Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior

Transcription

1 Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior Ehsan Azmoodeh University of Vaasa Finland 7th General AMaMeF and Swissquote Conference September 7 1, 215

2 Outline 1 Financial motivation Robust pricing models A no-arbitrage and robust-hedging result 2 Estimation Discrete observation Realized quadratic variation Continuous observation Randomized periodogram: semimartingale setup Randomized periodogram: non-semimartingale setup 3 Asymptotic behavior Malliavin calculus & The 4th moment theorem Main results: asymptotic normality Main results: the Berry Esseen bound 4 References

3 Robust pricing models We introduce a class of pricing models that is somehow "invariant" to the Black Scholes model pricing model. We say (Ω, F, (S t ), (F t ), IP) is in the model class M σ if 1 S takes values in C s,+,

4 Robust pricing models We introduce a class of pricing models that is somehow "invariant" to the Black Scholes model pricing model. We say (Ω, F, (S t ), (F t ), IP) is in the model class M σ if 1 S takes values in C s,+, 2 the pathwise quadratic variation S of S is of the form d S t = σ 2 S 2 t dt,

5 Robust pricing models We introduce a class of pricing models that is somehow "invariant" to the Black Scholes model pricing model. We say (Ω, F, (S t ), (F t ), IP) is in the model class M σ if 1 S takes values in C s,+, 2 the pathwise quadratic variation S of S is of the form d S t = σ 2 S 2 t dt, 3 for all ε > and η C s,+ we have the small ball property IP [ S η < ε] >.

6 Robust pricing models M σ contains non-semimartingale models. So, we cannot use stochastic integration theory for semimartingale. However, the forward integral is economically meaningful: t IP a.s. Φ r ds r = lim n t k πn t k t Φ tk 1 ( Stk S tk 1 ).

7 Robust pricing models M σ contains non-semimartingale models. So, we cannot use stochastic integration theory for semimartingale. However, the forward integral is economically meaningful: t IP a.s. Φ r ds r = lim n t k πn t k t Φ tk 1 ( Stk S tk 1 ). Even in the classical Black Scholes model one restricts to admissible strategies to exclude arbitrage. We shall restrict the admissible strategies a little more in a careful and elegant way that still many interesting options can be hedged.

8 A no-arbitrage and robust-hedging result Theorem (Bender, Sottinen & Valkeila - 28) There is no arbitrage with "allowed" strategies.

9 A no-arbitrage and robust-hedging result Theorem (Bender, Sottinen & Valkeila - 28) There is no arbitrage with "allowed" strategies. Theorem (Bender, Sottinen & Valkeila - 28) Suppose a continuous option G : C s,+ IR. If G(S) can be hedged in one model S M σ with an allowed strategy then G(S) can be hedged in any model S M σ.

10 A no-arbitrage and robust-hedging result Theorem (Bender, Sottinen & Valkeila - 28) There is no arbitrage with "allowed" strategies. Theorem (Bender, Sottinen & Valkeila - 28) Suppose a continuous option G : C s,+ IR. If G(S) can be hedged in one model S M σ with an allowed strategy then G(S) can be hedged in any model S M σ. Conclusion: In the Black Scholes model hedges for European, Asian, and lookback-options can be constructed by using the Black Scholes partial differential equation. These hedges hold for any model that is continuous, satisfies the small ball property, and has the same quadratic variation as the Black Scholes model.

11 Discrete observation Let X be a semimartingale. Then, it is well-known that: ( ) 2 [X, X] T = IP- lim Xtk X tk 1, π t k π where π = {t k : = t < t 1 < < t n = T } is a partition of the interval [, T ], π = max {t k t k 1 : t k π}.

12 Discrete observation Let X be a semimartingale. Then, it is well-known that: ( ) 2 [X, X] T = IP- lim Xtk X tk 1, π t k π where π = {t k : = t < t 1 < < t n = T } is a partition of the interval [, T ], π = max {t k t k 1 : t k π}. Statistically speaking, the sums of squared increments (realized quadratic variation) is a consistent estimator for the bracket as the volume of observations tends to infinity.

