The term structure and the forward rates
Let $P(t,T)$ denote the price at time $t$ of a bond paying one unit of currency at time $T$. If we see $P(t,T)$ as a function of $T$, then this will represent the term structure of bond prices available at time $t$.
Let's suppose now that at time $t$ we adopt the following strategy: we buy one bond expiring at $T_2$, costing us $P(t,T_2)$. To finance this transaction, we sell $P(t,T_2)/P(t,T_1)$ units of a bond expiring at $T_1$. Since the inflow coming from the sale of this bond is equal to the outflow due to the purchase of the $T_2$ expiring bond, this strategy has a zero initial cost.
The net result of this strategy is that we'll have an outflow of $P(t,T_2)/P(t,T_1)$ at time $T_1$ and an inflow of $1$ at $T_2$. This means that this strategy allows us to lock an interest rate of
\begin{equation}\label{eq:fwdrate}
\frac{1}{\Delta}\left(\frac{P(t,T_1)}{P(t,T_2)}-1\right)
\end{equation}
between $T_1$ and $T_2$ (here $\Delta$ represents the time in years between these two points in time). The set of rates given by \ref{eq:fwdrate} are known as forward rates, and will be denoted as $F(t,T_1,T_2)$ which must be read as the forward rate from $T_1$ to $T_2$ as seen at time $t$.
Let's now introduce the concept of instantaneous forward rates. We can think of it as the forward rate when the tenor is very small, that is
\[
f(t,T)=\lim_{\Delta \to 0}F(t,T,T+\Delta)
\]
We can find a very useful expression for this quantity by expanding the right hand side
\[
f(t,T)=\lim_{\Delta \to 0}\frac{1}{\Delta}\left(\frac{P(t,T)}{P(t,T+\Delta)}-1\right)=
\]
\[
=\lim_{\Delta \to 0}\frac{1}{P(t,T+\Delta)}\left(\frac{P(t,T)-P(t,T+\Delta)}{\Delta}\right)=-\frac{1}{P(t,T)}\frac{\partial P(t,T)}{\partial T}=
\]
\[
=-\frac{\partial \text{ln}P(t,T)}{\partial T}
\]
Tuesday, 1 March 2016
Monday, 29 February 2016
Short rate models part 1
Short rate models
In a short rate model, we suppose that the short rate $r(t)$ evoloves according to a predefined SDE. What might be surprising is that once we set a model for the short rate, then the knowledge of the value of the short rate at a time $t$ allows us to fully reconstruct the yield curve. To be convinced of this, let's recall that the bond prices might be obtained, using no arbitrage pricing, as follows \begin{equation}\label{eq:bond} P(t,T) = \mathbf{E}_t^Q\left(e^{-\int_t^Tr(u)du}\right) \end{equation} There are several short-rate models used in practice. Here we'll just focus on the simplest ones.
Ho-Lee Model
This is probably the simplest model for the short rate, and it is just a simple diffusion model \begin{equation}\label{eq:holee} dr(t) = \sigma dW(t) \end{equation} where $W(t)$ is the usual Brownian motion. We will now find a closed expression for bond prices in the Ho-Lee model. Integrating from $t$ to $u$ yields \begin{equation}\label{eq:cc} r(u) = r(t) + \sigma\int_u^tdW(s) \end{equation} Pluggin \ref{eq:cc} in \ref{eq:bond} yields \[ P(t,T) = \mathbf{E}_t^Q\left(e^{-r(t)(T-t)-\int_t^T\int_t^udW(s)du}\right) \] The double integral can be addressed using Fubini's theorem. Exchanging the order of integration between $u$ and $s$ we have \[ \int_t^T\int_t^u dW(s)du=\int_t^T\int_s^T du dW(s) = \int_t^T (T-s)dW(s) \] This implies that $-\int_t^Tr(u)du$ is normally distributed with mean $-r(t)(T-t)$ and variance \[ \frac{1}{3}\sigma^2(T-t)^3 \] This implies that $e^{-\int_t^Tr(u)du}$ is log-normal. Since the expectation of a log-normal variable is $e^{\mu+\frac{1}{2}\sigma^2}$, we can say that \begin{equation}\label{eq:holeebond} P(t,T)=\text{exp}\left(-r(t)(T-t)+\frac{1}{6}\sigma^2(T-t)^3\right) \end{equation} We can now derive the expression for the continuos yield, which is simply \[ y(t,T)=-\frac{\text{ln}P(t,T)}{T-t}=r(t)-\frac{1}{6}\sigma^2(T-t)^2 \] In the simple form \ref{eq:holee} it's clearly impossible to fit the term structure of discount bonds, since we only have two parameters to play with, $r(0)$ and $\sigma$. Let's now make the model more flexible by adding a deterministic function of time to the model \begin{equation}\label{eq:hl1} r(t) = r(0) + a(t) + \sigma W(t) \end{equation} The question now becomes: is it possible with this