Forward Price of a Dividend-Paying Stock: No-Arbitrage Proof with Detailed Procedures

To prove the forward price of a stock that pays dividends using the no-arbitrage argument, we need to construct a replicating portfolio that mimics the cash flows of the stock. Here are the detailed procedures:

1. Assume the current time is t, and we want to determine the forward price of the stock at time T.

2. Consider a portfolio consisting of the following components:

   • Long one unit of the stock
   • Short a risk-free bond with face value equal to the present value of the future dividends paid by the stock
   • This portfolio should replicate the cash flows of the stock until time T

3. Let's analyze how this portfolio value changes over time:

   • At time t, the value of the stock is S_t, and the value of the risk-free bond is e^(-r(T-t)), where r is the risk-free interest rate.
   • The total value of the portfolio at time t is S_t - e^(-r(T-t)) * D, where D is the present value of the future dividends.

4. Assume that the stock pays dividends continuously at a rate of q per annum, expressed as a percentage of the stock price on a continually compounded basis. The present value of the future dividends can be calculated as:

   • D = q * S_t * (1 - e^(-q(T-t))) / q = S_t * (1 - e^(-q(T-t)))

5. As time progresses from t to T, the stock price remains unchanged, but the present value of the future dividends decreases. The value of the risk-free bond component of the portfolio decreases as well, offsetting the decrease in the present value of the dividends.

6. At time T, the value of the risk-free bond component is e^(-r(T-T)) = 1, as the bond matures. The total value of the portfolio at time T is S_T - 1 * D.

7. To eliminate arbitrage opportunities, the forward price F(t, T) at time t should equal the initial value of the replicating portfolio, which is S_t - D.

8. Therefore, the forward price of a stock that pays dividends is given by: F(t, T) = S_t - D = S_t - S_t * (1 - e^(-q(T-t))) = S_t * e^(-q(T-t))

This formula shows that the forward price of a stock that pays dividends is equal to the current stock price multiplied by the exponential function of the dividend yield (q) and the time to maturity (T-t).