64
6 The Greeks and Risk Management
Calculating the VaR for investments that follow a normally distributed random
walk is straightforward, we just need to find the volatility for the ten-day time horizon,
that can be derived from the annual or daily volatility and scaling with the root of the
number of days. As the second ingredient we need to know that the tail of a Gaussian
distribution beyond 2.33 standard deviations contains 1% of the area. Thus, for an
initial investment of S 0 we have a 1% chance that the share price wanders beyond
S 0 − 2.33S 0 σ (10 days). The value at risk in that case is
VaR = 2.33S 0 σ (10 days)
(6.9)
and is easily estimated, even for complex portfolios, in which case the volatility for
the portfolio as a whole is used.
An alternative method uses historic data [1] and considers a number N of consecutive trading days, calculating gains and losses for each day-to-day variation and
ranking the losses. The 0.01 × N worst loss may serve as an estimate for the 1%
chance to exceed that loss.
It is apparent that the underlying quantity for all the previously discussed methods
are the daily varying share prices and in particular the volatility σ. In Chap. 7 we pay
closer attention to the characterization and modeling of such time series and how to
extract quantities, such as the volatility, directly from the raw data. But before doing
so, we will address methods to combine options and shares to build portfolios that
allow us to tailor our risk.
6.4 Tailoring Risk to One’s Desire
In this section we will discuss different linear combinations of options that address
certain expectations of the portfolio owner. To illustrate this, we introduce the profit
diagram, which is closely related to the payoff diagram of options encountered in
previous chapters, but has a subjective twist because the profit of one partner in a
trade is the loss of the other partner. In this sense their profit axes are inverted with
respect to each other. This is illustrated in Fig. 6.4, where we display the payoff
function of a call option on the left and the profit functions for the writer and the
owner of the option, respectively. The profit functions are shifted vertically because
the writer of the option makes an initial profit by selling the option and the purchaser
and later owner of the option has to pay for it, thus incurring a negative profit.
We first discuss portfolio insurance with a put option. Our portfolio = S + p
consists of one share S and one put option p with a strike price below the current
value of the shares. Note that we do not hedge in this scenario, we simply buy one
put option for each of the initially acquired shares at the same time. The put option
with the lower strike price will allow us to limit losses for the shares, should the share
price drop below the strike price K . In that case we can sell at K instead of some even
lower price to which the shares may have dropped. In the other case where the share
value increases, we had just incurred a small cost when purchasing the put option.
6 The Greeks and Risk Management
Calculating the VaR for investments that follow a normally distributed random
walk is straightforward, we just need to find the volatility for the ten-day time horizon,
that can be derived from the annual or daily volatility and scaling with the root of the
number of days. As the second ingredient we need to know that the tail of a Gaussian
distribution beyond 2.33 standard deviations contains 1% of the area. Thus, for an
initial investment of S 0 we have a 1% chance that the share price wanders beyond
S 0 − 2.33S 0 σ (10 days). The value at risk in that case is
VaR = 2.33S 0 σ (10 days)
(6.9)
and is easily estimated, even for complex portfolios, in which case the volatility for
the portfolio as a whole is used.
An alternative method uses historic data [1] and considers a number N of consecutive trading days, calculating gains and losses for each day-to-day variation and
ranking the losses. The 0.01 × N worst loss may serve as an estimate for the 1%
chance to exceed that loss.
It is apparent that the underlying quantity for all the previously discussed methods
are the daily varying share prices and in particular the volatility σ. In Chap. 7 we pay
closer attention to the characterization and modeling of such time series and how to
extract quantities, such as the volatility, directly from the raw data. But before doing
so, we will address methods to combine options and shares to build portfolios that
allow us to tailor our risk.
6.4 Tailoring Risk to One’s Desire
In this section we will discuss different linear combinations of options that address
certain expectations of the portfolio owner. To illustrate this, we introduce the profit
diagram, which is closely related to the payoff diagram of options encountered in
previous chapters, but has a subjective twist because the profit of one partner in a
trade is the loss of the other partner. In this sense their profit axes are inverted with
respect to each other. This is illustrated in Fig. 6.4, where we display the payoff
function of a call option on the left and the profit functions for the writer and the
owner of the option, respectively. The profit functions are shifted vertically because
the writer of the option makes an initial profit by selling the option and the purchaser
and later owner of the option has to pay for it, thus incurring a negative profit.
We first discuss portfolio insurance with a put option. Our portfolio = S + p
consists of one share S and one put option p with a strike price below the current
value of the shares. Note that we do not hedge in this scenario, we simply buy one
put option for each of the initially acquired shares at the same time. The put option
with the lower strike price will allow us to limit losses for the shares, should the share
price drop below the strike price K . In that case we can sell at K instead of some even
lower price to which the shares may have dropped. In the other case where the share
value increases, we had just incurred a small cost when purchasing the put option.