In this module which is our last module on fixed income derivatives pricing we're going to talk about the practice of fixed income derivatives pricing. We'll explain how it is really not much more than using observable liquid Security prices to extrapolate and compute the prices of non-observable security prices. We're going to discuss, again, the problem of calibration and how it is a difficult problem to solve in practice. So, in practice, more complex models than binomial models are used to price fixed income derivatives today. In fact, the models that are used today are a lot more complicated than the binomial models we've seen so far. However, the pricing philosophy is still very much the same. It works as follows. We specify a model under the risk mutual dynamics which we are calling q of theta and where theta is a vector of parameters that you need to choose. So, for example, the theta's could be the ai's and bi's that we have seen in the Ho-Lee and BDT models. So, we fixed our model. There is some unknown parameters theta. What we will do is we will price all securities with our new model using risk neutral pricing. And if you recall this is our risk neutral pricing framework. So, what we do is we say that Z t over B t where Z t is the price of the security in question, B t is the value of the cash account So that is equal to the expected value, conditional in time t information, under the risk neutral probabilities of the expected value of the security at time t + s divided by the cash account, plus all the intermediate coupons or cash flows that you receive between times t + 1 and t + s, again divided by the cash account value. This is our risk neutral pricing framework, we know that there cannot be any in the model. When we price all of our securities like this. So, that's step number two. Step number three is then to choose data so that the market prices of appropriate liquid securities agree with model prices of those securities. This is the model calibration procedure. And we've seen in the last module, some examples of how to do model calibration in the context of the Black–Derman–Toy model. And when our calibration securities [INAUDIBLE] coupon bond prices. The calibration problem typically requires minimizing a sum of squares of the following form. So remember, the Theta are the vector of parameters we want to choose in our model These could be the a i's or b i's in the black [INAUDIBLE] your [INAUDIBLE] model so the goal is to choose the thetas in such a way as to minimize the following. So we have a bunch of securities which are indexing by i here and we have a market price of the i security which is pi market and then we have our model price of the i security, pi model and this model price is calculated using this equation here, equation 14. This is our risk mutual pricing that we use in our model. Doing this ensures that our model prices are arbitrage free. So in our model, we can compute this price. We see this in the marketplace. And omega i is a weight, it's a positive weight reflecting the importance of the ith security in our calibration. Maybe omega i is large for some values or for some securities, and omega i is small for other values. Omega i could reflect, maybe, how certain we are about the market price that we see. Maybe we think this market price is stale, maybe it's a few minutes old, or maybe there's a wide bid ask on this market price, so we're not as confident of the true value of the security. Either way the omega i's can be used to reflect the importance of the i security in the calibration. We also have this additional term, which I won't spend too much time on. But, theta prev stands for theta of the previous calibration. So, the previous model calibration might have occurred the previous day or the previous hour or the previous half hour. What we want to do typically is make sure that our calibrated parameters don't move around too much. There are various reasons for this, but it helps to keep the model somewhat stable through time if we penalize Theta being too far away from the previously calibrated vector theta. And lambda will be a parameter that's greater than or equal to 0 which reflects the importance of this term to the overall calibration. Once this model has been calibrated we can use it to hedge and price more exotic or illiquid securities. One problem, however, is that this calibration problem is often very difficult to solve. This is typically a known convex optimization problem. Well, if you don't know what I mean by nonconvex basically the upshot of that is that there will be many local minima. And it would be very hard to find the true solution to this optimization problem. As market conditions change we often find that we need to recalibrate frequently, often many times a day. If the model wasn't right then actually you would only need to calibrate once. So, in fact your model typically in practice is not right it's never even close to being right. We always need to recalibrate it. So, this is very different to the situation on the physical sciences for example. Or typically if you have the right model then you don't need to recalibrate it very often. In finance unfortunately you do. So, derivatives pricing in practice then is little more than using observable market prices to interpolate and extrapolate to price non-observable security prices. But risk-neutral pricing at the model level at least implies that we can extrapolate and interpolate in an arbitrage-free manner. And in case you're wondering, we haven't said anything about the true probabilities in this section. We've discussed fixing con derivatives up to to now, and we've completely ignored true probabilities. We've only discussed risk-neutral probabilities, Q and one minus Q, and in fact in the binomial model, we set them equal to a half. So, does that mean that the derivative prices and practise don't depend on true probabilities? Well that is not true. Of course they depend on practise and the dependence actually enters into the calibration process because the true probabilities will enter the market's perception. Of market conditions, economic conditions, political conditions and so on. And so, the markets perception of the two probabilities will certainly enter in here into the market prices of these securities. So, implicitly when we do this calibration we're actually Including the market's views, the market's probabilities, the market's level of risk aversion, into the entire derivative's pricing exercise.
