ABSTRACT
With the numerous books on mathematical finance published each year, the usefulness of a new one may be questioned. This is the first book setting out the applications of advanced analytical and geometrical methods used in recent physics and mathematics to the financial field. This means that new results are obtained when only approximate and partial solutions were previously available. We present powerful tools and methods (such as differential geometry, spectral decomposition, supersymmetry) that can be applied to practical problems in mathematical finance. Although encountered across different domains in theoretical physics and mathematics (for example differential geometry in general relativity, spectral decomposition in quantum mechanics), they remain quite unheard of when applied to finance and allow to obtain new results readily. We introduce these methods through the problem of option pricing. An option is a financial contract that gives the holder the right but not the obligation to enter into a contract at a fixed price in the future. The simplest example is a European call option that gives the right but not the obligation to buy an asset at a fixed price, called strike, at a fixed future date, called maturity date. Since the work by Black, Scholes [65] and Merton [32] in 1973, a general probabilistic framework has been established to price these options. In this framework, the financial variables involved in the definition of an option are random variables and their dynamics follow stochastic differential equations (SDEs). For example, in the original Black-Scholes-Merton model, the traded assets are assumed to follow log-normal diffusion processes with constant volatilities. The volatility is the standard deviation of a probability density in mathematical finance. The option price satisfies a (parabolic) partial differential equation (PDE), called the Kolmogorov-Black-Scholes pricing equation, depending on the stochastic differential equations introduced to model the market. The market model depends on unobservable or observable parameters such as the volatility of each asset. They are chosen, we say calibrated, in order to reproduce the price of liquid options quoted on the market such as Euro-
historical data for movement of the assets can be used.