extended model to fit the initial term structure of discount bonds? And if so, what functional form $a(t)$ should have? If we assume the short rate follows \ref{eq:hl1}, that is, that the short rate process follows the SDE \begin{equation}\label{eq:hl2} dr(t) = a'(t)dt + \sigma dW(t) \end{equation} then following the same reasoning that led us to \ref{eq:holeebond}, will give us the expression for the bond prices at time $0$ in this new set-up, which is the same except for an additional factor \[ P_m(0,t)=\text{exp}\left(-r(0)t+\frac{1}{6}\sigma^2 t^3\right) \text{exp}\left(-\int_0^ta(u)du\right) \] where we have added the subscript $m$ to emphasize that these are the bond prices implied by the model, to distinguish it from the market observable bond prices $P(0,t)$. If we want to find the expression for $a(t)$ such that $P_m(0,t)=P(0,t)$ then we get, by taking the log of both sides of the equation we have \[ -\int_0^t a(u)du = r(0)t - \frac{1}{6}\sigma^2t^3 + \text{ln}P(0,t) \] Taking the derivative of both sides yields \[ a(t)=-\frac{\partial \text{ln}P(0,t)}{\partial t} - r(0) + \frac{1}{2}\sigma^2 t^2=f(0,t)-r(0)+\frac{1}{2}\sigma^2t^2 \]
In a short rate model, we suppose that the short rate $r(t)$ evoloves according to a predefined SDE. What might be surprising is that once we set a model for the short rate, then the knowledge of the value of the short rate at a time $t$ allows us to fully reconstruct the yield curve. To be convinced of this, let's recall that the bond prices might be obtained, using no arbitrage pricing, as follows \begin{equation}\label{eq:bond} P(t,T) = \mathbf{E}_t^Q\left(e^{-\int_t^Tr(u)du}\right) \end{equation} There are several short-rate models used in practice. Here we'll just focus on the simplest ones.
Ho-Lee Model
This is probably the simplest model for the short rate, and it is just a simple diffusion model \begin{equation}\label{eq:holee} dr(t) = \sigma dW(t) \end{equation} where $W(t)$ is the usual Brownian motion. We will now find a closed expression for bond prices in the Ho-Lee model. Integrating from $t$ to $u$ yields \begin{equation}\label{eq:cc} r(u) = r(t) + \sigma\int_u^tdW(s) \end{equation} Pluggin \ref{eq:cc} in \ref{eq:bond} yields \[ P(t,T) = \mathbf{E}_t^Q\left(e^{-r(t)(T-t)-\int_t^T\int_t^udW(s)du}\right) \] The double integral can be addressed using Fubini's theorem. Exchanging the order of integration between $u$ and $s$ we have \[ \int_t^T\int_t^u dW(s)du=\int_t^T\int_s^T du dW(s) = \int_t^T (T-s)dW(s) \] This implies that $-\int_t^Tr(u)du$ is normally distributed with mean $-r(t)(T-t)$ and variance \[ \frac{1}{3}\sigma^2(T-t)^3 \] This implies that $e^{-\int_t^Tr(u)du}$ is log-normal. Since the expectation of a log-normal variable is $e^{\mu+\frac{1}{2}\sigma^2}$, we can say that \begin{equation}\label{eq:holeebond} P(t,T)=\text{exp}\left(-r(t)(T-t)+\frac{1}{6}\sigma^2(T-t)^3\right) \end{equation} We can now derive the expression for the continuos yield, which is simply \[ y(t,T)=-\frac{\text{ln}P(t,T)}{T-t}=r(t)-\frac{1}{6}\sigma^2(T-t)^2 \] In the simple form \ref{eq:holee} it's clearly impossible to fit the term structure of discount bonds, since we only have two parameters to play with, $r(0)$ and $\sigma$. Let's now make the model more flexible by adding a deterministic function of time to the model \begin{equation}\label{eq:hl1} r(t) = r(0) + a(t) + \sigma W(t) \end{equation} The question now becomes: is it possible with this extended model to fit the initial term structure of discount bonds? And if so, what functional form $a(t)$ should have? If we assume the short rate follows \ref{eq:hl1}, that is, that the short rate process follows the SDE \begin{equation}\label{eq:hl2} dr(t) = a'(t)dt + \sigma dW(t) \end{equation} then following the same reasoning that led us to \ref{eq:holeebond}, will give us the expression for the bond prices at time $0$ in this new set-up, which is the same except for an additional factor \[ P_m(0,t)=\text{exp}\left(-r(0)t+\frac{1}{6}\sigma^2 t^3\right) \text{exp}\left(-\int_0^ta(u)du\right) \] where we have added the subscript $m$ to emphasize that these are the bond prices implied by the model, to distinguish it from the market observable bond prices $P(0,t)$. If we want to find the expression for $a(t)$ such that $P_m(0,t)=P(0,t)$ then we get, by taking the log of both sides of the equation we have \[ -\int_0^t a(u)du = r(0)t - \frac{1}{6}\sigma^2t^3 + \text{ln}P(0,t) \] Taking the derivative of both sides yields \[ a(t)=-\frac{\partial \text{ln}P(0,t)}{\partial t} - r(0) + \frac{1}{2}\sigma^2 t^2=f(0,t)-r(0)+\frac{1}{2}\sigma^2t^2 \]
