Short-Time Asymptotic Methods In Financial Mathematics

Transcription

1 Short-Time Asymptotic Methods In Financial Mathematics José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis School Of Mathematics, UMN March 14, 2019 Based on joint works with R. Gong, C. Houdré, C. Mancini, J. Nisen, and S. Ólafsson Research supported by NSF grants: DMS & DMS J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 1 / 45

2 Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 2 / 45

3 Overview and Applications Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 3 / 45

4 Overview and Applications General Formulation Of Problem 1 Consider a stochastic dynamic system evolving continuously in time; 2 We are interested in a variable of the system, whose time-t value is denoted by X t ; suppose that the paths t X t are right continuous with left limits (cádlág); denote the path of X on [0, t] by X [0,t] ; 3 Consider a functional F : D([0, t)) R on the space of cádlág function on [0, t]; 4 We want to study the asymptotic behavior of the expected value of F(X [0,t] ) when t 0; 5 Examples: E[f (X t )] }{{} Simple European Option E[X t inf X s], E 0 s t }{{} Drawup Process [ f ( 1 t t 0 X s ds )] } {{ } Asian Type Option, J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 4 / 45

5 Overview and Applications Applications of Short-Time Asymptotics Methods I II III IV V In Econometrics: Statistical Inference of continuous-time processes based on high-frequency (HF) sampling data. In Engineering: Optimal Change- or Breakpoint-Point detection in continuous-time. In Finance: To study option prices near expiration, as a numerical approximation, model selection, and/or calibration tool. Characterization of Optimal Trading Strategies in market based on Limit Order Book trading. Numerical Solutions and Simulation of Stochastic Dynamical Systems. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 5 / 45

6 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 6 / 45

7 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data HF Estimation of Integrated Variance In econometrics, an important problem is the estimation of the integrated variance, for a semimartingale model IV T := T dx t = a t dt + σ t dw t + dj t, 0 σ 2 s ds, (fixed T ), (W Standard Brownian Motion, J pure-jump process), based on a discrete record X t1,..., X tn of observations of X when mesh := max(t i t i 1 ) 0 i (high-frequency or infill estimation). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 7 / 45

8 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Key Estimators 1 Realized Quadratic Variation: n 1 ( QV n := Xti+1 X ) 2 mesh 0 t i i=0 T 0 σ 2 s ds + t T ( J t ) 2 2 Bipower Realized Variations (Barndorff-Nielsen and Shephard): n 2 BPV n := X ti+1 X ti X ti+2 X ti+1 mesh 0 i=0 2 π T 0 σ 2 s ds 3 Truncated Realized Variations (Mancini): n 1 ( TRV n (ε n ) := X ) 2 mesh 0 Xti+1 t i 1{ Xti+1 Xti ε n} i=0 T 0 σ 2 s ds J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 8 / 45

9 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Some Questions What are the infill asymptotic properties of the estimators? Asymptotic bias, variance, CLT, etc... Is it better to use BPV or TRV? How do you calibrate or tune up the truncation parameter ε n? How do you optimally estimate the spot volatility σ t? How do you account for potential observation errors (called microstructure noise)? J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 9 / 45

10 Overview and Applications Change-Point Detection In Continuous-Time Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 10 / 45

11 Overview and Applications Change-Point Detection In Continuous-Time Change-Point Detection Change-point detection is the problem of detecting, with as little delay as possible, a change in the statistical properties of a process that is being observed discretely or continuously in time. First began to emerge in quality control applications in the 1930 s Applications in various branches of science and engineering: Signal processing, statistical surveillance, climate monitoring, cybersecurity,... Applications in finance: (i) Active risk management (e.g., shifts in the parameters of credit risk models or the expected performance of an investment) (ii) Actuarial science (e.g., changes in mortality rates); (iii) Algorithmic trading strategies. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 11 / 45

12 Overview and Applications Change-Point Detection In Continuous-Time Lorden s Change-Point Detection Problem Sequentially observe a process (X t ) t T, whose statistical properties change abruptly at some unknown nonrandom point in time τ T { }. A sequential detection procedure is a stopping time T, w.r.t. the observed process (X t ) t T, at which a change" is declared. The design of an optimal detection procedure aims at minimizing the detection delay (T τ)1 {T >τ}, while controlling the false alarm rate" (P [T < t], t): inf sup ess sup E τ [(T τ) + F τ ] T : E [T ] γ ω τ T (Lorden s worst-worst case approach) where E τ (resp., E ) is the expectation given that the change happens at time τ (resp., never happens). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 12 / 45

