We now consider Stochastic interest rate models. In the first part, I'll provide the basics in Stochastic calculus, that is needed for developing Stochastic interest rate models. We will then look at short-rate models, where the dynamics of the short-rate is specified; and in a second step we'll look at so-called Heath-Jarrow-Morton models, where, the evolution of the entire forward curve is specified, as a Stochastic process. Eventually, we will see how these models can be used to price options on bonds. In this first part, I recap the basic notions of Stochastic calculus. That is: Brownian motion, the Stochastic integral Ito formula, the Girsanov theorem. Obviously we cannot go into the mathematical details. But the good news is, once you acquire the rules of Stochastic calculus, you can engineer any of the following interest rate models. I will also, in this part, give you the basics for pricing options, namely, the Arbitrage pricing theorem We always fix a Stochastic basis. Random variables are modeled on a probability space. This is a triple consisting of the set of samples Omega; and on this set of samples Omega we have a structure which is called a sigma algebra, which is denoted by a caligraphic F. This is the collection of the so-called measurable events. The probability measure P, acts on the sigma algebra, caligraphic F and it assigns a probability to any measuble event, A. The flow of information is modeled by a filtration. This is an increasing family of sigma algebras indexed by time, t. And F,t represents the information available by time, t. Let's look at following example which consists of a sample set consisting of four samples.p Suppose the probability assigns the same weight to each sample. We have two time periods. At t2, the true state of the nature is fully revealed However at t1, we can only tell whether we are in the set, cosisting of Omega 1 and Omega 2 or in the set consisting of Omega 3 and Omega 4. So the signma algebra F1 consists of the two sets: Omega 1, Omega 2 and Omega 3, Omega 4. At t0, the sigma algebra is trivial. We cannot distinguish any of the samples at time 0. A stochastic proces is a family of random variables indexed by time t. We usually suppress the argument omega. We call the stochastic process adapted if for any fixed time, t, the random variable Xt is Ft measurable. We call Xt a martingale if it is adapted and such that, at any time t. The conditional expectation of the future values at some time point, T, is just equal to the currently prevailing value. A martingale is a model for fair game. In the sense that on average, you cannot gain more than what you have today. As an example, consider again the Stochastic basis from the previous slide; and let's assume that Xt models the price of a financial asset at state T and assume that Xt is adapted. In the two period context it means at t2, we know the terminal price, which is a function of the state of the world. At t1, we cannot anticipate the future. So, the asked price must be the same in the state Omega 1, Omega 2. and also it must be the same in the two states Omega 3, Omega 4. At t0 we have no information yet, and the asked prices is a constant. The prototype of a martingale in continuous time, is a Brownian motion. A standard Brownian motion, W, is characterized by the following four properties: it is continuous and adapted process, it starts at t0, and its increments w(t2)-w(t1) are independent of the past and normally distributed with mean 0 and variants proportional to the time increment. As I already said, the Brownian motion is a continuous martingale. The figure on the right shows four trajectories simulated from a Brownian motion. All trajectories start at 0. As you can see, Brownian motion has rough paths. They are not smooth. They are not differentiable. And at any given point in time, the expectation of the Brownian motion is equal to 0. A Brownian motion is a Gaussian process in the following sets: We define a Stochastic process Z(t) to be a Gaussian process if its final dimensional distributions are multivariate Gaussian or normal distributed for any finite selection of time points t1 up to tn. Since the multivariate normal distribution is determined by its mean vector and the covariance matrix, it follows that the distribution of a Gaussian process Z(t) is determined by its mean function, that we call m(t), and the covariance function, which gives the pair of covariances of Zt1 and Zt2. For any adapted process sigma(t), whose trajectories are square integrable in time, we can define the Stochastic integral with respect to a Brownian motion. This is how we denote the stochastic integral. Even though the rigourous definition of the stochastic integral is a bit more involved, we can think of it as being the limit of remaining sums as shown here. Where we go over partitions of time and let the mesh tend to 0. It can be proved that the stochastic integral is a continuous martingale, if the integral satisfies a somewhat strong integrability condition. The stochastic integral will be the model for the risky part of the return of an asset. An ito process X(t) is an adapted process of the following form. It can be decomposed into an initial value, plus a drift term, plus a martingale term, which is a stochastic integral. Remember that the Brownian motion, and therefore the stochastic integral, has rough trajectories. The drift in contrast has smooth trajectories and models trends in the evolution of X. An ito process is thus the ideal model for the price process of a financial asset. We often use differential notation for X where we omit the initial value and write as shown here. Keep in mind, whenever we specify an ito process X in differential notation, there is always an initial value to be specified with it. Sigma is also called the volatility of the process X. It can be proved that if the initial value, the drift, and the volatility of X, are deterministic functions, then, the procss X(t) is a Gaussian process with the following mean and covariance functions, as shown here. It is easy to understand why the mean function is as shown here, because the martingale has zero expectations. We take for granted that the second moment of the martingale is given by the integal of sigma² ds. The core of stochastic calculus is the ito formula. It states that for a C²-function f(x), meaning, the function x is