Question

A stock price is currently $25. It is known that at the end of two months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose ST is the stock price at the end of two months. The derivative pays off ST*(ST-S0) at T.

Consider a portfolio consisting of long delta shares of stock and short 1 unit of derivative. What delta makes the portfolio risk-free?

	A.	100
	B.	25
	C.	1/25
	D.	0.5

Answer 1

Answer: D. 0.5

+Δ:shares

−1:derivative

The price of the portfolio will be 27Δ−729 or 23Δ−529 in two months.
If 27Δ−729=23Δ−529

That is: Δ=50

The price of the portfolio is sure to be 621. The portfolio is riskless for this value of delta. The present value of the portfolio is:

50×25−f  

where f is the price of the derivative. The portfolio should earn a risk-free interest rate.

(50×25−f)e0.10×2/12=621

This equals to: f=639.3

Therefore the value of the option is $639.3.