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Mathematics in Finance

Mathematics in Finance. Binomial model of options pricing. 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

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Mathematics in Finance

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  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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