0:00

Now, after we looked at how the optimal action or hedge is computed we can put

our two main formulas together in order to produce

an algorithmic solution to the problem of hedging and pricing of our option option.

So, we have the recursive relation for

the optimal Q-function which holds for any value of at.

It's given by the first equation here.

But we already computed the optimal action

a-star for the current time-step t which is given by the second equation here.

So, we can substitute the second equation into

the first one to get the recursive relation for

the optimal Q-function Q-star evaluated directly at the optimal action, a-star.

This is very convenient as it does not require

extra storage to keep the values of Q-star for other values of actions.

If our ultimate goal is to evaluate Q-star at time t equals to zero

that is now directly at the optimal action a-star,

such equation of the Q-star and a-star is all we need.

And this is indeed all we need if we want to find the optimal price

and hedge of the option now at time t equals zero.

This is because Q-star with a equal to

a-star is the same thing as V-star as we saw before.

But the V-star is

exactly the negative optimal option price

according to our definition of the value function

V. Therefore we recursively rank

these two formulas starting from time capital T minus one,

then capital T minus two and so on till the current time t equals zero,

and that each time we first compute the optimal action a-star from the second equation,

and then plug it in the right hand side of the first equation.

This gives us a simple recursive algorithm to compute both the optimal hedge

a zero star and optimal Q-function Q zero star at a- star.

And after flipping the sign the last number gives the option price.

So, to summarize this procedure we say the following,

we proceed with the backward recursion all the way back

to time t equal zero to get the optimal hedge and price of the option.

The whole calculation is semi-analytical because the optimization problems solved at

each time-step is quadratic and therefore

can be solved semi-analytically as we just did above.

The other point is that normally,

algorithms of dynamic programming and reinforcement learning

require some representation of an action space.

More to this they usually differ in whether they apply to

a discrete action space or a continuous action space or to both.

But in our setting because of availability of the analytical formula for

the optimal hedge a-star we can apply backward recursion to the Q-star function,

evaluate it directly at the value of the optimal hedge a- star.

Because we carry only one value of at in this calculation which is the optimal value,

there is no need to have any representation of the action space in this setting.

So, we just obtained an algorithmic solution in

our discrete time MDP model for the optimal option price and hedge.

Let's now compare it with the classical Black-Scholes model.

The first thing we can note is that now formulation,

both the hedge and the price are parts of the same expression for Q-star.

And this is different from the Black-Scholes model where we

have two separate formulas for the price and the hedge.

Moreover, in the Black-Scholes model we first compute

the option price by solving the Black-Scholes equation

and only then we compute the option hedge by

differentiating this price with respect to the stock price.

But now MDP formulation,

the order of these operations is exactly the opposite.

In each time-step t we first compute the optimal hedge and

then plug it into the Q-star function to roll it backwards in time.

This is intuitive and in fact corresponds to the market practice

of working with options that are imbalanced infrequently,

for example, once a month.

For such options risk is clearly present because it cant' be hedged

away and the option price is decided based on the analysis of such risk.

But if we follow the Black-Scholes model,

it assumes that we can hedge continuously in time.

This means that by following the Black-Scholes model and three-hedging every

second or a millisecond we can completely eliminate any risk in the option.

To achieve this learning you need to keep exactly the amount of

the Black-Scholes delta of the stock in our hedge portfolio at every millisecond.

This means that in a continuous time limit hedging becomes trivial and

the only non-trivial part of the whole problem is

pricing which is solved using the Black-Scholes equation.

We can break this disappearance of risk in the Black-Scholes model into two steps.

First, if we keep both lambda and delta t larger than zero we have

the MDP problem with a quadratic risk and a direct link between the objective function,

option price, and optimal hedge.

Now, if we set lambda to zero but still keep delta t finite,

then there is no direct link between the option price and optimal hedge anymore,

as there is no proper objective function that would relate them.

But risk is still there and we still can do quadratic risk hedge in this setting.

And finally, when both lambda and delta t

are zero we are back to the Black-Scholes model that

claims that options have no risk at all because it

all can be hedged away if only we can hedge all the time.

But this is of course,

totally unrealistic and losing any track of

actual risk in option is the price paid for the beauty of the Black-Scholes equation.

On the other hand,

discrete-time formulations similar to one I presented

here do not have closed warm expressions for option price,

but they are much more realistic because they explicitly model risk in options.

When both lambda and delta t are larger than zero the MDP formulation gives

a consistent hedging and pricing formulation

that takes the residual risk and options into account.

We just saw a simple algorithm that does that.

So, let's pause here for a minute and then see in the next video how we

can implement this algorithm.