13 Overview and Applications Change-Point Detection In Continuous-Time Change-Point Detection In Discrete Time Moustakides, 86: When X 1, X 2,... are i.i.d. with densities f and g before and after the change point τ, the optimal stopping time is the CUSUM Rule: t T h = inf{t : y t h}, y t := u t inf u s, u t := log g(x i) s t f (X }{{} i ) i=0 }{{} Drawup Process Log-Likelihood Ratio Process J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 13 / 45

14 Overview and Applications Change-Point Detection In Continuous-Time Change-Point Detection In Continuous Time By shifting Lorden s optimality criterion from E τ [(T τ) + F τ ] to [ ] E τ (ut u τ )1 {T τ} Fτ, optimality of the CUSUM rule in continuous time has been established for arbitrary processes with continuous paths (Chronopoulou & Fellouris 13) J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 14 / 45

15 Overview and Applications Change-Point Detection In Continuous-Time Change-Point Detection In Continuous Time By shifting Lorden s optimality criterion from E τ [(T τ) + F τ ] to [ ] E τ (ut u τ )1 {T τ} Fτ, optimality of the CUSUM rule in continuous time has been established for arbitrary processes with continuous paths (Chronopoulou & Fellouris 13) Results for processes with discontinuities are rare, except in special cases; e.g., changes in Cox processes (El Karoui et al. 15) J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 14 / 45

16 Overview and Applications Change-Point Detection In Continuous-Time Change-Point Detection In Continuous Time By shifting Lorden s optimality criterion from E τ [(T τ) + F τ ] to E τ [ (ut u τ )1 {T τ} Fτ ], optimality of the CUSUM rule in continuous time has been established for arbitrary processes with continuous paths (Chronopoulou & Fellouris 13) Results for processes with discontinuities are rare, except in special cases; e.g., changes in Cox processes (El Karoui et al. 15) F-L & Ólafsson 19 AAP: The CUSUM stopping time Th c is optimal to detect the change point τ from one Lévy process to another: dx t = dx 0 t 1 {t<τ} + dx 1 t 1 {t τ}, where (X 0 t ) t 0 and (X 1 t ) t 0 are two Lévy process with different Lévy triplets so that their laws are mutually equivalent. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 14 / 45

17 Overview and Applications Change-Point Detection In Continuous-Time Approach of Proof The proof of the theorem consists of two main steps and uses again short-time asymptotic methods: 1 For > 0 show that a discretized version of the CUSUM stopping time, Th c ( ) := inf{k 0 : y k h}, solves a hybrid problem where the change-point τ is restricted to values in the discrete set (k ) k 0. 2 Let 0 and (i) show that Th c ( ) Th c, (ii) use a limiting procedure to establish the optimality of Th c for the continuous-time change-detection problem. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 15 / 45

18 Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 16 / 45

19 Motivation Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 17 / 45

20 Motivation Why short-time asymptotics of option prices? New trading opportunities in short-term option market: weekly options. Important analytical tools for Model Selection and Calibration. Source of interesting probabilistic problems. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 18 / 45

21 Background Information Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 19 / 45

22 Background Information The Financial Market We consider a continuous-time market consisting of three types of assets: J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 20 / 45

23 Background Information The Financial Market We consider a continuous-time market consisting of three types of assets: Risk-free asset: Provides a secure" rate of return r > 0; that is, the price of the asset at a future time t (0, ) is known to be A t = e rt. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 20 / 45

24 Background Information The Financial Market We consider a continuous-time market consisting of three types of assets: Risk-free asset: Provides a secure" rate of return r > 0; that is, the price of the asset at a future time t (0, ) is known to be A t = e rt. Risky asset (such as a stock) whose price per shares at a future time t (0, ) is assumed random and denoted S t. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 20 / 45

25 Background Information The Financial Market We consider a continuous-time market consisting of three types of assets: Risk-free asset: Provides a secure" rate of return r > 0; that is, the price of the asset at a future time t (0, ) is known to be A t = e rt. Risky asset (such as a stock) whose price per shares at a future time t (0, ) is assumed random and denoted S t. Contingent claims written on S: J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 20 / 45

