In this module, we're going to discuss Delta-Hedging within the Black-Scholes model. Delta-Hedging allows to exactly replicate the payoff of an option. When we Delta-Hedge, we're following a self-financing trading strategy whose value at maturity is exactly equal to the value of the derivative that we're trying to replicate. So Delta-Hedging is possible, we're going to see how it works within the context of the Black-Scholes model. However, in practice, we cannot Delta-Hedge exactly datas because we cannot trade at every instance in time. And, because we actually don't know the true model and the true perimeters of the model that generate the security prices. So in practice, Delta-Hedging can only be done approximately. We're going to discuss Delta-Hedging in this model. Recall that the delta of a European call and put option, respectively, are given by the following terms here. So, Call-Delta is e to the minus cT N d1, and then using Put-Call Parity, we can easily see that the Put-Delta is given to us by the Call-Delta minus e to the minus c times T. Where T is the maturity of the option and k is the strike of the option. c is the dividend yield, r is the risk free interest rate, and S0 is the initial stock price. In the Black-Scholes model, an option can be replicated exactly by following a self-financing trading strategy. Now, just remind ourselves, first of all, what is the Black-Scholes model? Well, remember in the Black-Scholes model we've seen that the stock price follows the geometric Brownian motion. So, that means that St is equal to S0e to the mu minus sigma squared over 2 times t plus sigma times a Brownian motion Wt. So, this the geometric Brownian motion process that the security price follows. We also assume there's a risk free interest rate r, a dividend dlc. We assume that continuous trading is allowed. So, cts stands for continuous trading. And that short sales are also allowed. And of course, we also assume that we can borrow or lend at the risk-free rate of r. So this is the Black-Scholes model, here, also I should have mentioned, if it's implicit assumptions continues trading that there are no transactions costs. Another is we can trade continuously without having to pay a charge or fee for trading. So, no transactions costs. Alright. So, in the Black-Scholes model, an option can be replicated exactly by following a self-financing trading strategy. Now, we did this in the binomial model. So, if you follow the argument in the binomial model, you'll realize that we can replicate any security in the binomial model. Now, if you also recall, we mentioned that the binomial model can be viewed as an approximation to geometric Brownian motion. And indeed, as the number of periods n goes to infinity. We argued, or at least, we said that the binomial model converges in an appropriate sense to geometric Brownian motion. And therefore, it should not be surprising that it is also the case in the Black-Scholes model that every security can be replicated exactly by following a self-financing trading strategy. When we execute this self-financing trading strategy in practice, we often say that we are delta-hedging the option. And I will explain where this terminology delta-hedging comes from in a moment. But of course, and as I just said in the previous slide, the Black-Scholes model assumes we can trade continuously. However, this is not feasible in practice. In practice, you cannot trade continuously. And indeed, you have to pay transactions cost when you do trade. So what people do instead is they trade periodically, and in particular they hedge periodically. It also means we can no longer exactly replicate the option payoff. In fact, the best that we can do is hope to approximately replicate the option payoff. Anyway, let T be the option in exploration, so if you want to understand delta-hedging a margin that we are trying to replicate pay off of a call option. So remember, the pay off of the call option C capital 'T' is equal to the maximum of 0 and ST minus k.' And of course, we can price this at any time little t via the Black-Scholes formula. So, what we're going to do is, we're actually going to set a fixed number of times. T0 up to t little n. T0 is time 0. T little n is time capital T. We'll assume that Delta t is the length of any of these intervals, so Delta t equals ti plus 1 minus ti for all i. And what we're going to do is we're going to say V0 of S0 Sigma0 is the initial value of the option, so you can think of V0 of S0 Sigma0 as being the Black-Scholes price of the option. So we often write it as S0 or Ck sigma, and we call it sigma 0 now and t. So V0 of S0 sigma 0 is the Black Scholes price of the call option at time 0. And we're emphasizing here that it's going to depend on S0 and sigma 0. The other parameters over here we'll keep implicate, but we wont mention them explicitly. So what were going to do is, were going to define the following trading strategy. Where Vi plus 1 is going to be the value of the trading strategy at time I plus one. So we're going to see Vi plus 1 equals Vi plus delta i. So in the case of our call option, delta i is equal to at the minus c times T minus Ti times nd1. So, it's delta i times Si plus 1 plus Sic delta t minus Sic, plus another term over here. So what are we doing here? Just looking at equation one here, it's not really clear what's going on. So let's try and do this. So, at time Ti, we have, at time Ti, we have Vi dollars. So this is the value of the trading strategy at time i. So what we're going to do is we are going to hold delta i units of the stock at that time. And we are going to put the remainder of our portfolio value, which will be Vi minus delta i times Si into the cash account. So if we do that, then at time Ti plus 1. What will we have? Well, we can see we are going to have Vi. So Vi is what we have at the previous period at time Ti. And remember, we hold delta i units of the stock at time Ti, and we're going to hold them until trading at time Ti plus 1. And therefore, we're going to make or lose delta i times the value of the security at time i plus 1 minus the initial value of the security, which will