Near-expiration behavior of implied volatility for exponential Lévy models

Transcription

1 Near-expiration behavior of implied volatility for exponential Lévy models José E. Figueroa-López 1 1 Department of Statistics Purdue University Financial Mathematics Seminar The Stevanovich Center for Financial Mathematics The University of Chicago March 31st, 2011 Joint work with Martin Forde, Dublin City University

2 Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

3 Motivation Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

4 Motivation Empirical features of implied volatility Implied volatility 1 Throughout, S t denotes the time-t price of an underlying asset; 2 A European Call Option is a contract that promises the payoff X = max{s T K, 0} = (S T K ) +, at a predetermined time T for a premium Π t at time t < T ; 3 The implied volatility ˆσ := ˆσ t (K, T ) of a given premium Π t is defined by { ( ) } Π BS t (ˆσ, T, K, S, r) := e r(t t) E SeˆσW T t +(r ˆσ2 2 )(T t) K = Π t, + where W T t N (0, T t); 4 There is one (and only one) implied volatility ˆσ for each possible arbitrage-free price Π t ((S Ke r(t t) ) +, S); 5 Larger ˆσ means a higher premium and, hence, higher stock riskiness;

5 Motivation Empirical features of implied volatility Usage and empirical features 1 Implied vols are used as a quotation devise of options; 2 Under "normal conditions", observed market ˆσ ranges in (0.1, 0.6); 3 Market ˆσ(K, T ) varies with K in a very specific manner (smile effect): U-shape (smirk) with a minimum around K ; Decreasing (skew); 4 Interpretation: Market charges a premium for deep OTM puts (or deep ITM calls) above their BS price computed with ATM implied vol ˆσ(S 0, T ); 5 Smile flattens out as T ; 6 Significant skew for short-term options (t T ); 7 Less variability when expressed in terms of the moneyness m = K /S;

6 Motivation Close-to-expiration smile for continuous models Known results for continuous models 1 [Gatheral et al. (2010)]. In a local volatility model, ds t = S t {rdt + σ(s t )dw t } S t+h S t S t D N ( rh, σ 2 (S t )h ), the implied volatility satisfies: ( where σ 0 := σ 0 (K ) := ˆσ t (T, K ) = σ 0 + σ 1 (T t) + O((T t) 2 ), 1 ln(s/k ) K S ) 1. 1 uσ(u) du 2 In particular, as t T, implied volatility smile behaves like σ 0 (K ); 3 Similar behavior for other popular continuous model including Heston model, SABR model, etc.

7 Close-to-expiration option prices for exponential Lévy models (ELM) Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

8 Close-to-expiration option prices for exponential Lévy models (ELM) Generalities of the model Generalities 1 The model: (i) Risky asset S t following an exponential Lévy model: S t = S 0 e X t ; (ii) A risk-free asset with constant interest rate r: B t = e rt. 2 The market (i)-(ii) is arbitrage-free and typically incomplete; hence, there exist -many Equivalent Martingale Measures (EMM) Q; 3 In fact, there -many EMM Q such that S t = S 0 e Xt follows also an exponential Lévy model under Q; 4 Arbitrage-free pricing: Under any of this EMM Q, the premium process Π Lévy t (T, K, S, r) := e r(t t) E Q {(S T K ) + S t = S}, is arbitrage-free relative to the market (i)-(ii);

9 Close-to-expiration option prices for exponential Lévy models (ELM) Generalities of the model Generalities 1 The model: (i) Risky asset S t following an exponential Lévy model: S t = S 0 e X t ; (ii) A risk-free asset with constant interest rate r: B t = e rt. 2 The market (i)-(ii) is arbitrage-free and typically incomplete; hence, there exist -many Equivalent Martingale Measures (EMM) Q; 3 In fact, there -many EMM Q such that S t = S 0 e Xt follows also an exponential Lévy model under Q; 4 Arbitrage-free pricing: Under any of this EMM Q, the premium process Π Lévy t (T, K, S, r) := e r(t t) E Q {(S T K ) + S t = S}, is arbitrage-free relative to the market (i)-(ii);

