Option Pricing Modeling Overview

Transcription

1 Option Pricing Modeling Overview Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch) Stochastic time changes Options Markets 1 / 11

2 What is the purpose of building a model? 1 Interpolation and extrapolation (for market makers/broker dealers): The model value needs to match observed price as market makers cannot take big views. Interpolation example: Off-grid quotes; stripping forward rate curve Extrapolation example: Long-dated OTC trades based on short-dated exchange traded prices. 2 Alpha generation (for investors/hedge funds): The criterion for being a good model is not to generate small pricing errors, but to generate large, transient errors. 3 Risk management: The modeled risks are real risks and are the only risk sources. Once one hedges away these risks, no other risk is left. Daily calibration is problematic Parameters become risk sources. Both market making and investment need good risk management. Liuren Wu (Baruch) Stochastic time changes Options Markets 2 / 11

3 What is the purpose of building a model? 4 Understanding the underlying dynamics and market pricing of risk: This is mostly an academic endeavor (a good one), but related to risk management and investment. By observing the prices of various financial securities and derivatives, one can infer exactly what s going on for the dynamics and how market prices each source of risk. Long time series are hard to come by. The large cross sections of derivatives provide a partial replacement. Caveat: When a certain security (derivatives in particular) is traded by only a few people, its implied dynamics or market prices can be problematic (or even manipulated). Liuren Wu (Baruch) Stochastic time changes Options Markets 3 / 11

4 A discrete time analog to illustrate ideas Security return over one time period: R t+1 = µ t + σ t ε t+1 µ t captures return predictability: Usually not something you learn from options. Not the focus of option pricing. σ t denotes the return volatility level. It determines the level of the implied volatility surface. The distribution of the standardized variable ( return innovation ) ε t+1 determines the shape of the implied volatility smile at short maturities (one period): After the standardization, we often assume that εt+1 is iid return. Normal distribution implies a flat smile. To capture a smile, add non-normality to ε. At longer maturity, σ t and ε t+1 can interact to affect the distribution of aggregate return over a longer horizon, n i=1 R t+i, and hence the IV smile at longer maturities. Liuren Wu (Baruch) Stochastic time changes Options Markets 4 / 11

5 Design models to match evidence Security return over one time period: R t+1 = µ t + σ t ε t+1 IV smile at short maturity non-normal distribution for ε. IV smile at long maturity slow down the central limit theorem through persistent volatility or infinite return variance. IV varies over time σ t varies over time accordingly. Smile/skew shape changes over time Several modeling possibilities 1 Let the distribution of ε vary over time. This is often not that tractable and its breaks the iid assumption. 2 Use multiple return components, σ 1,t ε 1,t+1 + σ 2,t ε 2,t+1 + Maintain the iid assumption on the two innovations ε; use variation in the relative magnitude of σ 1,t versus σ 2,t to generate variation in skewness/kurtosis. Example: ε 1 is normal. ε 2 is negatively skewed. Example: ε 1 is positively skewed. ε 2 is negatively skewed. Liuren Wu (Baruch) Stochastic time changes Options Markets 5 / 11

6 Design models to match economic intuition/story 1 CAPM: R i,t+1 = β i σ m,t ε m,t+1 + σ i,t ε i,t+1. Link single-name IV surface to SPX (or sector ETF) IV surface. Evidence: SPX IV is more negatively skewed than single0name IV. ε m,t+1 tends to be more negatively skewed than ε i,t+1. Application: Estimate β i from options instead of stock returns. Beyond CAPM: How does σm,t and σ i,t interact with each other? Idiosyncratic return has systematic volatility. (wp) Is idiosyncratic return risk priced? Is idiosyncratic volatility risk priced? Distinguish between volatility of idiosyncratic return and idiosyncratic volatility. Similar models: Global CAPM linking IV surfaces from many equity indices. Liuren Wu (Baruch) Stochastic time changes Options Markets 6 / 11

