# Value of a forward contract proof

I am reading the book - "Mathematics for Finance: An Introduction to Financial Engineering" by Marek Capinski and Tomasz Zastawniak. I am going through the proof of Theorem 6.4 - For any $$t$$ such that $$0 ≤ t ≤ T$$ the time $$t$$ value of a long forward contract with forward price $$F(0, T)$$ is given by $$V(t) = [F(t, T) − F(0, T)]e^{−r(T−t)}$$.

I understood the case when the authors proved that $$V(t)$$ cannot be less than $$[F(t, T) − F(0,T)]e^{−r(T−t)}$$ by building an arbitrage strategy.

However, I am not able to understand the second case (Exercise 6.6) where the authors proved that $$V(t)$$ cannot be greater than $$[F(t, T) − F(0, T)]e^{−r(T−t)}$$ by building an arbitrage strategy at the end of the book. Their proof goes like this:

At time $$t$$ • borrow and pay (or receive and invest, if negative) the amount $$V(t)$$ to acquire a short forward contract with forward price $$F(0, T)$$ and delivery date $$T$$, • initiate a new long forward contact with forward price $$F(t, T)$$ at no cost.

Then at time $$T$$ • close out both forward contracts receiving (or paying, if negative) the amounts $$S(T) − F(0, T)$$ and $$S(T) − F(t, T)$$, respectively; • collect $$V(t)e^{r(T−t)}$$ from the risk-free investment, with interest. The final balance $$V(t)e^{r(T−t)} − [F(t, T) − F(0, T)] > 0$$ will be your arbitrage profit.

My question is if $$V(t)$$ is positive and if at time $$t$$, we are borrowing $$V(t)$$ to acquire a short forward contract with forward price $$F(0,T)$$, won't we have to return $$V(t)e^{r(T−t)}$$ to the bank? In that case in the final balance won't we have $$-V(t)e^{r(T−t)} − [F(t, T) − F(0, T)]$$ (a minus symbol is coming before $$V(t)$$ since you have to clear the loan with interest at time $$T$$)??

• Please try to use MathJax for formatting going forward. It's much easier to read. Sep 7, 2023 at 13:23

The question is not self contained and hard to answer if one does not know that book. With the notation

• $$S_t$$ asset price, $$K$$ forward price, $$T$$ maturity, $$r$$ riskless rate

and assuming zero dividends the PV of the forward contract is $$V_t=S_t-e^{-r(T-t)}K\,.$$ When we sell this initially for $$V_0$$ we can use that cash to set up a self-financing trading strategy that gives us the payoff $$H_T=S_T-K$$ at time $$T\,.$$ To do so we hold one unit of the stock at all times and $$\beta_t=\frac{V_t-S_t}{e^{rt}}=-e^{-rT}K$$ units of the money market account $$B_t=e^{rt}\,.$$ To see that this is self-financing is totally simple:

Since both portfolio weights are constant: $$dV_t=dS_t+d(\beta_tB_t)=dS_t+\beta_tdB_t\,.$$ Therefore this strategy is self-financing. From this relation it also follows that $$V_T-V_0=S_T-S_0-e^{-rT}K(B_T-B_0)=S_t-e^{-rT}K-V_0$$ that is: $$V_T=H_T\,.$$ The strategy replicates the payoff.

When $$V_0$$ was initially negative we short one unit of $$S_t$$ and invest the cash surplus into $$B_t\,.$$ The formulas stay the same.

• I have edited my question. Can you please answer my specific query? Sep 8, 2023 at 3:34
• @anthony . Why should we want to follow every "proof" in a literature when we can find our own proofs we understand better? In fact: mine can be drastically simplified. As the seller of a forward contract you are obliged to pay to the buyer $S_T-K$ at maturity. What else would you do to hedge that than holding the stock and lending the amount $Ke^{-rT}$ at time zero to receive $K$ at maturity? Look at those authors' proof from the same angle. I believe their basic idea is similarly simple. Sep 8, 2023 at 4:09
• I agree with @Kurt G. In addition it does seem like the author’s proof should have started with “receive positive V(t) and invest “ .
– dm63
Sep 10, 2023 at 13:37