Question

Question 6 - Chapter 7 Textbook:

The current share price of Elica plc is £2.26 per share. It offers a continuously compounded dividend yield of 2.00% per year. The volatility of its stock returns is 50% and risk free rate is 5%, both per annum with continuous compounding.

1. Using Black-Scholes-Merton (B-S-M) model, find the values of N(d1) and N(d2) for an option with a strike price of £2.00 and maturity in six months. (show all step-by-step calculations)
2. Determine the price of this call option using B-S-M model. (show all step-by-step calculations)
3. What would be the cost of ten call option contracts? (show all step-by-step calculations)
4. Put options on Elica plc with a strike price of £2.00 and maturity in six months are trading at £0.30 per option. Does this offer an arbitrage opportunity? If yes, how can one take advantage of this opportunity? Fully illustrate your answer using 10 option contracts. (show all step-by-step calculations)

Answer 1

Please see the table below. Please be guided by the llast column to understand the mathematics. The last few rows highlighted in yellow contain your answers. Figures in parenthesis, if any, mean negative values. All financials are in £. Adjacent cells in blue contain the formula in excel I have used to get the final output.

Please do round off the results as per your requirement.

Part (i)

N(d1) = 0.713924608
N(d2) = 0.583686431

Part (ii)

Value of the call option, C = £ 0.4589

Part (iii)

the cost of ten call option contracts = 10 x C = £ 4.589

Part (iv)

Based on call put parity equation for a dividend paying stock, no arbitrage price of a put option, should be

P = C - Se-d x t + Ke-r x t

= 0.4589 - 2.26 x e-2% x 0.5 + 2 x e-5% x 0.5

=  0.1720

However the current trading price of a put option is £0.30

So, clearly there is an arbitrage. The actual put option is pricey while the synthetic put option created as P = C - Se-d x t + Ke-r x t is cheaper. Hence, the strategy should be short the option and buy the synthetic put. For 10 options we need to create the following arbitrage portfolio:

• Short 10 put options
• Buy 10 call options
• Short 10 x e-d x t = 10 x e-2% x 0.5 = 9.9005 nos. of stocks today
• Lend Ke-r x t = 2 x e-5% x 0.5 = 1.9506 today @ risk free rate for 6 months

So, the cash flows today = 10 x (Trading price of put option - no arbitrage price of put option) = 10 x (0.3 - 0.1720) = £1.28 which is purely riskfree, riskless without any investment and without any liability in future.