CHAPTER 3The Mathematical Foundations

3.1 THE PRICING EQUATION

We will be interested in this book in pricing derivatives of a semi‐exotic nature using analytic formulae. We suppose that the derivative contract has a value at some time t greater-than-or-equal-to 0, with 0 the present time, dependent on the value of a number of risk factors left-parenthesis x Subscript 1 t Baseline comma x Subscript 2 t Baseline comma ellipsis comma x Subscript n t Baseline right-parenthesis equals colon bold-italic x Subscript t Baseline, stochastic processes which capture the fluctuation of market variables. We write the (stochastic) value of the derivative contingent on bold-italic x Subscript t Baseline equals bold-italic x symbolically as f left-parenthesis bold-italic x comma t right-parenthesis. In general we shall adopt the convention of representing stochastic processes as indexed with time as a subscript. Deterministic functions of time will be denoted in the standard way with time as an argument. The present value, or PV, of the derivative will be f left-parenthesis 0 comma 0 right-parenthesis, i.e. it is a deterministic function, the parameters on which the price depends being those specified in the derivative contract and through current market data, gleaned from available price or other observable data used for model calibration.

All pricing will be in ...

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