13 Discrete observation Let X be a semimartingale. Then, it is well-known that: ( ) 2 [X, X] T = IP- lim Xtk X tk 1, π t k π where π = {t k : = t < t 1 < < t n = T } is a partition of the interval [, T ], π = max {t k t k 1 : t k π}. Statistically speaking, the sums of squared increments (realized quadratic variation) is a consistent estimator for the bracket as the volume of observations tends to infinity. Barndorff-Nielsen & Shephard (22) studied precision of the realized quadratic variation estimator for a special class of continuous semimartingales. They showed that sometimes the realized quadratic variation estimator can be rather noisy estimator.

14 Discrete observation Let X be a semimartingale. Then, it is well-known that: ( ) 2 [X, X] T = IP- lim Xtk X tk 1, π t k π where π = {t k : = t < t 1 < < t n = T } is a partition of the interval [, T ], π = max {t k t k 1 : t k π}. Statistically speaking, the sums of squared increments (realized quadratic variation) is a consistent estimator for the bracket as the volume of observations tends to infinity. Barndorff-Nielsen & Shephard (22) studied precision of the realized quadratic variation estimator for a special class of continuous semimartingales. They showed that sometimes the realized quadratic variation estimator can be rather noisy estimator. Although the consistency result does not depend on a specific choice of the sampling scheme, the asymptotic distribution does Jacod (1994), Barndorff-Nielsen & Shephard (22), Fukasawa (21), Hayashi, Jacod & Yoshida (211), and the recent book by Aït-Sahalia & Jacod (214).

15 Continuous observation Dzhaparidze and Spreij [4] suggested another estimator of the bracket [X, X] using the spectral characterization. Let IF X be the filtration of X, and τ be a finite stopping time.

16 Continuous observation Dzhaparidze and Spreij [4] suggested another estimator of the bracket [X, X] using the spectral characterization. Let IF X be the filtration of X, and τ be a finite stopping time. For λ IR, define the periodogram I τ (X; λ) of X at τ I τ (X; λ) : = τ = 2 Re e iλs dx s 2 τ t e iλ(t s) dx sdx t + [X, X] τ (by Itô formula).

17 Continuous observation Dzhaparidze and Spreij [4] suggested another estimator of the bracket [X, X] using the spectral characterization. Let IF X be the filtration of X, and τ be a finite stopping time. For λ IR, define the periodogram I τ (X; λ) of X at τ I τ (X; λ) : = τ = 2 Re e iλs dx s 2 τ t e iλ(t s) dx sdx t + [X, X] τ (by Itô formula). Now, the idea is to randomize λ. To this end, let ξ be a symmetric random variable independent of IF X with a density g ξ, and a real-valued characteristic function ϕ ξ. For given L >, define the randomized periodogram IE ξ I τ (X; Lξ) = I τ (X; Lx) g ξ (x)dx. IR

18 IE ξ I τ (X; Lξ) IP [X, X] τ. Continuous observation Dzhaparidze and Spreij [4] suggested another estimator of the bracket [X, X] using the spectral characterization. Let IF X be the filtration of X, and τ be a finite stopping time. For λ IR, define the periodogram I τ (X; λ) of X at τ I τ (X; λ) : = τ = 2 Re e iλs dx s 2 τ t e iλ(t s) dx sdx t + [X, X] τ (by Itô formula). Now, the idea is to randomize λ. To this end, let ξ be a symmetric random variable independent of IF X with a density g ξ, and a real-valued characteristic function ϕ ξ. For given L >, define the randomized periodogram IE ξ I τ (X; Lξ) = I τ (X; Lx) g ξ (x)dx. Theorem (Dzhaparidze & Spreij ) If the characteristic function ϕ ξ is of bounded variation, then IR

19 Continuous observation Consider mixed Brownian-fractional Brownian motion X: X t = W t + B H t t [, T ], W is a standard Brownian motion & B H is a fractional Brownian motion with Hurst parameter H ( 1, 1), independent of W. 2

20 Continuous observation Consider mixed Brownian-fractional Brownian motion X: X t = W t + B H t t [, T ], W is a standard Brownian motion & B H is a fractional Brownian motion with Hurst parameter H ( 1, 1), independent of W. 2 Cheridito (21) showed that the process X is a IF X semimartingale, if H ( 3, 1), and for H ( 1, ] , X is not a semimartingale with respect to its own filtration IF X.

