Financial Calculus

Abstract

The rewards and dangers of speculating in the modern financial markets have come to the fore in recent times with the collapse of banks and bankruptcies of public corporations as a direct result of ill-judged investment. At the same time, individuals are paid huge sums to use their mathematical skills to make well-judged investment decisions. Here now is the first rigorous and accessible account of the mathematics behind the pricing, construction and hedging of derivative securities. Key concepts such as martingales, change of measure, and the Heath-Jarrow-Morton model are described with mathematical precision in a style tailored for market practitioners. Starting from discrete-time hedging on binary trees, continuous-time stock models (including Black-Scholes) are developed. Practicalities are stressed, including examples from stock, currency and interest rate markets, all accompanied by graphical illustrations with realistic data. A full glossary of probabilistic and financial terms is provided. This unique, modern and up-to-date book will be an essential purchase for market practitioners, quantitative analysts, and derivatives traders, whether existing or trainees, in investment banks in the major financial centres throughout the world.

Pricing a European Option in a Black-Scholes Quanto Market When Stock Price is a Semimartingale

“…The theory of pricing of the European call option (see [17]) and the Markovian property of t Y , the value of the European call option at time T is given by …”

Numerical integration of the Heath-Jarrow-Morton model of interest rates

“…Here we consider two examples: a one-factor model with deterministic exponential volatility function (Vasicek model, see, e.g. [2,4]) and a two-factor model with proportional volatility function (see, e.g. [13,9,21,11]).…”

“…L is the number of independent runs in the Monte Carlo simulation (see (3.34) It is known (see, e.g. [2,4,23]) that a caplet corresponds to a put option on a zerocoupon bond. In [14] analytic expressions for the European option prices on zero-coupon and coupon bearing bonds under the Vasicek model are derived.…”

Martingale option pricing

“…Girsanov's theorem is often stated in financial math texts [9,18] as transforming a Wiener process plus a drift term into another Wiener process. This is wrong: when the drift depends on a random variable x and is not merely t-dependent, then the resulting process is not Wiener.…”