section1 (1)

# section1 (1) - Continuous Time Finance Notes, Spring 2004...

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Continuous Time Finance Notes, Spring 2004 – Section 1. 1/21/04 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use in connec- tion with the NYU course Continuous Time Finance. This section discusses risk-neutral pricing in the continuous-time setting, for a market with just one source of randomness. In doing so we’ll introduce and/or review some essential tools from stochastic calculus, especially the martingale representation theorem and Girsanov’s theorem . For the most part I’m following Chapter 3 of Baxter & Rennie. ************* Let’s start with some general orientation. Our focus is on the pricing of derivative securities. The most basic market model is the case of a lognormal asset with no dividend yield, when the interest rate is constant. Then the asset price S and the bond price B satisfy dS = μS dt + σS dw, dB = rB dt with μ , σ ,and r all constant. We are also interested in more sophisticated models, such as: (a) Asset dynamics of the form dS = μ ( S, t ) Sdt + σ ( S, t ) Sdw where μ ( S, t )and σ ( S, t ) are known functions. In this case S is still Markovian (the statistics of dS depend only on the present value of S ). Almost everything we do in the lognormal case has an analogue here, except that explicit solution formulas are no longer so easy. (b) Asset dynamics of the form dS = μ t + σ t where μ t and σ t are stochastic processes (depending only on information available by time t ; more technically: they should be F t -measurable where F t is the sigma-algebra associated to w ). In this case S is non-Markovian . Typically μ and σ might be determined by separate SDE’s. Stochastic volatility models are in this class (but they use two sources of randomness, i.e. the SDE for σ makes use of another, independent Brownian motion process; this makes the market incomplete.) (c) Interest rate models where r is not constant, but rather random; for example, the spot rate r may be the solution of an SDE. (An interesting class of path-dependent options occurs in the modeling of mortgage-backed securities: the rate at which people reﬁnance mortagages depends on the history of interest rates, not just the present spot rate.) (d) Problems involving two or more sources of randomness, for example options whose payoﬀ depends on more than one stock price [e.g. an option on a portfolio of stocks; or an option on the max or min of two stock prices]; quantos [options involving a random exchange rate and a random stock process]; and incomplete markets [e.g. stochastic volatility, where we can trade the stock but not the volatility]. My goal is to review background and to start relatively slowly. Therefore I’ll focus in this section on the case when there is just one source of randomness (a single, scalar-valued Brownian motion). 1

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Option pricing on a binomial tree is easy. The subjective probability is irrelevant, except that it determines the stock price tree. The risk-neutral probabilities (on the same tree!) are determined by E RN t [ e - rδt s t + δt ]= s t .
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## This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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section1 (1) - Continuous Time Finance Notes, Spring 2004...

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