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 a great history in arithmetic. Key innovations corresponding 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 versions and tools. the ultimate bankruptcy introduces extra complex themes together with inventory rate types with jumps, and stochastic volatility. a number of workouts and examples illustrate how the equipment and ideas might be utilized to reasonable monetary questions.

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5 units of stock, which costs 50, and we borrow 35 in cash bonds. Suppose that S1 = 120. 25 units of stock, taking our total bond borrowing to 65. Suppose that S2 = 140. Now φ = (60 − 20)/(160 − 120) = 1, so we buy still more stock, to take our holding up to 1 unit and our total borrowing to 100 bonds. Finally, suppose that S3 = 120. The option will be in the money, so we must hand over our unit of stock for 100, which is exactly enough to cancel our bond debt. The table below summarises our stock and bond holding if the stock price follows another path through the tree.

Suppose that a stock pays dividends at discrete times, T0 , T1 , . . , Tn . Show that it can be optimal to exercise an American call on such a stock prior to expiry. 3 will pay a dividend of 5% of its value at time 2. As before, interest rates are zero and between times 2 and 3 the value of the stock will either increase or decrease by 20. Find the time zero price of an American call option on this stock with strike 100 and maturity 3. Is it ever optimal to exercise early? 1 at time (i + 1)δt.

This is certainly not true for the stock. In spite of our newfound freedom, for simplicity, we shall continue to suppose that the interest rate is the constant, r . 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.

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