A Course in Financial Calculus by Alison Etheridge

By Alison Etheridge

This article is designed for first classes in monetary calculus geared toward scholars with an outstanding heritage in arithmetic. Key strategies akin 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 least difficult monetary context. next chapters are dedicated to expanding the monetary sophistication of the types and tools. the ultimate bankruptcy introduces extra complicated subject matters together with inventory fee versions with jumps, and stochastic volatility. a lot of routines and examples illustrate how the tools and ideas will be utilized to practical monetary questions.

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1 the multiperiod binary model Replicating portfolios At first sight it is not clear that we can make progress with our new model. For a tree consisting of k time steps there are 2k possible values for the stock price. 5, this suggests that we need at least 2k stocks to be traded in our market if we want it to be complete. For k = 20, this requires over a million ‘independent’ assets, far more than we see in any real market. But things are not so bad. More claims become attainable if we allow ourselves to rebalance our replicating portfolio after each time period.

We specified the possible values that the stock price could take at time n, corresponding to prescribing the functions {X n }n≥0 , and superposed the probabilities afterwards. Even if we had a preconception of what the probabilities of up and down jumps might be, we then changed probability (to the risk-neutral probabilities) in order actually to price claims. This process of changing probability will be fundamental to our approach to option pricing, even in our most complex market models. 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.

Suppose that the process {X n }n≥0 is a P, {Fn }n≥0 -martingale, and that T is a bounded stopping time. Then E [ X T | F0 ] = X 0 , and hence E [X T ] = X 0 . Proof: The proof is a simple application of the calculation that we did above. e. E [ X T | F0 ] = X 0 . Taking expectations once again yields E [X T ] = X 0 . ✷ It is essential in this result that the stopping time be bounded. In practice this will be the case in all of our financial applications, but Exercise 15 shows what can go wrong. More general versions of the theorem are available; see for example, Williams (1991).

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