In preparation for the review sessions (which will try to concentrate high "bang for buck" ideas), I thought I would collect here the running list of deemed key themes:
1. [John Hull Chapter 4 & 6; Tuckman] We can extract forward rates from the spot rate curve and vice-versa. Tuckman uses semi-annual compounding while John Hull uses continous compounding. Under continuous compounding the math is elegant due to the "time-consistent" nature of log returns ("time consistent" is Linda Allen's term for the advantage of continuous frequency; a.k.a., time additive) . There are several instances in the FRM where variations on the following relationship are employed.
by "time additive" property of continuously compounded returns:
EXP[S(T+i)*(T+i)] = EXP[S(T)*T]*EXP[F(T,T+i)*i], so that:
S(T+i)*(T+i) = S(T)*T + F(T,T+i)*i
solving for S(T+i); e.g., extend LIBOR/Zero
S(T+i) = [S(T)*T + F(T,T+i)*i] / (T+i)
solving for implied forward rate F(T,T+i):
F(T,T+i) = [S(T+i)*(T+i) - S(T)*T] / (i)
Here is an illustration in Zoho XLS; this happens to be Hull's example 6.5 (extending the LIBOR zero curve using Eurodollar Futures) but the point is generic.
There are two ideas here:
1a. The time-consistent elegance of log returns let's us simply add returns from period to period (!)
1b. [This part does not require log returns; as Tuckman shows] The forward is implied by (can be extracted from) the spot, and vice-versa due to a no-arbitrage idea: at time zero you should be indifferent to (i.e., find no advantage between) (i) investing at the [T+i] spot rate or (ii) investing at the [T] spot rate and "rolling over" into the [T,T+i] forward rate.
1. [John Hull Chapter 4 & 6; Tuckman] We can extract forward rates from the spot rate curve and vice-versa. Tuckman uses semi-annual compounding while John Hull uses continous compounding. Under continuous compounding the math is elegant due to the "time-consistent" nature of log returns ("time consistent" is Linda Allen's term for the advantage of continuous frequency; a.k.a., time additive) . There are several instances in the FRM where variations on the following relationship are employed.
by "time additive" property of continuously compounded returns:
EXP[S(T+i)*(T+i)] = EXP[S(T)*T]*EXP[F(T,T+i)*i], so that:
S(T+i)*(T+i) = S(T)*T + F(T,T+i)*i
solving for S(T+i); e.g., extend LIBOR/Zero
S(T+i) = [S(T)*T + F(T,T+i)*i] / (T+i)
solving for implied forward rate F(T,T+i):
F(T,T+i) = [S(T+i)*(T+i) - S(T)*T] / (i)
Here is an illustration in Zoho XLS; this happens to be Hull's example 6.5 (extending the LIBOR zero curve using Eurodollar Futures) but the point is generic.
There are two ideas here:
1a. The time-consistent elegance of log returns let's us simply add returns from period to period (!)
1b. [This part does not require log returns; as Tuckman shows] The forward is implied by (can be extracted from) the spot, and vice-versa due to a no-arbitrage idea: at time zero you should be indifferent to (i.e., find no advantage between) (i) investing at the [T+i] spot rate or (ii) investing at the [T] spot rate and "rolling over" into the [T,T+i] forward rate.
