hello everyone and welcome to one pie it's high time we tackle the hjm framework and let me quickly explain what we are going to cover in this video we know that the hjm one is the dynamics of an instantaneous forward it reference is a fixed maturity time there's a continuum of such forwards but you can develop the whole framework by focusing on just one generic maturity say capital T like many stochastic differential equations that you see in finance it has got a drift term and volatility term the drift term has to satisfy certain conditions to avoid arbitrage so we're going to derive those conditions first under the recent neutral major and then under the forward major these conditions are the crux of the hjm framework and once you understand the drift conditions then the rest follow quite naturally we will then discuss the volatility term and we shall see that the volatility practically dictates what kind of stochastic process does the instantaneous forward take we will then discuss a few simplifications of the volatility that makes the model more tractable from theoretical and implementation perspective by which I mean Gaussian in Markovian really by the way we are assuming that the randomness is coming from just one Brownian and practice you will use a few but once you understand the framework using one Brownian an extension to multiple Brownian is a relatively straightforward the volatility could be a function of the instantaneous food as well but I'm not going to write it explicitly to avoid clutter once the volatility could be a function of the instantaneous forward the geometric brownian type specification warn world for some technical reasons and towards the end of the video I'm going to explain how those technical conditions come about so let's start with the dynamics under the risk-neutral major if we know the volatility of an asset process then we can easily write its dynamics under the risk-neutral major because we know the expected return under the risk-neutral major must be equal to the risk-free rate but the problem that we have is the instantaneous forward is not a traded asset not a big problem though because the zero coupon of the same maturity is a tragedy acid so we can write its dynamics under the residential major where R is the same R that we used in the definition of the bank account Sigma P is the volatility of the zero coupon which is different from Sigma F and we shall see very shortly how these two are linked so now we need to find a way to infer the dynamics of the instantaneous forward from the dynamics of the zero coupon we saw in the previous video that we can write the instantaneous forward as a function of the zero coupon we're interested in the differential so at least supply differential to both sides where we assume data we can interchange the derivative and the differential this is going to be legit for the cases that we are considering here so these assumptions is going to be harmless for now we are going to use this relationship to derive the dynamics of the instantaneous forward from the dynamics of the zero coupon so we need to determine the differential of log of P for which we will need the Ito's lemma which says that the differential of a function of a random process is equal to the first derivative times the differential plus half the second derivative times the quadratic term we have derived this formula under the stochastic calculus playlist so if you'd like a refresher then please do watch this video for log of X the first derivative is 1 over X and the second derivative is minus 1 over x squared so this becomes substitute the price of the zero coupon 4x we need to determine the two terms on the right hand side the first one is easy we will just shift P to the left hand side and for the quadratic term we know Delta T Square and the cross term between delta T and Delta W will become 0 and Delta W Square will become DT so we get Sigma square delta T substitute these into the differential of the log of P now need to take the derivative of this with respect to the capital T where I just used the fact that the derivative of Sigma square is equal to 2 Sigma so the 2's cancel laughter forward is equal to the negative of this so we will change the signs of the two terms and rearrange and now we have the dynamics of the instantaneous forward under the rescission measure but the differentials are representing the dynamics of the same instantaneous forward so it means the volatility terms must be equal and the drift terms must be equal some the equation on the left hand side for Sigma P we just have to integrate we see is the integration constant we know that the zero coupon pays one quit at maturity with zero volatility so the volatility at maturity is zero which means C is zero so now we have Sigma P - sign is down to the fact that the prices of the zero coupon and the instantaneous food are inversely related and the integral is down to the fact that zero-coupon spends the whole range of a forward or is remaining maturity now we can substitute 2 Sigma B and the derivative of Sigma B from the left hand side and to the drift term and now we have the full dynamics of the instantaneous food under Dilys Newton measure so everything is down to the volatility then and to build the term structure what you need to do is to get the initial four will and specify the volatility and hgm will take care of the rest and so now that we have the dynamics under the reason agent my jam let's see if we can translate these into the dynamics under the forward measure by the way the probability measure manifests itself through the Brownian so this w here will have the properties of the Brownian under the least neutral measure