21 Continuous observation Consider mixed Brownian-fractional Brownian motion X: X t = W t + B H t t [, T ], W is a standard Brownian motion & B H is a fractional Brownian motion with Hurst parameter H ( 1, 1), independent of W. 2 Cheridito (21) showed that the process X is a IF X semimartingale, if H ( 3, 1), and for H ( 1, ] , X is not a semimartingale with respect to its own filtration IF X. In both cases, since fbm has zero quadratic variation for H > 1/2, we have ( ) 2 [X, X] t = IP- lim Xtk X tk 1 = t, t [, T ]. π t k π If the partitions are nested, i.e. π (n) π (n+1), then the convergence can be strengthened to almost sure convergence. Hereafter, we always assume that the sequence of partitions are nested.

22 Continuous observation Similarly as in semimartingale setup, for mixed BfBm, we define the randomized periodogram as IE ξ I T (X; Lξ) = I T (X; Lx) g ξ (x)dx. IR

23 Continuous observation Similarly as in semimartingale setup, for mixed BfBm, we define the randomized periodogram as IE ξ I T (X; Lξ) = I T (X; Lx) g ξ (x)dx. IR Note that in general we are in non-semimartingale world, so one has to take care of stochastic integrals. Here, for any given λ IR, we define the periodogram of X at deterministic time T I T (X; λ) = e iλt dx t 2 = e iλt X T iλ X te iλt dt 2.

24 Continuous observation Similarly as in semimartingale setup, for mixed BfBm, we define the randomized periodogram as IE ξ I T (X; Lξ) = I T (X; Lx) g ξ (x)dx. IR Note that in general we are in non-semimartingale world, so one has to take care of stochastic integrals. Here, for any given λ IR, we define the periodogram of X at deterministic time T I T (X; λ) = Theorem (A & Valkeila - 213) e iλt dx t 2 = e iλt X T iλ X te iλt dt 2. Assume that X is a mixed Brownian-fractional Brownian motion, and IEξ 2 <. Then, IP IE ξ I T (X; Lξ) [X, X] T.

25 Continuous observation Similarly as in semimartingale setup, for mixed BfBm, we define the randomized periodogram as IE ξ I T (X; Lξ) = I T (X; Lx) g ξ (x)dx. IR Note that in general we are in non-semimartingale world, so one has to take care of stochastic integrals. Here, for any given λ IR, we define the periodogram of X at deterministic time T I T (X; λ) = Theorem (A & Valkeila - 213) e iλt dx t 2 = e iλt X T iλ X te iλt dt 2. Assume that X is a mixed Brownian-fractional Brownian motion, and IEξ 2 <. Then, IP IE ξ I T (X; Lξ) [X, X] T. Here, the main tools are: (1) a version of stochastic Fubini s theorem for fbm (2) using generalized Lebesgue Stieltjes integration theory of Zähle (1998) and developed further by Nualart & Rascanu (22) that makes stochastic

26 Malliavin calculus & The 4th moment theorem Let X = {X t} t [,T ] be a "nice" Gaussian process. The Wiener Itô chaotic decomposition tells us that for any F L 2 (σ(x t; t [, T ])) F = IE(F) + I p(f p), p 1 I p(f p) = f p(t 1,, t p)dx t1 dx tp, [,T ] p

27 Malliavin calculus & The 4th moment theorem Let X = {X t} t [,T ] be a "nice" Gaussian process. The Wiener Itô chaotic decomposition tells us that for any F L 2 (σ(x t; t [, T ])) F = IE(F) + I p(f p), p 1 I p(f p) = f p(t 1,, t p)dx t1 dx tp, [,T ] p The vector space H p generated by the elements I p(f ) is called the pth Wiener chaos associated to X. i.e. L 2 (Ω, σ(x t; t [, T ]), IP) = IR H 1 H 2 H p.