be Si. So, the value of the security of time I plus 1 will be Si plus 1 Plus any dividends we get. So what we're doing is here is we're assuming that we're going to get a dividend yield of c, so over the period delta t, we're going to get a dividend yield of c delta t times Si. So this represents the dividend payment in the period ti to ti plus 1. So , the total value of the, of the position at time i plus 1, or ti plus 1. Will be si plus 1 plus the dividend that we obtain. Therefore, the gain on the stock position will be this quantity here minus si. Likewise, we would have invested this quantity into the cash account at time i or time ti. And therefore, we will earn interest at the risk free rate between time ti and t plus 1 according to the rate of or. And so, we would obtain all of this quantity from our position in the cash account, time ti plus 1. Therefore, the value of the strategy times i plus 1 will be vi or initial value. Plus the gain or loss from holding delta i into the security, plus our position in the cash account. So if delta iSi is less then vi, we will be investing money in the cash account and earning some interest. If delta iSi is greater than vi, you would actually be borrowing from the cash account. This term would be negative, reflecting the fact that we owe interest to the cash account or to the bank that lent us the money. Now I mentioned here that this is a self-financing trading strategy and it's almost implicit that it is self-financing. And that is because we're seeing that the value of the portfolio time i plus 1 is equal to the value in the previous period, vi plus the gains or losses from trading. These are the gains or losses. This is the gain or loss from the stop position. This is the gain or loss from the cash account. Clearly, if you look at this, we're not injecting any new money into the strategy at time i plus 1, nor are we taking any money out. So in fact, the way we have defined this, value process for the trading strategy, it is actually a self financing trading strategy. So at time, t plus 1, Vi plus 1 equals Vi plus gains or losses from trading. And these gains or losses are given to us in x equation 1 here. So this is a self-financing trading strategy. Now, it is a particular type of soft announcing trading strategy. Because the number of units of the stalk, we hold the time I, is equal to the delta of the option. Remember delta I is short on for the delta we saw on the previous slide so it's this for a call option. It would be this for a put option. So we're going to hold that many units of the underlying security, at time i. So when we adapt this strategy, this self-financing trading strategy, we are delta-hedging the options. And if we let delta t go to 0. Remember, what we have is, we've got a time horizon, a capital T. Maybe capital T is 1 year, or 6 months, or 3 months. And we're breaking this down into a number of subperiods, t1, t2, t3 and so on up to tn minus 1. And the length of each interval here is delta t. So what were saying is, if we let delta t go to 0, in other words, if we let n go to infinity then this self-financing trading strategy will actually replicate the option payoff at time capital T. This is in fact what Black and Scholes showed in there original paper. Otherwise, if we don't let delta t go to 0, but we have to keep it fixed maybe delta t is equal to one day or one week. Then in that case we're only going to replicate the payoff of the option approximately. So Vn is equal to the option payoff at the time t approximately. This by the way assumes that the stigma perimeter that we use to price the option is. Correct. And this is why, returning to the previous slide, I emphasize the dependence of the option price on sigma 0. The value of sigma that we're using within the Black-Scholes formula over here. In practice, we can't expect to know sigma 0. We can only guess what the true volatility is, what the true volatility parameter is. So in fact, we can only expect to replicate the strategy exactly if we know the true sigma 0 and we let delta t go to, to 0. If we assume the wrong sigma 0, then V0 and all the delta i's will be wrong. Remember, V0's the initial value of the option, the call option in this example that's given to us by the Black-Scholes formula. But if we use the wrong sigma 0 inside here, we're going to get the wrong price. Not only will we get the wrong price, but all of these delta i's will also depend on sigma 0. If you look to the previous slide, you'll see that, in the case of a call option or delta i is equal to this expression here. And d1, as we see here, is a function of the sigma parameter. So if we get the wrong sigma, we're going to get the wrong d1. And we're going to get the wrong delta as well. And so, therefore, if we get the wrong sigma 0, we're going to have the wrong value v0, and we'd have the wrong delta I's. And so, we will not be able to replicate the option exactly even if delta t went to 0. In practice, as I said, we can't let delta t go to 0, that's because we don't trade continuously in practice. So delta t would be fine, maybe one day or half a day or one week and we can only hope to guess the right value of sigma. So what that means in total is that, we can't exactly replicate an option. We can't hope to replicate an option in practice, because we don't know sigma. And in fact, the true dynamics of the security prices don't follow geometric Brownian Motions or they are not geometric Brownian motions. So the concept of dynamic replication is really only a theoretical concept. That said, it is useful in practice, because we can hope to use these ideas to replicate option prices approximately, and replicate other, derivative payoffs approximately. Remember, this strategy here is self-financing. So we can actually always follow this strategy, we just have to guess what the correct sigma is to use inside N of d1 to get the correct delta. And we also hope that the initial value price was calculated correctly, i.e., that we used the right sigma 0. If we do that, we can hope to replicate the option approximately and sufficiently accurately for risk management purposes.