twice continuously differentiable, and for an ito process X(t) which is given in differential notation here, the composiition f applied to X, is again an ito process with the composition given by that. It is important to see that real calculus would give us a total derivative that is underlined. But this would only hold if X(t) would have differentiable trajectories. Recall that the stochastic integral has rough trajectories which are not differentiable, which is inherited by X(t); and that causes the appearance of a second order correction term. It also explains why this formula only works for C²-functions f. We need f to be twice continuously differential. There is also a multivariate version of ito's formula but I'll refrain from showing it here, with the following exception: The integration by parts formula. It states that for two ito processes X1 and X2, both indifferential, notation as shown here the product X1 times X2 is again an ito process, with the composition as shown here. Again we realize that this is the product rule that we are knowing from real calculus; but that holds only if X1 and X2 had differentiable trajectories. Since they do not, due to the stochastic integral, we have the second order correction term. As an example of ito's formula, let's look at the exponential function applied to an ito process X, which is given in differential notation here. The decomposition of e to the X is given by the sum of the derivative of e to the X, which is e to the x itself, dX, plus the second order correction term; which involves the second order derivative of the exponential function, which is again the exponential function itself. Again, we see here the fundamental differential equation that Y would satisfy if X(t) would have differential trajectories. There is a counter-part to the classical exponential function in stochastic calculus. It's called the stochastic exponential and it's denoted by a calligraphic E. It is defined as shown here. It is now an exercise for you applying ito formula. to show that the stochastic exponential actually satisfies the fundamental stochastic differential equation, which corresponds to the fundamental equation that would hold in real calculus if X would have differentiable trajectories. This property of the stochastic exponential is very useful for two reasons First of all, the stochastic exponential is positive by definition. This is useful because most price processes of financial assets are positive. And second, due to this fundamental stochastic differential equation, the stochastic exponential preserves the martingale property. That means if X is a martingale, Then the stochastic exponential of X is also a martingale. In sum, the stochastic exponential is the prototype of a positive martingale in stochastic calculus. In financial modeling, we often change the probability measure. This is why it is useful to review base rules. Let Q and P be equivalent probability measures with Radon-Nikodym density, denoted by dQ over dP. Base rule relates conditional expectations on the P and Q. For any bounded random variable X, the conditional expectation of X on the Q can be expressed as conditional expectation on the P, if we multiply the argument X by the Radon-Nikodym density of Q with respect to P; and normalize by the conditional expectation of the Radon-Nikodym density. If we vary little t, then this becomes a martingale. It's called a Radon-Nikodym density process. It is a positive martingale. The Girsanov theorem describes what happens to a Brownian motion on the P when we change to an equivalent probability measure Q. The statement is as follows: For an adapted process lambda, satifying some more technical conditions, the ito process called W* given by the expression as shown here becomes a Brownian motion under the equivalent probability measure, Q, which is determined by the Radon-Nikodym density process as shown here. This means if we are given a Brownian motion with a drift lambda dt under the measure P we can de-trend by changing the probability measure to an equivalent measure, Q, exactly when the Radon-Nikodym density of Q with respect to P is given in terms of the drift lambda in this way. Here is a very important application of the Girsanov theorem in financial modeling. It's the arbitrage pricing theorem. It states that any traded asset with a positive price process denoted by S must have a return as shown here. That means the return is decomposed into a drift component and a martingale component. The drift component models the expected return Its rate is the sum of the short rate r(t) plus a risk premium which is of the form sigma times lambda. Sigma is the volatility of the return. the larger sigma, the more risky the return is and lambda is the so-called market price of risk. In consequence, the more risky the asset is ie, the larger sigma is, the larger is going to be the excess expected rate of retun over the risk-free short rate Recall that the short rate is called the risk-free rate of return because this is what the investor would earn if he would simply invest in the savings account. We now see how the Girsanov theorem comes into the play. We can rearrange the terms, the excess expected rate of return, and denote its part, as show here, what we see here is the Brownian motion plus drift we use the equivalent probability measure, Q, as shown on the previous slide in Girsanov theorem where we learnt that this is a Brownian motion W* under the equivalent measure, Q. We can re-express the asset price return as follows. In terms of W* We now see that under the equivalent measure, Q, the expected rate of return is the risk-free rate, r while the volatility remains the same. We can thus re-state the arbitrage pricing theorem by saying that there must exist an equivalent measure, Q. which is called risk neutral-measure, under which the return of any traded asset has an expected rate, given by the risk-free short rate. Using the integration by parts and ito formula we can show that this differential notation is equivalent to what is shown here. It states two things. First, the discounted asset price process discounted by the risk-free numeraire the risk-free asset, the savings account is a martingale under the risk neutral-measure Q. And second, this martingale is given as stochastic exponential of the martingale part of the return of the price process.