28 Malliavin calculus & The 4th moment theorem Let X = {X t} t [,T ] be a "nice" Gaussian process. The Wiener Itô chaotic decomposition tells us that for any F L 2 (σ(x t; t [, T ])) F = IE(F) + I p(f p), p 1 I p(f p) = f p(t 1,, t p)dx t1 dx tp, [,T ] p The vector space H p generated by the elements I p(f ) is called the pth Wiener chaos associated to X. i.e. L 2 (Ω, σ(x t; t [, T ]), IP) = IR H 1 H 2 H p. Theorem (Nualart-Peccati - 25) Let {F n} n 1 = {I q(f n)} n 1 be a sequence of random variables in the qth Wiener chaos, q 2, such that IE(F 2 n ) σ 2. Then, the following asymptotic statements are equivalent: (i) F n converges in law to N (, σ 2 ). (ii) IE(F n) 4 3σ 4 = IE(N (, σ 2 )) 4, (κ 4 (F n) ). (iii) DF n 2 H converges in L 2 to q σ 2, (Var DF n 2 H ). (iv) f n r f n (2q 2r) for all r = 1,, q 1.

29 Main results: asymptotic normality Theorem Assume that ϕ ξ L 1 (IR). Then, as L, we have the following asymptotic statements: 1 if H ( 3 4, 1), then L (IE ξ I T (X; L ξ) [X, X] T ) law N (, σ 2 T ).

30 Main results: asymptotic normality Theorem Assume that ϕ ξ L 1 (IR). Then, as L, we have the following asymptotic statements: 1 if H ( 3 4, 1), then L (IE ξ I T (X; L ξ) [X, X] T ) law N (, σ 2 T ). 2 if H = 3, then 4 ( ) law L IE ξ I T (X; L ξ) [X, X] T N (µ, σt 2 ), where µ = 2α H T ϕ ξ (x)x 2H 2 dx, and α H = H(2H 1).

31 Main results: asymptotic normality Theorem Assume that ϕ ξ L 1 (IR). Then, as L, we have the following asymptotic statements: 1 if H ( 3 4, 1), then L (IE ξ I T (X; L ξ) [X, X] T ) law N (, σ 2 T ). 2 if H = 3, then 4 ( ) law L IE ξ I T (X; L ξ) [X, X] T N (µ, σt 2 ), where µ = 2α H T ϕ ξ (x)x 2H 2 dx, and α H = H(2H 1). 3 if H ( 1 2, 3 4 ), then L 2H 1( IEI T (X; L ξ) [X, X] T ) IP µ. Notice that when H ( 1, ) 3 2 4, we have 2H 1 < 1. 2

32 Main results: asymptotic normality Sketch of the proof: Using the generalized Lebesgue-Stieltjes integration theory to obtain t IE ξ I T (X; L ξ) = [X, X] T + 2 ϕ ξ (L(t s))dx s dx t (1) } {{ } :=u t where the iterated stochastic integral is pathwise, as limit of Riemann-Stieltjes sums.

33 Main results: asymptotic normality Sketch of the proof: Using the generalized Lebesgue-Stieltjes integration theory to obtain t IE ξ I T (X; L ξ) = [X, X] T + 2 ϕ ξ (L(t s))dx s dx t (1) } {{ } :=u t where the iterated stochastic integral is pathwise, as limit of Riemann-Stieltjes sums. Using the link between pathwise and the Skorokhod integrals to obtain u tdx t = u tδx t + α H D (BH ) s u t t s 2H 2 dsdt, where δx stands for the Skorokhod integral w.r.t mixed BfBm X, and D (BH ) denote the Malliavin derivative with respect to B H.

34 Main results: asymptotic normality Set ψ L (s, t) := ϕ ξ (L t s ), we have: IEI T (X; Lξ) [X, X] T = I2 X (ψ L ) + α H ϕ ξ (L t s ) t s 2H 2 dsdt.

35 Main results: asymptotic normality Set ψ L (s, t) := ϕ ξ (L t s ), we have: IEI T (X; Lξ) [X, X] T = I2 X (ψ L ) + α H To obtain the correct rate of convergence, observe that IE(I 2 X (ψ L )) 2 = 2 ψ L 2 H 2 1 L. ϕ ξ (L t s ) t s 2H 2 dsdt.

