Forward Price of a Stock Paying Dividends: No-Arbitrage Proof

To prove the forward price of a stock that pays dividends using the no-arbitrage argument, we can construct a portfolio that replicates the cash flows of the stock and the dividends. We assume that there are no transaction costs or restrictions on short-selling.

1. Start by constructing a portfolio consisting of:

   • Long 1 share of the stock
   • Short selling a forward contract on the stock with maturity T.

2. At time t (current time), the value of the portfolio is: V(t) = S(t) - F(t, T)

3. At time T (maturity of the forward contract), the value of the portfolio is: V(T) = S(T) - F(t, T)

4. We want to ensure that the value of the portfolio is the same at both time t and time T, regardless of the stock price and dividend payments. This is to prevent any arbitrage opportunities.

5. At time T, the stock will pay dividends. Let's assume that the dividends are paid continuously at a rate q, expressed as a percentage of the stock price on a continually compounded per annum basis.

6. The value of the stock at time T, after considering the dividend payments, is: S(T) = S(t)*e^[(r-q)(T-t)] - D(T)

   • S(t) is the stock price at time t
   • r is the risk-free interest rate
   • q is the dividend yield

7. The value of the portfolio at time T becomes: V(T) = S(t)*e^[(r-q)(T-t)] - D(T) - F(t, T)

8. To ensure no arbitrage, the value of the portfolio at time t should be equal to the value of the portfolio at time T: V(t) = V(T)

9. Substituting the values from steps 2 and 7, we get: S(t) - F(t, T) = S(t)*e^[(r-q)(T-t)] - D(T) - F(t, T)

10. Rearranging the equation, we can solve for the forward price F(t, T): F(t, T) = S(t)*e^[(r-q)(T-t)] - D(T)

This equation represents the forward price of a stock that pays dividends, where q is the known dividend yield expressed as a percentage of the stock price on a continually compounded per annum basis.

By constructing a portfolio with a long position in the stock and a short position in the forward contract, we ensure that the portfolio value remains the same at both time t and time T, eliminating any arbitrage opportunities.