Trading options is particularly risky due to the possibly high
random component. Advanced strategies to reduce and manage this
risk can be derived from Black-Scholes formula (6.24). To
illustrate this issue we consider an example and some traditional
strategies.
Example 7.1
A bank sells a European call option to buy 100000 shares of a
non dividend paying stock for 600000 EUR. The details of this
option are given in Table
6.3.
Table 6.3:
Data of the example
Current time  |
6 weeks |
Maturity  |
26 weeks |
Time to maturity
 |
20 weeks = 0.3846 |
Continuous annual interest rate  |
0.05 |
Annualized stock volatility  |
0.20 |
Current stock price  |
98 EUR |
Exercise price  |
100 EUR |
|
Applying Black-Scholes' formula (6.24) for a non dividend
paying stock,
gives a theoretical value of
480119 EUR, approximately 480000 EUR, of the above
option. That is, the bank sold the option about 120000 EUR
above its theoretical value. But it takes the risk to incur
substantial losses.
A strategy to manage the risk due to the option would be to do
nothing, i.e. to take a naked
position. Should the option be exercised at maturity the bank has
to buy the shares for the stock price prevailing at maturity.
Assume the stock trades at
EUR. Then an options'
exercise costs the bank
EUR, which is a multiple of what the bank received for selling
the derivative. However, if the stock trades below
EUR
the option will not be exercised and the bank books a net gain of
600000 EUR.
SFEBSCopt2.xpl
In contrast to the naked position it is possible to set up a covered position by buying 100000
shares at
EUR at the same
time the option is sold. In case
the option will be
exercised and the stocks will be delivered at a price of
EUR, which discounted to time
is
about 9800000 EUR. Thus the bank's net gain is equal to
600000 EUR, the price at which the option is sold. If the
stock price decreases to
EUR then the option will
not be exercised. However, the bank incurs a loss of 2000000
EUR due to the lower stock price, which is as above a multiple
of the option price. Note that from put-call parity for European
options (Theorem 2.3) it follows that the risk due to a
covered short call option position is identical to the risk due to
naked long put option position.
Both risk management strategies are unsatisfying because the cost
varies significantly between 0 and large values. According to
Black-Scholes the option costs on average around 480000 EUR,
and a perfect hedge eliminates the impact of random events such
that the option costs exactly this amount.
An expensive hedging strategy, i.e. a strategy to decrease the
risk associated with the sale of a call, is the so-called stop-loss strategy: The bank selling
the option takes an uncovered position as long as the stock price
is below the exercise price,
and sets up a covered
position as soon as the call is in-the-money,
The shares to be delivered in case of options exercise are bought as soon as the stock
trades above the exercise price
and are sold as soon as
falls below the
exercise price
Since all stocks are sold and bought at
after time 0 and at
maturity
either the stock position is zero,
or
the stocks are sold at
to the option holder,
this
strategy bears no costs.
Note that playing a stop-loss strategy bears a cost if
i.e. stocks are bought at
and sold at
costs of a stop-loss hedging strategy:
Because these costs are smaller than the Black-Scholes price
arbitrage would
be possible by running a stop-loss strategy. However, this reasoning ignores some
aspects:
- Buying and selling stocks bear transaction costs,
- Buying stocks before time
involves binding capital leading to renounce of interest rate
revenue,
- practically it is not possible to buy or sell stocks exactly at
rather stocks
are bought at
if stocks are increasing and stocks are sold at
if stocks are decreasing, for a
In practice, purchases and sales take place only after
time units. The larger
, the greater
in general, and the less transaction costs have to be
paid. Hull (2000) investigated in a Monte Carlo study with
simulated
stock price paths the stop-loss strategy's ability to reduce the risk associated with
the sale of a call option. For each simulated path the costs
caused by applying the stop-loss strategy are registered and their sample variance
is computed. Dividing the sample standard deviation by the call price measures the
remaining risk of the stop-loss hedged short call position
Table 6.4 shows the results. A perfect hedge would reduce the risk to zero,
i.e.
Table 6.4:
Performance of the stop-loss strategy
(weeks) |
5 |
4 |
2 |
1 |
 |
 |
 |
1.02 |
0.93 |
0.82 |
0.77 |
0.76 |
0.76 |
|
In order to reduce the risk associated with option trading more
complex hedging strategies than those considered so far are
applied. Let us have a look at the following example. Sell a call
option on a stock, and try to make the value of this portfolio for
small time intervals as insensitive as possible to small changes
in the price of the underlying stock. This is what is called delta
hedging. Later on, we consider further Greeks (gamma, theta,
vega, rho) to fine tune the hedged portfolio.
By the delta or the hedge
ratio we understand the derivative of the
option price with respect to the stock price. In a discrete time
model we use the differential quotient of the change in the option
price
with respect to a change in the stock price

