Black-Scholes Model for European vanilla options

Black-Scholes formulas for European vanilla options

Pricing American vanilla options

Pricing exotic options under Black-Scholes framework Multi-asset options Barrier options Asian options Lookback options Forward start option, shout option, compound options

Beyond the Black-Scholes World

Implied volatility The value for volatility that makes the theoretical option value and the market price the same

Volatility smile Finance.yahoo.com

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Improved models Local volatility model Stochastic volatility model Jump diffusion model Others: discrete hedging, transaction cost

Local volatility model

No closed form solution How to identify ?

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How to use the local volatility model Calibration of the model: Identify the volatility function from the market prices of vanilla options Price non-traded contracts by using the model

Stochastic Volatility Model

Option Pricing

Option pricing with non-traded underlying So far, the underlying is assumed to be a traded asset. The underlying is a consumption asset Oil Short selling is prohibited Pricing of forward contract on oil The underlying is a non-traded asset Volatility, interest rate Both long and short positions are prohibited No arbitrage pricing

Continued (stochastic volatility model)

Continued

The Market Price of Risk

Risk Neutral Processes

Two Named Models Hull White Heston

Example 1: Hull-White model

Example 2: Heston Model

Jump Diffusion Model Poisson process

Jump-diffusion Process

Hedging

Ito Lemma

Two special models Merton (1976) Wilmott et al. (1998) to hedge the diffusion only Wilmott et al. (1998) to hedge both jump and diffusion as much as we can

Merton’s Model (1976) Jump risks are diversified

Wilmott et al.’s Model Hedging strategy

Continued

Continued Under this best strategy, we let

Summary

Purpose Understand the market better Price options at the OCT market

Beyond the Black-Scholes World Local volatility model Stochastic volatility model Jump diffusion model

Parameters , J Local volatility model: =(S,t) Stochastic volatility model: Hull-White model (3 parameters) Heston model (2 parameters) Jump diffusion model , J

Option Pricing at the OTC Market Model calibration Extend the model to exotic options Solution