# How to understand wedge?

It is heard that trading wedges (cap/floor straddle - swaption) is actually trading the correlation btw forward rates. How to understand this? Either swaption or cap/floor seem to be insensitive to the correlation and that's one reason it is often suggested to calibrate correlation structure of LMM/HJM to CMS spread options. Pls let me know if I missed sth.

A ‘wedge’ as understood by interest rate options traders is a structure of the form : long a cap/floor straddle struck ATM for a period of 1 yr starting in N years / short a N year into 1 year swaption straddle also struck ATM. Usually the cap/floor underlying is 3mo Libor but nowadays it can be daily SOFR.

This transaction has two principal exposures (a) it is short the correlation between the short term rates (either quarterly Libor or daily SOFR ) and (b) it is long volatility , specifically the forward vols of the short term rates that remain after swaption has expired.

‘Wedge’ is indeed a term recognized by the trading community. If you draw a diagram of forward vol vs calendar vol , the region of exposure of this trade is represented by a triangle that looks like a ‘wedge’.

To answer the comment of @JUW: yes this is well expressed in HJM framework. In that model, correlations are defined as between pairs of short term rates. Therefore as you say , (a) comes only from the swaption. However (b) is the net exposure (eg cap/floor is long 120 units of Vega/ swaption is short 100 units).

• Thanks for your insights! So can we say that (a) comes from the swaption component and (b) comes from the cap/floor part? And which part plays the dominant role? Also, shall we expect the wedge price should be close to zero?
– JUW
Nov 29, 2022 at 1:25
• I like your diagram argument. Now I understand why the product is called as wedge. It seems that you are an HJM/LMM user. lol
– JUW
Nov 29, 2022 at 1:50

I have never seen the wedge term in the literature, where do you get it from?

Caps and floors are indeed insensitive to correlation, since they are baskets of options (caplets/floorlets), but this is not the case of swaptions. Indeed, their single payoff at $$T_\text{expiry}$$ is $$\left[\sum\limits_{i = 1}{P \left(T_\text{expiry}, T_i\right) \left[F \left(T_\text{expiry}, T_{i - 1}, T_i\right) - K\right]}\right]^+$$ Because they are options on a basket, they are sensitive to the correlation between the components of the underlying basket, which are the forward rates $$F \left(\cdot, T_{i - 1}, T_i\right)$$.

• Thanks for your answer! Yes, in theory swaptions shall have sensitivity to correlation. But it is an empirical observation that they are less sensitive to correlations compared with CMS spread options and mid-curve options. So it is suggested in the literature that one shall calibrate the correlation structure of fwd rates in a term-structure-model (like HJM, LMM) to the spread options.
– JUW
Nov 29, 2022 at 1:46