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Analysis of Financial Time Series, Third Edition by RUEY S. TSAY

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Appendix A: Integration of Black–Scholes Formula

In this appendix, we derive the price of a European call option given in Eq. (6.19). Let x = ln(PT). By changing variable and using g(PT) dPT = f(x) dx, where f(x) is the probability density function of x, we have

(6.36) 6.36

Because x = ln(PT) ∼ N[ln(Pt) + (r − σ2/2)(Tt), σ2(Tt)], the integration of the second term of Eq. (6.36) reduces to

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where CDF[ln(K)] is the cumulative distribution function (CDF) of x = ln(PT) evaluated at ln(K), Φ( · ) is the CDF of the standard normal random variable, and

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The integration of the first term of Eq. (6.36) can be written as

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where the exponent can be simplified to

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Consequently, the first integration becomes

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which involves the CDF of a normal distribution with mean ln(Pt) + (r + σ2/2)(Tt) and variance σ2(Tt). By using the same techniques as those of the second integration shown before, we have ...

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