European Swaptions

11. Interest-Rate Swaptions (Tech. Index 5/10)

We described interest-rate caps in the last chapter. A cap is a collection or basket of options (caplets), each written on an individual forward rate. A swaption, on the other hand, is one option written on a collection or a basket of forward rates, namely all the forward rates in a given forward swap’. The fact that the swap option is written on a composite object means that correlations between the individual forward rates are critical for swaptions.

Swaptions are European if there is only one exercise date, Bermudan if there are several possible exercise dates, and American if exercisable at any time. The forward swap into which the swaption exercises can be either a pay-fixed or a receive-fixed swap2.

Swaptions are usually based on Libor. Swaptions on other rates also exist.

I I

Swaption valued today to for exercise at

t*

into the forward swap between [t,,,, ,TMaf]

r I P Y E 2 - J

^Exercise

/41

The market-standard formula for a European swaption price $CSw, is the Black formula with a constant volatility. Swaption volatilities are quoted in this language in the market. The result is obtained by assuming that the forward swap breakeven (BE) rate RE, is lognormal43

'.

W e have, for a pay-fixed swaption with strike rate E ,

(1 1.1) Here,

(1 1.2) In ( R , , / E )

k

- 1 0 2 r * ]

d, =-

2

The kinematic factor $ X of notionals and discount factors corresponds t o the forward swap,

Black Swaption Formula and Consistency Issues: Jamshidian (ref. i) has proved that the Black formula is an exact result for European swaptions under certain assumptions.

Physically the BE rate corresponds to a swap with zero value which costs nothing. Hence the BE rate is not an asset, and therefore the swaption is given by the Black formula.

However there are consistency difficulties. Note that if the break-even rate is lognormal, individual forward rates cannot be exactly lognormal. Also, note that the break-even rates of all swaps cannot all consistently be lognormal. For example, the BE rates of the three swaps between successive time intervals (t, , t2). (t2, t3), and the total time interval (tl , t;) are related and cannot be independently lognormal.

4

Interest-Rate Swaptions I39

(11.3)

The discount factors serve to discount the in-arrears cash-flow payoffs in the forward swap back to the value date, today t o .

Note that deep in the money RE,

>>

E , the pay-fixed swaption approaches the pay-fixed forward swap. This means that exercise becomes highly probable if the swaption holder X can exercise into a swap with a large positive swap value, to X 's benefit.

Put-Call Parity and European Swaptions

We have a relation between options that follows simply from the fact that the sum of all probabilities of all paths starting at a given point has to be one. This becomes the statement of put-call parity for European options5. The probability sum rule is translated into the statement about the normal functions

N (x)

+

N (-x) = 1

.

We illustrate the result for European swaptions:

(11.4)

'Payers Swaption

-

'Receivers Swaption

- -

SPay-Fixed Swap

This equation also holds for the hedges ( A , y

,

vega).

European Swaption Volatility

In general, the volatility for a composite rate (the BE rate) will be less than the volatilities for individual rates. This is because correlations are less than perfect and some diversification occurs. The situation can be thought of similarly to the volatility for a basket of equities. The volatility of the basket of equities is lowered because some stock prices can go up while others go down. In a similar fashion, some forward rates can go up while others go down, keeping the BE rate relatively constant or less volatile.

In particular, if p I I , is the correlation between forward rate returns, if all dynamics are lognormal, and if we ignore the (relatively small) rate dependence of the discount factors, then the lognormal BE volatility o,, is approximately

1.1'

(1 1.5)

Put-Call Parity and American Options: Because of the complex nature of exercise, although the sum rule for probabilities of course remains, no simple relation exists for American or Bermuda options.

Here, a sort of “component weight”

5;

is given in terms of the ratio of the time- averages of the I”’ forward rate in the forward swap and the BE rate. This time average is indicated by brackets

(...).

It is to be taken over the same time window as that defining the volatility and correlations. To get this formula, factorization approximations were used for ratios of averages and for products of averages. Explicitly,

(1 1.6)

Hedging European Swaptions: Delta and Gamma (The Matrix)

European swaption A risk is hedged using the same techniques as previously described using ladders. Note that the A ladder will only be significant in the period of the forward swap. Before the start of the forward swap, small residual

A effects are present from the discount factors.

European swaption y risk needs to take into account the fact that gamma is a

matrix ₌

dl ^3;,zfl, I

mentioned in the section on swaps works reasonably well for swaptions to include the effects of the off-diagonal terms. W e emphasize again that this only holds deal-by-deal, i.e. for each swaption separately, not for a portfolio.

yll.. The approximate factorization relation

I Yl,, I

A Paradox, a Paradox, a Most Ingenious Vega Paradox6

European swaption vega risk is tricky and involves a little paradox. On the one hand, if the swaption is exercised all volatility dependence disappears. This is because the swap obtained by exercising the swaption has no volatility dependence. This would imply that the vega should be concentrated in the vega bucket containing the swaption exercise time t*

.

On the other hand, the forward rates f i individually making up the forward swap associated with the swaption live at times rr after r*

.

Hence this argues that the sensitivities to the individual forward rate volatilities

a,

should be spread out in the buckets at times t, after

t*

.

The paradox therefore is how to construct the vega risk report7.

‘

Musical Reference: Listen to the trio of Ruth, Frederic, and the Pirate King in The Pirates of Penzance, Gilbert and Sullivan (No. 19). The paradox in the operetta has to do with whether birthdays should appear or disappear, relative to Frederic’s ability to exercise an option to leave the pirates and subsequently exterminate them.

Vega Paradox: There is no good way out of this paradox. Different desks report vega using either of the two methods presented in the text. If the procedure spreading out vega

Interest-Rate Swapiions 141