BlackScholes Model for European vanilla options

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Black-Scholes Model forEuropean vanilla options
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Black-Scholes formulas forEuropean vanilla
options
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Pricing American vanilla options
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Pricing exotic optionsunder Black-Scholes
framework

• Multi-asset options
• Barrier options
• Asian options
• Lookback options
• Forward start option, shout option, compound
  options

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Implied volatility

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

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Volatility smile

• Finance.yahoo.com

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continued
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Improved models

• Local volatility model
• Stochastic volatility model
• Jump diffusion model
• Others discrete hedging, transaction cost

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Local volatility model
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A special case
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Identification of
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In general case

• No closed form solution
• How to identify ?

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continued
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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

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Stochastic volatility model
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Pricing model
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continued