Hotel Placetas

In this subsection we compare the Quasi-Gaussian model (6.9) to a swap MFM in terms of the calibration. In particular we consider the one-dimensional swap MFM with a Gaussian driving process together with a displaced diffusion marginal distributions of the swap rates{y_Ti,n+1−i

i ;i= 1, ..., n}.

The Quasi-Gaussian model can be calibrated in a similar way to the swap MFM. They can be both calibrated to the displaced diffusion type implied volatility skews of the co-terminal swaptions and the market correlations of swap rates. Recall that in MFMs, we discussed the one-step covariance type driving process which can be specified by calibrating to the market one-step covariance (correlation) of swap rates {y_Ti,n+1−i

i ;i = 1, ..., n}, and this is an example of parametrizations by time.

Therefore the Quasi-Gaussian model is also parameterized by time.

When the Quasi-Gaussian model and the swap MFM are calibrated to the same market prices of swaptions and market correlation structure, we are interested in the problem of whether or not the two models produce the same prices and hedges of, for example, a Bermudan swaption. We expect that the results could be different because the two models have different copulas for the driving processes. In particular the MFM has a Gaussian copula while the Quasi-Gaussian model has a non-Gaussian copula. Although they have the same marginal distributions of the swap rates{yi,n_T +1−i

i ;i= 1, ..., n}, different copulas for the drivers result in different

joint distributions of the swap rates {y_Ti,n+1−i

i ;i = 1, ..., n} which determines the

for a Bermudan swaption produced by the two models could be different. But we need to investigate if this leads to a significant difference in practice. Similar questions arise for the stochastic volatility versions of these models.

In this chapter we have studied the Quasi-Gaussian models via the Markov- functional model framework. This enables us to gain insight into how the mod- els are similar and how they are different. The Quasi-Gaussian model without stochastic volatility can be calibrated to the one-step correlation (covariance) of swap rates and its marginal distributions are approximately that of a displaced diffusion. The main difference from the standard Markov-functional model (with displaced diffusion marginal) is the driver which is non-Gaussian and requires us to track a two-dimensional process. Similarly the stochastic volatility Quasi-Gaussian model requires a three-dimensional driver whereas the stochastic volatility Markov- functional model developed in this thesis requires a two-dimensional driver and the choice of driver is motivated by empirical data rather than tractability considera- tions. Further numerical work is required to study the differences between these two models in terms of pricing and hedging path dependent derivatives.

6.A

Appendix: Markovian projection

We give the following Markovian projection from [5] and the proof can also be found there. Let us consider the following SDE

dXt=λtdWt, (6.31)

where W is a one-dimensional Brownian motion under some probability measure

P and the processλ is adapted and bounded such that (6.31) has a unique strong

solution. Define b(t, x) by

b(t, x)2=EP(λ 2

t|Xt=x).

Then the SDE

dYt=b(t, Yt)dWt, Y0 =X0,

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