## 金融代写|利率建模代写Interest Rate Modeling代考|MATH3075

2022年7月2日

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

As was seen in the last section, option pricing under the HJM model can be achieved through Monte Carlo simulations. In fact, for interest-rate derivatives pricing under the HJM model, the Monte Carlo method is the only method that has been developed so far. Although this method is flexible and widely applicable, it suffers from slow convergence, and thus it is usually not the choice of market participants for whom pricing in real time is necessary. Hence, fast pricing methods must be developed. An important device for speedy option pricing is by a proper change of measure. As a preparation, we first introduce forward contracts and the notion of forward prices.

Let us begin with forward contracts. Suppose that we want to enter a deal now to purchase an asset at a future time when both payment and delivery take place. What should be taken as the fair price for this transaction? This contract is called a forward contract. We will try to figure out the fair price for the transaction, if there is one, by arbitrage arguments. To ensure delivery, the seller must borrow money now and acquire certain units of the asset. Denote the current price of the asset by $S_{t}$, the current time by $t$, the delivery time by $T$, and the unknown fair transaction price by $F$. Assume, at first, for simplicity that the asset pays no dividend. To be able to deliver one unit of the asset, the seller then does the following transactions:

1. Short $S_{t} / P_{t}^{T}$ units of $T$-maturity zero-coupon bond.
2. Long 1 unit of the asset.

Note that 1 and 2 are a set of zero-net transactions at time $t$. At the delivery time, $T$, the seller will deliver the asset to the buyer for the price of $F$, and thus ends up with the following P\&L value,
$$V_{T}=F-\frac{S_{t}}{P_{t}^{T}}$$
If arbitrage is not possible, there must be $V_{T}=0$, giving the fair transaction price
$$F=\frac{S_{t}}{P_{t}^{T}}$$

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

Because the price of a zero-coupon bond equals par at maturity, a forward price equals its spot price at the delivery date, that is, $F_{T}^{T}=\hat{S}{T}=S{T}$. As a result, any options written on $S_{T}$ can equivalently be treated as an option on $F_{T}^{T}$. Next, we will try to price an option on $F_{T}^{T}$. For this purpose, we first need to derive the dynamics that $F_{t}^{T}$ follows.

As tradable assets, the price of a stripped-dividend asset and a zerocoupon bond are assumed to be, respectively,
\begin{aligned} \mathrm{d} \hat{S}{t} &=\hat{S}{t}\left(r_{t} \mathrm{~d} t+\boldsymbol{\Sigma}{S}^{\mathrm{T}}(t) \mathrm{d} \tilde{\mathbf{W}}{t}\right) \ \mathrm{d} P_{t}^{T} &=P_{t}^{T}\left(r_{t} \mathrm{~d} t+\mathbf{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}{t}\right) \end{aligned} By the quotient rule, the forward price satisfies \begin{aligned} \mathrm{d}\left(\frac{\hat{S}{t}}{P_{t}^{T}}\right) &=\frac{\mathrm{d} \hat{S}{t}}{P}-\frac{\hat{S}{t} \mathrm{~d} P}{P^{2}}-\frac{\mathrm{d} \hat{S}{t} \mathrm{~d} P}{P^{2}}+\frac{\hat{S}{t}(\mathrm{~d} P)^{2}}{P^{3}} \ &=\frac{\hat{S}{t}}{P}\left(r{t} \mathrm{~d} t+\boldsymbol{\Sigma}{S}^{\mathrm{T}} \mathrm{d} \tilde{\mathbf{W}}{t}-r_{t} \mathrm{~d} t-\boldsymbol{\Sigma}^{\mathrm{T}} \mathrm{d} \tilde{\mathbf{W}}{t}-\boldsymbol{\Sigma}{S}^{\mathrm{T}} \boldsymbol{\Sigma} \mathrm{d} t+\boldsymbol{\Sigma}^{\mathrm{T}} \boldsymbol{\Sigma} \mathrm{d} t\right) \ &=\frac{\hat{S}{t}}{P}\left(\boldsymbol{\Sigma}{S}-\boldsymbol{\Sigma}\right)^{\mathrm{T}}\left(\mathrm{d} \tilde{\mathbf{W}}_{t}-\boldsymbol{\Sigma} \mathrm{d} t\right) . \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|BLACK’S FORMULA FOR CALL AND PUT OPTIONS

In this section, we derive the price formula for both call and put options using the forward price and under its corresponding forward measure. The payoff of a call option on anset, $S_{t}$, is
$$V_{T}=\max \left(S_{T}-K, 0\right) \triangleq\left(S_{T}-K\right)^{+}$$