oder
The delta of other financial instruments is defined accordingly.
The stock itself has the value
Consequently it holds
A futures contract on a non dividend
paying stock has a value of
(see
Theorem 2.1) and thus its delta is
as well. Stocks and future contracts
can therefore be used equivalently in delta hedging strategies. If
the latter are available they are preferable due to lower
transaction costs.
Example 7.2
A bank sells calls on 2000 shares of a stock for a price of

EUR/share at a
stock price of

EUR/share. Let the call's delta be

To
hedge the sold call options

shares of the stock are added to
the portfolio. Small changes in the option value will be offset by corresponding changes
in the value of the portfolio's stock shares. Should the stock price increase by 1
EUR, i.e. the value of the stock position in the portfolio increases by 800 EUR,
the value of one call on 1 share increases by

EUR and following the value of the portfolio's short call position decreases by 800
EUR. That is, gains and losses offset because the delta of the option position is
neutralized by the delta of the stock position. The portfolio has a

and the
bank takes a
delta neutral position.
Since the delta of an option depends on the stock price and time,
among others, the position is only for a short period of time
delta neutral. In practice, the portfolio has to be re-balanced frequently in order to adapt to
the changing environment. Strategies to manage portfolio risk
which involve frequent re-balancing are known as dynamic
hedging. We point out that the
Black-Scholes differential equation (6.3) can be derived
by means of a dynamic hedge portfolio whose position is kept
continuously delta neutral. This approach is analogous to
reproducing the option by a duplicating portfolio.
Example 7.3
The price of the underlying stock rises within a week to 110 EUR. Due to the time
decay and the increased stock price the option delta increased to

In
order to reobtain a delta neutral position

shares of the
stock have to be bought.
From the Black-Scholes formulae for the value of European call and put options on non
dividend paying stocks it follows for the delta:
with
being defined in equation (6.25).
Figure 6.1 displays the delta (6.27) as a
function of time and stock price. For an increasing stock price
delta converges to 1, for decreasing stock prices it converges to
0. Put differently, if the option is deep in-the-money (ITM) it
will be exercised at maturity with a high probability. That is the
reason why the seller of such an option should be long in the
underlying to cover the exercise risk. On the other hand, if the
option is far out-of-the-money it will probably not be
exercised, and the seller can restrict himself to holding a
smaller part of the underlying.
Fig.:
Delta as a function of the stock price (right axis) and time to maturity (left axis).
SFEdelta.xpl
|
Furthermore, the probability
that an out-of-the-money (OTM)
option will be exercised and an ITM option will not be exercised
at maturity is higher the longer the time to maturity. This
explains why the delta for longer times to maturity becomes more
flat (linear).
Table 6.5 according to Hull (2000) shows (in
the same spirit as Table 6.4) the performance of the
delta hedging strategy contingent on the time increments
between re-balancing trades. If
is small enough the
risk associated with a sold call option can be managed quite well.
In the limit
continuous re-balancing
underlying the derivation of the Black-Scholes formula follows,
and the risk is perfectly eliminated
Table 6.5:
Performance of the delta-hedging strategy
(weeks) |
5 |
4 |
2 |
1 |
 |
 |
 |
0.43 |
0.39 |
0.26 |
0.19 |
0.14 |
0.09 |
|
The linearity of the mathematical derivative implies for the delta
of a
portfolio consisting of
contracts of
financial derivatives
with deltas
Example 7.4
Consider a portfolio consisting of the following USD options
- 1.
- 200000 bought calls (long position) with
exercise price 1.70 EUR
maturing in 4 months. The delta of an option on 1 Doller is
- 2.
- 100000 written calls (short position) with
exercise price 1.75 EUR
maturing in 6 months and a delta of
- 3.
- 100000 written puts (short position) with exercise
price
EUR
maturing in 3 months with
The portfolio's delta is (increases in values of written options have a negative impact
on the portfolio value):
The portfolio can be made delta neutral by selling 111000 USD
or by selling a corresponding future contract on USD (both have
a delta of

