Question

A share of stock is currently priced at $20 and will change with equal likelihood to either $40 or $10. A call option with a $20 exercise price is available on the stock. The interest rate is zero. Which of the following positions will provide (approximately) the same payoffs as the option?

Buy 0.667 shares and lend $6.67

Buy 0.667 shares and borrow $6.67

Buy 0.5 shares

Sell 0.667 shares and borrow $0.667

Answer 1

Payoff from the option

If stock price rise to $40, Payoff= (40- 20)= $20

If the stock price declines to $10, Payoff= 0

Position 1   (Buy 0.667 shares and lend $6.67)

If stock price rise to $40, Profit from stock= 0.667x (40-20)= $13.34

Since the interest rate is zero, the principal $6.67 will be returned.

Payoff= 13.34+ 6.67= $20.01 i.e. 20 approximately

If the stock declines to $10, Profit from stock= 0.667x (10-20)= -$6.67

Since the interest rate is zero, the principal $6.67 will be returned.

Payoff= -6.67+ 6.67= 0

Position 2 (Buy 0.667 shares and borrow $6.67)

If stock price rise to $40, Profit from stock= 0.667x (40-20)= $13.34

Since the interest rate is zero, the principal $6.67 will have to be paid back.

Payoff= 13.34- 6.67= $6.67

If the stock declines to $10, Profit from stock= 0.667x (10-20)= -$6.67

Since the interest rate is zero, the principal $6.67 will have to have to be paid back.

Payoff= -6.67- 6.67= -$13.34

Position 3 (Buy 0.5 shares)

If stock price rise to $40, Profit from stock= 0.5x (40-20)= $10

Payoff= $10

If the stock declines to $10, Profit from stock= 0.5x (10-20)= -$5

Payoff= -$5

Position 4 (Sell 0.667 shares and borrow $0.667)

If stock price rise to $40, Profit from stock= -0.667x (40-20)= -$13.34

Payoff= -13.34- 0.667= -$14.007

If the stock declines to $10, Profit from stock= 0.667x (10-20)= $6.67

Payoff= 6.67- 0.667= $6.003

So, we can see that Position 1 (Buy 0.667 shares and lend $6.67) approximately provide the same payoff as that of the option.