A Course in Financial Calculus - download pdf or read online

By Alison Etheridge

ISBN-10: 0521813859

ISBN-13: 9780521813853

This article is designed for first classes in monetary calculus geared toward scholars with a superb historical past in arithmetic. Key ideas equivalent to martingales and alter of degree are brought within the discrete time framework, permitting an obtainable account of Brownian movement and stochastic calculus. The Black-Scholes pricing formulation is first derived within the easiest monetary context. next chapters are dedicated to expanding the monetary sophistication of the types and tools. the ultimate bankruptcy introduces extra complex issues together with inventory expense versions with jumps, and stochastic volatility. a great number of routines and examples illustrate how the equipment and ideas should be utilized to reasonable monetary questions.

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Conditional expectation When we constructed the probabilities on paths through our binary (or binomial) trees, we first specified the probability on each branch of the tree. This was done in such a way that the expected value of e−r δt Sk+1 given that the value of the stock at time kδt is known to be Sk is just Sk . This condition specifies the probabilities on the two branches emanating from the node corresponding to Sk at time kδt. We should like to extend this idea, but first we need to remind ourselves about conditional expectation.

Stochastic processes To specify a (discrete time) stochastic process, we typically require not just a single σ -field, F, but an increasing sequence of them, Fn ⊆ Fn+1 ⊆ · · · ⊆ F. The collection {Fn }n≥0 is then called a filtration and the quadruple , F, {Fn }n≥0 , P is called a filtered probability space. A real-valued stochastic process is just a sequence of realvalued functions, {X n }n≥0 , on . We say that it is adapted to the filtration {Fn }n≥0 if X n is Fn -measurable for each n. 2 One can then think of the σ -field Fn as encoding all the information about the evolution of the stochastic process up until time n.

For simplicity suppose that the risk-free interest rate is zero and T is 1. Suppose that under the probability P, at each time step, stock prices go up with probability p and down with probability 1 − p. The conditional expectation Mn E[S N |Fn ], 1 ≤ n ≤ N, is a stochastic process. Check that it is a P-martingale and find the distribution of the random variable Mn . 11 (a) Find a Markov process that is not a martingale. (b) Find a martingale that is not a Markov process. 12 Show that a previsible martingale is constant.

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A Course in Financial Calculus by Alison Etheridge

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