10 Close-to-expiration option prices for exponential Lévy models (ELM) Generalities of the model Generalities 1 The model: (i) Risky asset S t following an exponential Lévy model: S t = S 0 e X t ; (ii) A risk-free asset with constant interest rate r: B t = e rt. 2 The market (i)-(ii) is arbitrage-free and typically incomplete; hence, there exist -many Equivalent Martingale Measures (EMM) Q; 3 In fact, there -many EMM Q such that S t = S 0 e Xt follows also an exponential Lévy model under Q; 4 Arbitrage-free pricing: Under any of this EMM Q, the premium process Π Lévy t (T, K, S, r) := e r(t t) E Q {(S T K ) + S t = S}, is arbitrage-free relative to the market (i)-(ii);

11 Close-to-expiration option prices for exponential Lévy models (ELM) Generalities of the model Generalities 1 The model: (i) Risky asset S t following an exponential Lévy model: S t = S 0 e X t ; (ii) A risk-free asset with constant interest rate r: B t = e rt. 2 The market (i)-(ii) is arbitrage-free and typically incomplete; hence, there exist -many Equivalent Martingale Measures (EMM) Q; 3 In fact, there -many EMM Q such that S t = S 0 e Xt follows also an exponential Lévy model under Q; 4 Arbitrage-free pricing: Under any of this EMM Q, the premium process Π Lévy t (T, K, S, r) := e r(t t) E Q {(S T K ) + S t = S}, is arbitrage-free relative to the market (i)-(ii);

12 Close-to-expiration option prices for exponential Lévy models (ELM) Description of the problem The problem 1 Goal 1: Analyze the behavior of the option prices close-to-expiration: Π Lévy t (T, K, S, r) := e r(t t) E Q ((S T K ) + S t = S). 2 Goal 2: Obtain close-to-expiration asymptotics for the implied volatility ˆσ := ˆσ t (T, K ) of Lévy option prices: 3 Applications: Π BS t (ˆσ, T, K, S, r) = Π Lévy t (T, K, S, r). Calibration of the model parameters to market option prices near expiration; Quick and stable pricing of option near expiration;

13 Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result Key Result 1 Small-time tail distributions of a Lévy process 1 Let (b, σ 2, s) be the triple of the Lévy process X with a smooth jump intensity function" s (aka Lévy density); 2 Theorem: [F-L & Houdré, SPA 2009] For any n 0 and x > 0, P(X t x) = d 1 (x)t + d 2(x) 2 t d n(x) t n + O(t n+1 ). n! 3 The coefficients: 1 d 1 (x) = lim t 0 t P(Xt x) = s(u)du; x { 2 d 2 (x) = lim 1 t 0 t t P(Xt x) d 1(x) } = σ 2 s (x) + 2bs(x) ( 2 s(u)du) + x 2s(x) ( x 2 s(u)du) + 2 x/2 ys(y)dy + 2s(x) 1 2 x< y <1 1 2 x x 1 2 x x 1 2 x x y x y s(u)s(y)dudy (s(u) s(x))s(y)dudy.

14 Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result Key Result 1 Small-time tail distributions of a Lévy process 1 Let (b, σ 2, s) be the triple of the Lévy process X with a smooth jump intensity function" s (aka Lévy density); 2 Theorem: [F-L & Houdré, SPA 2009] For any n 0 and x > 0, P(X t x) = d 1 (x)t + d 2(x) 2 t d n(x) t n + O(t n+1 ). n! 3 The coefficients: 1 d 1 (x) = lim t 0 t P(Xt x) = s(u)du; x { 2 d 2 (x) = lim 1 t 0 t t P(Xt x) d 1(x) } = σ 2 s (x) + 2bs(x) ( 2 s(u)du) + x 2s(x) ( x 2 s(u)du) + 2 x/2 ys(y)dy + 2s(x) 1 2 x< y <1 1 2 x x 1 2 x x 1 2 x x y x y s(u)s(y)dudy (s(u) s(x))s(y)dudy.