In terms of the forward price, $F_{t}^{T}=\hat{S}{t} / P{t}^{T}$, we also have
$$V_{T}=\left(F_{T}^{T}-K\right)^{+}$$
Under the $T$-forward measure, we know that the price is given by
$$V_{t}=P_{t}^{T} E^{\mathbb{Q}{T}}\left[\left(F{T}^{T}-K\right)^{+} \mid \mathcal{F}{t}\right]$$ The good news here is that $F{t}^{T}$ is a lognormal martingale under $\mathbb{Q}{T}$ : $$\mathrm{d} F{t}^{T}=F_{t}^{T} \boldsymbol{\Sigma}{F}^{\mathrm{T}} \mathrm{d} \hat{\mathbf{W}}{t}$$
where $\boldsymbol{\Sigma}{F}$ is the difference between the volatilities of the asset and the $T$-maturity zero-coupon bond: $$\mathbf{\Sigma}{F}=\mathbf{\Sigma}{S}-\mathbf{\Sigma}(t, T)$$ By repeating the procedure to derive the Black-Scholes formula, we obtain $$V{t}=\hat{S}{t} N\left(d{1}\right)-K P_{t}^{T} N\left(d_{2}\right)$$
where $N(\cdot)$ is the normal accumulative function,
$$N(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} \exp \left(-\frac{y^{2}}{2}\right) \mathrm{d} y$$
and
\begin{aligned} d_{1} &=\frac{\ln \left(\hat{S}{t} /\left(P{t}^{T} K\right)\right)+(1 / 2) \sigma_{F}^{2}(T-t)}{\sigma_{F} \sqrt{T-t}} \ d_{2} &=d_{1}-\sigma_{F} \sqrt{T-t} \end{aligned}
with
$$\sigma_{F}^{2}=\frac{1}{T-t} \int_{t}^{T}\left|\boldsymbol{\Sigma}{F}\right|^{2} \mathrm{~d} s=\frac{1}{T-t} \int{t}^{T}\left|\boldsymbol{\Sigma}_{S}(s)-\mathbf{\Sigma}(s, T)\right|^{2} \mathrm{~d} s$$

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

1. 短的 $S_{t} / P_{t}^{T}$ 单位 $T$ – 到期零息债券。
2. 多头 1 个单位的资产。
注意 1 和 2 是当时的一组零净交易 $t$. 在交货时， $T$ ，卖方将把资产交付给买方，价格为 $F$ ， 因此得到以下 P $\ \& L$ 值，
$$V_{T}=F-\frac{S_{t}}{P_{t}^{T}}$$
如果不能套利，就必须有 $V_{T}=0$ ，给出公平的交易价格
$$F=\frac{S_{t}}{P_{t}^{T}}$$

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

$F_{T}^{T}=\hat{S} T=S T$. 因此，任何写在 $S_{T}$ 可以等效地被视为一个选项 $F_{T}^{T}$. 接下来，我们将営 试为期权定价 $F_{T}^{T}$. 为此，我们首先需要推导出动力学 $F_{t}^{T}$ 跟随。

$$\mathrm{d} \hat{S} t=\hat{S} t\left(r_{t} \mathrm{~d} t+\mathbf{\Sigma} S^{\mathrm{T}}(t) \mathrm{d} \tilde{\mathbf{W}} t\right) \mathrm{d} P_{t}^{T}=P_{t}^{T}\left(r_{t} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}} t\right)$$

$$\mathrm{d}\left(\frac{\hat{S} t}{P_{t}^{T}}\right)=\frac{\mathrm{d} \hat{S} t}{P}-\frac{\hat{S} t \mathrm{~d} P}{P^{2}}-\frac{\mathrm{d} \hat{S} t \mathrm{~d} P}{P^{2}}+\frac{\hat{S} t(\mathrm{~d} P)^{2}}{P^{3}}=\frac{\hat{S} t}{P}\left(r t \mathrm{~d} t+\mathbf{\Sigma} S^{\mathrm{T}} \mathrm{d} \tilde{\mathbf{W}} t-r_{t} \mathrm{~d} t-\mathbf{\Sigma}^{\mathrm{T}} \mathrm{d} \tilde{\mathbf{W}} t-\mathbf{\Sigma} S^{\mathrm{T}}\right.$$

## 金融代写|利率建模代写Interest Rate Modeling代考|BLACK’S FORMULA FOR CALL AND PUT OPTIONS

$$V_{T}=\max \left(S_{T}-K, 0\right) \triangleq\left(S_{T}-K\right)^{+}$$

$$V_{T}=\left(F_{T}^{T}-K\right)^{+}$$

$$V_{t}=P_{t}^{T} E^{Q T}\left[\left(F T^{T}-K\right)^{+} \mid \mathcal{F} t\right]$$

$$\mathrm{d} F t^{T}=F_{t}^{T} \boldsymbol{\Sigma} F^{T} \mathrm{~d} \hat{\mathbf{W}} t$$

$$\boldsymbol{\Sigma} F=\boldsymbol{\Sigma} S-\boldsymbol{\Sigma}(t, T)$$

$$V t=\hat{S} t N(d 1)-K P_{t}^{T} N\left(d_{2}\right)$$

$$N(x)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{x} \exp \left(-\frac{y^{2}}{2}\right) \mathrm{d} y$$

$$d_{1}=\frac{\ln \left(\hat{S} t /\left(P t^{T} K\right)\right)+(1 / 2) \sigma_{F}^{2}(T-t)}{\sigma_{F} \sqrt{T-t}} d_{2}=d_{1}-\sigma_{F} \sqrt{T-t}$$

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

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