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Quantitative Finance by Matt Davison

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Chapter 22

Black Scholes PDE for Pricing Options in Continuous Time

22.1 CHAPTER SUMMARY

In this chapter, we use the idea of creating a delta hedged portfolio, the value of which does not depend on stock price fluctuations, and extend it to a continuous time model. Basically, we find a hedge that works at each instant in time, and rebalance our portfolio so that our hedge is always correct. Using stochastic calculus, we show that this is possible and results in a partial differential equation for the value of the option. We present the Black Scholes formula solution to this option, although we leave a complete derivation of a way in which that formula could be obtained until Chapter 23.

22.2 Introduction

In Chapter 18, we saw that a call option ...

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