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.