### 金融代写|利率建模代写Interest Rate Modeling代考|MTH5520

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## 金融代写|利率建模代写Interest Rate Modeling代考|The Hull–White Model

In the early days, many stochastic models were introduced to describe the dynamics of the short rate. As examples, see Cox et al. (1985; hereinafter,CIR), Ho and Lee (1986), Hull and White (1990), and Vasicek (1977), among others. A strong point of these models is their parsimoniousness. Additionally, these models are described by affine term structures. For details of affine models, readers are recommended to consult Duffie and Kan (1996), Björk $(2004)$, or Munk (2011).

It is known that the Ho-Lee model and the Hull-White model are special cases of the Gaussian HJM model. The Hull-White model, in particular, is one of the most popular models in many financial institutions. Following along these lines, this section introduces the Hull-White model as a special case of the HJM model.
Short rate process
Let us consider a one-dimensional process of the short rate $r(t)$ represented by
$$d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}$$
where $W_{t}$ is a one-dimensional Brownian motion under the real-world measure $\mathbf{P} ; \kappa$ and $\sigma$ are positive constants; $\theta(t)$ is a positive process; and $\varphi_{t}$ denotes the market price of risk.

It is empirically observed that the volatility of long-term interest rates is less than that of short term rates, reflecting a general phenomenon referred to as mean reversion. To model this feature, the rate at which $r(t)$ reverts to $\theta(t)$ is the speed $\kappa$, called the mean reversion rate.

The savings account $B_{t}=\exp \left{\int_{0}^{t} r(s) d s\right}$ is taken as a numéraire. We set $Z_{t}=\int_{0}^{t} \varphi_{s} d s+W_{t}$. By the Girsanov theorem, there exists a risk-neutral measure $\mathbf{Q}$ equivalent to $\mathbf{P}$ such that $Z_{t}$ is a Brownian motion under $\mathbf{Q}$. From these, the short rate $r(t)$ is represented under $\mathbf{Q}$ as
$$d r(t)=\kappa(\theta(t)-r(t)) d t+\sigma d Z_{t}$$
It is known that the price of a zero-coupon bond with maturity $T$ is given by
$$B(t, T)=\exp {-a(t, T)-b(T-t) r(t)}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|VaR Computed in the Real-world

This section studies the reason that the VaR should be computed using a real-world model. For this purpose, the valuation of the VaR depends on the choice of measure. We use the following simple example to illustrate this. For simplicity, we assume a null discount rate in the following argument (i.e. the forward price is equal to the present price).

Suppose a binary bond with expiry at time $T$ and with payoff $X$ at $T$ is given as follows.
$$\left{\begin{array}{l} \text { If } L>5 \% \text { at } T, \text { then } X=0 \ \text { If } L \leq 5 \% \text { at } T, \text { then } X=1.01, \end{array}\right.$$
where $L$ indicates the 6 -month LIBOR at $T$. Succinctly, the payoff is determined by the level of the 6 -month LIBOR at the expiry date.

The price of this security is computed by using some interest rate model under some risk-neutral measure $\mathbf{Q}$. For the model, we assume the probability distribution of $L$ as
$$\left{\begin{array}{r} \mathrm{Q}(L>5 \%)=0.09 \% \ \mathbf{Q}(L \leq 5 \%)=99.01 \% \end{array}\right.$$
With this distribution, the arbitrage price of this bond at $t=0$ is calculated by
$$(1.01 \times 0.9901+0 \times 0.0009) \times 1=1.00$$ because of the assumption of a null discount rate.
We buy this bond at price $1.00$. Let us valuate the $99 \% \mathrm{VaR}$ of this bond for holding period $T$. We can sell this for the price $1.01$ at time $T$ at a probability of more than $99 \%$. The $99 \% \mathrm{VaR}$ is valuated as the profit of $-0.01(=1.00-1.01)$ under Q.

Next, we assume that historical observation estimates for the 6-month LIBOR are
$$\left{\begin{array}{l} \mathbf{P}(L>5 \%)=2 \% \ \mathbf{P}(L \leq 5 \%)=98 \% \end{array}\right.$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Estimation of the Market Price of Risk

In empirical analysis concerning the term structure of interest rates, we are observing historical data under the real-world measure. To give an example, when we use the Hull-White model, the dynamics of the short rate is described from equation $(4.41)$ as
$$d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}$$
To calibrate this model such that this equation explains the historical dynamics of the short rate, we must estimate the parameters $\sigma$ and $\kappa$ and the market price of risk $\varphi_{t}$. In this way, we inevitably estimate $\varphi_{t}$ as part of fitting any interest rate model with the historical dynamics of the interest rates.

Along these lines, there are many studies on estimating the market price of risk in the field of economics. Some papers in this vein are Ahn and Gao (1999), Cheridito et al. (2007), Cochrane and Piazzesi (2010), De Jong (2000), and Duffee (2002). However, there are few papers that explicitly describe the method used in estimating the market price of risk in short rate models. It is even more difficult to find such papers that work with forward rate models.
In this section, we briefly describe three approaches to estimating the market price of risk in short rate models. For a more advanced treatment of this subject, we study theoretical methods for estimating the market price of risk in the forward rate model from Chapter $6 .$

## 金融代写|利率建模代写Interest Rate Modeling代考|The Hull–White Model

d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}

dr(吨)=ķ(θ(吨)−r(吨))d吨+σd从吨

## 金融代写|利率建模代写Interest Rate Modeling代考|VaR Computed in the Real-world

$$\左{ 如果 大号>5% 在 吨, 然后 X=0 如果 大号≤5% 在 吨, 然后 X=1.01,\正确的。$$

$$\left{ 问(大号>5%)=0.09% 问(大号≤5%)=99.01%\正确的。 在一世吨H吨H一世sd一世s吨r一世b在吨一世○n,吨H和一个rb一世吨r一个G和pr一世C和○F吨H一世sb○nd一个吨吨=0一世sC一个lC在l一个吨和db是 (1.01 \times 0.9901+0 \times 0.0009) \times 1=1.00$$ 因为假设零贴现率。

$$\left{ 磷(大号>5%)=2% 磷(大号≤5%)=98%\正确的。$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Estimation of the Market Price of Risk

d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}

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