All right. Now, after we verified the tower formulas, correctly reproduce the Black-Scholes model in a limit of zero lambda and Venetian time steps, let's go back to a more general case when lambda and delta t are both non-zero. So, let's look again at the Bellman optimality equation that we obtained. One thing that stands out here is that because the rewards RT is quadratic in AT, the Q-function is also quadratic in AT as can be seen in this equation. Therefore, it means that the Q-function can be maximized analytically. We just have to compute the derivative of the Q-function with respect to AT, and equate it to zero to find the optimal action AT. This is very easy to do as the whole expression is quadratic in AT. So, by differentiating it we get the second expression shown here. This should be set to zero and solved for AT in order to find the optimal action. So, by doing this simple rearrangement, we get our results in expression for the optimal hedge. It should already look familiar to us. The optimal hedge is given by a ratio of two conditional expectations: one, involving the next step portfolio value, pi sub T plus 1, and another one involving the square of the increments delta ST. We had a very similar expression in our discrete time Black-Scholes model. Let me remind you this expression. We wrote it before is the ratio of conditional covariance and conditional variance, but you can also rewrite it as a ratio of two conditional expectations as shown in the second form of the second equation on this slide. So, the only difference between these two expressions is in the second term in the numerator of the first formula. We can note several interesting things about this additional term. First, let's look at them from a mathematical perspective. If we take small time steps, then the expectation of delta ST will be proportional to the difference of MU minus R. This means that we can make this second term in the numerator vanish if we either set MU equal to R or we take a formal limit of lambda going to infinity in this expression. In this case, the optimal hedge obtained by maximization of the Q-function will coincide with the optimal hedge obtained in our discrete time Black-Scholes model. So, this is a formal mathematical statement. But what about a financial interpretation of these results? To answer this question, we can go back to our model formulation. Let's recall that we defined the value function we'd see as some of the expected portfolio value and the sum of discounted variances of the hedge portfolio at hedge times. As a consequence of these specifications, the expect that rewards in our formulation has two terms: proportional to the expected drift and the variance of the hedge portfolio as shown in a second equation here. When we maximize this expression with respect to AT, mathematically, it's identical to the setting of Markowitz portfolio analysis, which maximizes risk-adjusted portfolio returns for the case of stock analysis. So, the objective function here looks simultaneously at the risk and the returns of the hedge portfolio. And clearly, this is different from the discrete time Black-Scholes model, where we only looked at the risk of the hedge portfolio. But if we want to benchmark our model formulation against the discrete time Black-Scholes model, we can simply zero out this additional term by setting their MU equal to R or, alternatively, by taking the limit of Lambda going to infinity. Both with limits will achieve the same goal. That is, they will enforce a pure risk-based view of the hedging and therefore will be consistent with our previous discrete time Black-Scholes model. So, our hedge formula, it produces the pure risk-based hedge of the discrete time Black-Scholes model, but it can also be used when we not only look in at the risk in the option. For example, if options are used as investment vehicles, then we would need to look at both risk and the return of options. The classical Black-Scholes model is not able to address this case as it directly goes to the continuous time limit where everything concludes in all stocks and all options has the same risk created term R. Therefore, when people try to have a framework for speculation with options starting with the Black-Scholes model, they have to rely on various adhoc modifications of the original model. But here, we do not need any such adhoc modifications because our setting here is more general. And in the general case, it optimizes risk-adjusted return of the hedge portfolio rather than just risk. So, a pure-based hedging of the classical Black-Scholes model is just the special case here which can be reproduced by setting MU equal to R in the optimal hedge formula.
