Question

Rob wishes to buy a European put option on BioLabs, Inc., a non-dividend–paying common stock, with a strike price of $45 and six months until expiration. BioLabs' common stock is currently selling for $32 per share, and Rob expects that the stock price will either rise to $64 or fall to $17 in six months. Rob can borrow and lend at the risk-free EAR of 5 percent.

a. What should the put option sell for today? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)

b. What is the delta of the put? (Do not round intermediate calculations and round your final answer to 4 decimal places (e.g., 32.1246). Negative value should be indicated with a minus sign.)

c. How much would Rob have to lend to create a synthetic put? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)

d. How much does the synthetic put option cost? (Do not round intermediate calculations and round your final answer to 2 decimal places (e.g., 32.16).)

Answer 1

Let's create the replicating portfolio of stocks and bonds to match the put option:

• Buy delta, D nos. of stock
• Lend and amount L

All financials below are in $.

Current stock price, S₀ = 32, strike price, K = 45

Stock price in up state, S_u = 64

Stock price ion down state, S_d = 17

Time period, T = 6 months = 0.5 year

Risk free EAR = R = 5%

At t = T,

Payoff from the portfolio in up state = Payoff from the put option in up state

Hence, D x S_u + L x (1 + R x T) = max (K - S_u, 0)

Hence, D x 64 + L x (1 + 5% x 0.5) = max (45 - 64, 0)

Hence, 64D + 1.025L = 0 -------call it equation (1)

Payoff from the portfolio in down state = Payoff from the put option in down state

Hence, D x S_d + L x (1 + R x T) = max (K - S_d, 0)

Hence, D x 17 + L x (1 + 5% x 0.5) = max (45 - 17, 0)

Hence, 17D + 1.025L = 28 -------call it equation (2)

Subtract equation (2 ) from equation (1):

64D - 17D = 47D = - 28

Hence, D = -28 / 47 = - 0.5957

and hence, L = -64D / 1.025 =  37.20

Let's now get into the questions:

Part (a)

Price of the put option today, p = Value of the replicating portfolio today = D x S₀ + L = - 0.5957 x 32 + 37.20 = 18.13

Part (b)

Delta of the put option = D = -0.5957

Part (c)

Rob would have to lend, L = 37.20 to create a synthetic put

Part (d)

Cost of the synthetic put option = cost required to create the replicating portfolio = D x S₀ + L = - 0.5957 x 32 + 37.20 = 18.13 which is same as the sell price of the put option as determined in part (a)