5.4 Dynamic Hedging
55
c = N (d 1 )
for a European call option
p = N (d 1 ) − 1
for a European put option
(5.22)
Here N (x) is the normal distribution from (4.39) and d 1 is defined in (4.43).
We now illustrate how dynamic hedging works in detail by writing a simulation
of the process. See Appendices B.2 and B.3 for the MATLAB code. We assume that
I do not have any capital initially, but sell one option and receive the option price c 0
in cash that I deposit in a bank at the risk-free rate r f . Then I calculate the initial
hedge S = S∂c/∂ S, borrow that amount at the risk-free rate and purchase S
shares. During the remaining days until maturity, at the end of every trading day d
I calculate d using the Black-Scholes formula with the current share price S d and
the remaining time until maturity and use that information to calculate the updated
hedge value d S d . Depending on whether this value is larger or smaller than the
hedge value from the previous day, I either buy additional shares or sell from my
portfolio. Furthermore, I have to pay interest on the money borrowed from the bank
to pay for the shares. This procedure of re-calculating the hedge value every day
keeps the portfolio risk-free. As maturity approaches, either approaches zero or
unity as is obvious from the plot on the right-hand side in Fig. 5.1. If the option is in
the money, will approach unity and my position to provide shares is covered by
the hedge. Then I can hand over the shares in the hedge to the owner of the option but
only receive the strike price for the shares. That money I use to pay off the loan plus
accumulated interest from the bank. See the upper plot in Fig. 5.3 for the temporal
evolution of the share price, shown as the solid black line. The hedge is shown as the
dot-dashed red line and the money borrowed from the bank is displayed as a dashed
blue line. The cost for me, the writer of the option, is the borrowed money minus the
strike price, but that is covered by the initially received price of the option c 0 , that
was invested at the risk-free rate r f .
The lower plot in Fig. 5.3 shows the situation, where, at maturity, the option is
out of the money. In this case will approach zero and I will not need a hedge,
because the owner of the option will forfeit. We find that the initial hedge, shown
as a dot-dashed red line, is reduced to zero as time progresses towards maturity and
it becomes more and more obvious that the option will not be exercised. Therefore
approaches zero and I can sell off the shares in the hedge. At maturity no shares
are present and therefore all money was returned to the bank and I can cover the
accumulated interest from the initially received money for the option c 0 .
Note that in both cases discussed, I, the writer of the option, make a profit. This
is mostly due to the fact that the shares increased without fluctuating, whereas in the
calculation of the option price and the an annual volatility of 30% was assumed.
In reality I will always need to increase the hedge when the share price goes up and
sell shares to reduce the hedge when it goes down and therefore need to buy shares at
a high price and sell at a lower price. This will reduce my profits, but is, on average,
covered by the initially calculated option price c 0 .
55
c = N (d 1 )
for a European call option
p = N (d 1 ) − 1
for a European put option
(5.22)
Here N (x) is the normal distribution from (4.39) and d 1 is defined in (4.43).
We now illustrate how dynamic hedging works in detail by writing a simulation
of the process. See Appendices B.2 and B.3 for the MATLAB code. We assume that
I do not have any capital initially, but sell one option and receive the option price c 0
in cash that I deposit in a bank at the risk-free rate r f . Then I calculate the initial
hedge S = S∂c/∂ S, borrow that amount at the risk-free rate and purchase S
shares. During the remaining days until maturity, at the end of every trading day d
I calculate d using the Black-Scholes formula with the current share price S d and
the remaining time until maturity and use that information to calculate the updated
hedge value d S d . Depending on whether this value is larger or smaller than the
hedge value from the previous day, I either buy additional shares or sell from my
portfolio. Furthermore, I have to pay interest on the money borrowed from the bank
to pay for the shares. This procedure of re-calculating the hedge value every day
keeps the portfolio risk-free. As maturity approaches, either approaches zero or
unity as is obvious from the plot on the right-hand side in Fig. 5.1. If the option is in
the money, will approach unity and my position to provide shares is covered by
the hedge. Then I can hand over the shares in the hedge to the owner of the option but
only receive the strike price for the shares. That money I use to pay off the loan plus
accumulated interest from the bank. See the upper plot in Fig. 5.3 for the temporal
evolution of the share price, shown as the solid black line. The hedge is shown as the
dot-dashed red line and the money borrowed from the bank is displayed as a dashed
blue line. The cost for me, the writer of the option, is the borrowed money minus the
strike price, but that is covered by the initially received price of the option c 0 , that
was invested at the risk-free rate r f .
The lower plot in Fig. 5.3 shows the situation, where, at maturity, the option is
out of the money. In this case will approach zero and I will not need a hedge,
because the owner of the option will forfeit. We find that the initial hedge, shown
as a dot-dashed red line, is reduced to zero as time progresses towards maturity and
it becomes more and more obvious that the option will not be exercised. Therefore
approaches zero and I can sell off the shares in the hedge. At maturity no shares
are present and therefore all money was returned to the bank and I can cover the
accumulated interest from the initially received money for the option c 0 .
Note that in both cases discussed, I, the writer of the option, make a profit. This
is mostly due to the fact that the shares increased without fluctuating, whereas in the
calculation of the option price and the an annual volatility of 30% was assumed.
In reality I will always need to increase the hedge when the share price goes up and
sell shares to reduce the hedge when it goes down and therefore need to buy shares at
a high price and sell at a lower price. This will reduce my profits, but is, on average,
covered by the initially calculated option price c 0 .