26 Background Information The Financial Market We consider a continuous-time market consisting of three types of assets: Risk-free asset: Provides a secure" rate of return r > 0; that is, the price of the asset at a future time t (0, ) is known to be A t = e rt. Risky asset (such as a stock) whose price per shares at a future time t (0, ) is assumed random and denoted S t. Contingent claims written on S: A European contingent claim written on the asset S is a contract in which the buyer/holder receives the payoff X = Φ(S t ) at time t (Φ : R + R + ) and, in exchange, pays a premium Π 0 (t; X ) at time 0 to the seller/writer. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 20 / 45

27 Background Information The Financial Market We consider a continuous-time market consisting of three types of assets: Risk-free asset: Provides a secure" rate of return r > 0; that is, the price of the asset at a future time t (0, ) is known to be A t = e rt. Risky asset (such as a stock) whose price per shares at a future time t (0, ) is assumed random and denoted S t. Contingent claims written on S: A European contingent claim written on the asset S is a contract in which the buyer/holder receives the payoff X = Φ(S t ) at time t (Φ : R + R + ) and, in exchange, pays a premium Π 0 (t; X ) at time 0 to the seller/writer. Π 0 (t; X ) is called the price of the claim X at time 0. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 20 / 45

28 Background Information Examples of Claims (Prepaid) Forward contracts: X = S t J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 21 / 45

29 Background Information Examples of Claims (Prepaid) Forward contracts: X = S t Call Option: X = (S t K ) + = max{s t K, 0} J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 21 / 45

30 Background Information Examples of Claims (Prepaid) Forward contracts: X = S t Call Option: X = (S t K ) + = max{s t K, 0} K > S 0 Out-Of-The-Money (OTM); K = S 0 At-The-Money (ATM); K < S 0 In-The-Money (ITM) J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 21 / 45

31 Background Information Examples of Claims (Prepaid) Forward contracts: X = S t Call Option: X = (S t K ) + = max{s t K, 0} K > S 0 Out-Of-The-Money (OTM); K = S 0 At-The-Money (ATM); K < S 0 In-The-Money (ITM) Put Option: X = (K S t ) + = max{k S t, 0} J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 21 / 45

32 Background Information Risk-neutral Option Prices The only possible sensible price of a prepaid forward contract is S 0 : Π 0 (S t ) = S 0. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 22 / 45

33 Background Information Risk-neutral Option Prices The only possible sensible price of a prepaid forward contract is S 0 : Π 0 (S t ) = S 0. A risk-neutral pricing policy states that contingent claims ought to be priced as the expected present-values of their payoffs, but in such a way that (prepaid) forward contracts written on the asset are correctly priced: J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 22 / 45

34 Background Information Risk-neutral Option Prices The only possible sensible price of a prepaid forward contract is S 0 : Π 0 (S t ) = S 0. A risk-neutral pricing policy states that contingent claims ought to be priced as the expected present-values of their payoffs, but in such a way that (prepaid) forward contracts written on the asset are correctly priced: Π 0 (t; Φ(S t )) = e rt E [Φ(S t )] = e rt Φ(x) f t (x)dx, 0 J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 22 / 45

35 Background Information Risk-neutral Option Prices The only possible sensible price of a prepaid forward contract is S 0 : Π 0 (S t ) = S 0. A risk-neutral pricing policy states that contingent claims ought to be priced as the expected present-values of their payoffs, but in such a way that (prepaid) forward contracts written on the asset are correctly priced: Π 0 (t; Φ(S t )) = e rt E [Φ(S t )] = e rt Φ(x) f t (x)dx, where the probability density f t ( ) of S t must satisfy S 0 = e rt E [S t ] = e rt x f t (x)dx. 0 0 J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 22 / 45

36 Background Information Risk-neutral Option Prices The only possible sensible price of a prepaid forward contract is S 0 : Π 0 (S t ) = S 0. A risk-neutral pricing policy states that contingent claims ought to be priced as the expected present-values of their payoffs, but in such a way that (prepaid) forward contracts written on the asset are correctly priced: Π 0 (t; Φ(S t )) = e rt E [Φ(S t )] = e rt Φ(x) f t (x)dx, where the probability density f t ( ) of S t must satisfy S 0 = e rt E [S t ] = e rt x f t (x)dx. Fundamental Theorem of Asset Pricing: Risk-neutral prices are the only possible arbitrage-free prices of contingent claims. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 22 /