).
Using the delta to hedge an option position the option price is
locally approximated by a function which is linear in the stock
price
Should the time
passing by until the next
portfolio re-balancing be not very short this approximation is no
longer adequate (see Table 6.5). That is why a more
accurate approximation, the Taylor expansion of
as a function
of
and
is considered:
where (as we already saw in the demonstration of Theorem 6.1)
is of size
and the terms summarized in
are of size smaller than
Neglecting
all terms but the first, which is of size
the
approximation used in delta hedging is obtained:
Taking also the terms of size
into account it follows
where
is the options theta and
is the options gamma.
is also called the options time decay. For a call
option on a non dividend paying stock it follows from the
Black-Scholes formula (6.24):
and
 |
(7.28) |
where
is defined in equation (6.25).
Figure 6.2 displays the gamma given by equation
(6.28) as a function of stock price and time to
maturity. Most sensitive to movements in stock prices are
at-the-money options with a short time to maturity.
Consequently, to hedge such options the portfolio has to be
rebalanced frequently.
Fig.:
Gamma as a function of stock price (right axis) and time to maturity (left axis).
SFEgamma.xpl
|
Assuming a delta neutral portfolio gamma hedging consists of
buying or selling further derivatives to achieve a gamma neutral portfolio, i.e.
and thereby making the portfolio value even more insensitive to changes in the
stock price. Note that on the one hand neither stocks nor future contracts can be used
for gamma hedging strategies since both have a constant
and thus a zero gamma
On the other hand, however, those instruments can be used to make a gamma
neutral portfolio delta neutral without affecting the portfolio's gamma neutrality.
Consider an option position with a gamma of
. Using
contracts of an option
traded on a stock exchange with a gamma of
the portfolio's gamma is
By setting
the resulting gamma for the portfolio is
0.
Example 7.5
Let a portfolio of USD options and US-Dollars be delta neutral
with a gamma of

150000. On the exchange trades a
USD-call with

and

By
adding

contracts of this option
the portfolio becomes gamma neutral. Unfortunately, its delta will
be