and when you change the probability measure then you're rewriting the probabilities so process this Brownian under the new probability measure will be a different process and the main game now is to establish the connection between the two and once we have done that then we can substitute the new brownian for the old and calling dynamics under the forward merger now this is a classic learning exercise for one the technique called the change of numeraire approach so this approach we are going to use we know there is neutral valuation approach if you express the value of any asset and the units of the bank account then this process is a martingale under there is neutral major this approach is quite general so instead of using the bank account is the numerator you can use an other asset say the price of a zero coupon has the numerator then the value of any asset expressed and the units of the zero coupon will be martingale but under a different probability measure so each numeral and away induces a probability measure and the manager induced by using the capital T maturity zero coupon is called the t forward measure now we can shift the P and the denominator of the first equation to the right hand side and because it's known we can take it inside the expectation the same thing with the equation on the right inside right the expectation of a variable as the integral of the variable with respect to the probability measure so we get Delta Q on the left hand side and Delta P on the right hand side equations are representing the price of the same acid so the right-hand side must be equal and because this relationship must hold for any acid it means the following must be true can rearrange to get the derivative of the new probability measure with respect to the old one we need to determine the ratio of the bank account and the ratio of the zero coupon we were halfway through to the derivation of the first when we applied the Ito's lemma to the log of P so let's reproduce the results we can integrate she ate to get the ratio of the prices of the zero coupon at two different times we know the value of the bank account from the previous video can rearrange this to get the ratio of B naught and BT now if we substitute this expression into the derivative the integrals of our cancel and we are left with an expression it smells like the red on nikodem derivative discussed the radar nikodem derivative in the change of probability measure video so at least reproduce the main result of this video which is the girl sonnet theorem we counted the Kamran Martin goes on up theorem so if W is a Brownian motion under a probability measure Q and you introduce a new process then the Brownian motion adjusted by this process is a Brownian motion but under a different probability major which is defined by the radon nikodem derivative you can see the Y in the exponent of the radon Nikodim is linked to the Y in the adjustment to the Brownian you can also write the relationship between the two Brown ins and differential form if you compare the radar nikodem in the Doosan of theorem with a derivative that we have and we see that Y is equal to Sigma P so we can write the relationship between the Brownian motion and the forward major and the Brownian motion under the resolution major so we will just need to subtract this Sigma P we saw earlier that the Sigma P is just the integral of Sigma F so we can make this substitution reproduce the dynamics and the risk-neutral measure stitute for the whole brownian cancel each other and we are left with the stochastic term meaning the instantaneous forward is a martingale under the fourth major this is quite handy now so if you were to model an instantaneous forward of maturity capital T using the zero coupon of the same maturity as the numeral then you will get the martingale which means that you can apply the machinery of martingale Theory straightaway reality you'll be mostly tasked with the modeling multiple forwards at the same time say T one all the way to some terminal maturities this call it TF and the last thing you would want to do is to model each one of them under a different probability measure so normally you would see that people would select the longest maturities zero-coupon as the numeraire and an obvious advantage of using the longest maturity zero-coupon is that you're Nomura asset will be alive are meaningful for the whole length of your analysis period which is good because we know that the zero coupon becomes meaningless after its maturity and so now let's see what Donna means to beget for an instantaneous forward of maturity capital T which is smaller than TF so we can write the valuation formula under the risk-neutral measure and under the TF forward measure if we compare the right in size of the to equation we get the read on nicotine derivative which is very similar to the results we got earlier the only change is that we have TF and place a capital T which you would expect because we're using the longer maturity zero-coupon as the numeral now so the Brownian under the terminal forward will be linked to the Browning under the risk-neutral again by the Sigma P but now it's the volatility of pellagra maturity zero coupon so the integral will now run from T to t f reproduce the dynamics under the rescission measure substitute for the old brownian now we can split the last integral into two intervals and we are left with the residual so the drift term will now have the integral running from capital T which is the maturity of the forward that we are modeling and TF which is the maturity of the longer maturities zero-coupon that we are using as the numeral so in a nutshell then when we model the dynamics of the team maturity