37 Application To Model Selection: Jumps or not? BM or not? Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 23 / 45

38 Application To Model Selection: Jumps or not? BM or not? Model Selection Using Option Price Asymptotics Carr and Wu 03 studied the asymptotic behavior of European call and put option prices as time-to-maturity t shrinks to 0. Argued that prices are sharply different in short-time for a Purely Continuous, a Pure-Jump, or a Mixed (combination of both) Model. The following rough asymptotics for option prices were suggested: Model Out-of-the-money (OTM) At-the-money (ATM) Purely Continuous (PC) O(e c/t ), c > 0 O( t) Pure Jump (PJ) O(t) O(t p ), p (0, 1) Mixed Model (M) O(t) O(t p ), p (0, 1/2) The above asymptotics have been formalized and extended to higher order in several papers (Forde & Jaquier 09; Tankov 11; F-L & Forde 12; F-L, Gong, & Houdré 12, 16, and 17; F-L & Ólafsson 16). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 24 / 45

39 Application To Model Selection: Jumps or not? BM or not? Implied Volatility (IV) Asymptotics The Black-Scholes price Π BS is the risk-neutral price of a call or put option in an market where the stock price is assumed to be the exponential of a BM with drift; i.e., S t = S 0 e σwt +µt or log St S 0 N (µt, σ 2 t). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 25 / 45

40 Application To Model Selection: Jumps or not? BM or not? Implied Volatility (IV) Asymptotics The Black-Scholes price Π BS is the risk-neutral price of a call or put option in an market where the stock price is assumed to be the exponential of a BM with drift; i.e., S t = S 0 e σwt +µt or log St S 0 N (µt, σ 2 t). The IV ˆσ of an option price Π is the value of the BM s standard deviation σ needed for the B-S price Π BS (σ) to coincide with Π. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 25 / 45

41 Application To Model Selection: Jumps or not? BM or not? Implied Volatility (IV) Asymptotics The Black-Scholes price Π BS is the risk-neutral price of a call or put option in an market where the stock price is assumed to be the exponential of a BM with drift; i.e., S t = S 0 e σwt +µt or log St S 0 N (µt, σ 2 t). The IV ˆσ of an option price Π is the value of the BM s standard deviation σ needed for the B-S price Π BS (σ) to coincide with Π. The asymptotic behavior of option prices translates into asymptotics for the IV: Model Out-of-the-money (OTM) At-the-money (ATM) PC ˆσ t d > 0 ˆσ t σ 0 > 0, spot volatility PJ ˆσ t κ ˆσ t = O(t p ), p (0, 1 ] 2t log(1/t) 2 CM ˆσ t κ 2t log(1/t) ˆσ t σ 0 > 0, spot volatility J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 25 / 45

42 Application To Model Selection: Jumps or not? BM or not? Implications For The Implied Volatility Smile An IV smile today is a graph of today s Implied Volatilities of options with the same maturity against their strikes K or logmoneyness κ = ln( K S 0 ) Under the presence of jumps, we expect steep IV smiles" for short maturity options. But, also, under the presence of a continuous component, we expect the ATM IV to approach a positive value. These two behaviors are consistent with market-quotes as shown by the following IV smiles on Jan. 15, 14 (maturity from 8 days to 3 months). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 26 / 45

43 Main Results Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 27 / 45

44 Main Results Main Result: OTM Theorem (F-L & Forde SIFIN 12; F-L, Gong, & Houdré SPA 12) Consider a mixed model for the price of a risky asset of the form S t = S 0 e Xt = S 0 e Vt +Jt, where J is a Pure-Jump Lévy process (say, a compound Poisson process) and V is an independent stochastic volatility process: dv t = µ(y t )dt + σ(y t )dw t, V 0 = 0, dy t = α(y t )dt + γ(y t )db t, Y 0 = y 0, (e.g., a Heston or CIR process). Suppose the Lévy density, s( ), of J exists and is Cb 2 outside the origin. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 28 / 45