The delta neutrality can
be achieved by selling 65000 USD without changing the gamma.
Contrary to the evolution of the stock price the expiry of time is
deterministic, and time does not involve any risk increasing
randomness. If both
and
are 0 then the option
value changes (approximately risk free) at a rate
The parameter
is for most options negative,
i.e. the option value decreases as the maturity date approaches.
From Black-Scholes's formula (6.24) it follows for a delta neutral portfolio
consisting of stock options
with
denoting the portfolio value.
and
depend on each other in a
straightforward way. Consequently,
can be used instead of
to gamma
hedge a delta neutral portfolio.
Black-Scholes' approach proceeds from the assumption of a constant volatility
The appearance of smiles indicates that this assumption does not hold in practice.
Therefore, it can be useful to make the portfolio value insensitive to changes in
volatility. By doing this, the vega of a portfolio (in literature sometimes also
called lambda or kappa) is used, which is for a call option defined by
For stocks and future contracts it holds
Thus, in order to set up a vega
hedge one has to make use of traded options. Since a vega neutral portfolio is not
necessarily delta neutral two distinct options have to be involved to achieve
simultaneously
and
From Black-Scholes' formula (6.24) and the variable
defined in equation (6.25) it follows that the vega of a
call option on a non dividend paying stock is given by:
 |
(7.29) |
Since the Black-Scholes formula was derived under the assumption
of a constant volatility it is actually not justified to compute
the derivative of (6.24) with respect to
However, the above formula for
is quite similar to an
equation for
following from a more general stochastic
volatility model. For that reason, equation (6.29) can be
used as an approximation to the real vega.
Figure 6.3 displays the vega given by equation (6.29) as a function
of stock price and time to maturity. At-the-money options with a long time to maturity
are most sensitive to changes in volatility.
Fig.:
Vega as a function of stock price (right axis) and time to maturity (left axis).
SFEvega.xpl
|
Finally, the call option's risk associated with movements in interest rates can be
reduced by using rho to hedge the position:
For a call on a non dividend paying stock it follows from equation (6.24)
When hedging currency options domestic as well as foreign interest
rates have to be taken into account. Consequently, rho hedging
strategies need to consider two distinct values
and
A property of the
Black-Scholes formulae (6.22), (6.24) is that all
option parameters are known except the volatility parameter
In practical applications
is estimated from
available stock price observations or from prices of similar
products traded on an exchange.
Historical volatility is an estimator for
based on the variability
of the underlying stock in the past. Let
be
the stock prices at times
If the stock price
is modelled as Brownian
motion, the logarithmic relative increments
are independent and identical normally distributed random
variables.
is the increment
of the
logarithmic stock price
which as we saw in
Section 5.4 is in a small time interval of length
a Wiener process with variance
Consequently the variance of
is given by
A good estimator for
is the sample variance
with
being the
sample average.
is unbiased, i.e.
and the random variable
is
distributed (chi-square distribution with
degrees of freedom). In particular this implies that the mean
squared relative estimation error of
is given by
Since it holds
an estimator for the
volatility
based on historical stock prices is
By means of a Taylor expansion of the square root function and by
means of the known quantities
and
it follows that
is unbiased
neglecting terms of size
and that the mean squared relative estimation error of
is given by
again
neglecting terms of size smaller than
Thanks to this
relationship the reliability of the estimator
can
be evaluated.
Sample parameter selection:
- a)
- As data daily settlement prices
are often used. Since
is
in general expressed as an annualized volatility
corresponds to one day on a
yearly basis. Working with calender day count convention
Unfortunately, for weekends and holidays no data is available. The following empirical
argument favors to ignore weekends and holidays: If the stock dynamics behaved on Saturdays
and Sundays as it does on trading days even if the dynamics were not observed then
standard deviation of the change in the stock price from Friday to Monday would three
times as large as the standard deviation between two trading days, say Monday to
Tuesday. This follows from the fact that for the Wiener process
the
standard deviation of the increment
is
Empirical studies of stock markets show, however, that both standard deviations are
proportional with a constant of around 1 but in any case significantly smaller than
3. Put in other words, the volatility decreases on weekend days. A conclusion is that
trading increases volatility, and that the stock variability is not solely driven by
external economic influences. Estimating volatility should therefore be done by
considering exclusively trading days. Usually a year is supposed to have 252 trading
days, i.e.
Concerning monthly data,
is applied. In Section 3.3
we have calculated an annual volatility of
based on the monthly DAX
data.
SFEsumm.xpl
- b)
- Theoretically, the
larger
the more reliable
However,
empirically the volatility is not constant over longer time periods. That is to say
that stock prices from the recent past contain more information about the current
as do stock prices from long ago. As a compromise closing prices of the last
90 days respectively 180 days are used. Some authors advise to use historical data of a
period which has the same length as the period in the future in which the estimated
volatility will be applied. In other words, if you want to compute the value of a
call expiring in 9 months you should use closing prices of the past 9 months.
The implied volatility
of an option is computed from its market price observed on an
exchange and not from the prices of the underlying as it is case
for the historical volatility. Consider a European call on a non
dividend paying stock
which has a quoted market
price of
, then its implied volatility
is given by
solving
|
|
 |
|
|
|
|
|
|
with |
 |
|
is the value of the volatility which if substituted into the Black-Scholes
formula (6.24) would give a price equal to the observed market price
is implicitly defined as a solution of the above equation, and has to be
computed numerically due to the fact that the Black-Scholes formula cannot be inverted.
The implied volatility can be used to get an idea of the market view of the stock
volatility. It is possible to construct an estimator using implied volatilities of
options on the same stock but which are different in time to maturity
and exercise
price
A weighting scheme takes the option price dependence on the volatility into
account.
SFEVolSurfPlot.xpl
Example 7.6
Consider two traded options on the same underlying. One is
at-the-money (ATM) and the other is deep ITM with volatilities of

respectively

At-the-money the dependence of option price and volatility is
particular strong. That is, the price of the first option contains
more information about the stock volatility and

can
be considered a more reliable volatility estimate. Thus the
estimator combining both implied volatilities should attribute a
higher weight to

as for example
Some authors suggest to set

with

being the volatility of the option which is most
sensitive to changes in

i.e. the option with the highest
vega

in absolute terms.
In order to apply the concept of risk neutrality (see
Cox and Ross (1976)) the probability measure has to be
transformed such that the price process under this new measure is
a martingale. By doing this the absence of arbitrage opportunities
is guaranteed. In incomplete markets, however, a multitude of such
transformations exist (see Harrison and Kreps (1979)). In contrast
to complete markets the trader cannot build up a self-financing
portfolio reproducing the options payoff at maturity when the
market is incomplete. Therefore hedging is no more riskless, and
option prices depend on risk preferences. In this context we want
to point out that the lack of a perfect hedge is of great
importance in practice.