This situation usually occurs when the model has unhedgeable risk in other

This situation usually occurs when the model has

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consistent with no arbitrage. This situation usually occurs when the model has unhedgeable risk, in other words uncertainty associated to a quantity that is not traded. An example of this is stochastic volatility in the next section. 61
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13.11 Dividends Corporations often pay a portion of their earnings to shareholders in the form of dividend payments. Every time a firm pays a dividend to its shareholders, its cash holdings decrease by the sum of those dividends. Specifically, a company paying a dividend of $1 per share stays exactly the same after the dividend payment, except its cash holdings, and thereby the overall firm value, decrease by $1 per share. For example, if the stock price was $20 immediately before the dividend payment, it would instantly drop to $19 after the payment, reflecting that the shareholders received $1 per share from the company’s bank accounts. In general, dividend payments represent fixed amounts distributed at regular time inter- vals (e.g. at the end of each quarter). We consider a company, whose stock moves according to the Black-Scholes model, and that pays dividends continuously at a constant rate D 0 0 proportional to the share price S t . That is, a shareholder receives D 0 S t dt per each share held over an interval d t . If the stock’s expected growth rate is μS t d t over that same interval, the Black-Scholes model yields d S t = ( μ - D 0 ) S t d t + σS t d W t , (42) where σ is the volatility, and W t is a Brownian motion under the measure associated with the expected growth rate μ (if the firm were to not pay out dividends). This process is clearly equivalent to a geometric Brownian motion with drift μ * := μ - D 0 . Note that the dividends diminish the expected growth of the stock (as it represents cash leaving the firm). Pricing derivatives on the underlying stock S t then requires first rewriting d S t in the risk-neutral world, where the drift rate does not depend on the expected growth rate μ . One cannot simply take μ * to be the firm’s expected growth rate, since the dividend rate D 0 , just like the risk-free rate r , is unaffected by the value of μ (and every market participant has their own opinion of it). We use the no-arbitrage principle to determine the stock dynamics d S t in the risk-neutral world. Consider a portfolio Π t := P t - Δ t S t , which at time t is long in a derivative P t of the underlying stock S t , and short in Δ t > 0 shares of S t . Shorting one share of the stock requires borrowing it from a counter-party that has a long position in it. The counter-party expects to receive all dividends associated with that share. Just like a firm paying dividends to its shareholders, a short-seller of a stock must pay dividends to the counter-party taking the opposite long position. Hence the dynamics of the portfolio returns are t = d P t - Δ t d S t - Δ t D 0 S t d t, where D 0 S t d t is the dividend payment per share being paid out due to the short position in Δ t shares of S t . In the risk neutral world there exists a portfolio that is fully hedged (the rate of return is r ). In the following discussion we find the Δ
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  • Fall '11
  • Dividend, Mathematical finance, Black–Scholes

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