Liquidity and Basis Risk for Swaptions
As mentioned already, Bermudas and Americans are highly illiquid in the secondary market. Thus, considerable basis risk exists with European volatility.
Typically, limits will be set depending on the risk tolerance for this basis risk.
Fixed Maturity vs. Fixed Length Forward Swap
Generally, the forward swap arising from swaption exercise has a fixed maturity date. Sometimes the forward swap lasts for a fixed time period after the date of exercise. For European swaptions, this is the same thing. For Bermuda or American swaptions, it is different because there are several possible exercise dates.
Caplets and One-Period European Swaptions
A one-period payers swaption (e.g. exercising into 6M Libor at a given date) is theoretically the same as a caplet, since the definitions coincide. However, for CMT the equivalence is not exact. With some assumptions regarding the absence of volatility in the discount factors, the equivalence in the CMT case can however be shown to be approximately true.
Advance Notice
There is an advance-notice feature, where the swaption holder announces, before the actual exercise date, that he is going to exercise the swaption. This period is usually 30 days. The uncertainty of exercise is thus eliminated before the exercise date, and the advance notice stops the diffusion process.
Skew for Swaption Volatility
In previous chapters, we have discussed skew for equity and
FX
options extensively. Swaptions also have skew complications. The price of a swaption has an extra skew dependence on the strike or exercise rate E.
Specifically, if aInterest-Rate Swaptions I49
single volatility 0 is used to price two swaptions identical except for having different strike rates El and E 2 , the model prices C, and C, are not exactly market prices. To deal with this, the volatilities are made a function of the exercise rate to get an equivalent effective volatility, o ( E )
.
Skew here is the same idea as for equity orFX
options. Really all that is going on is that a simple model is adjusted to reproduce market prices for the securities by modifying the model parameters. Essentially, the model really just serves as an interpolation scheme using 0 ( E ).
Because volatility as quoted in the broker market is LN (lognormal) as described above, we can use 0, ( E ) with the Black formula. This 6, ( E ) can be either a parameterized function or a numerical look-up table with interpolations between the (sometimes sparse) market-implied data. In order to reproduce the market values reasonably, several parameters may have to be used.
Another procedure is to change the process from LN to “not-LN”. The skew effect as viewed from the LN model is replaced by the different dynamics from the not-LN model. For example, the “Lognorm Model” mixes LN and normal (Gaussian) processes. Because the rate changes and rate paths are different than in the LN model the probabilities of exercise become modified. Therefore, with a constant volatility oNo,-, for this “not-LN” model, we might hope to get market prices.
The behavior of skew can be inferred qualitatively. Say that the process that reproduces skew is some mix of Gaussian and lognormal. Because Gaussian rate probabilities are not suppressed at low rates, more Gaussian paths will go near r = 0 than in the LN model. For this reason, the equivalent LN volatility increases near r = 0. Therefore, for low strikes that probe the low-rate region, the effective LN volatility increases. In summary, the LN equivalent volatility with skew, 6, ( E ) , increases if strikes E decrease.
Since far-from-the-money swaptions do not trade regularly16, model prices are used. Sometimes quants believe that their models give the ”correct” prices when there are no market quotes-or sometimes even when the market quotes differ from the model”.
Far From the Money Options: Such options do not trade regularly, and so market quotes are often not available. In a sense, this is not terribly critical, since if an option is far ITM it will probably be exercised with a payoff independent of the volatility, while if it is far OTM it is not worth much regardless of the volatility. For transactions of far OTM options, a “nuisance” charge is sometimes applied because it costs money to monitor the option in addition to the small option value. A nuisance charge cannot be modeled by skew.
16
Aristotle’s Horse and Correct Models: While it may be difficult to obtain market quotes for illiquid instruments, it also puts the egoistic modeler in the same boat as the possibly apocryphal story about Aristotle. The story is that Aristotle theoretically (and
Model Dependence of Vega and
Risk
Aggregation ProblemsBecause volatility parameters in non-LN models are not standard LN volatilities, the volatility dependence (vega) for risk management is also model dependent''.
This difference in vega between models for the same Bermuda swaption can be on the order of 10%. This can lead to aggregation problems for volatility risk between desks that use different models.
One way of avoiding such aggregation problems is to demand that desks uniformly carry out the same risk procedure. This procedure could be to move the input LN volatilities for European swaptions by 1 %, revalue the whole swaption book using whatever non-LN model the desk has, and then take the difference of the revalued book with the value of the book using the original input LN European swaption volatilities.
Theta
The measurement of theta 6 (time decay) for swaptions can be monitored.
Revaluations can be done one day apart. In performing the revals, it must be specified whether rates are held constant or whether the rates are allowed to slide down the yield curve, as mentioned in previous chapters.
References
' Black Swaption Formula
Jamshidian, F., Sorting Out Swaptions. Risk, 3/96 and W B O R and swap market models and measures. Finance and Stochastics Vol 1, 1997.
" Swaption Example
Bloomberg News, N. Y. Transit Agency Planning $360 Million Swaption. 1/29/99.
incorrectly) deduced the number of teeth in a horse, and then refused to look at a real horse to count the teeth because he said his idea had to be correct. The decisive test for a model is whether the model price is near the price received if the option is sold.
Misunderstanding Vega: The fact that vega is model dependent can lead to misunderstandings. Comprehension of the technical fine points of volatility in complicated models outside a quant group is generally shaky. Therefore, the problem can exist but go unnoticed.