We consider a portfolio of bonds $V_i$. Each bond $V_i(t,y)$ depends on the same stochastic variable $y$. We interpret $y$ as the instantaneous short rate (continuously compounded risk-free rate at time $t$). To derive the partial differential equation (PDE) satisfied by each bond under this one-factor model, we follow a similar procedure as in the derivation for the [[Black-Scholes PDE]]. >[!note:] >In [[Black-Scholes PDE]] we denoted $V$ as the financial derivative and $S$ as the stock price, which is modeled as the stochastic variable. Now instead, $V$ is the bond price and the short rate $y$ is the stochastic variable. Note that in this setting the stochastic variable is not a tradable asset of the portfolio, that we are building. Using [[Itô Processes and Itô's Lemma#Itô's Lemma for $F(t, X_t)$|Itô's Lemma]] for bond $V_i$ (a function of an Itô process) where we collect all terms in $dt$. $ dV_i= \left (\frac{\partial V_i}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_i}{\partial y^2} \right)\, dt+ \frac{\partial V_i}{\partial y} \, dy$ ## Forming a Risk-Free Portfolio We form a portfolio $\pi$ of (for simplicity) two bonds, where $q_1$ and $q_2$ are the quantities of the bonds $V_1$ and $V_2$ that we hold. $ \pi= q_1 V1 + q_2V_2$ Since $y$ is our single source of randomness in this one-factor model, we must choose the $q_i$ terms such that the portfolio sensitivity to $y$ is zero. $ \frac{\partial \pi}{\partial y} = q_1\frac{\partial V_1}{\partial y}+q_2\frac{\partial V_2}{\partial y} = 0$ Solving this equation for $\frac{q_1}{q_2}$ gives: $ \frac{q_1}{q_2} = -\frac{\partial V_2 / \partial y}{\partial V_1 / \partial y}$ ## Risk-Free Rate Condition The change in the portfolio is: $ d\pi = q_1 \, dV_1+q_2 \, dV_2 $ Substituting the expansions from for $dV_1$ and $dV_2$ we get: $ d \pi = \sum_{i=1,2} \, q_i\left(\frac{\partial V_i}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_i}{\partial y^2} \right)\, dt+\sum_{i=1,2} q_i \frac{\partial V_i}{\partial y} dy $ Since we have chosen $q_i$ such that the portfolio change $d \pi$ is not impacted by changes in $y$, we can cancel out the $dy$ terms. $ d \pi = \sum_{i=1,2} \, q_i\left(\frac{\partial V_i}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_i}{\partial y^2} \right)\, dt $ Because this portfolio $\pi$ is now risk-free, it can only earn the risk-free rate to avoid arbitrage. The risk-free return is the instantaneous rate $y$ multiplied by the period length $dt$ and the portfolio nominal value $\pi$. $ d \pi = y \pi \,dt = y\overbrace{(q_1 V_1 + q_2 V_2)}^{\pi} \, dt$ Equating both expressions for $d\pi$ we get: $ \sum_{i=1,2} \, q_i\left(\frac{\partial V_i}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_i}{\partial y^2} \right)\, dt = y(q_1 V_1 + q_2 V_2) \, dt $ ## Choosing $q_1$ and $q_2$ Since the current setting of $\frac{q_1}{q_2}$ allows for infinite solutions, we fix it to one specific solution out of the possible ones: $ q_1 = \frac{1}{\partial V_1 / \partial y}, \quad q_2=-\frac{1}{\partial V_2 / \partial y} $ Now we take the above two equations and try to separate the $V_1$ and $V_2$ terms: $ q_1\left(\frac{\partial V_1}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_1}{\partial y^2}\right)\, dt + q_2\left(\frac{\partial V_2}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_2}{\partial y^2}\right)\, dt =(q_1V_1y+q_2V_2y) \,dt $ Plugging these into the risk-free condition, we arrive at a key relationship. First we cancel out the $dt$, then we separate the terms: $ \begin{align} q_1\left(\frac{\partial V_1}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_1}{\partial y^2}\right)\, dt + q_2\left(\frac{\partial V_2}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_2}{\partial y^2}\right)\, dt &=(q_1V_1y+q_2V_2y) \,dt\\[4pt] q_1\left(\frac{\partial V_1}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_1}{\partial y^2}\right)-q_1V_1y &= - q_2\left(\frac{\partial V_2}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_2}{\partial y^2}\right) + q_2V_2y\\[4pt] q_1\left(\frac{\partial