36 Main results: asymptotic normality Set ψ L (s, t) := ϕ ξ (L t s ), we have: IEI T (X; Lξ) [X, X] T = I2 X (ψ L ) + α H To obtain the correct rate of convergence, observe that IE(I 2 X (ψ L )) 2 = 2 ψ L 2 H 2 1 L. ϕ ξ (L t s ) t s 2H 2 dsdt. Now, using 4th moment criterion to infer that for any H > 1/2: L I X 2 (ψ L ) law N (, σt 2 ), σt 2 := 2 T ϕ 2 ξ(x)dx.

37 Main results: asymptotic normality Set ψ L (s, t) := ϕ ξ (L t s ), we have: IEI T (X; Lξ) [X, X] T = I2 X (ψ L ) + α H To obtain the correct rate of convergence, observe that IE(I 2 X (ψ L )) 2 = 2 ψ L 2 H 2 1 L. ϕ ξ (L t s ) t s 2H 2 dsdt. Now, using 4th moment criterion to infer that for any H > 1/2: L I X 2 (ψ L ) law N (, σt 2 ), σt 2 := 2 T ϕ 2 ξ(x)dx. And for the deterministic correction term we have if H > 3/4, L αh ϕ ξ (L t s ) t s 2H 2 dsdt µ if H = 3/4, + if H < 3/4, where µ = 2α H T ϕ ξ (x)x 2H 2 dx.

38 Main results: the Berry Esseen bound The following general Berry-Esseen type estimate is obtained by combining the Stein s method for normal approximation with Malliavin calculus. Theorem Let {F n} n 1 be a sequence of elements in the second Wiener chaos such that IE(F 2 n ) σ 2 and Var DF n 2 H. Then, F n law Z N (, σ 2 ) and sup IP(F n < x) IP(Z < x) 2 Var DF n 2 IE(F n 2 H ) + 2 IE(F n 2 ) σ 2 max{ie(f n 2 ), σ 2 }. x IR

39 Main results: the Berry Esseen bound Using this result, in the semimartingale case we obtain: Proposition Let H ( 3, 1), and Z N (, σ 2 4 T ). Then there exists a constant C (independent of L) such that for sufficiently large L, we have ( ( ) ) sup IP L IEξ (I T (X; L ξ) [X, X] T < x IP (Z < x) Cρ(L) x IR where { ρ(l) = max L 2 3 2H, L } ϕ 2 ξ(tz)dz.

40 Main results: the Berry Esseen bound Using this result, in the semimartingale case we obtain: Proposition Let H ( 3, 1), and Z N (, σ 2 4 T ). Then there exists a constant C (independent of L) such that for sufficiently large L, we have ( ( ) ) sup IP L IEξ (I T (X; L ξ) [X, X] T < x IP (Z < x) Cρ(L) x IR where { ρ(l) = max L 2 3 2H, L } ϕ 2 ξ(tz)dz. In many cases of interest the leading term in ρ(l) is the polynomial term L 3 2 2H. For example, if ϕ ξ admits an exponential decay, i.e. ϕ ξ (t) C 1 e C 2t. Example: if ξ is a standard normal random variable with ϕ ξ (t) = e t2 2 is a standard Cauchy random variable with ϕ ξ (t) = e t. or if ξ

41 Main results: the Berry Esseen bound Thank you for your attention!

42 Azmoodeh, E., Sottinen, T., Viitasaari, L. (215) Asymptotic normality of randomized periodogram for estimating quadratic variation. Modern Stochastics: Theory and Applications. 2(1), Azmoodeh, E., Valkeila, E. (213) Characterization of the quadratic variation of mixed Brownian fractional Brownian motion. Stat. Inference Stoch. Process. 16 (2), Bender, C., Sottinen, T., Valkeila, E. (28) Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12 (4), Dzhaparidze K, Spreij P. (1994) Spectral characterization of the optional quadratic variation processes. Stoch Procces Appl. 54: