Forward Contracts on Investment Assets with Known Yield
(Continuous Payment)

How many shares of a stock do you need to buy at t = 0 such that you will end up having exactly one share of the stock at t = T, but without accumulating dividends? The stock will pay dividend continuously at a rate of d per year.

The answer to this question is buy e^-δT shares at t = 0.

Note:

• The cost of obtaining a share of the stock that pays S_T at t = T is: S₀ e^-δT
• Therefore, F₀ = S₀ e^-δT e^rT = S₀ e^(r-δ)T

Proof:

Assume dividends are paid equal amount m times a year at an annual rate of δ % at an equal interval:

q₁ = q₂ = ... q_m = δ/m
Then the number of shares to purchase at t = 0 is:
(1 - q₁) (1 - q₂) ... (1 - q_m) = (1 - δ/m)^mT shares

If m becomes infinitely large, it is equivalent to the instantaneous (continuous) dividend payment:
⇒ lim_{m→ ∞} (1 - δ/m)^mT = e^-δT (remember lim_{x→ ∞} (1 + 1/x)^x = e)

δ = the instantaneous (or average) yield during the life of the contract
Thus, the cost of obtaining a share of the stock that pays S_T at t = T is:

S₀ e^-δT at t = 0

The forward price: F₀ = (S₀ e^-δT) e^rT = S₀ e^(r-δ)T at t = 0