45 Main Results Main Result: OTM Theorem (F-L & Forde SIFIN 12; F-L, Gong, & Houdré SPA 12) Then, for any fixed κ > 0 (Out-Of-The-Money), the European Option price a Π 0 (t) = E (S t K ) + = E ( S 0 e Xt S 0 e κ) +, is such that 1 S 0 Π 0 (t) = t (e x e κ ) + s(x)dx + t 2 ( σ s(κ)eκ b κ ) e x s(x)dx + o(t 2 ), where σ 0 = σ(y 0 ) (spot volatility) and b = (e x 1 x1 x 1 )s(x)dx. a (t is the maturity, K = S 0 e κ is the strike, and short-rate r = 0 for simplicity) J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 29 / 45

46 Main Results Remarks As seen above, the presence of the continuous component increases the price of the OTM option price by t 2 σ s(κ)e κ (1 + o(1)). It can be extended to a more general class of jump processes such as J t = t 0 R\{0} h s (x) (N(ds, dx) s(x)dxds), where N is a Poisson random measure with mean measure µ(dx, dt) = s(x)dxdt. In-The-Money (ITM) options (when κ < 0) can be handled by call-put parity. In practice, the most liquid or tradable options have log-moneyness κ close" to 0, so it is important to consider At-The-Money (ATM) options (namely, those with κ = 0). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 30 / 45

47 Main Results The Problem and Assumptions We aim to determine the asymptotic behavior of Π 0 (t) = S 0 E ( e Xt 1 ) +. Assume that the jump component J of X is Lévy with Lévy density s( ): J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 31 / 45

48 Main Results The Problem and Assumptions We aim to determine the asymptotic behavior of Π 0 (t) = S 0 E ( e Xt 1 ) +. Assume that the jump component J of X is Lévy with Lévy density s( ): For a J of bounded variation, Π 0 (t) d 1 t. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 31 / 45

49 Main Results The Problem and Assumptions We aim to determine the asymptotic behavior of Π 0 (t) = S 0 E ( e Xt 1 ) +. Assume that the jump component J of X is Lévy with Lévy density s( ): For a J of bounded variation, Π 0 (t) d 1 t. However, based on empirical studies, the most interesting case is when J is of unbounded variation. No general results are known. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 31 / 45

50 Main Results The Problem and Assumptions We aim to determine the asymptotic behavior of Π 0 (t) = S 0 E ( e Xt 1 ) +. Assume that the jump component J of X is Lévy with Lévy density s( ): For a J of bounded variation, Π 0 (t) d 1 t. However, based on empirical studies, the most interesting case is when J is of unbounded variation. No general results are known. The problem admits a solution under the following small-jump stable-like condition that includes most models of the literature: lim x ±0 s(x) = C(±1) (0, ), Y (0, 2), x Y 1 1 e x s(x)dx <. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 31 / 45

51 Main Results Comments Y is referred to as the index of jump activity. For Y (0, 1), the process is of bounded variation. For Y (1, 2), the process is of unbounded variation, and E X t < (this is more relevant for financial applications based on empirical evidence). If s(x) = C x Y 1, we recover a Stable Lévy Process (up to drift or mean shifts, the only self-similar Lévy process). When { s(x) = x Y 1 C(1)e Mx 1 {x>0} + C( 1)e G x 1 {x<0} }, for some M, G > 0, the process is called Tempered Stable Process (TSP). If, in addition, C(1) = C( 1), the process is called CGMY process. Both are widely used in finance. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 32 / 45

52 Main Results Main Result: ATM Theorem (F-L & Ólafsson F&S 16a) Let J be a pure-jump Lévy process satisfying the small-jump stable-like condition and V a stochastic volatility such that dv t = µ(y t )dt + σ(y t )dw t, V 0 = 0, dy t = α(y t )dt + γ(y t )db t, Y 0 = y 0, independent of J such that σ 0 := σ(y 0 ) > 0. Then, E ( e Xt 1 ) + ( = E e J t +V t 1 ) ) + = d1 t d2 t 3 Y 2 + o (t 3 Y 2, (t 0), where d 1 := σ 0 2π, d 2 := C(1) + C( 1) 2Y (Y 1) σ 1 Y 0 E ( W 1 1 Y ). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 33 / 45

53 Main Results Remarks The first-order term d 1 is the same as in the Black-Scholes model with volatility σ 0 : E ( e Jt +Vt 1 ) + E ( e σ 0W t σ t 1) + = O ) (t 3 Y 2, The second-order term d 2 incorporates information on the degree of jump activity Y, and the intensity of small jumps as measured by the quantity C(1) + C( 1); J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 34 / 45

