金融代写|利率建模代写Interest Rate Modeling代考|MATH4513

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金融代写|利率建模代写Interest Rate Modeling代考|Equity Options under the Hull–White Model

To price either a call or a put option by Black’s formula, we need to calculate Black’s volatility of the forward price (Equation 4.97). In applications, asset volatilities are often given in the form of a scalar instead of a vector, and asset correlations are given explicitly. In such a situation, the (square of) Black’s volatility, Equation 4.97, takes a different form. In this section, we present Black’s volatility for both the Ho-Lee model and the HullWhite model, and we work out two examples of option pricing under these models.

Consider first the pricing of an equity call option under the Ho-Lee model for interest rates. The forward-rate volatility under the Ho-Lee model is $\sigma_{0}$, resulting in the volatility of the zero-coupon bond being $\boldsymbol{\Sigma}(t, T)=-\sigma_{0}(T-t)$. Assume that the local volatility of the underlying asset is a constant, $\sigma_{S}$, and the correlation between the asset and the zero-coupon bond is $\rho$. Then, Black’s volatility of the forward price can be calculated as
\begin{aligned} \sigma_{F}^{2} &=\frac{1}{T-t} \int_{t}^{T}\left|\boldsymbol{\Sigma}{S}(u)-\boldsymbol{\Sigma}(u, T)\right|^{2} \mathrm{~d} u \ &=\frac{1}{T-t} \int{t}^{T}\left(\left|\boldsymbol{\Sigma}{S}(u)\right|^{2}-2 \boldsymbol{\Sigma}{S}^{\mathrm{T}}(u) \boldsymbol{\Sigma}(u, T)+|\boldsymbol{\Sigma}(u, T)|^{2}\right) \mathrm{d} u \ &=\frac{1}{T-t} \int_{t}^{T}\left(\sigma_{S}^{2}-2 \rho \sigma_{S} \sigma_{0}(T-u)+\sigma_{0}^{2}(T-u)^{2}\right) \mathrm{d} u \ &=\sigma_{S}^{2}-\rho \sigma_{S} \cdot \sigma_{0}(T-t)+\frac{1}{3} \sigma_{0}^{2}(T-t)^{2} . \end{aligned}
Note that a positive correlation reduces Black’s volatility. Let us wit ness the effect on the price of asset-interest rate correlation in the following example.

金融代写|利率建模代写Interest Rate Modeling代考|Options on Coupon Bonds

Options on coupon bonds actually belong to the first generation of fixedincome derivatives. Options on Treasury bonds are liquidly traded. In Section 4.6, we have studied the pricing of coupon bonds using Monte Carlo simulations. Here, we instead introduce a methodology for approximate pricing of options on coupon bonds.

As was already presented in Section 4.6, the payoffs of call options on coupon bonds take the form
$$V_{T}=\left(\sum_{i=1}^{N} \Delta T c P\left(T_{0}, T_{i}\right)+P\left(T_{0}, T_{N}\right)-K\right)^{+}$$
where $T_{0}$ is the maturity of the option. Let $B_{t}^{c}$ denote the bond price at time $t$. Then the $T_{0}$-forward price of the coupon bond is
\begin{aligned} F_{t}^{T_{0}} &=\frac{B_{t}^{c}}{P\left(t, T_{0}\right)} \ &=\sum_{i=1}^{N} \Delta T c \frac{P\left(t, T_{i}\right)}{P\left(t, T_{0}\right)}+\frac{P\left(t, T_{N}\right)}{P\left(t, T_{0}\right)} \end{aligned} $$=\sum_{i=1}^{N} \Delta T c \frac{P\left(0, T_{i}\right)}{P\left(0, T_{0}\right)} M_{i}(t)+\frac{P\left(0, T_{N}\right)}{P\left(0, T_{0}\right)} M_{N}(t)$$
Here,
\begin{aligned} M_{i}(t)=& \exp \left(\int_{0}^{t}-\frac{1}{2}\left|\boldsymbol{\Sigma}\left(s, T_{i}\right)-\boldsymbol{\Sigma}\left(s, T_{0}\right)\right|^{2} \mathrm{~d} s\right.\ &\left.+\left(\boldsymbol{\Sigma}\left(s, T_{i}\right)-\boldsymbol{\Sigma}\left(s, T_{0}\right)\right)^{\mathrm{T}} \mathrm{d} \hat{\mathbf{W}}{s}\right) \end{aligned} is a martingale under the $T{0}$-forward measure. For convenience we now rewrite Equation $4.117$ as
$$\frac{F_{t}^{T_{0}}}{F_{0}^{T_{0}}}=\sum_{i=1}^{N} \omega_{i} M_{i}(t)$$
where
$$\omega_{i}= \begin{cases}\frac{\Delta T c P\left(0, T_{i}\right)}{B_{0}^{c}}, & i<N \ \frac{(1+\Delta T c) P\left(0, T_{N}\right)}{B_{0}^{c}}, & i=N\end{cases}$$

