Transcript Forward Start Options - National University of Singapore

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
continued
Improved models
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Local volatility model
Stochastic volatility model
Jump diffusion model
Others: discrete hedging, transaction cost
Local volatility model
• No closed form solution
• How to identify
?
continued
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)
– 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
Black-Scholes world
Beyond the Black-Scholes World
• Local volatility model
• Stochastic volatility model
• Jump diffusion model
Parameters
• 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