4:Swaptions - Pandemonium

Mini Chapter Four

Swaptions

Swaptions – are options on interest rate swaps either with a right (but not an obligation) to enter a received or a paid position at a specific strike at the end of the option expiry, hence common reference is to a European swaption. Market lingo for a 3y forward 5y swaption addresses 3y as the expiry and 5y as the tail. Depending on the use and direction of hedge taken on interest rates there are two types of swaptions – receiver and payers; related strategies are a mix of these vanilla exposures in different variations (to be covered later).

As an example a corporate borrower keen to tap capital markets for fund raising 6 months down the road can hedge their interest rate exposure by buying a 6-month expiry payer swaption for a duration equivalent to the funding tenor at a strike of K%. This would cap their deferred borrowing cost to K% as a worst case outcome but with no obligation to exercise if funding rates are below K% at the end of 6 months.

Swaption vs a cap for a borrower’s hedge – key difference between the two option types is that a cap is a series of call options on the underlying caplets while a swaption is a series of annuity cash flows but still within a single option. Hence it’s important to think about whether a borrower wants to hedge a fixed or a floating rate liability via rate options. As per our example above a future fixed rate liability has been effectively hedged by a payer swaption, but if it were a current floating rate liability a single payer swaption wouldn’t protect against all the floating resets risks of the borrower’s interest obligation. To effectively hedge a floating rate obligation then, a cap on the reference rate can be purchased by the borrower for the full tenor of the loan hedging each of the reset risk.

A corporate which issues a 10y bond with an embedded call option at the end of 3 years (issuer’s right to call back the bond) is long a 3y call option on a 7y residual life bond or long a 3y receiver swaption on 7y swap rate.

We often hear/read about duration mismatches for Lifers (usually on account of liabilities extending duration as life expectancies go up) which keeps them perpetually bid for duration or long dated fixed rate assets. They are also observed selling payer swaptions on interest rates (put option on bonds) using the premium to enhance their overall portfolio returns. In case rates go higher and they are required to pay up on the swaption upon exercise, there is also relief alongside as their cash can be reinvested at higher rates thereafter. There’s often some subjectivity around mark to market losses in case of higher rates, but investments that bridge duration mismatches also create risk capital relief for lifers. Thus selling payer swaptions is a passive long rates exposure (in addition to the active portfolio exposure), where the premium earned gains value if rates stay constant or go lower.

- Valuation of a swaption – much like any other option on an underlying asset (interest rate in this case), valuation of a swaption is made up of both its intrinsic (moneyness) and extrinsic (time-dependent) values.

- As an example consider a payer swaption where,
  P is the USD notional of the swaption,
  T is the option’s time to maturity
  K is the strike on the underlying swap
  R_T is the current (implied) forward rate
  n is the tenor of the underlying swap
  And lastly, m is the compounding frequency of the swap i.e. it exchanges cash flows ‘m’ times per year.

Payoff of this payer swaption can be denoted as a series of payments (akin to annuities – discussed later):

\frac{P}{m} × Max (R_{T} - K, 0)

at each interest payment date of the swap which resembles the cash flow of a long call option on R_T (interest rate at option expiry) at strike K.

Let’s simplify the above and for the time being let’s avoid multi-period compounding i.e. assume annual compounding (m = 1). Think of a Bank that would like to hedge its locked-in fixed rate lending commitment to a borrower 1 year later for a period of 3 years by entering into a 1y forward 3y payer swaption (option expiry T = 1 year) at a strike of 10.50% for a notional of USD 25 mio.

Payoff of this swaption (at maturity) would be the sum of the discounted annuities over a 3 year period from when the option expires (residual maturity of the option).

If at inception below are the zero coupon forward rates:

1y forward 1y = 9.25%
1y forward 2y = 10.50%
1y forward 3y = 12%

Recall the valuation of an interest swap is same as that for a fixed rate bond with coupon equal to the par swap rate. Bootstrapping the zero rates above to arrive at the 1y forward 3y par swap rate R_T we get 11.78%. This would be the observed implied forward swap rate at the time of initiating the swaption, but referencing the zero rates here is important to connect the dots later.

ATMF 1y forward 3y swap at 11.78% amounts to a 128bps per annum gain on a USD 25 mio payer swaption at a strike of 10.50%. That’s an annuity cash flow of USD 320,000 (call it annuity A) to be discounted by the projected rates above:

Swaption payoff= \frac{A}{(1 + 9.25 %)} + \frac{A}{{(1 + 10.50 %)}^{2}} + \frac{A}{{(1 + 12 %)}^{3}}

Let’s compute the respective discount factors above separately as below:

D f_{1} = \frac{1}{(1 + 9.25 %)} = 0.9153

D f_{2} = \frac{1}{{(1 + 10.50 %)}^{2}} = 0.8190

D f_{3} = \frac{1}{{(1 + 12 %)}^{3}} = 0.7118

Swaption payoff or the PV of the cashflows received at expiry = USD 320,000 x (0.9153 + 0.8190 + 0.7118) = USD 782,752

Note that the vol of the underlying as the forward swap ages needs to be considered for valuing the swaption and the discounted cash flows above are only a sanity check/approximation for it. Time value of a swaption that’s determined by its time to expiry, volatility of the underlying in that horizon and the distance between the implied forward and strike would be the difference between the swaption value before and at expiry.

