So, let's start with the discrete time version of the Blach-Scholes model.

In this case, the problem of option hedging and pricing

in this formulation amounts to sequential risk minimization.

Risks that we have to minimize in this setting is the risk of mis-hedging,

that is the risk of mis-balance between your replication portfolio and your option.

The main open question is how to define risk in an option.

Here, we choose a local risk minimization approach that

was pioneered in the work of Follmer and Schweizer.

What I will be presenting next is a version of

this approach that was developed by physicists Potters and Bouchaud,

whose modifications suggested in a Ph.D. thesis by Grau.

In this approach, we take the view of a seller of a European option.

For example we put option with maturity T and the terminal payoff of H

of S of T at maturity that depends on the final stock price ST at that time.

Now, to hedge this option,

the seller uses the proceeds of the sale to set up

replication or hedge portfolio P sub T made of the stock,

ST and a risk-free bank deposit BT.

Let's call the value of this portfolio pie of T. If the stock position at time T is UT,

then the value of hedge portfolio at any time T less

than T will be equal to UT times ST plus BT.

Now, let's see how we should work with this replication portfolio.

As usual, the replication portfolio tries to exactly

match the option price in all possible future states of the world.

If we started maturity T when the option position is closed,

the hedge UT should be closed at the same time.

Therefore, we set U of T equals zero for T equal capital T. And because of that,

we have that pie of capital T should be equal to the cash position

BT which should be equal to the option payoff H sub T of S sub T,

which sets a terminal condition for the option and also for

the cash account that should hold in all future states of the world at time capital T.

Now, to find an amount needed to be held in

the bank account at previous times small T less than capital T,

we impose the self-financing constraint.

This constraint requires that all future changes in the hedge portfolio should be

funded from the initial set bank account,

without any cash infusion or withdrawals over the lifetime of the option.

This implies the following relation that ensures a consideration of the portfolio value

by a re-hedge at time T. In the left hand side of this equation,

we have the portfolio value that we have immediately before a re-hedge.

And on the right hand side,

we have a value that is obtained after a re-hedge.

Now, we can express it as a recursive relation that

can be used to calculate the amount of money to keep in

the bank account to hedge the option at any time small T less than capital T,

using its value at the next time instance.

And it's given by the expression shown here for the bank account.

Now, if we plug this into the definition of the portfolio and rearrange terms,

we obtain a recursive relation for pie of T in terms of its values at later times,

which is shown in this equation.

We can solve this equation backwards in time,

starting from T equal capital T without terminal condition and time

capital T and continue all the way to the current time T equals zero.

But note that the last two equations imply that both B sub T

and pie sub T are not measurable at any time T less than capital T,

as they depend on the future.

This means that their values today,

B zero and pie zero,

will be random quantities with some distributions.

For any given hedging strategy UT,

these distributions can be estimated using Monte Carlo simulation,

which first simulates N paths for the stock price.

And then evaluate pie of T going backwards on each path.

Please note that because the choice of

hedge strategy does not affect the evolution of the underlying,

such simulation of forward paths would be only needed once and then reused

for future evaluation of the hedge portfolio under different hedge strategies.

So, to summarize, the Monte Carlo simulation works in the following way.

First, we do the forward paths by simulating

the stock price ST all the way to the future T of time T. Then,

we do the backward paths using the recursive relation

for pie of T that takes a prescribed hedge strategy UT,

and back propagates uncertainty in the future into uncertainty today.

What makes such back propagation of uncertainty

possible is the self-financing constraint that we imposed on the portfolio,

which serves as a time machine for risk.

This propagation of future errors back to the present is exactly what

the dealer wishes the seller of the option needs,

as she has to set the price of option today.

This can be done, for example,

by setting the option price to be equal to the mean of

the distribution of pie zero plus some premium for risk,

that would be reflected by a variability of this distribution.

But all these should obviously come only

after the seller decides on the hedging strategy UT to

be used in the future that will be applied in the same way

as a mapping for any future scenarios for the stock price.

In the next video,

we will talk about the choice of this optimal strategy UT.