# Swap

## Definition

A swap is a contract in which two parties agree to exchange a stream of cash flows of one type against a stream of cash flows of another type. An interest rate swap (IRS), often referred as a plain vanilla swap, is an agreement to exchange cash flows determined by a floating interest rate against cash flows determined by a fixed interest rate.

A basic interest rate swap can be interpreted as a linear combination of forward rate agreements.

## Pricing

The value of a forward rate agreement with settlement day $T$ and maturity $T + \Delta$ is given by

$\displaystyle{V_{FRA}(t) = \left[\left(P(t,T) - P(t,T + \Delta)\right) - P(t,T + \Delta) L \Delta\right]F }$

for a fixed rate $L$, the so-called swap level, and a notional amount $F$.

Since the payoff structure of a basic swap is equal to the cash flows generated by a series of FRAs, the value on an arbitrage-free market must be

$\displaystyle{\sum_{i=0}^{N-1}V_{FRA}(t, T_{i}, T_{i+1})\,, }$

where $T_{0},...,T_{N-1}$ are the reset dates and $T_{i+1} = T_{i} + \Delta_{i}$. The period $\Delta_{i}$ is the tenor of the i-th payment. $T_{0} - t$ will be called the lead time and $T_{N}$ the maturity.

It follows that the value equals

$\displaystyle{\sum_{i=0}^{N-1}\left[\left(P(t,T_{i}) - P(t,T_{i+1})\right) - P(t,T_{i+1}) L \Delta\right]F \,. }$

The sum can be represented as the sum of two sums. The expression

$\displaystyle{\sum_{i=0}^{N-1}\left(P(t,T_{i}) - P(t,T_{i+1})\right)F = \left(P(t,T_{0}) - P(t,T_{N})\right)F }$

is the value of the floating leg and does not depend on the intermediate reset dates.

The value of the fixed leg is simply the sum of the discounted fixed payments, which are known in advance:

$\displaystyle{-\sum_{i=0}^{N-1} P(t,T_{i+1}) L \Delta_{i} F \,.}$

The value of the swap is simply the sum of the fixed leg and the floating leg value and henceforth given by

$\displaystyle{\left[\left(P(t,T_{0}) - P(t,T_{N})\right) - \sum_{i=0}^{N-1} P(t,T_{i+1}) L \Delta_{i} \right]F \,.}$

## Swap Yield

The swap yield is the fixed rate $L$ at which the contract has a value of $0$. Thus, this level $L^{*}$ is given by

$\displaystyle{ L^{*} = \frac{P(t,T_{0}) - P(t, t_{N})}{\sum_{i=0}^{N-1} P(t, T_{i+1}) \Delta_{i}} \,. }$

If the lead time $T_{0} - t$ is greater than $0$, then the swap contract is called a forward swap. If $T_{0} = t$, then $P(t,T_{0})$ simplifies to the payoff at maturity and equals $1$.

There are two representations of the swap yield which give further insight into the structure of swap contracts, both starting with a reformulation of the floating leg. First note that the value of the floating leg can be written as

$\displaystyle{\sum_{i=0}^{N-1}\frac{1}{\Delta_{i}}\left(\frac{P(t,T_{i})}{P(t,T_{i+1})} - 1\right)\,P(t,T_{i+1})\Delta_{i}F }$

or equivalently by

$\displaystyle{\sum_{i=0}^{N-1}f(t,T_{i},T_{i+1})P(t,T_{i+1})\Delta_{i}F \,. }$

Hence, the swap yield can be either represented as a linear combinations of simply compounded forward rates,

$\displaystyle{L^{*} = \sum_{j=0}^{N-1} v_{j} f(t,T_{j},T_{j+1}) \,,}$

where the weights

$\displaystyle{v_{j} = \frac{P(t, T_{j+1})}{\sum_{i=0}^{N-1} P(t, T_{i+1})\Delta_{i}}}\Delta_{j} \,, }$

depend on the term structure of discount factors (or equivalently, zerobonds), or as a linear combination of zerobonds,

$\displaystyle{L^{*} = \sum_{j=1}^{N} w_{j} P(t, T_{j}) \,,}$

where the weights are

$\displaystyle{w_{j} = \frac{f(t,T_{j-1}, T_{j})}{\sum_{i=1}^{N} P(t, T_{i})\Delta_{i-1}}}\Delta_{j-1} \,.}$

## Codebook

### Constructors

The class IRSwap provides a factory method to create interest rate swaps:

double notional = 1000000;
ILevelTS ts_nelson_siegel = SvenssonTS(0.06, -0.02, 0.02, 1.0);
ts_nelson_siegel.setExternalCompounding(Compounding.SIMPLE);
double tenors = {0.25, 0.25, 0.25, 0.25, 0.25};
IRSwap swap1 = IRSwap.create(notional, ts_nelson_siegel, lead_time, tenors);


The fixed rate is set to the swap yield during creation. Thus, the value of the contract will always be 0 after calling this method. It is, however, possible to change the fixed rate at any time.

### Methods

Most users will apply the IRSwap class to pricing problems. Since there are payments if a swap is established, both parties must have assigned a value of 0 to the contract. Thus, the most important question is how to choose the fixed rate, such that value is really 0, i.e. how to determine the swap yield. If an IRSwap object has just been created, you can call the fixedRate() method, since the initial fixed rate is set to the swap yield. Later on, if the fixed rate might have changed, the swap yield can be obtained by calling the swapYield() method:

double swap_yield = swap1.swapYield();   // swap_yield = in this example


Another question that might arise is to determine the price of a contract that has been established at a given swap level. The value can be determined by calling the value() method

swap1.setFixedRate(0.05);   // sets the fixed rate to 5 %
double value = swap1.value();   // value = in this example.