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Mathematics in Finance

This overview explores the binomial model of options pricing, which allows investors to evaluate options based on underlying asset movements. It details the concepts of call and put options, payoff functions, and cash settlements, highlighting the role of exchanges and key parameters such as strike price, volatility, and interest rates. Additionally, it addresses how options can be priced using methods like the Black-Scholes model, solving partial differential equations, and Monte Carlo simulations. The document also briefly explains the algorithm for the binomial method, its numerical implementation, and extensions.

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Mathematics in Finance

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Presentation Transcript

  1. Mathematics in Finance Binomial model of options pricing.

  2. Derivatives - Options Give the holder the right to buy or sell the underlying at a certain date for a certain price. (European options) • Right to buy  call option • Right to sell  put option • Payoff function • Cash settlement • Exchanges: AMEX, CBOT, Eurex, LIFFE, EOE, ...

  3. IV Derivatives - Options Example 1: Long Call on stock S with strike K=32, maturity T, price P=10. Payoff function: f(S) = max(0,S(T) – K)

  4. underlying maturity strike volatility Option value Interest rate dividends

  5. Derivatives - Options

  6. Problem: How can options be priced? • Modelling • Black-Scholes • Solving partial differential equations • Monte-Carlo simulation • ...

  7. Replicating portfolio

  8. Binomial one period method

  9. Binomial one period method

  10. Binomial one period method

  11. Binomial n-period method

  12. Binomial n-period method

  13. Binomial n-period method

  14. Binomial n-period method Algorithm for binomial method

  15. 164.38 120.09 80 85.37 52.92 26 33.46 13.59 0 Example 234.38 187.5 150 150 120 120 96 58.91 96 76.8 61.44

  16. Numerical implementation

  17. Some versions of binomial model

  18. Extensions of binomial model

  19. Black-Scholes formula

  20. Conclusions

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