23. Consider the following information for an individual stock:• Current share price is $10• Risk-free rate is 5% pa compounded continuously• Volatility of the stock returns (σ) is 30% pa• Strike price is $12• Time to maturity of the option is 9 mths• The firm is expected to pay dividend estimated at $1 per share in 2 mths, $1.40 pershare in 8 mths and $1.80 per share in 14 mths.Use the closed-form Black-Scholes model to price the European put option with the abovecharacteristicsa) 3.96b) 5.13c) 1.25d) None of the above.
Question
- Consider the following information for an individual stock:• Current share price is 12• Time to maturity of the option is 9 mths• The firm is expected to pay dividend estimated at 1.40 pershare in 8 mths and $1.80 per share in 14 mths.Use the closed-form Black-Scholes model to price the European put option with the abovecharacteristicsa) 3.96b) 5.13c) 1.25d) None of the above.
Solution
The Black-Scholes model is used to price European options and is given by the formula:
C = S0e^-qt * N(d1) - Xe^-rt * N(d2)
P = Xe^-rt * N(-d2) - S0e^-qt * N(-d1)
where:
C = Call option price P = Put option price S0 = Current price of the underlying asset X = Strike price of the option r = Risk-free interest rate t = Time to expiration q = Dividend yield N = Cumulative standard normal distribution function d1 = [ln(S0/X) + (r - q + σ^2/2)t] / σ√t d2 = d1 - σ√t
Given the information:
S0 = 12 r = 5% = 0.05 σ = 30% = 0.3 t = 9/12 = 0.75 years q = (1/10 + 1.4/10 + 1.8/10) / 3 = 0.42
We can calculate d1 and d2 as follows:
d1 = [ln(10/12) + (0.05 - 0.42 + 0.3^2/2)0.75] / (0.3√0.75) = -0.449 d2 = -0.449 - 0.3*√0.75 = -0.749
Then we can calculate the put option price:
P = 12e^-0.050.75 * N(-0.749) - 10e^-0.420.75 * N(-0.449) = $1.25
Therefore, the answer is c) 1.25.
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