Standard market model expression (this is a common academic reference) of a single period payoff (extending from Black’s model) would be:

\frac{P}{m} × D f (T_{i}) × {R_{T} × N (d_{1}) - K × N (d_{2})}

where Df (T_i) is the discount factor for cash flows for the period (0 to T_i). Note that the period T_i is longer than the option expiry T as ‘i’ signifies the swap cash flow period that appears m times per year. Hence for an n-year swap, i ranges from 1 to mn.

Summing up for all periods then, price of the payer swaption would be:

Σ_{i = 1 t o mn} \frac{P}{m} × D f (T_{i}) × {R_{T} × N (d_{1}) - K × N (d_{2})}

where d₁ and d₂ are familiar notations from before:

d_{1} = \frac{{\log (\frac{R_{T}}{K}) + \frac{σ^{2}}{2} × T}}{σ √ T}

d_{2} = \frac{{\log (\frac{R_{T}}{K}) - \frac{σ^{2}}{2} × T}}{σ √ T}

where

T is time to expiration of the option, expressed in years
𝜎² is the variance of the underlying Swap rate
N is the cumulative standard normal distribution function

A payer swaption is like buying a put option on a bond while a receiver swaption is like buying a call option on a bond. This can be clearer if we extend from our example earlier and assume a notional of 1 dollar, let’s first define the pay-off as:

(Swaption Notional) × Max (R_{T} - K, 0) × Σ_{i = 1 t o n} (D f_{i})

Now consider a 1y put option on a 3y bond at an exercise price of par (let’s define as 1 dollar) bearing a coupon value same as strike K; 10.50% as in the example above though calling it K serves our purpose. Market price P_T (at the end of 1 year) of this bond can be computed as:

P_T = K x Df₁ + K x Df₂ + (1+K) x Df₃

P_T = K x (Df₁+ Df₂+ Df₃) + Df₃

1-P_T = (1-Df₃) – K x (Df₁+ Df₂+ Df₃)

P_T = K^* Df₁ + K^* Df₂ + (1+K)^* Df₃

P_T = K^* (Df₁+ Df₂+ Df₃) + Df₃

1-P_T = K^* (1-Df₃) – k^*(Df₁+ Df₂+ Df₃)

Plugging the above in the payoff of the put option on the bond (assuming 1 dollar notional):

(Notional) x Max { 1-P_T, 0 } = Max {(1 – Df₃) – K x (Df₁+ Df₂+ Df₃), 0}

Max {\frac{(1 - D f_{3})}{(D f_{1} + D f_{2} + D f_{3}) - K, 0}} × \sum_{i = 1 t o 3} (D f_{i})

Taking the discount factor summation out, we arrive at:

Max {\frac{(1 - D f_{3})}{(D f_{1} + D f_{2} + D f_{3}) - K, 0}} \times Σ_{i = 1 t o 3} (D f_{i})

Max {\frac{(1 - D f_{3})}{(D f_{1} + D f_{2} + D f_{3})} - K, 0} \times Σ_{i = 1 t o 3} (D f_{i})

The expression \frac{(1 - D f_{3})}{(D f_{1} + D f_{2} + D f_{3})}

The expression

\frac{(1 - D f_{3})}{(D f_{1} + D f_{2} + D f_{3})}

is same as the bootstrapped implied 3y par swap rate at the end of 1y i.e. R_T which takes us back to the payer swaption payoff.

I hope this also helps reaffirm/clarify the intuition earlier on the pay-off for caplets that are call option on a single period swap to be the same as for a put option for a same tenor coupon bearing bond (coupon same as the cap rate) with exercise price at par.

Receiver/Payers Parity – the parity equation makes a loyal appearance yet again for the swaptions world too. Assuming European options of course – receivers and payers on the same strike, with the same expiries and underlying swap tenors:

P V_{LongReceivers} - P V_{ShortPayers} = P V_{ReceivedForwardSwap}

An In-the-money receiver and hence an out of the money payer would net off on their respective time value and the remaining intrinsic value would be the same as that of a received forward starting swap at the same strike. Conversely, an out of the money receiver/in the money payer would similarly reflect the negative PV of the received forward starting swap.

Related Resources

For best reading experience please use devices with the latest versions of macOS or Windows on a chrome browser.

Mini Chapter Four

Swaptions

Share

Related Resources

Interest Rate Options Demystified

1: BSM vs Black

2: Embedded Bond Options

3: Interest Rate Cap and Floor

5: Annuity and Convexity

6: Rate Option Strategies

7: Rate Option Strategies…Conditional Spread

quick links

Varda Pandey