54 Main Results CGMY Model: s(x) = C x Y 1 ( e M x 1 x>0 + e G x 1 x<0 ) General CGMY MC Prices 1st order Approx. 2nd order Approx. ATM Call Option Prices Time to maturity, T (in years) Figure: Comparisons of ATM call option prices (computed by Monte-Carlo simulation) and the first- and second-order approximations for a CGMY model with a Brownian component. Parameter values taken from Andersen et al. 12. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 35 / 45

55 Applications to Parameter Calibration Outline 1 Overview and Applications Inference Of Stochastic Processes Using High-Frequency Data Change-Point Detection In Continuous-Time 2 Motivation Background Information Application To Model Selection: Jumps or not? BM or not? Main Results Out-Of-The-Money (OTM) Options At-The-Money (ATM) Options Applications to Parameter Calibration J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 36 / 45

56 Applications to Parameter Calibration Relevance in Calibration I In principle, the asymptotics ) Π t = d 1 t d2 t 3 Y 2 + o (t 3 Y 2, d 1 = σ 0, 2π provide us with an approach to calibrate the spot vol. σ 0, given a sample Π t 1,..., Π t k of short-time option prices, by considering the linear regression model: Π t i = d 1 t 1 2 i + ε i, i = 1,..., k. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 37 / 45

57 Applications to Parameter Calibration Relevance in Calibration I In principle, the asymptotics ) Π t = d 1 t d2 t 3 Y 2 + o (t 3 Y 2, d 1 = σ 0, 2π provide us with an approach to calibrate the spot vol. σ 0, given a sample Π t 1,..., Π t k of short-time option prices, by considering the linear regression model: Π t i = d 1 t 1 2 i + ε i, i = 1,..., k. Similarly, if we also had an estimate Ŷ of Y, we could calibrate the parameter C(1) + C( 1) in d 2 := C(1)+C( 1) 2Y (Y 1) considering the regression model: Π t i = d 1 t 1 2 i σ 1 Y 0 E ( W 1 1 Y ) by + d 2 t 3 Ŷ 2 i + ε i, i = 1,..., k. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 37 / 45

58 Applications to Parameter Calibration Relevance in Calibration II Unfortunately, the 2nd order approx. is not accurate enough for calibration since at any given date, there are few (about 10-15) maturities t i of ATM option prices (see below ATM SPX prices on Jan. 2, 14): Call Mid Price time-to-maturity (t) It is desirable to have higher order approximations. For a CGMY model, the 3rd order asymptotics is (F-L, Gong, and Houdré AMF 18): Π t = d 1 t d2 t 3 Y 2 + d 31 t + +d 32 t 5 2 Y +... J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 38 / 45

59 Applications to Parameter Calibration Relevance in Calibration III In that case, we can run the following regression model: Π t i := d 1 t 1/2 i + d 2 t 3 Ŷ 2 i + d 31 t i + d 32 t 5 2 Ŷ i + ε i, where t 1 < < t k are the available maturities and Π t 1,..., Π t k observed ATM option prices. Recall that are the d 1 = σ 0 C, d 2 = 2π Y (Y 1) σ1 Y 0 E ( W 1 1 Y ). Remark: An estimate of Ŷ can be obtained using HF stock data (cf. Aït-Sahalia & Jacod, 09) or short-time asymptotics for ATM IV skew (cf. J-F & Ólafsson, F&S 16b). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 39 / 45

60 Applications to Parameter Calibration Calibration of σ 0 and C for simulated data CGMY ATM option prices were computed for several realistic maturities t i for C = , G = 0.41, M = 1.93, Y = 1.5, σ = 0.1. These are shown on the right panel. ATM Call Option Price ATM Prices 1st order approx. 2nd order approx. High order approx time-to-maturity (t) Estimates of σ and C were then obtained for different values of Ŷ : Comparisons of Estimators of sigma (Y=1.5) Comparisons of Estimators of C Estimate of Sigma True Value = 0.1 Based on 1st order approx. Based on 2nd order approx. Based on higher order approx. Estimate of C True Value = Based on 2nd order approx. Based on higher order approx Value of Y Value of Y J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 40 / 45

