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Quantitative Finance by Matt Davison

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Chapter 23

Solving the Black Scholes PDE

23.1 CHAPTER SUMMARY

In the first section of this chapter, we solve the Black Scholes partial differential equation for the European call, from first principles, step by step, by introducing coordinate transformations, which reduce it to the random walk or diffusion equation of Chapter 19. In the second section, we quickly repeat the derivation, but for a general option payoff, and show that European options can be priced as the present value of the expected value of their payoffs, provided that this present value is taken using a probability density function corresponding to a “risk-neutral” stochastic process, that is, one with μ changed to r. If you are happy to work with the Black Scholes formula ...

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