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.
Igor Halperin
