# 金融代写|金融模型代写Modelling in finance代考|FI307

#### Doug I. Jones

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## 金融代写|金融模型代写Modelling in finance代考|Variation on a Theme

Up to now, we have worked with generic forward curves $F_X^{\mathrm{CPN}, j}(t, u, v)$ and we have not discussed their implementation. If there were infinitely many market instruments, one for each starting date $u$ – or even better, one for each starting time – the general curve would be a good enough description of the full economy. The market would provide the forward rate for each possible date and the curve description would merely be a data storage and not a data modelling tool.

In practice, there are a lot less market instruments on which we can build the forward curves; there are a lot less than one instrument for each time $u$. Often one has monthly or quarterly information on the short part of the curve, provided by FRA, futures or short term swaps, and annual information at best on the long part of the curve, provided by swaps or swap futures. One needs to resort to a modelling mechanism of some sort to find the intermediary value, often combined with interpolation. Chapter 4 discusses the subject of interpolation.

In the next two sections, we outline two ways to describe the curves; the two methods lead to different implementations. Each of them can be combined with different interpolation schemes. This dual approach was initially described in Henrard (2012c). A third approach is proposed in Fries (2013) based on coupons present values.

The first implementation uses pseudo-discount factors. This is certainly the most commonly used approach in practice; in a lot of literature and software packages it is even the definition of multi-curve framework. Starting from the description of the multi-curve framework we proposed in the previous chapter, it may seems strange to use such a convoluted approach to describe the curves. This is mainly due to the evolution of the framework from the one-curve framework. It allows us to have the same description of the curves for the discounting curve and the forward curves. As we will see in the next chapter, a lot of the curve interpolation literature focuses on the impact of interpolation in that specific framework.

The second implementation, which in many ways should really be the first, is to describe the forward curve directly by modelling the forward rates themselves. As we will see later in this chapter and in the next one, many of the drawbacks of simple interpolation, such as linear interpolation, can be reduced by using the direct curve description in terms of forward rates. Some similarities between discounting and forward curves are lost. As the theoretical descriptions of the two types of curve are very different, having different implementations is not a problem. On the positive side, the intuition of the forward curves is more direct and not obtained through a ratio of discount factors.

## 金融代写|金融模型代写Modelling in finance代考|Forward curves through pseudo-discount factors

The pseudo-discount factor forward curves are defined as follows.
Definition 3.1 (Coupon pseudo-discount factor curves). The forward curve $P_X^{\mathrm{CDF}, j}$ $(t, s)$ is the continuous function defined for $t \leq s$ such that $P_X^{\mathrm{CDF}, j}(t, t)=1, P_X^{\mathrm{CDF}, j}(t, s)$ is an arbitrary strictly positive function for $t \leq s<\operatorname{Spot}(t)+j$, and for $t_0 \geq t, u=$ $\operatorname{Spot}\left(t_0\right)$ and $v-u+j$ one has
$$F_X^{\mathrm{CPN}, j}(t, u, v)=\frac{1}{\delta}\left(\frac{P_X^{\mathrm{CDF}, j}(t, u)}{P_X^{\mathrm{CDF}, j}(t, v)}-1\right) .$$
The origin of the above definition can be traced back to the one-curve world, where the forward rate was written with a similar formula described by Equation (2.1). Nevertheless the substance of the two formulas is very different. The one-curve formula is a result obtained from different hypotheses and from no arbitrage condition. The above formula is merely a definition. The pseudo-discount factor should be viewed as the ‘wrong number used in the wrong formula to obtain the correct result’ type of approach. The formula is better understood through the evolution than through the foundations.

Definition 3.1, which refers to an arbitrary function, is itself arbitrary in more than one way. The definition fixes the first $j$-period as the arbitrary part; one could instead fix any other $j$ period and deduce the rest of the curve from there.

In Baviera and Cassaro (2012) the authors propose to use the arbitrariness on the second period by imposing a specific interpolation in that period and building the first period in a way similar to our definition. One could even take an arbitrary decomposition of the $j$-period time interval into subintervals and distribute those subintervals arbitrarily on the real axis in such a way that, modulo the $j$-periods, they recompose the initial $j$-period. As another arbitrary choice, one could also change the value of $P_X^{\mathrm{CDF}, j}(t, t)$ to any value different from 1 . As only the ratios between two values are used and never a value on its own, the choice of initial value has no impact on the end results. One could also impose an arbitrary value for $P_X^{\mathrm{CDF}, j}$ in $(t, \operatorname{Spot}(t))$ instead of in $(t, t)$. As the curve is used only for forward computation, it is used only with time $s>\operatorname{Spot}(t)$; the value for shorter times is irrelevant.

This arbitrariness of the pseudo-discount factors makes it very difficult to design a model for those values. All the relevant models I’m aware of model the value $F_X^{C P N}, j(t, u, v)$ and not the pseudo-discount factors $P_X^{\mathrm{CDF}, j}(t, u)$. This modelling problem with pseudo-discount factors is discussed further in Section 7.1.

# 金融模型代写

## 金融代写|金融模型代写金融建模代考|通过伪贴现因子正向曲线

$$F_X^{\mathrm{CPN}, j}(t, u, v)=\frac{1}{\delta}\left(\frac{P_X^{\mathrm{CDF}, j}(t, u)}{P_X^{\mathrm{CDF}, j}(t, v)}-1\right) .$$

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