forward under the risk-neutral major the drift will have the integral with the remaining maturity of the zero coupon when we model the dynamics under the t4 word major the drift will be zero and when we model the dynamics under a longer maturity zero-coupon then the drift will have the integral running from the maturity of the forward that we are modeling and the maturity of the zero coupon that we using as the numeral so whatever probability manager you work with volatility is the king because it pretty much drives the dynamics of the instantaneous forward given the importance of the volatility let's discuss how the volatility drives the dynamics of the instantaneous forward lists reproduce the hjm dynamics under the risk-neutral measure let's integrate to see how the solution looks like said that the volatility could be a function of the instantaneous forward so in general the EDM will give you the dynamics of the instantaneous forward it are non Markovian there's nothing wrong with the non Markovian processes but the implementation is hard so let's discuss a couple of simplifications the first one mrs. azim that the volatility is a deterministic function of time in maturity only then we know from the properties of the ito integral that the instantaneous Ford will be Gaussian roughly you can see that the Brownian increments are normally distributed so you can interpret the last time as a linear combination of normals with deterministic weights the first time is deterministic so the whole expression is like a linear transformation of a combination of normals so the result must be normal and Gaussian process is really the equivalent of normal when you switch from random variable to random process now let's see how can we make this a Markov process so remember from the introduction to the stochastic processes video a process is Markov if the distribution of the future values depends only on the current value of the process and not the history so knowing the current value is as good as knowing the entire history of the process and you can see if that would be quite handy because you won't need to keep track of the entire history of the process when you're modeling it the first term is deterministic so it's not going to dictate whether the process is Markov Yin so everything is going to be down to the stochastic term so at least focus on the stochastic term and consider its increments from small T to capital T now we need to check whether the expected value of these increments require the past knowledge of the process or does the information that we have at small T suffice they split the first integral into two so that one of the interval gets aligned to the integral on the right hand side and then we can combine the two integrals were the smaller interval you know the expected value of the first time is zero because it is an integral of a deterministic function with respect to the Brownian the second term is not zero da because this value is fixed by small T so the Brownian values that we have here would already have been realized what it means is the process is not Markov if we assume that the process is separable in the sense that we can write it as a product of the function of time in maturity we can show that the process is Markov we can make the substitution in the stochastic term and into the implement we can take the H out of the integral because it doesn't depend on the integrating variable affected value of the first term will be zero because it is the integral of a deterministic function with respect to the Brownian for the second term if you divide and multiply by edge then you get DT so now the expected value of the increment of the process from small T to capital T only requires the value of the process at small T which means that this is now a Markov process and now for the icing on the cake at least see why the lognormal type specification won't work so Elise reproduced at gem dynamics under the risk-neutral Naja and let's assume that the volatility is equal to Sigma times F this is how you drag the volatility term of the lognormal process right we can substitute the volatility the volatility and eff out of the integral because they don't depend on the integrating variable evaluate the integral which is a just replacing the integral with the length of the interval really and now let's focus on the deterministic component we can solve this equation through the separation of variables approach so essentially you separate each variable on one side and then integrate the left-hand side is like the derivative of minus 1 over X and the right-hand side is more like the derivative of X square we can evaluate the expression at the upper and lower integration limits and we can separate one more F on the left hand side and we can combine the terms on the right hand side we can invert this to get the F T now the denominator has got one minus something which means at some point it can become zero and if the denominator becomes zero it means the instantaneous forward will become infinity which means the price of the zero coupon will be zero so you can get a zero coupon which pays one quit at maturity for zero meaning this arbitrage this the reason that the log normal type specifications won't work but this is only a problem for the continuous settings and once you move to the discrete settings like the LIBOR market model then this is no longer an issue right so I think we covered on the topics that we had on the list so I hope you enjoyed this video and I look forward to seeing you in the next when we use this framework to derive the short rate models of holy and the hollwyood extended was a checked model and we will leave the Cir as an exercise