Saturday, 2 January 2016
Girsanov's theorem
Girsanov's theorem
Let's consider two equivalent probability measures $\textbf{P}$ and $\textbf{P}^\theta$ on the measurable space $(\Omega,\cal{F})$. According to Radon-Nikodym's theorem, we know that there exists a martingale $\zeta^\theta$ such that \[ \zeta^\theta(t)=\mathbf{E}^\mathbf{P}_t\left(\frac{d\mathbf{P}^\theta}{d\mathbf{P}}\right) \] where $\frac{d\mathbf{P}^\theta}{d\mathbf{P}}$ is the Radon-Nikodym derivative linking the two probability measures. Let's now suppose that $\zeta^\theta$ satisfies the SDE \[ \frac{d\zeta^\theta}{\zeta^\theta}=-\theta(t)dW(t) \] This equation can be easily solved using Ito's lemma, yielding \[ \zeta^\theta = \text{exp}\left[-\int_0^t \theta_s dW_s -\frac{1}{2}\int_0^t \theta^2_s ds \right] \] Now, this is not necessarily a martingale. A sufficient (however difficult to verify) condition is the Novikov condition. For sake of simplicity, we'll assume that $\zeta^\theta$ is indeed a martingale. Under the above assumptions, if $W(t)$ is a Brownian motion under $\textbf{P}$, then the process \[ dW^\theta_t = dW_t + \theta_t dt \label{a}\tag{1} \] is a martingale under $\textbf{P}^\theta$.
Let's consider two equivalent probability measures $\textbf{P}$ and $\textbf{P}^\theta$ on the measurable space $(\Omega,\cal{F})$. According to Radon-Nikodym's theorem, we know that there exists a martingale $\zeta^\theta$ such that \[ \zeta^\theta(t)=\mathbf{E}^\mathbf{P}_t\left(\frac{d\mathbf{P}^\theta}{d\mathbf{P}}\right) \] where $\frac{d\mathbf{P}^\theta}{d\mathbf{P}}$ is the Radon-Nikodym derivative linking the two probability measures. Let's now suppose that $\zeta^\theta$ satisfies the SDE \[ \frac{d\zeta^\theta}{\zeta^\theta}=-\theta(t)dW(t) \] This equation can be easily solved using Ito's lemma, yielding \[ \zeta^\theta = \text{exp}\left[-\int_0^t \theta_s dW_s -\frac{1}{2}\int_0^t \theta^2_s ds \right] \] Now, this is not necessarily a martingale. A sufficient (however difficult to verify) condition is the Novikov condition. For sake of simplicity, we'll assume that $\zeta^\theta$ is indeed a martingale. Under the above assumptions, if $W(t)$ is a Brownian motion under $\textbf{P}$, then the process \[ dW^\theta_t = dW_t + \theta_t dt \label{a}\tag{1} \] is a martingale under $\textbf{P}^\theta$.
A simple exercise
Let's suppose we have an economy with an asset whose dynamics in the physical measure is
\[
dX_t=\mu X_t dt + \sigma X_t dW_t
\label{b}\tag{2}
\]
and a bank account $B_t$ with the usual dynamics $dB_t/B_t = r dt$. What is the dynamics of $X$ under the risk-neutral measure?
We can start by reminding that under the risk-neutral measure, the discounted asset price $X_t/B_t$ is a martingale. Let's find the dynamics of this process under the physical measure. Since \[ d\left(\frac{X_t}{B_t}\right)=\frac{dX_t}{B_t}-\frac{X_t dB_t}{B_t^2} \] we can write \[ d\left(\frac{X_t}{B_t}\right)/\frac{X_t}{B_t}=\frac{dX_t}{X_t}-\frac{dB_t}{B_t}=(\mu-r)dt+\sigma dW_t \] If we change measure to $\textbf{P}^\theta$, we get, according to Girsanov's theorem \[ d\left(\frac{X_t}{B_t}\right)/\frac{X_t}{B_t}=(\mu-r-\sigma \theta)dt+\sigma dW_t^\theta \] We need now to choose the market price of risk $\theta$ in order to remove the drift of the process. We easily get \[ \theta = \frac{\mu-r}{\sigma} \] With this choice, the normalized asset price is a martingale, therefore this is the risk-neutral measure. In order to find out what is the dynamics of the asset price (non normalised) we simply substitute \ref{a} in \ref{b} yielding \[ \frac{dX_t}{X_t}=r dt + \sigma dW_t^\theta \] which is the known result stating that asset prices grow at the risk-free rate in the risk-neutral world.
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