Why does the valuation of the floating leg of a swap only use the next payment?

Question

At time $t=0$, swap has zero cost. In fact, both parties may have valued the swap differently based on their zero swap curve-but somehow they agreed. Once a swap is agreed upon it cannot be dissolved because it is an OTC contract.

Even if the first floating payment is known after the first reset, surely the floating payments after that are not known. It seems that one would need to estimate the evolution of the forward rates in time. No book talks about that.

They just assume that the forward rates will be realized; post a ficticious payment at the end; make them look like bonds; find discount rate that matches the value of both the legs. No simulation. I must be missing something here. I will appreciate an explanation.

Answer 1

The reason why you can price a swap without a model is because you can replicate the payoff using only zero-coupon bonds.

For the fixed leg this is trivial.

For the floating leg,

• at $T_0$ invest $1$ at Libor,
• at $T_1$ you get $1/B(T_0,T_1) = 1 + \tau L(T_0,T_1)$,
• you pay the floating coupon $\tau L(T_0,T_1)$
• reinvest $1$ at Libor
• etc...
• at $T_{n}$, you get $1/B(T_0,T_1) = 1 + \tau L(T_0,T_1)$,
• you pay the floating coupon $\tau L(T_0,T_1)$ and you keep the $1$.

So you replicated the floating leg payment and all you needed was $1$ at $T_0$ and you get back $1$ at $T_n$. So the PV of the float leg at time $t$ is $B(t,T_0) - B(t,T_n)$.

Note that I assumed the discount curve is the Libor curve. Things are actually a bit more complicated in a multicurve framework.

Answer 2

Adding to the excellent answer by @AFK, you can show the same algebraically:

Suppose your benchmark (LIBOR) rates are $r_1,r_2,\dots,r_T$ for time $1,2,\dots,T$. If the principal is $1$, the floating leg pays $r_1\cdot1$ at time $1$, $r_2\cdot1$ at time $2$, and so on. In the end the floating leg also pays the principal $1$ at time $T$ (in addition to $r_T$). Then the value $V$ of the floating leg at time $0$ is $$ V = \frac{r_1}{1+r_1} + \frac{r_2}{(1+r_1)(1+r_2)}+\dots+\frac{r_T+1}{(1+r_1)\dots(1+r_T)}. $$ Notice that $r_T+1$ payment at time $T$. Now by simplifying numerator and denominator for the last sum member you would get $$ \frac{1}{(1+r_1)\dots(1+r_{T-1})}, $$ which is that payment of $1$ at time $T-1$. Working terms all the way, you would get that $V=1$.

Rates $r_1,\dots,r_n$ are not even need to be known at time $0$, they can be realized and this is enough.

Answer 3

The currently existing interest rate curve (expressed for example in terms of the present value of 1 USD paid at various future dates, or equivalently as a list of forward rates for all future periods) is sufficient to price swaps by arbitrage. That is why no model, no simulation of future interest rates is necessary to value a swap (OTOH interest rate options like a caplet are a different kettle of fish). [Edited: Thanks AFK].

Answer 4

If I got your question correct, i guess this is what you are referring to -

Value of the floating leg = (K + 1)/(1+r) where k = interest payment based on the LIBOR during the previous reset point, r = spot rate until the next reset.

Question is why we are considering the immediately next coupon payment date for valuation of floating rate..

Answer - the coupon rate on the floating rate is reset on every reset date so as to make the value of floating leg = 1 or par on that date.

So the amount =1 in the above equation represents the PV of all the future cashflows as on the next reset date. Add to that the coupon payment on the next payment date essentially gives you the value of the floating leg as on the next reset date. Discount it today's date to give the value of the floating leg as of today.