Derivatives

CHAPTER 48Black–Scholes PDE

Aims

• To show how a no-arbitrage ‘replication portfolio’ can be constructed from stocks and options which results in a deterministic partial differential equation (PDE) for the dynamics of an option price.
• Solving this PDE for a European option gives the Black–Scholes closed-form solutions for call and put premia.
• To show how European options can be priced by solving the Black–Sholes PDE numerically, using finite difference methods.

We show how a stochastic differential equation (SDE) for and for the option premium , can be ‘combined’ to give a purely deterministic partial differential equation (PDE) for the option premium – this is the ‘Black–Scholes PDE’. This PDE can be solved (by various methods) to give an analytic solution for European option prices – this is the famous Black–Scholes–Merton closed-form solution for call (or put) premia.

Then we show how finite difference methods can be used to solve the PDE numerically and provide an estimate of call and put premia. After completing this chapter, the reader should have a good basic knowledge of the continuous time approach and feel confident in consulting more advanced texts in this area.

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