Plain Vanilla Interest Rate Swap

Question

I'm trying to build an intuitive understanding of the following

The price of the replicating portfolio at time $t$ of the floating rate receiver is

$P_t^{swap}=P_{t,t_0}-P_{t,t_N}-\bar{R}\sum_{n=1}^N(t_n-t_{n-1})P_{t,t_n}$.

(Some notation: $\bar{R}$ is the fixed rate. $P_{t,t_n}$ is the value at time $t$ of a zero coupon bond with maturity $t_n$. And we have future times $t_0,…,t_N$.)

My understanding of this is still very young and I have several questions as a result:

So is $P_t^{swap}$ essentially the money it would take to buy the side of the swap (at time $t$) that receives the floating rate, and therefore pays out the fixed rate? e.g. if $P_t^{swap}=0$, you wouldn't make money or lose money in entering this swap.

As we're the floating rate receiver here, we have to pay out the fixed rate every $t_n$ and hence the final term in the expression? It's just been modelled as a sum of zero coupon bonds?

What does $P_{t,t_0}-P_{t,t_N}$ really mean? The value of a zero coupon bond maturing at time $t_0$ minus the value of a zero coupon bond maturing at time $t_n$ (I hope that's right to say) it surely always $>0$, as who would rather buy a zero coupon bond that matures at a later time?

And finally, how is the combination of these three terms the value of the floating rate receiver's replicated portfolio?

I hope it's clear what these questions mean and apologies for anything I've missed.

Answer 1

There are several ways to understand how to price a swap. One way is to see it as a sum of Forward Rate Agreements that you can price individually. This is more or less what Probilitator explained.

A simpler way imho is this: if you are receiver of floatting leg the value of the swap at $t\leq T_0$
$$ Swap_t = Leg_{Float,t} - Leg_{Fixed,t} $$

I think you already understand how to price the fixed leg (it is a sum of coupons so its price is just the sum of the discounted coupons) so let's look at the floating leg. Simply put you just have to roll a dollar from one date of payment of the floatting leg to the next.

I will write $L(T,\delta)$ for the linear rate at time $T$ for maturity $T+\delta$ so $P(T,T+\delta) = (1+\delta L(T,\delta))^{-1}$. More precisely consider the following simple strategy:

• at time $t$, buy a ZCB with maturity $T_0$ and you sell a ZCB with maturity $T_N$ (so you will have to pay $1$ at time $T_N$).
• at time $T_0$, you receive $1$. Use it to buy ZCB's with maturity $T_1$. You can buy $1/P(T_0,T_1)$ such ZCB. (You still have to pay $1$ at time $T_N$).
• at time $T_1$, you receive $1/P(T_0,T_1) = 1 + (T_1-T_0)L(T_0,T_1-T_0)$. You pay $(T_1-T_0)L(T_0,T_1-T_0)$ and you still have $1$. Once again you use it to buy ZCB's with maturity $T_2$. (You still have to pay $1$ at time $T_N$).
• continue until the last date of the swap $T_N$.
• at time $T_N$, you receive $1/P(T_{N-1},T_N) = 1 + (T_{N}-T_{N-1})L(T_{N-1},T_{N}-T_{N-1})$. You use the $(T_{N}-T_{N-1})L(T_{N-1},T_{N}-T_{N-1})$ to pay the floating rate and you use the $1$ to pay the person you sold the ZCB with maturity $T_N$ to. You have replicated the floating leg and all you needed to start with is the money to buy one ZCB while selling another one so by absence of arbitrage $$ Leg_{Float,t} = P(t,T_0) - P(t,T_N) $$

Hope this answers your question.

Answer 2

To explain it I will need some preliminaries. A forward starting payer swap (or receiver swap of the floating leg) is an instrument where the holder pays fixed and receives floating at some predetermined points in time in the future. (The payment/exhange dates of fixed and floating could differ - e.g. the fixed leg is paid annualy and the floating is paid semi-annualy)

Now we introduce the simply compounded forward interest rate for $T>S$, $F(t,T,S)=\frac{1}{T-S}(\frac{P(t,S)}{P(t,T)}-1)$. Thus the rate prevailing at time $t$ for the expiry $S>t$ and maturity $T>t$. (The rate you would receive if you decided at $t$ to lend money at $S$ for a duration of $T-S$) These are the floating rates.

The value of a Swap is $leg_{fixed}-leg_{floating}$. Assuming a fixed rate of $K$, a notional $N=1$ and that fixed and floating are paid at the same times $(t_1,\dots t_n)$ you have:

Formula $(*)$: $P^{swap}_t=\sum_{i=1}^{n}(t_i-t_{i-1}) P(t,t_i)(F(t,t_i,t_{i+1})-K)$

Note: the first rate to be swaped is fixed at $t_0$ but the first payment is due at $t_1$

Above formula is very straight forward: You are just discounting the differences between the fixed rate and the floating rates - for this is the payoff of the swap seen at time $t$

Now we insert the formula for the $F(t,t_i,t_{i+1})$ that we have defined above.

$P^{swap}_t=\sum_{i=1}^{n}(t_i-t_{i-1}) P(t,t_i)(\frac{1}{t_i-t_{i-1}}(\frac{P(t,t_{i-1})}{P(t,t_i)}-1)-K)$ $P^{swap}_t=\sum_{i=1}^{n}[P(t,t_{i-1})-P(t,t_i)]-K\sum_{i=1}^{n}(t_i-t_{i-1})P(t,t_i)$

The telescopic sume $\sum_{i=1}^{n}[P(t,t_{i-1})-P(t,t_i)]$ simplifies to $P(t,t_0)-P(t,t_n)$ and you arrive at

Formula $(**)$: $P^{swap}_t=P(t,t_0)-P(t,t_n)-K\sum_{i=1}^{n}(t_i-t_{i-1})P(t,t_i)$ which is the formula in your question.

Also note that $P(t,t_n)+K\sum_{i=1}^{n}(t_i-t_{i-1})P(t,t_i)$ is the value of a bond with coupon $K$.

Whereas Formula $(*)$ helps in understanding the products functionality Formumla $(**)$ gives you the hedge.

As you have already note the $K$ will be selected so that $P(t,t_0)-(P(t,t_n)+K\sum_{i=1}^{n}(t_i-t_{i-1})P(t,t_i))=0$

Thus to hedge the floating receiver swap one goes short a zero bond $P(t,t_0)$ and holds a long position in a coupon bearing bond with coupon $K$. As the fixed rate payer you will be perfectly hedged for the coupon bond will always give you the necessary fixed payments $K$