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Now, let's see how the formulas that we obtained in the last video,

become the formulas of

the celebrated Black-Scholes option pricing model

in the continuous time limit when delta_t goes to zero.

First, the Black-Scholes model assumes a very specific model

for evolution of the stock price ST. Namely,

it assumes that the stock dynamics in continuous time are described by

Geometric Brownian motion with drift mu and volatility sigma,

which is shown here,

and Wt here stands for a standard Brownian motion.

Now, we can check that what happens

with our hedging and pricing formulas in this limit and with these dynamics.

Let's start with hedges.

For the optimal hedge,

we already obtained the formula shown here.

While in the second equation,

I just wrote it in terms of the future value C sub t plus one,

of the mean option value using the same law of

conditional expectations that we used in the previous video.

Now, when delta_t goes to zero,

we can simply substitute the first order tailored

expansion for C sub T plus one in this formula.

And when we plug this second equation into the first one,

then we see that variance of delta_S cancels out in the numerator and denominator.

So, in this limit,

the optimal hedge is just a derivative of

the mean option price with respect to the stock price.

And this is the classical result of the Black-Scholes model.

So, our framework reproduces the Black-Scholes hedges in the limit of continuous time.

Now, let's see what happens with the option price in the same limit.

First, there is no more risk premum in this limit

because risk is instantaneously eliminated in the Black-Scholes model.

So, what he means,

is only the mean option price C_hat.

We had the recursive formula for the mean option price given by the first equation here.

So, we have to compute the small delta_C limit of both terms in this relation.

The limit of the second term is easy to find

using the definition of the log normal stock dynamics.

Computing it, we get the expression shown here,

which is proportional to the difference between the stock if mu,

and risk free rate R. Next,

we compute the limit of the first term here.

This calculation is shown here.

So, what we do.

For the first term in the recursive formula for C,

hat_C we need to use the second order of Taylor expansion of C_ha at time T plus one.

And then plug it into the expression for

a dStm as shown in the second line of equation 20.

Now, when we have both prices for both pieces,

we can substitute them back in the recursive formula for C_hat.

And when we do this,

we set the mean value of the increment dWt of the Brownian motion to zero,

and the mean of its square Wt squared, to one.

Then as you can check yourself,

a miracle happens and the stock drift mu,

drops out from the problem in this limit.

If we now simplify what we obtained,

we see that it coincides with the celebrated Black-Scholes equation for the option price.

Now, the reason why stock drift mu drops out in the limit

of various small time steps is easy to understand.

This is simply because for very small variants of delta_T,

the square root of delta_T is much larger than delta_T itself.

But in the lognormal equation for the stock price when you

discretize it to small but finite time steps delta_T,

they turn proportional to the Brownian motion scales as the square root of delta_T,

while the drift term scale says delta_T.

But this exactly means that the stock drift mu has no chance to impact

anything when the re-hedging frequency delta_T goes to zero.

The hedges just happened way too fast.

So, the model cannot differentiate between mu and these three rate R in this limit.

So, we conclude that our discrete R formulation correctly reproduces

the Black-Scholes model if time steps are very small and the world is lognormal.

And on this we round up our second lesson of this week.

And your homework for this week,

you will implement the discrete time Black-Scholes model that we constructed here,

and compare the hedgers and option prices that

you obtained with this model with the price and

hedgers of the original Black-Scholes model.

In the next week,

we will see how this whole thing can

be reformulated in the language of dynamic programming

and the investment growing.