Market Risk Analysis Volume I: Quantitative Methods in Finance

I.5

Numerical Methods in Finance

I.5.1 INTRODUCTION

An analytic solution to a problem is a solution that can be expressed as an explicit formula in terms of well-known functions.¹ Numerical methods are applied when there is no analytic solution. To give just a few common examples:²

• The Black–Scholes–Merton (Black and Scholes, 1973; Merton, 1973) model gives an analytic solution for the price of a standard European option under certain (rather unrealistic) assumptions about the behaviour of asset prices. However, it is not possible to invert the Black–Scholes–Merton formula so that we obtain an analytic solution for the implied volatility of the option. In other words, the implied volatility is an implicit function, not an explicit function of the option price (and the other variables that go into the Black–Scholes–Merton formula such as the strike and the maturity of the option). So we use a numerical method to find the implied volatility of an option.
• The allocations to risky assets that give portfolios with the minimum possible risk (as measured by the portfolio volatility) can only be determined analytically when there are no specific constraints on the allocations such as ‘no more than 5% of the capital should be allocated to US bonds’.
• The value at risk (VaR) of a portfolio has an analytic solution only under certain assumptions about the portfolio and its returns process. Otherwise we need to use a numerical method – usually simulation – to compute the VaR of a portfolio. ...