In short rate models, we specify the dynamics of the short rates. In the Heath-Jarrow-Morton framework we take a different approach. We specify evolution of the entire forward curve directly. As a consequence, the initial forward curve enters as an exogenous parameter. So there are no tiers with fitting the initial forward curve as we had then with the short rate models. We will see that the drift of the forward rate dynamics is going to be fully determined by the volatility of the forward rate dynamics. This is the famous HJM drift condition. The HJM framework is very broad and it contains all interest rates models which are driven by a finite number of Brownian motions. Formally as ingredients, again, we fix filtered probability space which is now carrying a d dimensional Brownian motion. A d dimension Brownian motion is simply composed of independent standard scale of Brownian motions W1 to Wd. In turn, we need the market prizes of risks that we stack into the d dimensional market price of risk, back to the Lambda and the Girsanov theorem still applies under the risk neutral measure Q which is equivalent to P with the Radom-Nikodym density process as given here. Under this measure, the pros is W* which is the Brownian motion and the P plus the market price of risk drift is a Brownian motion under Q. The Heath-Jarrow-Morton framework or HJM framework is very broad and contains all interest rate models driven by a finite number of Brownian motions. We specify directly the evolution on the Q of the entire forward curve. Formally this means, |for every maturity capital T, we the forward rate process as a process in small T follow an Ito process with drift alpha. Hence, volatility sigma which are both indexed by the maturity capital T but otherwise are processes in small t as we're used to in the case of generic Ito processes. We will see later on that the drift process, alpha is going to be fully determined by sigma. That is going to be a consequence of the Arbitrage pricing theory. keep in mind that Sigma is now a vector valued process. We stack the d component sigma 1 up to sigma d into this row vector that we call sigma. We're also given as an exogenous parameter and initial forward curve f(0), capital T. Bond prices are explicit in terms of the forward curve. This is in contrast to the short rate models where bond prices were given implicit in terms of the integrated short rates via the conditional expectation under the risk neutral measure. Hence in contrast to short rate models, here, we can directly infer the dynamics of the bond prices in terms of dynamics of the forward rates by using Ito's formula. Let's do this in several steps. We first start with the process that is in the exponent of the right hand side. We see that the running time small T showing up at two places, as an integration boundary and the time argument of the forward rate process. Formally differentiating under the integral sign gives us the integrant at the lower integration boundary, f, small t, small t which by the way just to short rate, r(t) as we're going to, right here, minus the integral of the differential of the integrant. We then plug in the differential of the forward rates and change the order of integration dt, du and the same with dW*(t), du and we arrive at the second line here where we use short hand notation and define v(t, T) as minus the integrated forward rate volatility. Summarizing this first step, we shown that the T bond price is given by the exponential of an Ito process whose the composition is given here. We can now apply the Ito formula. The Ito formula applied to the exponential function gives us what we see here on the first slide. Notice that I have divided on both sides by T bond price. Plugging in the expression for the Ito composition of this process here from the previous slide, we arrive at what is shown here. This is the T bond return decomposition into drift component and martingale component. From the Arbitrage Pricing Theorem, we know that the expected rate of return to drift of the T bond under the measure Q has to be equal to the short rate. But this implies that minus the integral of alpha du plus one half square of normal v has to be 0. This must true for all maturities capital T. So if we differentiate this equality in capital T, we arrive at this HJM drift condition. The drift of the forward rates dynamics alpha is fully determined in terms of the forward rate volatility signal. Summarizing, we arrive at this HJM theory. The Q dynamics of the forward rates is fully determined by the forward rate volatility sigma because the drift that we use to label alpha is given in terms of sigma as shown here. The corresponding risk neutral bond price return dynamics is of this form. it has a risk free expected rate of return and a volatility v which again, is defined as minus the integrated forward rate volatility. The minus sign comes from the fact that an upward shock in the Brownian motion W* drives forward rates up. Rates going up means bond prices go down. The HJM framework contains all the short rates models we have seen so far. So which HJM model, sigma (t, T) corresponds to the Vasicek short rate model. To answer this question, we recap the Vasicek short rate model dynamics and we recall that the Vasicek short rate model implies forward rates which are f Phi functions in the prevailing short rates. The functions A and B are given in closed form. They solve a system of Riccatic equations, the first order derivative of B is e to the minus kappa, cap T minus small t. This is enough applying integration by parts to determine the martingale component of the forward rate dynamics which is as shown here. We could as well by integrating by parts we apply stochastic calculus, derive the drift of this implied forward rate dynamics. But we don't need to go through these calculations because the HJM theorem already tells us what the HJM drift here has to to look like. It is given explicitly in terms of the volatility, the volatility of the forward rates.