introduction to stochastic calculus with application

As you can see, Brownian motion has rough paths. The figure on the right shows four trajectories simulated from a Brownian motion. As I already said, the Brownian motion is a continuous martingale. 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. The prototype of a martingale in continuous time, is a Brownian motion. At t0 we have no information yet, and the asked prices is a constant. and also it must be the same in the two states Omega 3, Omega 4. So, the asked price must be the same in the state Omega 1, Omega 2. In the two period context it means at t2, we know the terminal price, which is a function of the state of the world.

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 sense that on average, you cannot gain more than what you have today. The conditional expectation of the future values at some time point, T, is just equal to the currently prevailing value. We call Xt a martingale if it is adapted and such that, at any time t. We call the stochastic process adapted if for any fixed time, t, the random variable Xt is Ft measurable. A stochastic proces is a family of random variables indexed by time t. We cannot distinguish any of the samples at time 0. So the signma algebra F1 consists of the two sets: Omega 1, Omega 2 and Omega 3, Omega 4. 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. 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. And F,t represents the information available by time, t. This is an increasing family of sigma algebras indexed by time, t. The flow of information is modeled by a filtration. The probability measure P, acts on the sigma algebra, caligraphic F and it assigns a probability to any measuble event, A. This is the collection of the so-called measurable events. 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. Random variables are modeled on a probability space. I will also, in this part, give you the basics for pricing options, namely, the Arbitrage pricing theorem We always fix a Stochastic basis. But the good news is, once you acquire the rules of Stochastic calculus, you can engineer any of the following interest rate models. Obviously we cannot go into the mathematical details. That is: Brownian motion, the Stochastic integral Ito formula, the Girsanov theorem. In this first part, I recap the basic notions of Stochastic calculus. Eventually, we will see how these models can be used to price options on bonds. 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.

In the first part, I'll provide the basics in Stochastic calculus, that is needed for developing Stochastic interest rate models. We now consider Stochastic interest rate models.

Introduction to stochastic calculus with application