61 Applications to Parameter Calibration Calibration of σ 0 and C for real data The same procedure as above was applied to ATM SPX option data on Jan. 2, 14: Comparions of Estimators of sigma Comparions of Estimators of C Estimate of Sigma Based on 1st order approx. Based on 2nd order approx. Based on higher order approx. Estimate of C Based on 2nd order approx. Based on d1, d2, d31, and d32 Based on d1, d2, and d31 Based on d1, d2, and d Value of Y Value of Y The estimates of σ based on the 3rd order approx. are relatively stable on Ŷ, while those based on the 1st and 2nd order approx. are much higher or volatile. The estimates of C is more sensitive to Ŷ (it ranges from to ), but, per our numerical experiment above, it is expected that an accurate estimate of Ŷ would be able to yield an accurately estimate of C. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 41 / 45

62 Applications to Parameter Calibration Ongoing and Future Research 1 Extend the 3rd order expansions to more general models and develop a general expansion formula of arbitrary order. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 42 / 45

63 Applications to Parameter Calibration Ongoing and Future Research 1 Extend the 3rd order expansions to more general models and develop a general expansion formula of arbitrary order. 2 Small-time asymptotics of options written on Leveraged Exchange-Traded Fund (LETF). LETF is a managed portfolio, which seeks to multiply the instantaneous returns of a reference Exchange-Traded Fund (ETF). J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 42 / 45

64 Applications to Parameter Calibration Ongoing and Future Research 1 Extend the 3rd order expansions to more general models and develop a general expansion formula of arbitrary order. 2 Small-time asymptotics of options written on Leveraged Exchange-Traded Fund (LETF). LETF is a managed portfolio, which seeks to multiply the instantaneous returns of a reference Exchange-Traded Fund (ETF). 3 Besides the ATM IV ˆσ (0, t), the ATM slope and convexity of the IV are very important in trading and, thus, its asymptotic behavior is of great interest. We have results for the skew but not the convexity. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 42 / 45

65 Applications to Parameter Calibration Ongoing and Future Research 1 Extend the 3rd order expansions to more general models and develop a general expansion formula of arbitrary order. 2 Small-time asymptotics of options written on Leveraged Exchange-Traded Fund (LETF). LETF is a managed portfolio, which seeks to multiply the instantaneous returns of a reference Exchange-Traded Fund (ETF). 3 Besides the ATM IV ˆσ (0, t), the ATM slope and convexity of the IV are very important in trading and, thus, its asymptotic behavior is of great interest. We have results for the skew but not the convexity. 4 There is no known results for American options. It is expected that the early exercise feature won t affect the leading order terms of the asymptotics but they should contribute to the high-order terms. J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 42 / 45

66 Further Reading I J.E. Figueroa-López and S. Ólafsson, Change-point detection for Levy processes. Annals of Applied Probability, Vol. 29, No. 2, p , J.E. Figueroa-López, and M. Forde, The small-maturity smile for exponential Lévy models. SIAM Journal on Financial Mathematics 3(1), 33-65, J.E. Figueroa-López, R. Gong, and C. Houdré, High-order short-time expansions for ATM option prices of exponential Lévy models. Mathematical Finance, 26(3), , J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 43 / 45

67 Further Reading II J.E. Figueroa-López and S. Ólafsson, Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility. Finance & Stochastics 20(1), , 2016a. J.E. Figueroa-López and S. Ólafsson, Short-time asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps. Finance & Stochastics 20(4), , 2016b. J.E. Figueroa-López, R. Gong, and C. Houdré, Third-Order Short-Time Expansions for Close-to-the-Money Option Prices Under the CGMY Model. Journal of Applied Mathematical Finance, 24, p , J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 44 / 45

68 Further Reading III J.E. Figueroa-López & J. Nisen. Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models. Stochastic Processes and their Applications 123(7), , J.E. Figueroa-López & C. Mancini. Optimum thresholding using mean and conditional mean square error. The Journal of Econometrics, Vol. 208, issue 1, p , J.E. Figueroa-López, C. Li, & J. Nisen. Optimal iterative threshold-kernel estimation of jump diffusion processes. In preparation, J.E. Figueroa-López (WUSTL) Short-Time Asymptotics in Financial Mathematics Mathematics, UMN 45 / 45