A gambler oers you the following three payos each for

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Unformatted text preview: screte time, we can guess that the option price is its expected value after changing the probabilities to make the stock price a martingale. 2 Theorem 6.10. Write E ⇤ for expected values when µ = r /2 in (6.26). ⇤ rT g (ST ). The value of a European option g (ST ) is given by E e 200 CHAPTER 6. MATHEMATICAL FINANCE Proof. We prove this by taking limits of the discrete approximation. The risk neutral probabilities, p⇤ , are given by h 1 + rh d . ud p⇤ = h (6.28) Using the formulas for u and d in (6.23 and recalling that ex = 1+ x + x2 /2+ · · · , p p2 1 u = 1 + µh + h + (µh + h) + . . . 2 p h + ( 2 /2 + µ)h + . . . =1+ p d = 1 + µh h + ( 2 /2 + µ)h + . . . so from (6.28) we have p h + (r µ ⇤ p ph ⇡ 2h 2 /2)h = 1r + 2 µ 2 2 (6.29) /2 p h h h If X1 , X2 , . . . are i.i.d. with h P (X1 = 1) = p⇤ h h P (X1 = 1) = 1 p⇤ h then the mean and variance are h EXi = 2p⇤ = h (r h var (Xi ) = 1 2 µ /2) p h h (EXi )2 ! 1 To apply the central limit theorem we note that t/h t/h pX h pX h h Xm = h (Xm m=1 h E Xm...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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