V_1}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_1}{\partial y^2}-V_1y\right) &= q_2\left(\frac{\partial V_2}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_2}{\partial y^2} - V_2y\right) \end{align} $ Inserting the above choice for $q_1$​ and $q_2$​ yields: $ \frac{\left(\frac{\partial V_1}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_1}{\partial y^2}-V_1y\right)}{\frac{\partial V_1}{\partial y}} = \frac{\left(\frac{\partial V_2}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_2}{\partial y^2} - V_2y\right)}{\frac{\partial V_2}{\partial y}} = f(t,y)$ We successfully separated all parameters of the two bonds $V_1$ and $V_2$ from each other and established their equality. Since we do not see any bond specific parameters (e.g. maturity $T$) explicitly, we can form this equality between any two bonds (irrespective of their bond specifics). ## Resulting PDE Because the ratio is the same for any two bonds, each $V_i$ individually must satisfy: $ \frac{\left(\frac{\partial V_1}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V_1}{\partial y^2}-V_1y\right)}{\frac{\partial V_1}{\partial y}} =f(t,y)$ Rewriting for a generic bond $V\equiv V_i$ we have: $ \boxed{\left(\frac{\partial V}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V}{\partial y^2}-Vy\right) - f(t,y)*\frac{\partial V}{\partial y}= 0} $ Thus, one says that every bond is governed by the same function $f(t,y)$. Once $f(t,y)$ is determined, this PDE can be solved with the appropriate terminal (or maturity) condition. For a zero-coupon bond maturing at $T$ with face value $1$, the boundary condition is: $ V(T,y)=1$ ## Market Price of Risk We can interpret the unspecified function $f(t,y)$ in terms of the risk premium for the bond $V$. From the PDE above (boxed equation), we know that: $ \frac{\partial V}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V}{\partial y^2} = f(t,y)\frac{\partial V}{\partial y}+Vy $ Filling the right side into Itô's Lemma for $dV$: $ \begin{align} dV&= \left (\frac{\partial V}{\partial t} + \frac{b^2}{2}\frac{\partial^2 V}{\partial y^2} \right)\, dt+ \frac{\partial V}{\partial y} \, dy \\[2pt] &= \left(f(t,y)\, \frac{\partial V}{\partial y}+Vy\right) dt+ \frac{\partial V}{\partial y} \, dy \end{align} $ Since we model $y$ as an Itô process, we can break it down into its components, where $a$ is the deterministic trend, and $dB$ the [[Brownian Motion]]. $ dy = a \, dt + b \, dB$ Substituting $dy$ into the expression for $dV$ gives: $ \begin{align} dV &=\left(f(t,y)\, \frac{\partial V}{\partial y}+Vy\right) dt+ a \,\frac{\partial V}{\partial y} dt + b\, \frac{\partial V}{\partial y} dB \tag{1}\\[4pt] dV - Vy \, dt &=\frac{\partial V}{\partial y} \Big(f(t,y) \, dt + a\, dt + b\, dB\Big) \tag{2}\\[4pt] \frac{dV - Vy \, dt}{\frac{\partial V}{\partial y}} &= f(t,y) \, dt + a\, dt + b\, dB \tag{3}\\[4pt] \frac{dV - Vy \, dt}{b\frac{\partial V}{\partial y}} &= \underbrace{\frac{f(t,y) + a}{b}}_{\eta}\,dt+dB \tag{4} \end{align} $ where: - (1) Fill in the separate components of $dy$. - (2) Move $Vy \,dt$ to the left side, which now shows the excess return beyond the risk-free rate. - (3-4) Divide both sides by $b \frac{\partial V}{\partial y}$ which is the sensitivity of the bond to the stochastic variable, multiplied by the coefficient $b$. Thus, the fraction tells us the excess return per unit of risk, which comes from the exposure to the randomness. Since $dB$ is just the Brownian motion (which has zero expectation), the market price of risk per over $dt$ is just $\eta$. $ \eta \equiv \frac{a+f(t,y)}{b}, \quad f = b \eta -a $ where: - $a$ is the real-world drift of the short-rate - $b$ is the volatility coefficient of the short rate - $f(t,y)$ is the additional drift adjustment to ensure no-arbitrage. Also called "risk-premium function". Since this measure of market price of risk is not directly observable as a tradable asset, we can estimate it, either by making assumptions about its functional form, or by fitting other observable market parameter.