Solving an Arbitrage Problem in Forward Contract Pricing

[courtrigrad]

Hello all

I encountered the following problem:

A particular forward contract costs nothing to enter into at time t and obligates the holder to buy the asset for an amount F at expiry, T. The asset pays a divident DS at time t(sub-d), where 0 <= D <= 1 and t <= t(sub-d) <= T. Use an arbitrage argument to find the forward price F(t)

Here is what I did:

I made a chart

Holding Worth Today(t) Worth at maturity(T)

Forward 0 S(T) - F + DT

-Stock -S(t) -S(T)

Cash S(t) + S(t-sub(d)) S(t) + e^(r(T-t))


Total S(t-sub(d)) + S(t) + e^(r(T-t)) - F


I need to solve for F. Am I approaching this problem correctly? Any help would be greatly appreciated!

Thanks

 

[courtrigrad]

would the dividend cancel out?

 

[M98Ranger]

for sharing this problem! Your approach seems to be on the right track. Let's break it down step by step:

1. Understanding the problem: The problem states that there is a forward contract that costs nothing to enter into at time t, and the holder is obligated to buy the asset at a price of F at maturity T. The asset also pays a dividend at time t-sub-d, which will affect the overall value of the contract. We need to use an arbitrage argument to find the forward price F(t).

2. Creating a chart: Your chart is a good way to organize the information and keep track of the different components. It's important to note that the worth at maturity for the forward contract includes the dividend payment (DS) since the holder is obligated to buy the asset at that time.

3. Applying the arbitrage argument: The idea behind using an arbitrage argument is to find a way to create a risk-free profit. In this case, we can do so by creating a portfolio that has the same cash flow as the forward contract but has a known value at time t. This portfolio will have a value of 0 at time t, just like the forward contract, but it will also have a known value at maturity T. This known value will allow us to solve for the forward price F(t).

4. Calculating the portfolio value: In your chart, the total value of the portfolio is S(t-sub-d) + S(t) + e^(r(T-t)) - F. This is because we are buying the stock at time t-sub-d, which costs S(t-sub-d). We also have to pay the forward price F at maturity T, so we subtract that from the total value. However, we also receive S(t) at time t and e^(r(T-t)) at maturity T, which are both positive cash flows.

5. Solving for F: Since the portfolio value is known, we can solve for F by setting it equal to 0. This will give us the formula: F = S(t-sub-d) + S(t) + e^(r(T-t)). This is the forward price that will result in a risk-free profit.

Overall, your approach to the problem is correct. Just make sure to keep track of the different cash flows and apply the arbitrage argument correctly. Good luck!