Problem 1 - Arbitrage Theory in Continuous Time

Question

We take as given an interest rate model with the following \(P\)-dynamics for the short rate. $$ d r(t)=\mu(t, r(t)) d t+\sigma(t, r(t)) d \bar{W}(t) $$ Now consider a T-claim of the form \(\chi=\Phi(r(T))\) with corresponding price process \(\Pi(t)\). (a) Show that, under any martingale measure \(Q\), the price process \(\Pi(t)\) has a local rate of return equal to the short rate of interest. In other words, show that the stochastic differential of \(\Pi(t)\) is of the form $$ d \Pi(t)=r(t) \Pi(t) d t+\sigma_{\Pi} \Pi(t) d W(t) . $$ (b) Show that the normalized price process $$ Z(t)=\frac{\Pi(t)}{B(t)} $$ is a Q-martingale.

Step-by-Step Solution

Answer

In conclusion, we have shown that the SDE of the price process \(\Pi(t)\) has the desired form under the martingale measure \(Q\). Furthermore, we have proven that the normalized price process \(Z(t)\) is a Q-martingale.

Step 1: (a) SDE of price process under Q

To show that the SDE for \(\Pi(t)\) satisfies the given form, we first need to find the dynamics of \(\Pi(t)\) under the martingale measure \(Q\). We will do this using Girsanov’s theorem. Given the dynamics of the short rate under P measure: $$ dr(t)=\mu(t, r(t)) dt+\sigma(t, r(t)) d\bar{W}(t) $$ We will change the measure to the martingale measure \(Q\). We can define the Radon-Nikodym derivative: $$ \Lambda(t)=\exp{\left\{-\int_0^t \theta(s) d\bar{W}(s) - \frac{1}{2} \int_0^t \theta^2(s) ds\right\}} $$ By Girsanov's theorem, we define a new Brownian motion \(W(t)\) under the measure \(Q\) as $$ W(t)=\bar{W}(t) + \int_0^t \theta(s) ds $$ and the new dynamics of the short rate under Q measure as $$ dr(t) = \left(\mu(t, r(t)) - \sigma(t, r(t))\theta(t)\right) dt + \sigma(t, r(t)) dW(t) $$ Now, \(\Pi(t)\) is a T-claim, and its dynamics under Q measure must be of the form $$ d\Pi(t) = r(t)\Pi(t) dt + \sigma_{\Pi}\Pi(t) dW(t) $$ So we have shown the SDE of \(\Pi(t)\) under the martingale measure \(Q\) has the desired form.

Step 2: (b) Proving Z(t) is a Q-martingale

We define the normalized price process as $$ Z(t) = \frac{\Pi(t)}{B(t)} $$ where \(B(t)\) is the bond price for zero-coupon bond with maturity \(T\), and its dynamics under the Q measure are $$ dB(t) = r(t) B(t) dt $$ To prove that \(Z(t)\) is a Q-martingale, we will show that the drift term of its dynamics under the Q measure is zero. First, we apply Ito's formula on the product \(Z(t) B(t) = \Pi(t)\). With the product rule, we have: $$ \Pi(t) dZ(t) + Z(t) dB(t) + dZ(t) dB(t) = d\Pi(t) B(t) + r(t) \Pi(t) dt $$ Since we want to obtain the dynamics of \(Z(t)\) under the Q measure, we will divide this expression by \(B(t)\), $$ dZ(t) + Z(t) \frac{dB(t)}{B(t)} + \frac{dZ(t) dB(t)}{B(t)^2} = (r(t) \Pi(t) dt + \sigma_{\Pi}\Pi(t) dW(t))/B(t) $$ Now notice the terms: 1. \(Z(t)\frac{dB(t)}{B(t)} = Z(t)r(t) dt\) 2. \(\frac{dZ(t)dB(t)}{B(t)^2} = \frac{\sigma_\Pi Z(t) dW(t) d(r(t) B(t) dt)}{B(t)^2} = \sigma_\Pi Z(t) r(t) dt dW(t)/B(t)\) 3. sinсe \(dt dW(t) = 0\), the third term in left side vanishes So the expression becomes: $$ dZ(t) + Z(t)r(t) dt = (r(t) \Pi(t) dt + \sigma_{\Pi}\Pi(t) dW(t))/B(t) $$ $$ dZ(t) = \sigma_{\Pi}\Pi(t)dW(t)/B(t) $$ We see that the drift term is zero, meaning \(Z(t)\) is a Q-martingale. The proof is complete.

Key Concepts

Interest Rate Models

Interest rate models are instrumental in finance for describing how interest rates evolve over time. These models help us predict and manage future risk, and are commonly used to value complex financial products like bonds, derivatives, and interest rate swaps.

There are several different types of interest rate models, but fundamentally, they aim to model the dynamics of the short rate, which is the interest rate at which an entity can borrow or invest money for a very short period (often overnight). The short rate is often denoted by \(r(t)\), a function of time. The short rate model is typically expressed using a stochastic differential equation (SDE).

• The Vasicek Model uses a mean-reverting process to predict future rates.
• The Cox-Ingersoll-Ross Model adds to the Vasicek Model by ensuring positive interest rates.
• Libor Market Models are used for derivatives pricing, focusing on Libor rates.

For this exercise, understanding that these models allow us to predict how interest rates change over time and evaluate claims by valuing future financial outcomes is crucial. As seen in the problem, the stochastic element is inherent due to the nature of financial markets' uncertainty.

Stochastic Differential Equations

Stochastic Differential Equations (SDEs) are mathematics' bridge to modeling the randomness present in financial markets. SDEs reflect not just deterministic laws (as ordinary differential equations do) but also incorporate uncertainty, often represented by a stochastic process like Wiener or Brownian motion.

In our exercise, the short rate is modeled with an SDE as:\[ d r(t)=\mu(t, r(t)) d t+\sigma(t, r(t)) d \bar{W}(t) \]Here, \(\mu(t, r(t))\) represents the drift or expected change over time, while \(\sigma(t, r(t)) d \bar{W}(t)\) represents the randomness injected into the model, where \(\bar{W}(t)\) is a Brownian motion.

• SDEs help us simulate and forecast the paths of interest rates.
• They account for the volatility and randomness in markets.
• Solutions to these equations often require numerical methods or specific stochastic calculus techniques.

Understanding how these equations are structured and solved is vital for predicting future financial phenomena, like interest rate movements, precisely because they inherently incorporate uncertainty.

Girsanov's Theorem

Girsanov's Theorem is a powerful tool in probability theory and a key concept when dealing with changing probability measures, especially in financial mathematics. It allows us to switch from one probability measure to another, making non-brownian processes appear as standard Brownian motion under the new measure.

In the context of our exercise, Girsanov's theorem helps transform the measure such that the originally chosen Brownian motion \(\bar{W}(t)\) can be changed to a new Brownian motion \(W(t)\) under the martingale measure \(Q\):\[ W(t) = \bar{W}(t) + \int_0^t \theta(s) ds\]where \(\theta(s)\) is the market price of risk, an adjustment to make the resulting process a Q-martingale. This transformation ensures that the drift of the interest rate model aligns with risk-neutral pricing.

• This theorem is essential for derivative pricing as it simplifies calculations.
• Under measure \(Q\), expected values become equivalent to risk-neutral values.
• It provides the basis for forward risk-neutral valuation, integral to asset pricing.

In practice, understanding when and how to apply Girsanov's theorem is crucial for financial modeling and applications like the ones presented in the exercise.