (PhD 2009, UCSB), we propose in Calibration of Stock Betas from Skews of Implied Volatilities (Applied Mathematical Finance, 2010): ˆβ a = ( V a,ǫ 3 V M,ǫ 3 ) 1/3 = ( a a,ǫ a M,ǫ ) 1/3 ( b a b M )

47 LMMR fits (2/18/2009): S&P500 and Amgen, beta estimate is S&P AMGN Implied Vol Implied Vol LMMR LMMR

48 LMMR fits (2/19/2009): S&P500 and Goldman Sachs, beta estimate is S&P GS Implied Vol Implied Vol LMMR LMMR

49 THANKS FOR YOUR ATTENTION