In addition to the five factors discussed in the chapter, dividends also affect

Question

In addition to the five factors discussed in the chapter, dividends also affect the price of an option. The Black–Scholes option pricing model with dividends is:

All of the variables are the same as the Black–Scholes model without dividends except for the variable d, which is the continuously compounded dividend yield on the stock.

A stock is currently priced at $82 per share, the standard deviation of its return is 56 percent per year, and the risk-free rate is 3 percent per year, compounded continuously. What is the price of a call option with a strike price of $78 and a maturity of six months if the stock has a dividend yield of 3 percent per year? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Explanation / Answer

S0 = underlying price $82 E = strike price $78 = volatility (% p.a.) 0.56 ^2 = variance (% p.a.) 0.3136 r = continuously compounded risk-free interest rate (% p.a.) 0.03 d = continuously compounded dividend yield (% p.a.) 0.03 t = time to expiration (years in %) 0.5 C = S × e–dt × N(d1) – E × e–Rt × N(d2) d1 = [ln(S/E) + (R – d + 2 / 2) × t] / ( × sqrt(t)) d2 = d1 – × sqrt(t) d1 = [ln($82/$78) + (.03 - .03 + .3136/2) × .05] / (.56 × sqrt (.5 ) 0.162142641 d2 = 0.1621 - .56 x sqrt(.5) -0.233837156 Normal Distribution N( d1) using NormDIST 0.564403236 Normal Distribution N(d2) 0.407555701 e-dt 0.98511194 e-rt 0.98511194 C = $82 x .98511 x 0.5644 - $78 x .98511 x .40755 $14.28

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