Setting Up the Experiment

Let us try to understand how to manage a book of derivatives.

Suppose we know the the stock is following the BS dynamics.

We could then analyse how the delta hedge procedure works.

If this theory is solid, we should be able to eliminate all risks while hedging.

Example:

- IR = 2%

- non-dividend paying stock that follows a geometric BM with µ = 10% and σ = 20%. S(0) = 100€.

- ATM Call option: maturity T = 0.1

- Delta hedging at regular time step dt = 0.005 --> 20 steps.

We will generate sample paths of the stocks by generating standard normal deviates and use the known formula.

From the time series, we can calculate back the realised volatility after the experiment --> RV = 17%.

First day

Trader sells the option at a price = 2.62€, corresponding to an IV = 20%.

He will also use this IV for hedging purpose, which prescribed a delta = 52.52% --> buy 0.5252 stocks.

He does this by borrowing (0.5252*100€ - 2.62€) = 49.90€.

Instruments in his book:

• a sold call option = 2.62€
• a loan with PV = 49.90€
• a stock ptf = 52.52€.

Second day

- The outstanding loan is increased because of the interest effect. In this case, the effect is about 1 cent.

- The stock has decreased by 10 cents. Because of the delta-position, his stock ptf loses about 5 cents.

- The sold option loses more than just 5 cents.

All of this brings the trader's book at a positive value.

After the change in the market, the delta has changed to 0.4183%.

The trader will thus sell part of his stock ptf at the current level. With this cash, he pays back part of the loan.

The trader is in a new delta neutral position to start the next day.

The theory of BS is prescribing that the hedging cost leads to the option price.

However, we notice a profit coming into the book after just one day of hedging.

Why?

The reason is that the theory assumes that you are hedging continuously through time, rather than once a day.

So the error is induced by the discrete nature of the hedging strategy.

Another way of looking at this is by analysing the random number that was used to generate the value of the second day.

This number is a draw from the standard normal distribution. This one is a small number.

One could say that the volatility of the first day, because of this small draw, is smaller than actually predicted by the distribution.

It is smaller than the "typical" number in a standard normal distribution.

Which random number one has to use in order to see no profit and no loss in this first day?

Z = -1 or +1 --> one std deviation out of the mean.

CCL:

Every day where the move of the stock is less than anticipated, based on vol, trade will have a gain in his book.

Every day where the move of the stock is bigger than anticipated, based on vol, trader will have a loss in his book.

Because of the statistical nature and the assumed independence of the moves on a daily basis, his losses and gains will average each other out, at least if the volatility that realises corresponds to the pricing volatility at the start.

Q. What happens if right after the price agreement has been done, the market volatility drops to 5%?

This means that the trader will see a nice profit in his book.

However, this profit will die out if trader hedges the option during the lifetime, because the final P&L will depend on how the RV turns out.

The smart trader would try to buy back this option in the market if possible and hence lock in this margin.

As we explained in this delta-hedging procedure, there are discrete effects.

Up to now, we only have a qualitative understanding of those.

No worries, the introduction of gamma and theta will help us quantify them.