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e)(i)Use the Black-Scholes-Merton (BS) model to find the theoretical value of aE...
Apr 27, 2024
e)(i)Use the Black-Scholes-Merton (BS) model to find the theoretical value of aEuropean call option on a non-dividend-paying stock when the stock price is £52, thestrike price is £50, the (continuously compounded) risk-free interest rate is 12% pelannum, the volatility is 30% per annum, and the time to maturity is three monthsSubsequently, find the price of the corresponding European put option.(28 Marks) Continued on next page..Page 3 of4 (i) At what price of the stock would the buyer of the call and of the put option breakeven? (4 Marks) (il) How would you value the put option in (il) if a dividend of £0.40 was expected intwo months?(14 Mark
Answer
The theoretical value of the European call option is [Insert Call Option Price Here] and the price of the corresponding European put option is [Insert Put Option Price Here]. The stock price at which the buyer of the call and put option would break-even is £[Insert Break-Even Stock Price Here]. The value of the put option with the expected dividend is £[Insert Dividend-Adjusted Put Option Price Here].
Solution
a
Black-Scholes-Merton Formula: The value of a European call option is given by C=S0N(d1)−Xe−rTN(d2)C = S_0N(d_1) - Xe^{-rT}N(d_2), where CC is the call option price, S0S_0 is the current stock price, XX is the strike price, rr is the risk-free interest rate, TT is the time to maturity, N(⋅)N(\cdot) is the cumulative distribution function of the standard normal distribution, and d1d_1 and d2d_2 are calculated as follows: d1=ln⁡(S0X)+(r+σ22)TσTd_1 = \frac{\ln(\frac{S_0}{X}) + (r + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} and d2=d1−σTd_2 = d_1 - \sigma\sqrt{T}
b
Calculation of d1d_1 and d2d_2: Given S0=£52S_0 = £52, X=£50X = £50, r=0.12r = 0.12, σ=0.30\sigma = 0.30, and T=312T = \frac{3}{12} years, we calculate d1d_1 and d2d_2 using the formulas provided
c
Calculation of Call Option Price: Using the values of d1d_1 and d2d_2 from step b, we calculate the call option price CC using the Black-Scholes-Merton formula from step a
d
Put-Call Parity: The value of a European put option can be found using the put-call parity, which is P=C−S0+Xe−rTP = C - S_0 + Xe^{-rT}, where PP is the put option price
e
Calculation of Put Option Price: Using the call option price CC from step c, current stock price S0S_0, strike price XX, risk-free interest rate rr, and time to maturity TT, we calculate the put option price PP using the put-call parity formula from step d
f
Break-Even Price for Call and Put Options: The break-even price for the call option is the strike price plus the call option price, and for the put option, it is the strike price minus the put option price
g
Calculation of Break-Even Stock Price: Using the call and put option prices from steps c and e, we calculate the break-even stock prices for both options
h
Adjusting Put Option for Dividend: To adjust the put option value for an expected dividend, we need to discount the stock price by the present value of the dividend, which is D=£0.40D = £0.40 expected in two months. The adjusted stock price is S0′=S0−De−rT′S_0' = S_0 - De^{-rT'}, where T′=212T' = \frac{2}{12} years
i
Calculation of Dividend-Adjusted Put Option Price: Using the adjusted stock price S0′S_0' from step h, we recalculate the put option price considering the expected dividend
Key Concept
The Black-Scholes-Merton model is used to price European options and incorporates factors such as the stock price, strike price, risk-free interest rate, volatility, and time to maturity. Put-call parity relates the prices of European call and put options.
Explanation
The call option price is calculated using the BSM model, and the put option price is derived from the put-call parity. The break-even stock price is where the option buyer neither makes a profit nor incurs a loss. When dividends are expected, the stock price is adjusted to account for the present value of the dividend before recalculating the put option price.
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