Forward Prices with Different Borrowing and Lending Rates

# Replication Proofs and No-Arbitrage Proofs


# Replication Proofs and No-Arbitrage Proofs












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?