![]() This is one of the most interesting and easiest reads in the discipline a gem of a book." - D. "The main results are reinforced with simple special cases, and only when the intuitive foundations are laid does the author resort to the formalism of probability. Cassano ( see the full review from the Journal of Finance) By focusing solely on Brownian motion, the reader is able to develop an intuition and a feel for how to go about solving problems as well as deriving results." - Mark A. a book that is a marvelous first step for the person wanting a rigorous development of stochastic calculus, as well as its application to derivative pricing. We can then finally use a no-arbitrage argument to price a European call option via the derived Black-Scholes equation.Stochastic Calculus and Financial Applications Reviews and Comments on the Text Stochastic Calculus and Financial Applications Some Reviewer Comments 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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