Seminar20050602

# Seminar20050602 - Mathematics in Finance Numerical solution...

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Mathematics in Finance Numerical solution of free boundary problems: pricing of American options Wil Schilders (June 2, 2005)

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Contents American options The obstacle problem Discretisation methods Matlab results Recent insights and developments
1. American options American options can be executed any time before expiry date, as opposed to European options that can only be exercised at expiry date We will derive a partial differential inequality from which a fair price for an American option can be calculated.

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Bounds for prices (no dividends) ) , ( ) , ( t S C t S C t E t A = K t S C t S P S Ke t A t A t t T r - + - - ) , ( ) , ( ) ( t t E t T r t S t S C Ke S - + - - ) , ( ) ( ) ( ) ( ) ( ) , ( ) ( t T r t E t t T r Ke t S P S Ke - - + - - - K t S P S Ke t A t t T r - + - - ) , ( ) ( ) ( For American options: For European options: ) ( ) , ( ) , ( t T r t E t E t Ke t S C t S P S - - = - + Reminder: put-call parity
Why is ? Suppose we exercise the American call at time t<T Then we obtain S t -K However, Hence, it is better to sell the option than to exercise it Consequently, the premature exercising is not optimal ) , ( ) , ( t S C t S C t E t A = K S Ke S t S C t t T r t t A - - - - ) ( ) , (

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What about put options? For put options, a similar reasoning shows that it may be advantageous to exercise at a time t<T This is due to the greater flexibility of American options
American options are more expensive than European options Comparison European-American options

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An optimum time for exercising…. (1) Statement: There is S f such that premature exercising is worthwhile for S<S f , but not for S>S f . Proof: Let be a portfolio. As soon as , the option can be exercised since we can invest the amount at interest rate r. For it is not worthwhile, since the value of the portfolio before exercising is , but after exercising is equal to . S P + = π S K S K P - = - = + ) ( K S S K = + - = ) ( + - ) ( S K P K S S K S P + - + = + ) ( K
An optimum time for exercising…. (2) The value S f depends on time, and it is termed the free boundary value . We have This free boundary value is unknown, and must be

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Seminar20050602 - Mathematics in Finance Numerical solution...

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