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

# This situation usually occurs when the model has

• Notes
• 9

This preview shows page 6 - 8 out of 9 pages.

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

Subscribe to view the full document.

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
• Fall '11
• COULON
• Dividend, Mathematical finance, Black–Scholes

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes