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# L1.T4.5. Black-Scholes with dividends

#### David Harper CFA FRM

##### David Harper CFA FRM
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David's ProTip: We already have enough option pricing formulas to remember, what do we do about dividends? Just remember that dividends tend to manifest as a reduction on the stock price. Like dividends reduce delta, as N(d1) becomes N(d1)*exp(-qT), so to in Black-Scholes. But also, don't forget, in put-call parity: p + S(0) = c + K*exp(-rT). Replace S(0) with [S(0) - D]: p + S(0) - D = c + K*exp(-rT), and you've got parity with a dividend. The intuition? Assume for a given TSR (TSR = capital appreciation + dividend), higher dividend lowers the expected appreciation component. For example, BSM: the long option holder forgoes dividends, and for a given TSR, higher dividends imply lower option value as this implies lower appreciation. (true BSM does not use expected appreciation, I'm just referring to an intuition here!)

AIMs: Explain how dividends affect the early decision for American call and put options. Compute the value of a European option using the Black‐ Scholes‐ Merton model on a dividend‐paying stock.

Questions:

5.1. In the absence of dividends, Hull shows that an American-style call option should never be exercised early. However, if the American-style call option instead does pay dividends, which of the following is true?
a. It is still never optimal to early exercise an American call option
b. It may be optimal to early exercise an American call option immediately after the ex-dividend date
c. It may be optimal to early exercise an American call option immediately before the ex-dividend date
d. It is always optimal to early exercise an American call option immediately before the ex-dividend date

5.2. For purposes of option valuation, how is "dividend" defined?
a. The increase in the stock price on the ex-dividend date arising from any dividends declared
b. The reduction in the stock price on the ex-dividend date arising from any dividends declared
c. The increase in the stock price on the declaration date arising from any dividends declared
d. The reduction in the stock price on the declaration date arising from any dividends declared

5.3. A European call option has a time to maturity of six months on a stock with ex-dividend dates in two and five months. Each dividend pay $1 per share. The current stock and strike prices are both$50. The volatility of the stock is 18% per annum. The risk free rate is 4%. What is the price of the call option?
a. $2.00 b.$2.75
c. $3.08 d.$3.16

5.4. A European call option has a time to maturity of six months on a stock with a 2% dividend yield. The current stock and strike prices are both $50. The volatility of the stock is 18% per annum. The risk free rate is 4%. What is the price of the call option? a.$2.00
b. $2.75 c.$3.08
d. $3.16 5.5. A one-year European call option on the Euro has an exercise price of$1.40 when the current exchange rate is EUR/USD $1.34. The risk-free rate in the United States is 4% and the Eurozone riskfree rate is 3%. The volatility of the spot exchange rate is 30% per annum. What is the price of the call option? a.$0.136
b. $0.297 c.$0.355
d. \$0.425