15 Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result Key Result 1 Small-time tail distributions of a Lévy process 1 Let (b, σ 2, s) be the triple of the Lévy process X with a smooth jump intensity function" s (aka Lévy density); 2 Theorem: [F-L & Houdré, SPA 2009] For any n 0 and x > 0, P(X t x) = d 1 (x)t + d 2(x) 2 t d n(x) t n + O(t n+1 ). n! 3 The coefficients: 1 d 1 (x) = lim t 0 t P(Xt x) = s(u)du; x { 2 d 2 (x) = lim 1 t 0 t t P(Xt x) d 1(x) } = σ 2 s (x) + 2bs(x) ( 2 s(u)du) + x 2s(x) ( x 2 s(u)du) + 2 x/2 ys(y)dy + 2s(x) 1 2 x< y <1 1 2 x x 1 2 x x 1 2 x x y x y s(u)s(y)dudy (s(u) s(x))s(y)dudy.

16 Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result Key Result 1 Small-time tail distributions of a Lévy process 1 Let (b, σ 2, s) be the triple of the Lévy process X with a smooth jump intensity function" s (aka Lévy density); 2 Theorem: [F-L & Houdré, SPA 2009] For any n 0 and x > 0, P(X t x) = d 1 (x)t + d 2(x) 2 t d n(x) t n + O(t n+1 ). n! 3 The coefficients: 1 d 1 (x) = lim t 0 t P(Xt x) = s(u)du; x { 2 d 2 (x) = lim 1 t 0 t t P(Xt x) d 1(x) } = σ 2 s (x) + 2bs(x) ( 2 s(u)du) + x 2s(x) ( x 2 s(u)du) + 2 x/2 ys(y)dy + 2s(x) 1 2 x< y <1 1 2 x x 1 2 x x 1 2 x x y x y s(u)s(y)dudy (s(u) s(x))s(y)dudy.

17 Close-to-expiration option prices for exponential Lévy models (ELM) A fundamental preliminary result Key Result 1 Small-time tail distributions of a Lévy process 1 Let (b, σ 2, s) be the triple of the Lévy process X with a smooth jump intensity function" s (aka Lévy density); 2 Theorem: [F-L & Houdré, SPA 2009] For any n 0 and x > 0, P(X t x) = d 1 (x)t + d 2(x) 2 t d n(x) t n + O(t n+1 ). n! 3 The coefficients: 1 d 1 (x) = lim t 0 t P(Xt x) = s(u)du; x { 2 d 2 (x) = lim 1 t 0 t t P(Xt x) d 1(x) } = σ 2 s (x) + 2bs(x) ( 2 s(u)du) + x 2s(x) ( x 2 s(u)du) + 2 x/2 ys(y)dy + 2s(x) 1 2 x< y <1 1 2 x x 1 2 x x 1 2 x x y x y s(u)s(y)dudy (s(u) s(x))s(y)dudy.

18 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Prices of out-the-money Call Options 1 WLOG, we assume that r = 0; 2 Equivalent formulation: Π Lévy t := E Q [ (S T K ) + St = S ] [ = E Q ( S0 e X T K ) ] S + t = S [ = E Q ( St e X T X t K ) ] [ S + t = S = E Q ( Se X T X t K ) ] S + t = S [ = E Q ( Se X T X t K ) ] [ = SE Q ( + e X τ e κ) ], + where τ = T t (time-to-maturity) and κ = log(k /S) (log-moneyness). 3 Next, as it is done when deriving the B-S formula, E Q (e Xτ e κ ) + = E Q (e Xτ e κ )1 Xτ κ = E Q (e Xτ 1 Xτ κ) e κ Q(X τ κ) = Q (X τ κ) e κ Q(X τ κ), where Q (A) := E Q {1 A e Xt } if A F t (Esscher or Share measure);

19 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Prices of out-the-money Call Options 1 WLOG, we assume that r = 0; 2 Equivalent formulation: Π Lévy t := E Q [ (S T K ) + St = S ] [ = E Q ( S0 e X T K ) ] S + t = S [ = E Q ( St e X T X t K ) ] [ S + t = S = E Q ( Se X T X t K ) ] S + t = S [ = E Q ( Se X T X t K ) ] [ = SE Q ( + e X τ e κ) ], + where τ = T t (time-to-maturity) and κ = log(k /S) (log-moneyness). 3 Next, as it is done when deriving the B-S formula, E Q (e Xτ e κ ) + = E Q (e Xτ e κ )1 Xτ κ = E Q (e Xτ 1 Xτ κ) e κ Q(X τ κ) = Q (X τ κ) e κ Q(X τ κ), where Q (A) := E Q {1 A e Xt } if A F t (Esscher or Share measure);

20 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Prices of out-the-money Call Options 1 WLOG, we assume that r = 0; 2 Equivalent formulation: Π Lévy t := E Q [ (S T K ) + St = S ] [ = E Q ( S0 e X T K ) ] S + t = S [ = E Q ( St e X T X t K ) ] [ S + t = S = E Q ( Se X T X t K ) ] S + t = S [ = E Q ( Se X T X t K ) ] [ = SE Q ( + e X τ e κ) ], + where τ = T t (time-to-maturity) and κ = log(k /S) (log-moneyness). 3 Next, as it is done when deriving the B-S formula, E Q (e Xτ e κ ) + = E Q (e Xτ e κ )1 Xτ κ = E Q (e Xτ 1 Xτ κ) e κ Q(X τ κ) = Q (X τ κ) e κ Q(X τ κ), where Q (A) := E Q {1 A e Xt } if A F t (Esscher or Share measure);

21 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Prices of out-the-money Call Options 1 Conclusion: Π Lévy t (K, T, S) = SQ (X τ κ) Se κ Q(X τ κ) 2 Key result 2: Under Q, {X t } is again a Lévy process with triple (b, σ 2, s ): s (x) = e x s(x) and b = b + x 1 x (e x 1) s(x)dx + σ 2. 3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail distributions of Lévy processes (Key result 1).

22 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Prices of out-the-money Call Options 1 Conclusion: Π Lévy t (K, T, S) = SQ (X τ κ) Se κ Q(X τ κ) 2 Key result 2: Under Q, {X t } is again a Lévy process with triple (b, σ 2, s ): s (x) = e x s(x) and b = b + x 1 x (e x 1) s(x)dx + σ 2. 3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail distributions of Lévy processes (Key result 1).

23 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Prices of out-the-money Call Options 1 Conclusion: Π Lévy t (K, T, S) = SQ (X τ κ) Se κ Q(X τ κ) 2 Key result 2: Under Q, {X t } is again a Lévy process with triple (b, σ 2, s ): s (x) = e x s(x) and b = b + x 1 x (e x 1) s(x)dx + σ 2. 3 Hence, when κ > 0 (OTM), we can apply the asymptotic result for the tail distributions of Lévy processes (Key result 1).

24 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Near-expiration option prices for ELM Theorem (F-L & Forde, 2010) For κ := log K /S > 0 (OTM options), Π Lévy t (T, K, S) = τs (e x e κ ) + s(x)dx + τ 2 2 S[ d 2 e κ d 2 ] + O(τ 3 ), where τ = T t > 0, d 2 := d 2 (κ; b, σ 2, s), d 2 := d 2(κ; b, σ 2, s ), and. ( ( 2 ) 2 x d 2 (x; b, σ 2, s) = σ 2 s (x) + 2bs(x) s(u)du) + s(u)du x x/2 1 2 x x +2 s(u)s(y)dudy 2s(x) x y +2s(x) 1 2 x 1 2 x x x y (s(u) s(x))s(y)dudy. ys(y)dy 1 2 x< y <1

25 Close-to-expiration option prices for exponential Lévy models (ELM) Our main result Near-expiration option prices for ELM Theorem (F-L & Forde, 2010) For κ := log K /S > 0 (OTM options), Π Lévy t (T, K, S) = τs (e x e κ ) + s(x)dx + τ 2 2 S[ d 2 e κ d 2 ] + O(τ 3 ), where τ = T t > 0, d 2 := d 2 (κ; b, σ 2, s), d 2 := d 2(κ; b, σ 2, s ), and. ( ( 2 ) 2 x d 2 (x; b, σ 2, s) = σ 2 s (x) + 2bs(x) s(u)du) + s(u)du x x/2 1 2 x x +2 s(u)s(y)dudy 2s(x) x y +2s(x) 1 2 x 1 2 x x x y (s(u) s(x))s(y)dudy. ys(y)dy 1 2 x< y <1

