Let’s play a game. We flip a coin ten times (here is a particularly nice way to do this – you can take the Greek 2 Euro coin for example, it has Εὐρώπη (both the goddess and the continent) on it). If it is heads I pay you one Euro. If it is tails you pay me two. Oh and if you should win more than three times while we are playing we just stop the whole thing, ok ?
A fx tarf is a sequence of fx forward trades where our counterpart pays a strike rate. If the single forward is in favour of the counterpart she or he executes it on the structure’s nominal (so she or he is long a call). If it is in our favour we execute it on twice the nominal (so we are long a put on twice the nominal). And if the sum of the fixings in favour of the counterpart, with denoting the fx fixing, exceeds a given target, the remaining forwards expire without further payments. In such structures there are several usances for the coupon fixing which triggers the target: Either the full amount for this fixing is paid, or only part of the coupon necessary to reach the target, or no coupon at all.
The valuation of fx tarfs in general depend on the fx smiles for each component’s fixing. The whole smile is important here: Both the strike of the trade and the target minus the accumulated amount are critical points on the smile obviously. Since the accumulated amount is itself a random quantity after the first fixing, the whole smile will affect the structure’s value. In addition the intertemporal correlations of the fx spot on the fixing dates play an important role.
In this and probably one or more later posts I want to write about several things:
- how a classic, fully fledged monte carlo pricing engine can be implemented for this structure
- how an approximate npv for market scenarios and time decay assumptions can be calculated very quickly
- how this can be implemented in a second pricing engine and how this is related to the first engine
- how QuantLib’s lucent-transparent design can be retained when doing all this
Obviously fast pricing is useful to fill the famous npv cube which can then be used to calculate XVA numbers like CVA, DVA etc.
Today’s post is dedicated to some thoughts on the methodology for fast, approximate pricings. I am heavily inspired by a talk of some Murex colleagues here who implemented similar ideas in their platform for CVA and potential future exposure calculations. Moreover the idea is related to the very classic and simple, but brilliant paper by Longstaff and Schwartz Valuing American Options by Simulation: A Simple Least-Squares Approach, but has a slightly different flavour here.
Let’s fix a specific tarf termsheet. The structure has payment dates starting on 15-Nov-2014, then monthly until 15-Oct-2015. The fx fixing is taken to be the ECB fixing for EUR-USD two business days prior to each payment date. The nominal is 100 million Euro. Our counterpart holds the calls, we the puts and our puts are on 200 million Euro so leveraged by a factor of two. The strike is 1.10, so the counterpart’s calls were in the money at trade date.
The valuation date is 28-Apr-2015 and the remaining target is 0.10. The fx spot as of the valuation date is 1.10. The implied volatility for EUR-USD fx options is 20% (lognormal, not yet a problem in fx markets 😉 …), constant over time and flat and we assume equal Euro and USD interest rates, so no drift in our underlying Garman-Kohlagen process. The payoff mode is full coupon. Of course the assumptions on market data are only partly realistic.
The idea to approximate npvs efficiently is as follows. First we do a full monte carlo pricing in the usual way. Each path generates a npv. We store the following information on each grid point of each path
( # open Fixings , fx spot, accumulated amount so far , npv of the remaining fixings )
The hope is then that we can do a regression analysis of the npv on these main price drivers, i.e.fx spot, the already accumulated amount and the number of open fixings.
Note that this approach implies that the fx spot is taken from the “outside” XVA scenario set, but everything else (the interest rate curves and the volatility) is implied by the pricing model. This is slightly (or heavily) inconsistent with a XVA scenario set where rate curves and maybe also the volatility structure is part of the scenarios.
