Question

(Option leverage; straddle payoffs; replication; % margin) In the one-period binomial model, the current stock price of CAT (Caterpillar) is $90. Robert expects that in one year the stock price of CAT will be either $108 (up move) or $75 (down move). The exercise price of one-year European call (or put) option of CAT=$100 and risk-free rate r=2% per annum. Robert would like to construct a portfolio with the stock and cash to replicate the payoff of 1,000 units of “straddle” of CAT, where one unit of straddle is the combination of long one call option and long one put option with the same strike price.

(a) What are the gross payoffs ($) of 1,000 units of CAT straddle in the up and down move, respectively?

(b) How many shares of CAT does Robert need to buy/short now?

(c) How much money does Robert need borrow/save now?

(d) Calculate the percentage margin.

[Note: percentage margin=(equity)/(short position) or percentage margin=(equity)/(value of stock)]

(e) Calculate the current price of CAT straddle (per unit) in the one-period binomial setting.

Answer 1

(a) What are the gross payoffs ($) of 1,000 units of CAT straddle in the up and down move, respectively?

Gross payoff = N x [max (S - K, 0) + max (K - S, 0)]

Gross payoff in the up state = 1,000 x [max (108 - 100, 0) + max (100 - 108, 0)] = 1,000 x [8 + 0] = $ 8,000

Gross payoff in the down state = 1,000 x [max (75 - 100, 0) + max (100 - 75, 0)] = 1,000 x [0 + 25] = $ 25,000

(b) How many shares of CAT does Robert need to buy/short now?

Number of shares = (Payoff in up state - payoff in down state) / (Su - Sd) = (8,000 - 25,000) / (108 - 75) = -515.15

Since, there is a minus sign, hence, Robert needs to short 515.15 shares of CAT now.

(c) How much money does Robert need borrow/save now?

Since Robert has to take a short position, hence he needs to save now an amount = (515.15 x Su + Cu) / (1 + r) = 515.15 x 108 + 8,000) / (1 + 2%) = $ 62,388.59

(d) Calculate the percentage margin.

Short position = 515.15 x x Current stock price = 515.15 x 90 = $ 46,363.64

Equity = $ 62,388.59 - 46,363.64 = $ 16,024.96

Hence, percentage margin=(equity)/(short position) = 16,024.96 / 46,363.64 = 34.56%

(e) Calculate the current price of CAT straddle (per unit) in the one-period binomial setting.

the current price of CAT straddle per unit = 16,024.96 / 1,000 = $ 16.025