# Valuation of a swap where both parties can cancel (not settle at market) with accrual method instead of present-value?

Consider a single-name total return swap (TRS) on some reference asset $S$. For concreteness, suppose the length of the contract is one year with quarterly resets, and the performance of $S$ is exchanged for LIBOR.

Then the TRS value resets at $0$ at each reset date, so for some $t$ in some period that ends at time $T$, the value of the agreement assuming there is no cancellation feature is simply $$V_{t}=\mp(S_{t}-P_{t}^{T})\pm P_{t}^{T}S_{0}L_{0}^{T}T$$ where $P_{t}^{T}$ is the discount factor observed at time $t$ for the period $[t,T]$ and $L_{0}^{T}$ is the period LIBOR (simple) rate set at time $0$ (beginning of the period) for the period $[0,T]$.

However, if at any time in the performance period of the contract either party can terminate the agreement (not settle at the market value $V_{t}$), I have seen it claimed that an accrual valuation method is appropriate and that the contract can be valued for any $t$ as $$V_{t}=\mp(S_{t}-S_{0})\pm S_{0}L_{0}^{T}(T-t).$$

I don't understand the justification for this formula. Since the bi-cancellable feature can be modeled as a long (short) call and short (long) put position (depending on the side of the contract you are on), some sort of put-call parity should be applicable, which leads to the accrual formula above, but I could not get this to quite work out (perhaps someone with more practice with these types of derivations would more easily be able to do it).

Another argument is that the hedging strategy employed by a desk selling this TRS would be to overnight repo $S$, and thus the cumulative borrowing cost upto time $t$ is in some sense $S_{0}L_{0}^{T}(T-t)$, and so if the counterparty wanted to cancel, the desk could just not roll over the repo and sell the asset to cover the position and repo loan. But besides the assumption of constant interest rates, if the term of the contract is set to $T$ at the beginning of the performance period, then ostensibly the desk takes out a $T$-period loan of $S_{0}$, charges the client the interest on this loan, and then liquidates the position at time $T$ to cover the loan and the performance gain or loss on $S$ (i.e., $S_{T}-S_{0}$). This hedging argument is how the first valuation formula is obtained (although it can also be obtained in a straight forward manner using risk-neutral pricing methods).

Any time that a contract is cancellable by either party, it will be cancelled. That's because it is always to one party's advantage to cancel rather than carry on. The exception is that the contract is worth exactly zero, which has effectively zero probability. Therefore, the value of the contract is whatever will get paid out on cancellation. I.e. The accrual value.

First little point, maybe your pricing formula is, for the libor part, L_0 * t, and not L_0 * (T-t) as when you break, you receive the accrued.

Analysis of the formula : The formula has no forecast term, it's like the call is exercised immediately at the valuation date.

Practical knowledge : From what I know, very little breakable swaps are cancelled the day they start. So this valuation method seems to be wrong, moreover it's not because it's possible to break that the break will occur, if we applied the same method for american option, they would have no time value and just intrinsic value ... quite gross.

Now we know the formula is wrong.

Call criteria proposition : I want to tell a little more by defining a reason to call. There are many reasons, I've chosen a marked to market criteria : if the forward value of the deal gets big for a counterparty, and too small for the other one, then the call occurs and the payoff is (by definition) the wrong formula, plus a 0 marked to market payoff. So it's like you have american down out/up out deal on an underlying that is the forward part of the marked to market. Moreover, that forward part is evanescent : it converges to 0 as the time passes.

Now you just have to diffuse your marked to market, you can use a backward method as you have an american criteria. You may specify the dynamic of your marked to market and the boundaries of each counterparty (unknown parameters ... !). You'll then get your new value.

Special examples :

1. if the market to market is constant and equal to 0, there'll be no call, the swap is non breakable then valued as a ... VANILLA one !

2. if 1) and moreover if you have no specified maturity, it'll last forever !

3. In general cases, the value depends on the expected time it lasts, so from the volatility, drift of the diffusion parameters and the up and down boundaries.

To resume : it's very difficult to get an accurate model as we have to make behaviour hypothesis for the call. But for sure, the accrued style value is wrong. For short maturities the swap may be more vanilla, for long ones he may be model dependent. Thank you very much for your interest and remarks on my reflection.

P.S. : an analogy is when you can break a loan without paying interests, you don't break it as soon as you can earn one dollar you need more, maybe 1000$$, 3000$$, this has a value.