金融代写|利率建模代写Interest Rate Modeling代考|NUMERAIRES AND CHANGES OF MEASURE

A major achievement so far in this chapter is to take zero-coupon bonds as numeraires and price options under their corresponding forward measures. Mathematically, this is merely a technique of changing the numeraire asset, followed by taking the expectation of the option payoffs under the martingale measures of the numeraire assets. In this section, we discuss this technique in a general context.

Let $\mathbb{Q}{A}$ be the martingale measure associated with reference asset $A{t}$, meaning that, for any traded asset $V_{t}$, its price relative to that of asset $A_{t}$,
$$\frac{V_{t}}{A_{t}},$$
is a $\mathbb{Q}{A}$-martingale. Consider another asset, $B{t}$, and its associated martingale measure, $\mathbb{Q}{B}$. According to the one-price principle, the value of any traded asset at time $t, V{t}$, satisfies
$$V_{t}=A_{t} E_{t}^{\mathbb{Q}{A}}\left[A{T}^{-1} V_{T}\right]=B_{t} E_{t}^{\mathbb{Q}{B}}\left[B{T}^{-1} V_{T}\right]$$

From the above equation, we obtain
$$E_{t}^{Q_{B}}\left[B_{T}^{-1} V_{T}\right]=\frac{A_{t}}{B_{t}} E_{t}^{\mathbb{Q}{A}}\left[\frac{B{T}}{A_{T}}\left(B_{T}^{-1} V_{T}\right)\right] .$$
Let $\zeta$ be the Radon-Nikodym derivative between $\mathbb{Q}{B}$ and $\mathbb{Q}{A}$ :
$$\left.\frac{\mathrm{d} \mathbb{Q}{B}}{\mathrm{~d} \mathbb{Q}{A}}\right|{\mathcal{F}{t}}=\zeta_{t} .$$
Then,
$$E_{\mathrm{t}}^{\mathbb{Q}{B}}\left[B{T}^{-1} V_{T}\right]=\zeta_{t}^{-1} E_{t}^{\mathbb{Q}{A}}\left[\zeta{T}\left(B_{T}^{-1} V_{T}\right)\right] .$$
Subtracting Equation $4.128$ from Equation 4.130, we obtain
$$0=E_{t}^{Q_{A}}\left[\left(\frac{\zeta_{T}}{\zeta_{t}}-\frac{B_{T} / A_{T}}{B_{t} / A_{t}}\right)\left(B_{T}^{-1} V_{T}\right)\right] .$$
Note that Equation $4.131$ holds for prices of any tradable assets, so we can argue that
$$\zeta_{t}=\left.\frac{\mathrm{d} \mathbb{Q}{B}}{\mathrm{~d} \mathbb{Q}{A}}\right|{\mathcal{F}{t}}=\frac{B_{t} / A_{t}}{B_{0} / A_{0}} \quad \text { a.s. }$$

