Ito’s lemma

Introducing the Radon-Nikodym derivative C(To) and applying Ito s lemma leads to... 

Along the lines of HIM it can be shown that the existence of an arbitrage-free setup implies that the drift ji t,T) is fully determined by the volatility function cr (f, T) and the stochastic state variable V (f). Applying Ito s lemma to the bond price... 

We noted at the start of the chapter that the price of an option is a function of the price of the underlying stock and its behaviour over the life of the option. Therefore, this option price is determined by the variables that describe the process followed by the asset price over a continuous period of time. The behaviour of asset prices follows a stochastic process, and so option pricing models must capture the behaviour of stochastic variables behind the movement of asset prices. To accurately describe financial market processes, a financial model will depend on more than one variable. Generally, a model is constructed where a function is itself a function of more than one variable. Ito s lemma, the principal instrument in continuous time finance theory, is used to differentiate such functions. This was developed by a mathematician, Ito (1951). Here we simply state the theorem, as a proof and derivation are outside the scope of the book. Interested readers may wish to consult Briys et al. (1998) and Hull (1997) for a background on Ito s lemma we also recommend Neftci (1996). Basic background on Ito s lemma is given in Appendices B and C. 

What we have done is taken the stochastic differential equation (SDE) for S, and transformed it so that we can determine the SDE for/,. This is absolutely priceless, a valuable mechanism by which we can obtain an expression for pricing derivatives that are written on an underlying asset whose price can be determined using conventional analysis. In other words, using Ito s formula enables us to determine the SDE for the derivative once we have set up the SDE for the tmderl5dng asset. This is the value of Ito s lemma. 

We substitute these values into Ito s lemma given in Equation (2.29) and this gives us... 

So, we have moved from df to dS using Ito s lemma, and Equation (2.31) is a good representation of the asset price over time. 

We have 6x, and we require dP. This is done by applying Ito s lemma. We require... 

So, using Ito s lemma, we have transformed the SDE for the bond yield into an... 

Ito s lemma, it can be shown that a function fofx and t will foUow the following process ... 

If the reader has followed this through, he or she has arrived at Ito s lemma. We can apply this immediately. Consider a process... 

A default-free zero-coupon bond can be defined in terms of its current value imder an initial probability measure, which is the Wiener process that describes the forward rate dynamics, and its price or present value under this probability measure. This leads us to the HJM model, in that we are required to determine what is termed a change in probability measure , such that the dynamics of the zero-coupon bond price are transformed into a martingale. This is carried out using Ito s lemma and a transformatiOTi of the differential equation of the bmid price process. It can then be shown that in order to prevent arbitrage, there would have to be a relationship between drift rate of the forward rate and its volatility coefficient. 

This is done by using Ito s lemma to transform the stochastic differential equation of the price process and then determine the change in the Brownian differential AW so that there remains no drift term. The first step is to consider the differential of P t, T). We express this in the form... 

Following HJM, by applying Ito s lemma, the model obtains the following result for Z(t, T), shown in Equation (4.32) ... 

To determine the volatility of the zero-coupon bond price v(f) at time t, it can be shown that applying Ito s lemma to Equation (4.34), we obtain... 

Use Ito s lemma to express the dynamics of the bond price in terms of the short rate. 

By applying Ito calculus, i.e., Ito s lemma, (4.17) can be transformed to solve for the price of an asset. Taking the integral of expression (4.17) results in equation (4.18), which derives the forward rate,... 

Market practitioners armed with a term-structure model next need to determine how this relates to the fluctuation of security prices that are related to interest rates. Most commonly, this means determining how the price / of a zero-coupon bond moves as the short rate r varies over time. The formula used for this determination is known as Ito s lemma. It transforms the equation describing the dynamics of the bond price P into the stochastic process (4.5). 

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