26 Close-to-expiration implied volatility smile for ELM Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

27 Close-to-expiration implied volatility smile for ELM Small-time asymptotics for Implied Volatilities 1 ˆσ τ (k) is the implied volatility at log-moneyness κ and time-to-maturity τ under the exponential Lévy models; 2 First-order approximation for ˆσ t (κ): [Tankov (2009), F-L & Forde (2010)] [τ log(τ 1 )] 1 2 ˆστ (κ) κ / 2; (κ > 0, τ 0); Hence, (rescaled) implies volatility is V-shaped independent of s; 3 Correction term or Second-order approximation: 1 2 κ2 ˆσ τ 2 [ 1 (κ) = 1 + τ log( 1 τ ) V1 (τ, κ) + o( log 1 )] (τ 0), τ where, denoting a 0 (κ) := (ex e κ ) + s(x)dx, [ 4 πa 0 (κ)e κ/2 V 1 (τ, κ) = 1 log( 1 τ ) log κ [ log ( )] ] 3/2 1. τ

28 Close-to-expiration implied volatility smile for ELM Small-time asymptotics for Implied Volatilities 1 ˆσ τ (k) is the implied volatility at log-moneyness κ and time-to-maturity τ under the exponential Lévy models; 2 First-order approximation for ˆσ t (κ): [Tankov (2009), F-L & Forde (2010)] [τ log(τ 1 )] 1 2 ˆστ (κ) κ / 2; (κ > 0, τ 0); Hence, (rescaled) implies volatility is V-shaped independent of s; 3 Correction term or Second-order approximation: 1 2 κ2 ˆσ τ 2 [ 1 (κ) = 1 + τ log( 1 τ ) V1 (τ, κ) + o( log 1 )] (τ 0), τ where, denoting a 0 (κ) := (ex e κ ) + s(x)dx, [ 4 πa 0 (κ)e κ/2 V 1 (τ, κ) = 1 log( 1 τ ) log κ [ log ( )] ] 3/2 1. τ

29 Close-to-expiration implied volatility smile for ELM Small-time asymptotics for Implied Volatilities 1 ˆσ τ (k) is the implied volatility at log-moneyness κ and time-to-maturity τ under the exponential Lévy models; 2 First-order approximation for ˆσ t (κ): [Tankov (2009), F-L & Forde (2010)] [τ log(τ 1 )] 1 2 ˆστ (κ) κ / 2; (κ > 0, τ 0); Hence, (rescaled) implies volatility is V-shaped independent of s; 3 Correction term or Second-order approximation: 1 2 κ2 ˆσ τ 2 [ 1 (κ) = 1 + τ log( 1 τ ) V1 (τ, κ) + o( log 1 )] (τ 0), τ where, denoting a 0 (κ) := (ex e κ ) + s(x)dx, [ 4 πa 0 (κ)e κ/2 V 1 (τ, κ) = 1 log( 1 τ ) log κ [ log ( )] ] 3/2 1. τ

30 Numerical examples Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

31 Numerical examples Variance Gamma Model Variance Gamma Model Approximation of Implied Volatility with k= "True" implied volatility 1st order approx. 2nd order approx. 0.7 Implied Volatility Time to maturity (in Days) Figure: Term structure of implied volatility approximations for the Variance Gamma model (i.e. s(x) = α x e x/β+ 1 x>0 + α x e x /β 1 x<0 and σ = 0) with κ = 0.2.

32 Numerical examples CGMY model 0 Approximation of Implied Volatility for VG Term Structure of Relative Error Relative Error (! a! ) /! st, k=0.3 2nd, k=0.3 1st, k=0.2 2nd, k=0.2 1st, k=0.1 1st, k=0.1 Figure: Relative errors Variance Gamma Model ˆστ (κ) στ (κ) σ τ (κ) Time to Maturity (in Days) of the implied volatility approximations for the

33 Numerical examples CGMY model CGMY Model Approximation of Implied Volatility with k= "True" implied volatility 1st order approx. 2nd order approx. Implied Volatility Time to maturity (in Days) Figure: Term structure of implied volatility approximations for the CGMY model (i.e. s(x) = C e x/m 1 x Y +1 x>0 + C e x /G 1 x 1+Y x<0 and σ = 0) with κ = 0.2.

