I was reading the book Stochastic Calculus for Finance II by Shreve and I read the proof that the forward price for the underlying $S$ at time $t$ with maturity $T$ is given by $$ For_S(t,T) = \frac{S(t)}{B(t,T)}, $$ where $S(t)$ is the Stock at time $t$ and $B(t,T)$ is the price of a ZCB at time $t$ maturing at time $T$.

The proof assumes no underlying model and simply argues that if the price would not fulfil this equation, we would have an arbitrage opportunity.

On page 242 and 243, the author (upon deriving the notion of futures contracts) calculates the value of a long forward position (startet at $t_k$) at some future date $t_j > t_k$. Using the risk neutral pricing formula he derives $$ V_{k,j} = S(t_j) - S(t_k) \cdot \frac{B(t_j,T)}{B(t_k,T)}. $$

I was curious if I can derive this equation using a no arbitrage argument: So, assume that $V_{k,j} > S(t_j) - S(t_k) \cdot \frac{B(t_j,T)}{B(t_k,T)}$. If this is the case, I could "borrow" a long forward position and sell it, yielding $V_{k,j}$. I can enter a new forward contract (at time $t_j$), for which I have to pay $S(t_j)/B(t_j,T)$. Finally I borrow $$ \frac{S(t_k)}{B(t_k,T)} - \frac{S(t_j)}{B(t_j,T)} + \frac{S(t_j)}{B(t_j,T)^2} $$ in Bonds being worth $$ \frac{S(t_k)}{B(t_k,T)}B(t_j,T) - S(t_j) + \frac{S(t_j)}{B(t_j,T)} $$ today and sell it. Effectively I gained $V_{k,j} - \frac{S(t_j)}{B(t_j,T)} + \frac{S(t_k)}{B(t_k,T)}B(t_j,T) - S(t_j) + \frac{S(t_j)}{B(t_j,T)} = V_{k,j} + \frac{S(t_k)}{B(t_k,T)}B(t_j,T) - S(t_j)> 0$ at time $t_j$. At time $T$ I still owe the $S(T) - S(t_k)/B(t_k,T)$ from the forward I shorted and I owe $\frac{S(t_k)}{B(t_k,T)} - \frac{S(t_j)}{B(t_j,T)} + \frac{S(t_j)}{B(t_j,T)^2}$ from the bonds I borrowed. From the forward I get $S(T) - S(t_j)/B(t_j,T)$. In total I owe $$ S(T) - S(t_k)/B(t_k,T) + \frac{S(t_k)}{B(t_k,T)} - \frac{S(t_j)}{B(t_j,T)} + \frac{S(t_j)}{B(t_j,T)^2} - S(T) + S(t_j)/B(t_j,T)\\ = \frac{S(t_j)}{B(t_j,T)^2} > 0. $$ I tried to vary my strategy at time $t_j$, however I am not able to make an riskfree profit. Where is my mistake or is it simply not possible to show this by a no-arbitrage argument without risk neutral pricing?

Thanks in advance!



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