# (Self-study) Futures, bonds, and arbitrage

I'm currently self studying futures, so I'm sorry if this questions comes off a bit stupid. I'm currently reading a book by Walsh, J.B. Knowing the Odds: An Introduction to Probability.

I quote this part of the text:

I want to understand how the bond's worth at time $$t$$, $$(P-S_0)e^{rt}$$ came about. If I understood it right, in b) $$S_0-P$$ was invested, so how come at time $$t$$, the bond's worth is not $$(S_0-P)e^{rt}$$.

• It appears to be a "sign error" and I believe the author meant to write $(S_0-P)e^{rt}$ Jul 15 '20 at 5:08
• So good to hear. Thank you for confirming. Jul 15 '20 at 6:10
• This is not a sign error.
– Tosh
Jul 15 '20 at 14:03

So at $$t=0$$, you short the expensive side, $$S_0$$.
Use proceed to buy the cheaper side, $$P$$.
You will invest the difference, $$(S_0 - P)$$ at the risk free rate, where you multiply by $$e^{rt}$$ due to time value of money, which grows at time $$t$$.
Now at time $$t=t$$, You close the position, i.e if you have gone short at $$t = 0$$, you will go for long (liquid) position at $$t = t$$. Hence you should get a net of $$(S_0 - P) + (P-S_0)$$. This where you get this rissoles profit of $$(P-S_0)e^{rt}$$.