金融代写|利率建模代写Interest Rate Modeling代考|Equity Options under the Hull–White Model

$\sigma_{0}$ ，导致零息债券的波动率为 $\boldsymbol{\Sigma}(t, T)=-\sigma_{0}(T-t)$. 假设标的资产的局部波动率是一个

$$\sigma_{F}^{2}=\frac{1}{T-t} \int_{t}^{T}|\boldsymbol{\Sigma} S(u)-\boldsymbol{\Sigma}(u, T)|^{2} \mathrm{~d} u \quad=\frac{1}{T-t} \int t^{T}\left(|\boldsymbol{\Sigma} S(u)|^{2}-2 \boldsymbol{\Sigma} S^{\mathrm{T}}(u) \mathbf{\Sigma}(u, T)+|\boldsymbol{\Sigma}(u, T)|^{2}\right) \mathrm{d} u$$

金融代写|利率建模代写Interest Rate Modeling代考|Options on Coupon Bonds

$$V_{T}=\left(\sum_{i=1}^{N} \Delta T c P\left(T_{0}, T_{i}\right)+P\left(T_{0}, T_{N}\right)-K\right)^{+}$$

\begin{aligned} F_{t}^{T_{0}} &=\frac{B_{t}^{c}}{P\left(t, T_{0}\right)}=\sum_{i=1}^{N} \Delta T c \frac{P\left(t, T_{i}\right)}{P\left(t, T_{0}\right)}+\frac{P\left(t, T_{N}\right)}{P\left(t, T_{0}\right)} \ &=\sum_{i=1}^{N} \Delta T c \frac{P\left(0, T_{i}\right)}{P\left(0, T_{0}\right)} M_{i}(t)+\frac{P\left(0, T_{N}\right)}{P\left(0, T_{0}\right)} M_{N}(t) \end{aligned}

$$M_{i}(t)=\exp \left(\int_{0}^{t}-\frac{1}{2}\left|\boldsymbol{\Sigma}\left(s, T_{i}\right)-\mathbf{\Sigma}\left(s, T_{0}\right)\right|^{2} \mathrm{~d} s \quad+\left(\boldsymbol{\Sigma}\left(s, T_{i}\right)-\mathbf{\Sigma}\left(s, T_{0}\right)\right)^{\mathrm{T}} \mathrm{d} \hat{\mathbf{W}} s\right)$$

$$\frac{F_{t}^{T_{0}}}{F_{0}^{T_{0}}}=\sum_{i=1}^{N} \omega_{i} M_{i}(t)$$

$$\omega_{i}=\left{\frac{\Delta T c P\left(0, T_{i}\right)}{B_{0}^{c}}, \quad i<N \frac{(1+\Delta T c) P\left(0, T_{N}\right)}{B_{0}^{c}}, \quad i=N\right.$$

金融代写|利率建模代写Interest Rate Modeling代考|NUMERAIRES AND CHANGES OF MEASURE

$$\frac{V_{t}}{A_{t}},$$

$$V_{t}=A_{t} E_{t}^{\mathbb{Q} A}\left[A T^{-1} V_{T}\right]=B_{t} E_{t}^{Q B}\left[B T^{-1} V_{T}\right]$$

$$E_{t}^{Q_{B}}\left[B_{T}^{-1} V_{T}\right]=\frac{A_{t}}{B_{t}} E_{t}^{Q A}\left[\frac{B T}{A_{T}}\left(B_{T}^{-1} V_{T}\right)\right] .$$

$$\frac{\mathrm{d} \mathbb{Q} B}{\mathrm{dQ} A} \mid \mathcal{F} t=\zeta_{t} .$$

$$E_{\mathrm{t}}^{\mathbb{Q} B}\left[B T^{-1} V_{T}\right]=\zeta_{t}^{-1} E_{t}^{\mathbb{Q} A}\left[\zeta T\left(B_{T}^{-1} V_{T}\right)\right] .$$

$$0=E_{t}^{Q_{A}}\left[\left(\frac{\zeta_{T}}{\zeta_{t}}-\frac{B_{T} / A_{T}}{B_{t} / A_{t}}\right)\left(B_{T}^{-1} V_{T}\right)\right] .$$

$$\zeta_{t}=\frac{\mathrm{d} \mathbb{Q} B}{\mathrm{~d} \mathbb{Q} A} \mid \mathcal{F} t=\frac{B_{t} / A_{t}}{B_{0} / A_{0}} \quad \text { a.s. }$$

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