L14 - Lecture 14 Introduction to Black-Scholes Theory The...

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Lecture 14 Introduction to Black-Scholes Theory The celebrated Black-Scholes (BS) theory in financial economics lays a foundation for three important aspects: a probability model, an option pricing formula and a statistical inference proce- dure for volatilities. The past few decades have seen a great deal of progress in various extensions of the BS theory. This lecture contains an informal introduction to these three elements, which paves a way for more general discussions in Lecture 15. 14.1 BS SDE — the geometric Brownian motion In the simplest form, the (continuous-time) BS market contains only two underlying assets: a riskless asset B (bank account) with a constant interest rate r > 0 such that for t IR + , B t = B 0 e rt , where B 0 = 1; (14.1) and a risky asset S (stock) that follows the SDE dS t /S t = μ dt + σ dW t , (14.2) where S t represents a stock price at time t IR + , the constants μ IR and σ > 0 represent the expected rate of return and volatility respectively, and { W t } is a standard Brownian motion. Recall from Example 13.2 that (14.2) has a solution S t = S 0 exp[( μ - σ 2 / 2) t + σW t ] . (14.3) The general SDE theory (we skip it) assures that (14.3) is the unique solution of (14.2) in some sense. To model a more realistic market, there should be more risky assets, each following its own stochastic dynamics. That would give us more freedom to hedge various kinds of risk. 14.2 BS option pricing formulas
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L14 - Lecture 14 Introduction to Black-Scholes Theory The...

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