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
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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
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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
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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
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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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