I received some complaints that my blog lacks seriousness here and there (female compilers), contains misleading references (Bourianochevsky and Jarulamparabalam formula) or presents non-quant-related topics (Brad Mehldau’s CDs). The title of the blog itself is offensive. Quant work is neither fun nor a game. I apologize for these incidents. That won’t happen again.
Pippi Langstrumpf has many things in common with us quants.
She has superpowers. She is incredibly rich. And if she doesn’t like the real world as it is she just changes it. That’s what we do too. Pippi calls that “ich mach mir die Welt, widewide wie sie mir gefällt”.
For example we do not like negative rates, no problem we just pretend as if they were one percent (EUR swaptions as of April 2015) or three percent (now the shift for EUR caps!) higher as they actually are. That was a topic of one of my recent posts.
And we do other incredible things. We can change the probability of events. For example we can change the probability of winning in the lottery from way-too-low to say 90% or 95% (we don’t actually do this, because we are notoriously overpaid and have enough money already). That’s pretty close to Pippi “zwei mal drei macht vier”.
The only thing we can not do is turn impossible events into possible ones. All this is thanks to the Radon-Nikodym theorem, the Girsanov Theorem, the Novikov condition. All these names sound Russian, but Nikodym was Polish and Radon Austrian in fact.
This changing of measures is totally usual in pricing. But there are some places where the result of a computation still depends on the choice of measure. One such place is the computation of potential future exposure. Another is the extraction of probabilities for the exercise of calls.
I would like to illustrate the latter point. Let’s fix a setting and write
as a pricing formula under the so called T-forward-measure for a payoff . What is always happening in pricing is that you express the value of the payoff in terms of a numeraire, you choose a measure that makes this expression a martingale and then take the expectation to get the price. Again in terms of the numeraire, so you have to multiply by the numeraire to get the price in the usual metric. The numeraire in the T-forward-measure is the zero bond with maturity T.
Now assume represents a long option with expiry t. Sometimes people compute
and label this exercise probability for the option. Does that make sense? Example:
We consider the valuation date 30-04-2013 and price an european at the money payer swaption with expiry in years and underlying swap of years.
The implied volatility is and the yield curve is flat forward at (continously compounded rate). The mean reversion in a Hull White
model is set to and the volatility is calibrated to in order to reprice the market swaption.
Now we compute for different choices of . We get 48% for T=16, 36% for T=60, 21% for T=250. Of course we use QuantLib for this purpose. I will not give code examples today…
Obviously the measures associated to each T are very different.
One can try to fix this. We define a measure-invariant exercise probability as
which is a forward digital price for a digital triggered by . This formula is for an arbitraty numeraire . The naive exercise probability in this notation would be
on the other hand, a quantity dependent on as denoted by the subscript. Actually both notions coincide for the special choice of the t-forward-measure:
We compare and for different T-forward-measure in the example above.
T \nu p \pi 16 0.0290412 0.478684 0.481185 30 0.0290385 0.435909 0.484888 60 0.0290381 0.364131 0.480800 100 0.0290412 0.30056 0.480948 250 0.0290415 0.211921 0.479532
First of all, is the same up to numerical differences under all pricing measures, as expected. The naive exercise probability on the other hand, is obviously not well defined. And different measures do not imply slight differences in the results, they are completely different. The alternative definition again is measure independent.
The following picture shows the underlying npv as a function of the Hull White’s state variable (expressed in normalized form, i.e. with zero expectation and unit variance).
For bigger T the payoff is shifted to the right, and the option is exercised less often (only for bigger z’s).
How is that made up for in pricing. This picture shows the factor that is multiplied with before taking the expectation to get the final NPV.
The correction is rather mild for T=16 i.e. around 1. For T=250 on the other hand model states belonging to bigger z’s are weighted with much higher factors.
So can we take instead of as a definition for the exercise probability ? Not really, because for a bermudan option the exercise probabilities for the different expiries and the probability for no exercise at all will not add up to one. This is evident as well, because we use different measures for the single expiries namely the t-forward-measures (using the representation for as the t-forward above).
Sure we could fix that as well by renormalizing the vector of ‘s again. But I would just stop here for the moment. Similarly a potential future exposure defined as the quantile of an underlying npv distribution will not be well defined under pricing measures.