7 Design models to match economic intuition/story 1 CAPM 2 From pricing kernels to exchange rates: S fh t+h S fh t = Mf t,t+h M h t,t+h Directly model the pricing kernel of each economy to derive the currency return dynamics. If each kernel has negatively skewness, ln M j t,t+h = σj tεt + h j, j = h, f, the skewness of the currency return will depends on the relative magnitude of σ f t versus σ h t. Variations in ( σ f t versus σ h t ) lead to stochastic skew in currency IV. Bonds and stocks take expectations on the pricing kernel Expectation operation amounts to censorship: Information is smoothed before released. By contrast, currency links to unfiltered information from the kernel. Application: Learn the behavior of the pricing kernel across multiple economies from currency option prices. Application: Price options on both primary and cross exchange rates consistently. Liuren Wu (Baruch) Stochastic time changes Options Markets 7 / 11

8 Design models to match economic intuition/story 1 CAPM 2 From pricing kernels to exchange rates. 3 One market with multiple exposures: Stock prices drop to zero (or very low level) upon default Stock price is subject to both market risk and credit risk. CAMP needs to be extended further. Is default contagious? How much of it is contagious? How default contagion affects the joint modeling of stock volatility surfaces? 4 The same risk affects multiple markets cross-market linkages/trades Credit enters both stock (currency for sovereign credit) options and CDS. Contagion again: Low-frequency, large, negative events lead to an increase in the probability of having more of these events: The credit spread (default probability) for company A increases after company B defaults. All these provide linkages of one IV surface to another, thus providing a modeling base for trading across IV surfaces. Liuren Wu (Baruch) Stochastic time changes Options Markets 8 / 11

9 From discrete to continuous time 1 How to model iid return innovations ε? The continuous time analog is called Lévy process. Process is more re-fined than distribution. Different processes can generate the same terminal distribution, but have quite different implications for hedging risk, or pricing path-dependent options. The Brownian motion W is a Lévy process: It generates iid normally distributed return innovations over equal-distant, non-overlapping sampling intervals. Merton (1976) s compound Poisson jump (plus diffusion) is also a Lévy process: It generates iid non-normal return innovation distributions. The innovation distribution is a mixture of normals, with the mixture probability given by the Poisson distribution. Liuren Wu (Baruch) Stochastic time changes Options Markets 9 / 11

10 From discrete to continuous time 1 How to model iid return innovations ε? The continuous time analog is called Lévy process. There is only one continuous Lévy process Brownian motion, but there are many different ways of specifying jumps: Poisson: Jump arrives following Poisson distribution (exponentially distributed time). Jump size is 1. Compound Poisson: Jump arrives via Poisson. Once the jump is arrived, the jump size is drawn from an independent distribution. Merton (76) assumes a normal distribution for the return jump size. Kou assumes double-exponential. You can also jumping to a fixed place (say zero, in case of default). Higher-frequency jumps: Arrive rate approaches infinity as jump sizes approaches zero. Examples include variance gamma, dampened power law, inverse gaussian... The choice of jump types is made according to situation and evidence. Liuren Wu (Baruch) Stochastic time changes Options Markets 10 / 11

11 From discrete to continuous time 1 How to model iid return innovations ε? 2 How to model stochastic σ t? It is often more tractable to model variance instead of volatility. For jumps, it is often more tractable to let the jump arrival rate be stochastic, instead of the jump size distribution. Both variance and arrival rate are linear in time. A smaller instantaneous variance rate over a longer time generates the same distribution as a larger variance rate but over a shorter time. Same for jump arrival rate. Stochastic time change captures this idea beautifully as it embeds the variance/arrival rate variation into variation in time. T t t 0 v sds, with v denotes either variance rate or arrival rate of jumps or both. We call it activity rate as it captures how much activity is happening per unit time. One tractable activity rate specification: The square-root process of Heston (1993), or Cox, Ingersoll, Ross (1984). dv t = κ(θ v t )dt + ω v t dw t. Liuren Wu (Baruch) Stochastic time changes Options Markets 11 / 11