Floor vs Receiver Swaption with Equal Strike

Question

Let's say we have the following two instruments.

1. A 5x10 floor (5-year floor, five years forward) with a 4% strike on 1-year SOFR and
2. A 5 into 5 European receiver swaption (right to enter into a 5-year swap, starting in 5 years) with a 4% strike on 1-year SOFR.

In other words, the instruments are otherwise identical (strike, underlying and maturity), except one is a floor and one is a receiver.

Can we say definitively which one is worth more?

Intuitively, it makes sense that the floor should be worth at least as much as the receiver swaption. But can we say for example that the floor is always worth more?

Answer 1

This is a classic question and has been asked/well-addressed several times in this forum in prior answers. Suffice it to say, a $K$-strike receiver swaption $\leq$ a $K$-strike floor and this inequality is strict if the underlying fwd rates are uncorrelated, as shown by no-arb arguments going all the way back to Merton (1973).

The issue is that these markets trade separately for historical reasons - mainly because the two products serve different purposes for investor's rate hedge requirements. So that dishevels their intimate mathematical similarities.

The more relevant question is how much more the floor should be worth than the rec swaption? One can go about answering this in a couple of different ways ways:

1. The academic/technical way is to analyze the short rate fwd/fwd (the state variable for pricing floors) correlation structure and, crucially, how these short rate fwds are related to the swap fwds (the state variable for pricing swaptions) - and ultimately contextualizing these correlations with the volatilities of each of these products... etc.
2. On the other hand, the market/practitioner way is to just outright trade the floor/receiver wedges as a single product in its own right.

Note that 1. usually employs some form of historical analysis of the fwd rates leading to what can be thought of as "realized" fwd/fwd correlations, while 2. provides the corresponding "implied" correlations. A thorough understanding warrants both approaches.

Such correlation analyses/markets have far reaching consequences to understanding forward volatility markets (and hence Bermudan swaption valuation / optimal exercise criteria), mid-curve swaptions, LMM calibration techniques and hence rates exotics in general.

So what seems like a relatively simple question actually has quite deep implications.

A good (if somewhat dated) reference is The Relative Valuation of Caps and Swaptions: Theory and Empirical Evidence by Longstaff, Santa-Clara and Schwartz, J. Fin. 2001. Also see Stephen Blyth: An Introduction to Quantitative Finance, OUP (2014), Chapter 12.

Answer 2

I'd say the floor should definitely always be worth more than the swaption.

The vol on the swaption is an average of the expected vol of forwards (averaged to some extent). Intuitively it makes sense that the unless the fwds have a correlation of 1, the swaption premia should be lower than the floor premia.

That is, a wedge should always trade positive. Obviously, supply/demand liquidity considerations apply. As the cap floor market is by far less liquid than the swaption vol market.

Answer 3

For single curve pricing (same funding/discounting and forward curve), with index $L(T_{i-1},T_i)$ and its $T_i$-forward expectation $F(T_m,T_{i-1},T_i)$ (forward rate of the index as of $T_m$), we do have (for times $T_m\leq T_{m+1}\leq \ldots \leq T_n$):

$$ \left(\sum_{i=m+1}^n P(T_m,, T_i)\tau_i(K-F(T_m,T_{i-1},T_i))\right)^+ $$ $$ \leq\sum_{i=m+1}^n P(T_m,, T_i)\tau_i(K-F(T_m,T_{i-1},T_i))^+ $$ $$ \leq \sum_{i=m+1}^n P(T_m,, T_i)\tau_i\mathbb{E}^{T_i}_{T_m}\left[(K-L(T_{i-1},T_i))^+\right], $$ where the first term is the $T_m$-time present value of the receiver swaption (physical delivery), and the last term is the one of the floor. The inequalities come from the convexity of the $^+$ function ($x^+ = \max(x,0)$) and Jensen's inequality.

Present values after $T_m$ may get messier on the floor side, if by 'SOFR' we mean the backward-looking rate based on compounding ON SOFR index (when valuation date falls between $T_{i-1}$ and $T_i$), instead of, say, the forward-looking rate based on Term SOFR index.