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Homework answers / question archive / A 1-year European call option on a stock with a continuous dividend rate of 2%

A 1-year European call option on a stock with a continuous dividend rate of 2%

Finance

A 1-year European call option on a stock with a continuous dividend rate of 2%. You are given:
The current price is 50 and the strike price is 55.
At the end of a year, the stock price will be either 40 or 60.
The risk-free rate is 5%.
Calculate the option premium.

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~ "u" = upmove factor = Stock price in upmove/Current stock price = 60/50 = 1.20

~ "d" = downmove factor = Stock price in downmove/Current stock price = 40/50 = 0.80

~ "P" = probability of upmove:

= (e(r-div)t - d) / (u - d)

where,

e = 2.7183

r = risk free rate

div = dividend yield

t = time to maturity

= (2.7183(0.05-0.02)x1 - 0.80) / (1.20 - 0.80)

= 0.23045 / 0.40

P = 0.576

~ (1 - P) = Probability of downmove:

= 1 - 0.576

(1-P) = 0.424

~ "fu" = payoff on upmove = pay off if stock price is 60 = Stock price-Exercise price = 60-55 = 5

~ "fd" = payoff on downmove = pay off if stock price is 40 = Not exercised = Payoff is zero.

~ Option premium:

= [ P x fu + (1-P) x fd ] x e-(r-div)t

= [0.576x5 + 0.424x0] x 2.7183-(0.05-0.02)x1

= [2.88 + 0] x 0.970445

= 2.7949

2.79

Answer: Option premium = 2.79