Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging

CHAPTER 4 Risk-Neutral Valuation

4.1 Introduction

Every sincere attempt to value financial derivatives needs to be grounded on a sound theory, formally represented in general by some kind of market model. A market model embodies a simplifying mathematical description of a real financial market. A priori, it is not clear what features a market model should have. These are mainly dictated by the market under observation and the tasks to be accomplished (e.g. pricing, trading, hedging, risk management). However, there is a minimum set of requirements a market model should obey. The most important are the absence of arbitrage opportunities (NA) and no free lunches with vanishing risk (NFLVR).

A central result in mathematical finance is the Fundamental Theorem of Asset Pricing which relates, for a given market model, the conditions of NA or NFLVR to the existence of an equivalent martingale measure (EMM) making all discounted stochastic processes of the market model martingales. A martingale is a stochastic process that does not change its value on average (under some suitable conditions). An important corollary of this result is that the (discounted) price processes of attainable, i.e. redundant, options are also martingales giving rise to a pure probabilistic approach to option pricing. Namely, the value of a European option maturing at some date in the future is simply its expected payoff at that date under the EMM discounted back to today by the risk-free short rate.

The market-based ...