Deriving Continuously Compounded Forward Rates from FRA Pricing
Summary
The document derives a continuously compounded forward rate from an FRA whose floating payment is based on the growth of a zero-coupon bond over the accrual period. It relates the bond price to the rate through exponential accumulation, then writes the FRA payoff at the end of the period in terms of the bond price, fixed rate, and notional.
The apparent equivalence between the continuous and simple forward formulas comes from comparing contracts with the same end-of-period payment structure. The included explanation expresses the simple forward through a bond-price ratio and the continuous forward as the logarithm of that ratio divided by the accrual period. Thus the continuous rate is lower than the simple rate for the same bond prices. The note suggests using continuous compounding on the fixed leg as well for a more consistent comparison. It offers an algebraic explanation, not market data or a numerical example, and leaves the detailed consequences of changing the contract convention unexplored.
Key ideas
- An FRA payoff can be expressed using the zero-coupon bond price over its accrual period.
- The continuous forward rate is the logarithm of the relevant bond-price ratio divided by the accrual period.
- Simple and continuous forward rates can imply the same bond prices while having different numerical values.
- A fair compounding comparison requires consistent conventions for the fixed and floating legs.
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Full text
# continuously compound forward rate formula
# continuously compound forward rate formula
I want to derive the continuously compound forward rate formula according to FRA.
fixed rate is $K$ and notional is $N$, $\delta=T_1-T_0$.
$t<T_0<T_1$, the FRA holder at time $T_1$ need to pay fixed $N\delta K$ and recieve floated $N(e^{y(T_0,T_1)\delta}-1)$
$P(T_0,T_1)$ is the value of zero-coupon bond at $T_0$ which pays $1$ at $T_1$, then $$P(T_0,T_1)e^{y(T_0,T_1)\delta}=1$$
so the payoff of FRA at $T_1$ is $$N(e^{y(T_0,T_1)\delta}-1)-N\delta K=N(\frac{1}{P(T_0,T_1)}-1-\delta K)$$ this is the same as the simple rate case(refer to page3 -page7, this equation is the same with the second line in page5)
so the continuously compound forward rate = simple forward rate, this is obvious wrong, but I can't find the mistake.
simple rate case:
## Answer by Magic is in the chain (score 4, accepted)
https://quant.stackexchange.com/a/42216
In the simple case, you have as per first equation on your last slide:
$\frac{P(t,T_0)}{P(t,T)}=1+\delta F(t,T_0, T)$
The continuous time equivalent, assuming constant piecewise rate, as per your question, is:
$\frac{P(t,T_0)}{P(t,T)}=e^{y (T_0,T) \delta}$
Taking log of both sides, and rearranging:
$\frac{1}{\delta} \ln {\frac{P(t,T_0)}{P(t,T)}}=y (T_0,T) $
The continuous forward should be lower than the simple forward rate. The reason you are getting the same price for both is because both of your contracts exchange payments at the end, and the bond prices are fixed. Essentially the continuous forward is compounded ‘more frequently’ but it has a lower rate. If you use the same forward rates in both simple and continuous compounding then you would get diffferent prices.
To make the continuous time case more consistent, a simple approach would be to assume that the fixed rate k is also continuously compounded over the tenor. Then k would be on the same basis as the floating and you will get more interesting result.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.