# 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?

• 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. Feb 28 at 9:02