Fortalecimiento del Sistema de Alerta Temprana (SAT)

We now consider the Quasi-Gaussian model (6.9) using the Markov-functional ap- proach to make clear the relationship between the Quasi-Gaussian models and MFMs. In particular we specify a swap MFM under the terminal measure Fn+1

corresponding to the numeraire D.,Tn+1. By choosing a particular combination of driving process, pre-model and marginals, the resulting swap MFM is expected to be close to the Quasi-Gaussian model (6.9).

Driving process

In order to specify a MFM, we first consider the choice of driving process. In Section 3.2, we specified a MFM by taking a Gaussian process as the driver for the sake of efficient implementation. Note that we are not forced to choose a Gaussian driving process. In order to match the Quasi-Gaussian model (6.9), we take the process (x, q) of the model (6.9) as the driver. However we will specify the MFM under the terminal measure while the process (x, q) is given under the risk-neutral measure. Thus following the change of numeraire technique we write down SDEs of (x, q) under the terminal measure.

Proposition 10. Under the terminal measureFn+1 the process (x, q) in the Quasi-

Gaussian model (6.9) satisfies

dxt= (qt−κtxt−G(t, Tn+1)λ2r(t)(αr(t) +br(t)xt)2)dt (6.28)

+λr(t)(αr(t) +br(t)xt)dWtn+1

dqt= (λ2r(t)(αr(t) +br(t)xt)2−2κ(t)qt)dt,

where the Brownian motionWn+1 under the terminal measure is defined by

dW_tn+1 :=dWQ t +G(t, Tn+1)λr(t)(αr(t) +br(t)xt)dt. Proof. Define ρt by ρt:= dFn+1 dQ _F t = DtTn+1/D0Tn+1 Bt/B0 ,

lation, we have that

dρt=−ρtG(t, Tn+1)λr(t)(αr(t) +br(t)xt)dWtQ.

From the Girsanov theorem we have that the process dW_tn+1:=dWQ t − dρt ρt ·dWQ t =dWQ t +G(t, Tn+1)λr(t)(αr(t) +br(t)xt)dt.

is a Brownian motion in the terminal measureFn+1 corresponding to the numeraire

D.,Tn+1. The result then follows from SDE (6.9).

Usually when using the Markov-functional approach for a low-dimensional MFM, the driving process is chosen to model the level of rates. For instance for the stochastic volatility MFM we developed in Chapter 3, we choose a SABR type model as the driver because Kaisajuntti and Kennedy [44] identified a SABR style model as an appropriate choice for the level of interest rates by investigating an extensive set of market data. The processx of (6.28) can be seen as modelling the level of interest rates while carrying the processq with it.

We note that the process x itself is not Markovian but (x, q) is a two- dimensional Markov process. Thus we have to set up a two-dimensional MFM although there is only one factor (Brownian motion) driving the whole economy. From the modelling perspective, this does not look a likely choice for a driver. In particular, in the Quasi-Gaussian model the form of the driver (6.3) and the neces- sity to carry around the finite variation processqis forced upon us as this is required to avoid arbitrage in the separable HJM framework. In that sense we do not truly choose a driver and we are just stuck with it. Though the driver (6.3) leads to a non- Gaussian copula, there are other choices of non-Gaussian copula, e.g. CEV driver, which would not require a two-dimensional process for a one-factor model and which maybe better from a modelling point of view. Thus the Markov-functional approach allows for more flexibility in the choice of driver.

The Markov-functional approach is closer in spirit to market models. In par- ticular the Quasi-Gaussian model is obtained from the separable HJM model while the MFM with particular choices of driver and marginals is found to be very similar to the corresponding separable market model. The HJM model models the instan- taneous forward rate which is not observable in the market while market models model the market interest rate directly which is more transparent.

Pre-model

Remember that under multi-dimensional MFMs we retain the univariate and mono- tonicity properties by introducing the idea of pre-model. Here we apply the same technique and choose a pre-model. Recall that a pre-modely_bTi,n+1−i

i :R 2 _→ Ris a function of (xTi, qTi): b y_Ti,n+1−i i (xTi, qTi) =f i_(x Ti, qTi).

In order to match the Quasi-Gaussian model (6.9), we choose the pre-model to be the functionyi,n_T +1−i

i (xTi, qTi) which is given in the Quasi-Gaussian model.

Since ZCBs are deterministic functions (6.5) of the process (x, q) in the Quasi- Gaussian model (6.9), the co-terminal swap rates can be expressed in the form of

yi,n_t +1−i(xt, qt) =

DtTi(xt, qt)−DtTn+1(xt, qt) Pn

j=iαjDtTj+1(xt, qt)

, (6.29)

where the functional form DtT(xt, qt) is given by (6.5). We choose the pre-model

b yi,n_T +1−i i (xTi, qTi) as ˆ y_Ti,n+1−i i (xTi, qTi) :=y i,n+1−i Ti (xTi, qTi). (6.30) Marginal distribution

Having chosen the driving process and the pre-model for our MFM, we now con- sider the problem of how to choose the marginal distribution of the swap rates {yi,n_T +1−i

i ;i = 1, ..., n}. In order to specify a MFM which is similar to the Quasi-

Gaussian model (6.9), we choose the marginals implied from the Quasi-Gaussian model. As we discussed in the Quasi-Gaussian model (6.9), the marginal distribu- tions of {y_Ti,n+1−i

i ;i = 1, ..., n} are approximately given by the displaced diffusion

model (6.19). Therefore we feed the displaced diffusion marginal distribution im- plied from (6.19) into our MFM.

Remark 13.Given a driving process for a MFM, one can feed in any exact marginal distributions of swap rates. This separation of the driver and marginals gives a MFM more flexibility. In contrast, in Quasi-Gaussian models, in order to fit the marginal distributions of swap rates we applied complicated approximations (see Section 6.2.3). Once the driving process of a Quasi-Gaussian model is given, the marginal distributions of swap rates are determined via the explicit formula for ZCBs (6.5).

pre-model and marginal distributions and we arrive at a MFM which is expected to be similar to the Quasi-Gaussian model (6.9). However we have flexibility to choose another combination and this will lead to a MFM different from the Quasi-Gaussian model (6.9). In this sense the Quasi-Gaussian model can be viewed as a special case of the MFM class.

In similar way, the stochastic volatility Quasi-Gaussian model (6.26) can also be viewed from the Markov-functional approach perspective by choosing an appropriate driver ((x, q) process (6.28) together with stochastic volatility process z(6.25)), pre-model (6.30) and the marginals implied from the stochastic volatility model (6.27).

Remark 14. In the stochastic volatility Quasi-Gaussian model, one has to keep track of three processes. In the stochastic volatility MFM introduced in Chapter 3 however we only need to deal with a two-dimensional Markov process.