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On pg. 7 of Financial Calculus: An Introduction to Derivative Pricing by Martin Baxter & Andrew Rennie, it states:

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My Understanding

I did try to derive this picture. But I don't understand how this outcome that is $S_0\exp(\mu -0.5\sigma^2)$ is derived. Could I get any hint of it?

Jessie
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user13232877
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  • Out of curiosity, which book did you take this from? It might be helpful to the other users if you made this reference in your question. – Jessie Mar 11 '21 at 12:03
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    @Laufen Google says it may be from Martin Baxter & Andrew Rennie's Financial Calculus: An Introduction to Derivative Pricing, Cambridge University Press, 1996, page 7 Google Books – CiaPan Mar 11 '21 at 12:45

2 Answers2

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The stated law is the all-too-familiar definition of $\Bbb E(h(X))$. Since$$X=\ln S_T-\ln S_0=\ln\tfrac{S_T}{S_0},$$we have $S_T=S_0e^X$. This function of $X$ has mean $\int_{-\infty}^\infty\frac{S_0}{\sigma\sqrt{2\pi}}e^{S_0x-(x-\mu)^2/(2\sigma)^2}$, which evaluates to $\Bbb E(S_T)=S_0e^{\mu+\tfrac12\sigma^2}$.

J.G.
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The quick answer is that $Y=e^X\sim \text{Lognormal}$ thus its mean is well known

If you want to do all the calculation with the gaussian distribution, it is not difficult; try, it is a good exercise

tommik
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