![]() ![]() ![]() The Sobolev spaces D, with s is an arbitrary real number, are introduced following Watanabe’s work. The case where H is an L -space is trated in detail aft- s,p wards (white noise case). The presentation of the Malliavin calculus has been slightly modi?ed at some points, where we have taken advantage of the material from the lecturesgiveninSaintFlourin1995(seereference).Themainchanges and additional material are the following: In Chapter 1, the derivative and divergence operators are introduced in the framework of an isonormal Gaussian process associated with a general 2 Hilbert space H. In preparing this second edition we have taken into account some of these new applications, and in this spirit, the book has two additional chapters that deal with the following two topics: Fractional Brownian motion and Mathematical Finance. ![]() Since then, new applications and developments of the Malliavin c- culus have appeared. We can then finally use a no-arbitrage argument to price a European call option via the derived Black-Scholes equation.There have been ten years since the publication of the ?rst edition of this book. In order to price our contingent claim, we will note that the price of the claim depends upon the asset price and that by clever construction of a portfolio of claims and assets, we will eliminate the stochastic components by cancellation. We will form a stochastic differential equation for this asset price movement and solve it to provide the path of the stock price. A geometric Brownian motion is used instead, where the logarithm of the stock price has stochastic behaviour. A standard Brownian motion cannot be used as a model here, since there is a non-zero probability of the price becoming negative. A vanilla equity, such as a stock, always has this property. For this we need to assume that our asset price will never be negative. In the subsequent articles, we will utilise the theory of stochastic calculus to derive the Black-Scholes formula for a contingent claim. The derivative of a random variable has both a deterministic component and a random component, which is normally distributed. The fundamental difference between stochastic calculus and ordinary calculus is that stochastic calculus allows the derivative to have a random component determined by a Brownian motion. Ito's Lemma is a stochastic analogue of the chain rule of ordinary calculus. A fundamental tool of stochastic calculus, known as Ito's Lemma allows us to derive it in an alternative manner. The Binomial Model provides one means of deriving the Black-Scholes equation. This process is represented by a stochastic differential equation, which despite its name is in fact an integral equation. The physical process of Brownian motion (in particular, a geometric Brownian motion) is used as a model of asset prices, via the Weiner Process. The main use of stochastic calculus in finance is through modeling the random motion of an asset price in the Black-Scholes model. In quantitative finance, the theory is known as Ito Calculus. Instead, a theory of integration is required where integral equations do not need the direct definition of derivative terms. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions. Many stochastic processes are based on functions which are continuous, but nowhere differentiable. Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems.
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