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I am now working with forward rates and have somehow been asked to use an "intuitive" formula for forward rates.

$$ \frac{F(0,s,T)}{F(0,t,T)} = \frac{F(s,s,T)}{F(s,t,T)} $$

I can understand the logic behind it but i am failling at proving/disproving it. I've tried to rewrite it in term of Zero Coupon Bond Price, in short term rates, but the equation are not working.

Is it because the previous equation does not hold ? Or is this because I am lacking some argument ?

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can you add what specifically $(t,t,T)$ stand for? I assume $s<t$? –  emcor Jul 7 at 21:35

3 Answers 3

up vote 4 down vote accepted
+50

Note that $\frac{F(0,s,T)}{F(0,t,T)} = \frac{T-t}{T-s}\frac{B(0,s)-B(0,T)}{B(0,t)-B(0,T)}$ and $\frac{F(s,s,T)}{F(s,t,T)} = \frac{T-t}{T-s}\frac{B(s,s)-B(s,T)}{B(s,t)-B(s,T)}$. Multiplying the numerator and denominator of the last expression with $B(0,s)$ and noting that $B(0,s)B(s,u)=B(0,u)$ (investing one Dollar for $s$ years and then for another $u-s$ years is equivalent to investing one Dollar for $u$ years) leads to the required expression.

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That was the trick I was looking for. How does it work if you add a dynamic $dF(s,t,T) = \sigma F(s,t,T) dWs$ ? –  lmorin Jul 12 at 17:43
    
Can we think of this formula as two ways to express the inflation between s and t, once seen from 0 and once from s ? –  lmorin Jul 12 at 17:45
    
This formula is derived from the definition of the forward rate. The SDE describing the forward rate process is not taken into account. Therefore, it is also valid in the case that the dynamics are as you describe them. –  KaapstadKwant Jul 15 at 13:30

1) Your question is a bit vague, are this single-curve Forward? dual (OIS + Libor)?

If s=0, this is fine (trivial). Otherwise, you lack an expectation w.r.t sigma-algebra at time-0. Also, the expectation is probably w.r.t t-forward measure here, not r.n.

without expectation - this should not be true, you cannot express unrealized, future ratio of forward rates using current time-0 data. Unless, the rates are deterministic like a constant short rate.

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He means the simple-compounded forward interest rate at times 0 (in the left hand side) and time s (in the right hand side). This rate is per definition the fixed rate in a prototypical forward rate agreement that renders the contract to be fair. –  DoubleTrouble Jul 8 at 11:46

Here is a simple (trivial) non-finance answer for the case of flat interest rates. Multiply both the denominator $F(s,t, T)$ and the numerator $F(s,s,T)$ with $e^{rs}$.

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Why not better just "multiply both the denominator and numerator with $3x+1-\cos(x)^2$"? –  emcor Jul 8 at 12:18
    
Because $e^{-rs}$ has a financial interpretation. –  KaapstadKwant Jul 8 at 12:24
    
Then you have a typo "$e^{rs}$" in either your answer or comment. And how can you discount it by the riskfree rate without no-arbitrage proof? Note that $S_s\neq e^{-rs}S_t$ –  emcor Jul 8 at 12:27
    
$e^{rs}$ was a typo. I did not intend this as the best answer to the question but rather as a very simple attempt in a special case as a first go. –  KaapstadKwant Jul 8 at 12:42

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