Chapter 7: Beyond Black-Scholes

Black-Scholes Model for vanilla options

Implied volatility and volatility smile

Continued

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

Local volatility model

A special case: Identification of

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

Pricing model

Continued

The Market Price of Risk

Risk neutral processes

Derivatives on a single underlying variable

Pricing equation

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

Merton’s Model (1976) Jump risks are diversified

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