Question

On June 15, you took a long forward contract (delivery on December 15) on a dividend-paying stock when the stock price was $30 and the risk-free interest rate (with discrete compounding) is 12% per annum. The amount of the dividends were known as $0.75 on Aug 15, and Nov 15. It is now September 15 and the current stock price and the risk-free interest rate are, respectively, $31 and 10%. What is the value of your long forward position now? Assume the forward contract prices are arbitrage free prices.

Answer 1

Step 1: Calculating Forward Price on June 1

On June 15 , Spot price = S₀ = $30, Risk free rate = r = 12% p.a..

Dividend on Aug 15 = D1 = 0.75, time to first dividend = t1 = (2/12) and dividend on Nov 15 = D2 = 0.75, time to second dividend = t2 = (5/12) years

Present value at June 1 of dividends = PVD = D1 / (1+r)^t1 + D2 / (1+r)^t2 = 0.75 / (1+12%)^2/12 + 0.75 / (1+12%)^5/12 = 0.7359 + 0.7154 = 0.75 / (1.12)^2/12 + 0.75 / (1.12)^5/12 = $1.4513

Time to delivery = t = (6/12) years

Forward Price = F = (S₀ - PVD) (1+r)^t = (30-1.4513)(1+12%)^6/12 = 28.5487 x 1.0583 = 30.2130

Step 2 . Calculating Value of Forward position on Sep 15

Spot price of Sep 15 = S_T = $31 , Risk free rate = r = 10%

After Sep 15 there will be only one dividend on Nov 15

Time to dividend on Nov 15 = t3 = (2/12) years

Present value at Sep 15 of dividend = PVD = D2 / (1+r)^t3 = 0.75 / (1+10%)^2/12 = 0.7381

Time to delivery = t = (3/12) years

Value of long forward position = S_T - PVD - F/(1+r)^t = 31 - 0.7381 - 30.2130 / (1+10%)^3/12 = 31 - 0.7381 - 30.2130 / (1.10%)^3/12 = 31 - 0.7381 - 29.5016 = 0.7603

Value of long forward position now = $0.7603