Question

A stock price is currently $20. It is known that at the end of one month that the stock price will either increase to 25 or decrease to 18. The risk-free interest rate is 12% per annum with continuous compounding. The hedge portfolio is a long position in Δ shares of stock plus one short Euorpean call option with strike price of $15 and expiration in 1 month. Using the no-arbitrage method, what is the present value of this hedge portfolio at time 0?

Answer 1

Present value of the Hedge portfolio using the provided information canbe calculated as follows

Stock Price (K) = $ 20

Payoff from the stock when the value goes up = $ 25

Payoff from hedge portfolio (Δ) = (Up Value of Stock Δ) - max (Stock up price - Stock price, 0)

= 25Δ - (25-20,0) = 25Δ - 5

Payoff from the stock when the value goes down = $ 18

Payoff from hedge portfolio (Δ) = (Down Value of Stock Δ) - max (Stock down price - Stock price, 0)

As per the formula only the positive values will be considered.

= 18Δ - (18-20, 0) = 18Δ

Since it's a hedge portfolio, payoff from the portfolio in either state should be the same.

25Δ - 5 = 18Δ

25Δ - 18Δ = 5   

7Δ = 5 ; Δ = 5/7

So, the value of Hedge Value portfolio is = 18Δ

= 18 * 5/7 = $ 12.85

Since it is a hedge portfolio, it must earn the risk-free rate of interest.

Therefore the Present Value of Hedge Portfolio is

Rate (r) = 12%

Time (t) = 1/12

= 12.85 * e^{(-12%*1/12) }

= 12.85 * 0.99 = $ 12.72

So the present value of hedge portfolio comes out to be $ 12.72 .