62
6 The Greeks and Risk Management
Note that Vega is not a character of the Greek alphabet, but rather an oversized
letter ν. The volatility σ was also assumed constant in the previous chapters, but
in practice depends dynamically on the trading activities of the underlying shares.
For example, at times of uncertainty, with impending strikes or political unrest, the
volatility increases. Under such circumstances it is beneficial to make a portfolio
insensitive to changes of the volatility σ, namely minimizing V.
A portfolio consisting of shares and other derivatives is a derivative itself and
as such it must obey the Black-Scholes differential equation, given by (5.6), thus
−
∂∂
∂t
=
1
2
σ
2 S
2 ∂
2
∂ S 2 + r f S
∂∂
∂ S
− r f .
(6.6)
After substituting the definitions of the Greeks we obtain
− =
1
2
σ
2 S
2
+ r f S − r f
(6.7)
In case the portfolio is perfectly hedged, or risk-balanced with = 0 we have
= r f −
1
2
σ
2 S
2
.
(6.8)
Furthermore, if the risk-free rate is small we can ignore the term with r f . If, at the
same time, is large we see that and are anti-correlated and we can monitor
one as a proxy for the other [1]. This anti-correlation is also visible by comparing
in Fig. 6.1 and on the left plot in Fig. 6.2. One is the very closely the negative of
the other.
6.2 Volatility Smile
The option pricing formula derived in the previous sections are simply guidelines
to value options. In practice the sellers of options can set a price to anything and
the market, or more accurately, the buyer decides, whether to accept the price or to
buy somewhere else. Following the market crash of 1987, option sellers realized that
some of the options that had a very low probability of ever being exercised, all of a
sudden were actually exercised and became very expensive for the option seller. The
reason was that the shares dropped to very low values in the course of the crash and,
for example, put options became extremely expensive. The low selling price of the
option was due to the assumptions that large market variations are extremely rare in
a system described by fluctuation that are distributed according to a Gaussian model.
The tails of the Gaussian, which describe large fluctuations, become exponentially
small. This caused the unlikely events to be under-valued.
6 The Greeks and Risk Management
Note that Vega is not a character of the Greek alphabet, but rather an oversized
letter ν. The volatility σ was also assumed constant in the previous chapters, but
in practice depends dynamically on the trading activities of the underlying shares.
For example, at times of uncertainty, with impending strikes or political unrest, the
volatility increases. Under such circumstances it is beneficial to make a portfolio
insensitive to changes of the volatility σ, namely minimizing V.
A portfolio consisting of shares and other derivatives is a derivative itself and
as such it must obey the Black-Scholes differential equation, given by (5.6), thus
−
∂∂
∂t
=
1
2
σ
2 S
2 ∂
2
∂ S 2 + r f S
∂∂
∂ S
− r f .
(6.6)
After substituting the definitions of the Greeks we obtain
− =
1
2
σ
2 S
2
+ r f S − r f
(6.7)
In case the portfolio is perfectly hedged, or risk-balanced with = 0 we have
= r f −
1
2
σ
2 S
2
.
(6.8)
Furthermore, if the risk-free rate is small we can ignore the term with r f . If, at the
same time, is large we see that and are anti-correlated and we can monitor
one as a proxy for the other [1]. This anti-correlation is also visible by comparing
in Fig. 6.1 and on the left plot in Fig. 6.2. One is the very closely the negative of
the other.
6.2 Volatility Smile
The option pricing formula derived in the previous sections are simply guidelines
to value options. In practice the sellers of options can set a price to anything and
the market, or more accurately, the buyer decides, whether to accept the price or to
buy somewhere else. Following the market crash of 1987, option sellers realized that
some of the options that had a very low probability of ever being exercised, all of a
sudden were actually exercised and became very expensive for the option seller. The
reason was that the shares dropped to very low values in the course of the crash and,
for example, put options became extremely expensive. The low selling price of the
option was due to the assumptions that large market variations are extremely rare in
a system described by fluctuation that are distributed according to a Gaussian model.
The tails of the Gaussian, which describe large fluctuations, become exponentially
small. This caused the unlikely events to be under-valued.