# 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 $$N$$, $$\delta=T_1-T_0$$.

$$t, the FRA holder at time $$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.

• I am unable to open the link you provided. – Charles Fox Oct 15 '18 at 21:25
• @CharlesFox I just added some related pictures from the link. – Lookout Oct 16 '18 at 3:20

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.

• your result is correct. But I want to know why my solution is wrong. You get the result from the simple case, I try to get the result from scratch following the idea in slides. – Lookout Oct 16 '18 at 11:52
• Thanks, I think I understand now, and have edited the answer. – Magic is in the chain Oct 16 '18 at 13:03
• Thanks for your answer, but I want to make it clear: we can get the the formula of forward rate by FRA, the FRA holder pays at a fixed rate $K$ and receive a floating rate. The forward rate is $K$ that makes FRA have zero value. When the floating rate is simple, then $K$ is calculated following the steps in the slides. When the floating rate is continuously compounded, how to calculate $K$ following the same idea in the slides? – Lookout Oct 16 '18 at 13:19
• Added more text! – Magic is in the chain Oct 16 '18 at 15:13