Let’s fix the simplest case of only one open fixing left, i.e. we put ourselves at a point in time somewhere after the second but last and the last fixing. Also we set the target to (i.e. we ignore that feature) for the time being and assume a leverage of one. Our structure collapses to a single, vanilla fx forward. We do 250k monte carlo paths and plot the results (the npv is in percent here):
Do you see what is happening ? We get a point cloud that – for fixed accumulated amount – conditioned on the spot averages averages to a line representing the fx forward npv. See below where I do this in 2d and where it gets clearer. What we note here as a first observation is that the position of the cloud depends on the accumulated amount: Lower spots are connected with lower accumulated amounts and higher spots with higher accumulated amounts. This is quite plausible, but has an impact on the areas where we have enough data to do a regression.
The next picture shows the same data but projecting along the accumulated amount dimension.
Furthermore I added a linear regression line which should be able to predict the npv given a spot value. To test this I added three more horizontal lines that estimate the npv for spot values of 1.0, 1.1 and 1.2 by averaging over all generated monte carlo data within buckets [0.99,1.01], [1.09,1.11] and [1.19,1.21] respectively. The hope is that the horizontal lines intersect with the regression line at x-values of 1.0, 1.1 and 1.2. This looks quite good here.
Let’s look at a real TARF now, i.e. setting the target to 0.15.
What is new here is that the cloud is cut at the target level, beyond collapsing simply to a plane indicating a zero npv. Quite clear, because in this area the structure is terminated before the last fixing.
Otherwise this case is not too different from the case before since we assume a full coupon payment and only have one fixing left, so we have a fx forward that might be killed by the target trigger before. More challenging is the case where we pay a capped coupon. Excluding data where the target was triggered before, in this case we get
We want to approximate npvs for spots 1.0, 1.1 and 1.2 and an accumulated amount of 0.05 (bucketed by 0.04 to 0.06) now. The target feature introduces curvature in our cloud. I take this into account by fitting a quadratic polynominal instead of only a linear function.
Furthermore we see that the npvs are limited to 15 now and decreasing for higher spots. Why this latter thing ? Actually until now I only used the fixing times as simulation times (because only they are necessary for pricing and the process can take large steps due to its simplicity), so the spot is effectively the previous fixing always. And if this is above 1.1 it excludes the possibility for higher coupons than its difference to 1.1.
Let’s add more simulation times between the fixings (100 per year in total), as it is likely to be the case in the external XVA scenarios asking for npvs in the end:
The approximation works quite ok for spot 1.0 but not to well for 1.1 and 1.2 any more (in both cases above). Up to now we have not used the accumulated amount in our npv approximation. So let’s restrict ourselves to accumulated amounts of e.g. 0.02 to 0.08 (remember that we want a prediction conditioned on an accumulated amount of 0.05, I choose a bigger bucket for the regression though to have “more” data and because I think I don’t want to compute too many regression functions on two little data sets in the end).
Better. Let’s move to the original termsheet now (leverage 2, remaining target 0.1, full coupon) and to 5 open fixings instead of only 1 to see if all this breaks down in a more complex setting (my experience says, yes, that will happen). The accumulated amount we want to approximation for is now 0.01:
Quite ok, puh. We see a new artefact now however: The quadratic regression function starts to fall again for spots bigger than 1.25. This is of course not sensible. So we have the necessity not only to compute different regression functions for different accumulated amounts (and different numbers of open fixings), but also for different spot regions. Let’s compute another quadratic regression for spots bigger than 1.2 for example (the blue graph):
That would work for higher spots.
To summarize the experiments, the approach seems sensible in general, but we have to keep in mind a few things:
The number of time steps in the simulation should be larger than for pure pricing purposes, possibly the grid’s step size should be comparable to the XVA simulation.
The regression function can be assumed to be quadratic, but not globally. Instead the domain has to be partitioned by
- the number of open fixings, possibly even
- the number of open fixings and the distance to the last fixing,
- the already accumulated amount
- the fx spot
The next task would be to think of an algorithm that does a sensible partition automatically. One idea would be to require just a certain minimum percentage of the data generated by the initial monte carlo simulation for pricing available in each partition.
Next post will be on the implementation of the two pricing engines then !