I've just started studying quantitative finance and have had questions closed on this forum for being too basic; if that's the case for this one please let me know a more suitable place to ask.

Assume the lending and borrowing interest rates are both a fixed constant $r$; let $V_K(t,T)$ be the value at time $t$ of being long a forward contract with maturity $T$ and delivery price $K$, and let $F(t,T)$ be the forward price at $t$ of a forward contract with maturity $T$. Assume the asset being traded pays no income, and let $S_t$ be the price of the asset at $t$.

We can compute $F(t,T)$ in two ways: using a replication proof or a zero-arbitrage proof. These proofs are taken from Blyth's Introduction to Quantitative Finance:

Replication. At current time $t$, we let portfolio $A$ consist of one unit of the asset and portfolio $B$ consist of long one forward contract with delivery price $K$, plus $Ke^{–r(T–t)}$ of cash which we deposit at the interest rate $r$. It's easy to verify that both portfolios have the same value at $T$, so they must have the same value at $t$; i.e., $$ S_t = V_K(t,T) + Ke^{-r(T-t)} $$ $F(t,T)$ is the delivery price $K$ such that $V_K(t,T) = 0$. Substituting for $K$ in the above equation gives us $F(t,T) = S_te^{r(T-t)}$.

No-Arbitrage. Assume $F(t,T) > S_te^{r(T-t)}$. Then starting with an empty portfolio, we can obtain a guaranteed profit by:

  • At $t$: Going short a forward contract at the forward price and borrowing $S_t$ to buy one unit of the asset.
  • At $T$: Executing the forward contract and repaying the loan.

It can be easily verified that this nets $F(t,T) - S_te^{r(T-t)}$ in profit. Assuming no-arbitrage, it follows that $F(t,T) \leq S_te^{r(T-t)}$.

Now assuming $F(t,T) < S_te^{r(T-t)}$, we can obtain a guaranteed profit by:

  • At $t$: Going long a forward contract at the forward price and selling one unit of the asset, depositing the proceeds $S_t$.
  • At $T$: Withdrawing the money and executing the forward contract. This guarantees a profit $S_te^{r(T-t)} - S_t$, and so we conclude that $F(t,T) = S_te^{r(T-t)}$.

My question is this: What happens if we instead assume that the borrowing rate is $r_{OFF}$ and the lending rate is $r_{BID}$, with $r_{BID} < r_{OFF}$? The no-arbitrage proof easily generalizes to a proof that $$S_te^{r_{BID}(T-t)} \leq F(t,T) \leq S_te^{r_{OFF}(T-t)}.$$

To my knowledge these bounds are the best we can do. But doesn't the replication proof also generalize? In portfolio $B$, the interest rate on the deposit is $r_{BID}$, so we should obtain $F(t,T) = S_te^{r_{BID}(T-t)}$. This is not a set of bounds, but rather a precise value, which shouldn't be right. Where am I going wrong?

  • 1
    $\begingroup$ To Replicate you have to actually carry out transactions (and it is different if you are replicating a long or short position), and so you have to take into account the bid ask spread. It is really no different from the Arbitrage. $\endgroup$
    – nbbo2
    Feb 28 at 9:02


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Browse other questions tagged or ask your own question.