34 Time-changed Lévy Model 0.1 Approximation of Implied Volatility for the CGMY Term Structure of Relative Error Relative Error (! a! ) /! st, k=0.3 2nd, k=0.3 1st, k=0.2 2nd, k=0.2 1st, k=0.1 2nd, k=0.1 Figure: Relative errors CGMY model ˆστ (κ) στ (κ) σ τ (κ) Time to maturity (in Days) of the implied volatility approximations for the

35 Time-changed Lévy Model Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

36 Time-changed Lévy Model OTM option prices for time-changed Lévy models 1 Set-up: We assume that under Q, 2 Key features: S t = S 0 e Xt, X t = Z T (t), T (t) = }{{} Clock t 0 Y }{{} u du, Y Z ; Speed 1 Can exhibit jumps ("Sudden big" changes in the price level); 2 Log returns with heavy-tails and high-kurtosis distributions; 3 Volatility clustering (intermittency). 3 Theorem: [F-L & Forde (2010)] 1 t EQ (S t K ) + = S 0 EY 0 (e x e κ ) + s(x)dx S 0 EY0 2 [ d 2 (κ) e κ d 2 (κ) ] t + O(t 2 ), where κ := log K S 0 > 0 and d 2, d2 as before.

37 Time-changed Lévy Model OTM option prices for time-changed Lévy models 1 Set-up: We assume that under Q, 2 Key features: S t = S 0 e Xt, X t = Z T (t), T (t) = }{{} Clock t 0 Y }{{} u du, Y Z ; Speed 1 Can exhibit jumps ("Sudden big" changes in the price level); 2 Log returns with heavy-tails and high-kurtosis distributions; 3 Volatility clustering (intermittency). 3 Theorem: [F-L & Forde (2010)] 1 t EQ (S t K ) + = S 0 EY 0 (e x e κ ) + s(x)dx S 0 EY0 2 [ d 2 (κ) e κ d 2 (κ) ] t + O(t 2 ), where κ := log K S 0 > 0 and d 2, d2 as before.

38 Time-changed Lévy Model OTM option prices for time-changed Lévy models 1 Set-up: We assume that under Q, 2 Key features: S t = S 0 e Xt, X t = Z T (t), T (t) = }{{} Clock t 0 Y }{{} u du, Y Z ; Speed 1 Can exhibit jumps ("Sudden big" changes in the price level); 2 Log returns with heavy-tails and high-kurtosis distributions; 3 Volatility clustering (intermittency). 3 Theorem: [F-L & Forde (2010)] 1 t EQ (S t K ) + = S 0 EY 0 (e x e κ ) + s(x)dx S 0 EY0 2 [ d 2 (κ) e κ d 2 (κ) ] t + O(t 2 ), where κ := log K S 0 > 0 and d 2, d2 as before.

39 Time-changed Lévy Model OTM option prices for time-changed Lévy models 1 Set-up: We assume that under Q, 2 Key features: S t = S 0 e Xt, X t = Z T (t), T (t) = }{{} Clock t 0 Y }{{} u du, Y Z ; Speed 1 Can exhibit jumps ("Sudden big" changes in the price level); 2 Log returns with heavy-tails and high-kurtosis distributions; 3 Volatility clustering (intermittency). 3 Theorem: [F-L & Forde (2010)] 1 t EQ (S t K ) + = S 0 EY 0 (e x e κ ) + s(x)dx S 0 EY0 2 [ d 2 (κ) e κ d 2 (κ) ] t + O(t 2 ), where κ := log K S 0 > 0 and d 2, d2 as before.

40 Open problems and future directions Outline 1 Motivation Empirical features of implied volatility Close-to-expiration smile for continuous models 2 Close-to-expiration option prices for exponential Lévy models (ELM) 3 Close-to-expiration implied volatility smile for ELM 4 Numerical examples 5 Time-changed Lévy Model 6 Open problems and future directions

41 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

42 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

43 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

44 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

45 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

46 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

47 Open problems and future directions Open problems and future directions 1 The numerical results show that the second order significantly improves the first order approximation for relatively large values of κ (say, κ.2); 2 For κ > 0.2, it seems that τ has to be extremely small for the approximations to work well; 3 How to improve the performance of the approximations? 4 Two possible approaches to this problem: 1 Look for third or upper order approximations (see Gao and Lee (2011)). 2 Note that V 0,α := 1 2 κ2 /(τ log(α/τ)) also satisfies στ 2 (κ) V 0,α (τ, κ), (τ 0). Why to pick α = 1? Choose α to minimize an approximation of the relative error of V 0,α. 5 Devise efficient methods to compute the 2nd term in the option price approximation.

