# 金融代写|金融模型代写Modelling in finance代考|BUS-F541

#### Doug I. Jones

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## 金融代写|金融模型代写Modelling in finance代考|Curve calibration

The curve calibration is the subject of Chapter 5. In this section we describe a very simple curve calibration to emphasise the difference between pseudo-discount factor curve description and the direct rate curve description of the next section.
A relatively standard way to calibrate the curves $P_X^D$ and $P_X^{C D F, j}$ is to select a set of market instruments for which the market quotes are known and an equal number of node points. An interpolation scheme is selected and the discount factors (or the associated rates) on the node points are calibrated to reproduce the market quotes. The market forward rates $F_X^{\mathrm{CPN}, j}(0, u, u+j)$ can be computed from the forward pseudo-discount factors curve using Formula (3.1). The pseudo-discount factors are calibrated using the formula and the instruments are priced from the pseudo-discount factors using the same formula. Everything is coherent and we could stop here. The big difference with a direct approach is the interpolation. Do you interpolate the pseudo-discount factors or the forward rates?

A typical forward rate curve using the pseudo-discount factors approach is displayed in black in Figure 3.1. The swap data used to calibrate the curve are those proposed in (Andersen and Piterbarg, 2010, Section 6.2) in a one-curve setting. The interpolation scheme is linear on (continuously compounded) rates. For our calibration, the swaps are fixed versus three months Ibor. We use a discounting curve with flat OIS market rates at $4 \%$. The discounting curve is mostly irrelevant for the discussion in this section.

The familiar sawtooth pattern can be seen. There are two angles in the curve for each node point. One when the fixing period end date is on a node and one when the fixing period start date is on the node. One of the reasons for this unpleasant shape is probably that we have an intuition on a market quantity – the forward rate – but model it indirectly through a ratio of discount factors where our intuition is diluted. The graph represents the forward rate, which is of interest to the trader and risk manager, but the data is stored and calibrated using a different mechanism.

## 金融代写|金融模型代写Modelling in finance代考|Direct forward curves

In this approach we do not need an intermediary function. We work directly with the curve $F_X^{C P N, j}(0, u, v)$, with $u>\operatorname{Spot}(0)$ the variable of this one-dimensional function. This approach, which seems the natural approach in the multi-curve framework, has not been documented until recently and software implementations seems to mostly ignore it.

Note that in this framework, the Ibor discounting is impossible as there is no discount factor associated to the Ibor curves. There is no longer an arbitrary part to the curve. The curve is defined unambiguously (as long as the corresponding market instruments exist) for all $u \geq \operatorname{Spot}(0)$.

Modeling the forward rate by constructing the discount or spot rates is similar to fixing a pair of eyeglasses while wearing gloves. […] Instead, if one models the forward curve directly and embeds all of the desired properties into it, then by construction the spot and discount rates will have the characteristics that the modeler requires.

The advantage of this approach is that the market rates on which we have some intuition are modelled directly. In some senses, and borrowing a well known name, it could be called the Libor Market Model of forward curve description – not of curve dynamic as its namesake.

There is no longer a requirement for an arbitrary part like in Definition $3.1$ of the pseudo-discount factor approach. The interpolation and constraints can be imposed directly on the market quantities. In Figure 3.1, the forward rate using the same data as the previous approach and the same linear interpolation scheme, but with the interpolation directly on the forward rate, is presented in dotted line.
It is to each market maker or risk manager to decide which one he prefers. With the reported data, the forward rates display less sawtooth effect with the direct forward rate approach. With some other market rates, the picture may be different. In Table $3.1$ we give the figures of the maximum variation of the three months rates three months apart. It is expected that the view of three months rates at some distant time in the future will not change too widely for relatively small differences in dates. Larger variations may indicate a problem with the way we estimate forward curves.

The three month period was chosen as this is the problematic period artificially introduced by the specific form of Definition 3.1. As can be seen from the numbers,the interpolation on pseudo-discount factors introduces a significantly larger variation. The figures using the linear interpolation on the direct forward rate are not very different from the cubic spline interpolation on the pseudo-discount factors.
In Figure 3.2, we did the same comparison as in the previous figure, this time using a smoother interpolation mechanism. The interpolation scheme is natural cubic spline.

# 金融模型代写

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