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# 数学代写|期权定价理论代写Option Pricing Theory代考|CORPFIN2504 THE BLACK–SCHOLES TERM STRUCTURE MODEL

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## 数学代写|期权定价理论代写Option Pricing Theory代考|THE BLACK–SCHOLES TERM STRUCTURE MODEL

The Black-Scholes model of Section $2.1$ is unrealistic, in that it does not allow for any term structure of interest rates (or volatility, for that matter). For that reason the model in (2.1) can be extended to
$$\mathrm{d} S_t=\mu_t S_t \mathrm{~d} t+\sigma_t S_t \mathrm{~d} W_t,$$
where both $\mu_t$ and $\sigma_t$ are deterministic processes. By risk neutrality, as in Section 2.7, we obtain
$$\mu_t=r_t^d-r_t^f .$$

From (2.71), we obtain
$$\mathrm{d} S_t / S_t=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t .$$
As in Section 2.4, we let $X_t=f\left(S_t\right)$, defined by $f(x)=\ln (x)$. Following (2.19a) we obtain
\begin{aligned} \mathrm{d} X_t &=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2} \ &=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t-\frac{1}{2} \sigma_t^2 \mathrm{~d} t . \end{aligned}
This integrates to give
$$X_T=X_0+\int_0^T \mu_s \mathrm{~d} s-\frac{1}{2} \int_0^T \sigma_s^2 \mathrm{~d} s+\int_0^T \sigma_s \mathrm{~d} W_s .$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|BREEDEN-LITZENBERGER ANALYSIS

Suppose that we have a continuum of prices available for call options with strike $K$ (all with the same time to expiry $T$ ). It was originally shown in Breeden and Litzenberger (1978) that this information is equivalent to an implied distribution. The argument is basically this.
We know that a call price is
\begin{aligned} C(K, T) &=\mathrm{e}^{-r^d T} \mathbf{E}^d\left[\left(S_T-K\right)^{+}\right] \ &=\mathrm{e}^{-r^d T} \int_0^{\infty}(s-K)^{+} f_{S_T}^d(s) \mathrm{d} s \ &=\mathrm{e}^{-r^d T} \int_K^{\infty}(s-K) f_{S_T}^d(s) \mathrm{d} s, \end{aligned}
where $f_{S_T}^d(s)$ is the probability distribution function for spot $S_T$ under the domestic risk-neutral measure.

Taking the first derivative with respect to $K$ follows by differentiating under the integral sign. The result we use is the following. Let $F(x)$ be defined by the following:
$$F(x)=\int_{a(x)}^{b(x)} f(x, s) \mathrm{d} s .$$
We then have
$$\frac{\mathrm{d}}{\mathrm{d} x} F(x)=f(x, b(x)) b^{\prime}(x)-f(x, a(x)) a^{\prime}(x)+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, s) \mathrm{d} s .$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|THE BLACK-SCHOLES TERM STRUCTURE MODEL

The Black-Scholes model of Section $2.1$ is unrealistic, in that it does not allow for any term structure of interest rates (or volatility, for that matter). For that reason the model in ( $2.1$ ) can be extended to
$$\mathrm{d} S_t=\mu_t S_t \mathrm{~d} t+\sigma_t S_t \mathrm{~d} W_t,$$
where both $\mu_t$ and $\sigma_t$ are deterministic processes. By risk neutrality, as in Section $2.7$, we obtain
$$\mu_t=r_t^d-r_t^f .$$
From (2.71), we obtain
$$\mathrm{d} S_t / S_t=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t .$$
As in Section 2.4, we let $X_t=f\left(S_t\right)$, defined by $f(x)=\ln (x)$. Following (2.19a) we obtain
$$\mathrm{d} X_t=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2} \quad=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t-\frac{1}{2} \sigma_t^2 \mathrm{~d} t$$
This integrates to give
$$X_T=X_0+\int_0^T \mu_s \mathrm{~d} s-\frac{1}{2} \int_0^T \sigma_s^2 \mathrm{~d} s+\int_0^T \sigma_s \mathrm{~d} W_s$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|BREEDENLITZENBERGER ANALYSIS

Suppose that we have a continuum of prices available for call options with strike $K$ (all with the same time to expiry $T$ ). It was originally shown in Breeden and Litzenberger (1978) that this information is equivalent to an implied distribution. The argument is basically this.
We know that a call price is
$$C(K, T)=\mathrm{e}^{-r^d T} \mathbf{E}^d\left[\left(S_T-K\right)^{+}\right] \quad=\mathrm{e}^{-r^d T} \int_0^{\infty}(s-K)^{+} f_{S_T}^d(s) \mathrm{d} s=\mathrm{e}^{-r^d T} \int_K^{\infty}(s-K) f_{S_T}^d(s) \mathrm{d} s,$$
where $f_{S_T}^d(s)$ is the probability distribution function for spot $S_T$ under the domestic risk-neutral measure.
Taking the first derivative with respect to $K$ follows by differentiating under the integral sign. The result we use is the following. Let $F(x)$ be defined by the following:
$$F(x)=\int_{a(x)}^{b(x)} f(x, s) \mathrm{d} s .$$
We then have
$$\frac{\mathrm{d}}{\mathrm{d} x} F(x)=f(x, b(x)) b^{\prime}(x)-f(x, a(x)) a^{\prime}(x)+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, s) \mathrm{d} s .$$

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