48 Appendix Bibliography For Further Reading I Figueroa-Lopez & Houdré. Small-time expansions for the transition distributions of Lévy processes. Stochastic Processes and Their Applications, 119: , Figueroa-López and Forde. The small-maturity smile for exponential Lévy models Preprint, Figueroa-López, Gong, and Houdré. Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jump Preprint, 2010.

49 Appendix Exponential Lévy model From Black-Scholes to Exponential Lévy Model 1 Key assumptions of the Black-Scholes model: (i) The (log) return on the asset over a time period [t, t + h], i.e. R t,t+h := log S t+h S t = log S t+h log S t, is Gaussian with mean µh and variance σ 2 h (independent of t); (ii) Log returns on disjoint time periods are mutually independent; (iii) The price path t S t is continuous. 2 Key assumptions of an Exponential Lévy Model (ELM): (i ) The (log) return on the asset over a time period [t, t + h] has a common distribution F h (independent of t); (ii ) Log returns on disjoint time periods are mutually independent; (iii ) The paths t S t exhibit only discontinuities of first kind. 3 In that case, the "log return process" X t := log S t /S 0 is a Lévy process;

50 Appendix Exponential Lévy model From Black-Scholes to Exponential Lévy Model 1 Key assumptions of the Black-Scholes model: (i) The (log) return on the asset over a time period [t, t + h], i.e. R t,t+h := log S t+h S t = log S t+h log S t, is Gaussian with mean µh and variance σ 2 h (independent of t); (ii) Log returns on disjoint time periods are mutually independent; (iii) The price path t S t is continuous. 2 Key assumptions of an Exponential Lévy Model (ELM): (i ) The (log) return on the asset over a time period [t, t + h] has a common distribution F h (independent of t); (ii ) Log returns on disjoint time periods are mutually independent; (iii ) The paths t S t exhibit only discontinuities of first kind. 3 In that case, the "log return process" X t := log S t /S 0 is a Lévy process;

51 Appendix Exponential Lévy model From Black-Scholes to Exponential Lévy Model 1 Key assumptions of the Black-Scholes model: (i) The (log) return on the asset over a time period [t, t + h], i.e. R t,t+h := log S t+h S t = log S t+h log S t, is Gaussian with mean µh and variance σ 2 h (independent of t); (ii) Log returns on disjoint time periods are mutually independent; (iii) The price path t S t is continuous. 2 Key assumptions of an Exponential Lévy Model (ELM): (i ) The (log) return on the asset over a time period [t, t + h] has a common distribution F h (independent of t); (ii ) Log returns on disjoint time periods are mutually independent; (iii ) The paths t S t exhibit only discontinuities of first kind. 3 In that case, the "log return process" X t := log S t /S 0 is a Lévy process;

52 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back

53 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back

54 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back

55 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back

56 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back

57 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back

58 Appendix Exponential Lévy model Distributional properties and parameters 1 Two important features of a Lévy process t X t : X X 0,..., X n X (n 1) are independent identically distributed; Law of {X t} t 0 is uniquely determined by the distribution of X 1. 2 Fundamental examples: Wiener process {W t} t 0 : W 1 N (0, 1) Poisson process {N t} t 0 : Compound Poisson: N 1 Poisson(λ) X t = N t i=1 ξ i, where ξ i i.i.d. p( ) 3 Parameters (σ, γ, ν): Volatility σ, "Drift" γ, and measure ν on R\{0}: X t = γt + σw t + Pure jump process", Number of jumps of magnitude between a and b, occurring before time t, is Poisson with intensity tν([a, b]). 4 Typically, ν(dx) = s(x)dx, where s : R\{0} R + is called the Lévy density. Intuitively: s( ) is a "jump intensity function". Back