Chapter 6
The Greeks and Risk Management
Abstract This chapter introduces the Greeks as derivatives of the option price with
respect to the variables and parameters of the Black-Scholes equation. Their use
in hedging risks and assessing the resilience of a hedge with respect to parameter
variations, is discussed. A discussion of the volatility smile and the concept of valueat-risk follows. Finally, combinations of simple options, such as bull-spreads or
straddles, are introduced and their use to adapt one’s investment to one’s expectations
are briefly touched upon.
After covering the derivation of the pricing formulas for the options we now proceed and investigate properties of the solutions, especially their robustness. Then we
discuss the use of options in order to create a portfolio insurance as well as other
financial instruments to bet on some expected behavior of the market.
6.1 The Greeks
Usually, options are used to hedge the risk of shares in a portfolio. The number
of shares that are needed to balance one option is given by the condition that the
portfolio , given by (5.3), is zero. In that case any variation in the share value S
is balanced by a corresponding opposite variation in the value of the option. The
proportionality constant
=
∂c
∂ S
(6.1)
is called—to everybodies surprise—Delta and is the first member of the quantities
to characterize the stability of a portfolio named the Greeks.
Electronic supplementary material The online version of this chapter
(https://doi.org/10.1007/978-3-030-63643-2_6) contains supplementary material, which is
available to authorized users.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_6
59
The Greeks and Risk Management
Abstract This chapter introduces the Greeks as derivatives of the option price with
respect to the variables and parameters of the Black-Scholes equation. Their use
in hedging risks and assessing the resilience of a hedge with respect to parameter
variations, is discussed. A discussion of the volatility smile and the concept of valueat-risk follows. Finally, combinations of simple options, such as bull-spreads or
straddles, are introduced and their use to adapt one’s investment to one’s expectations
are briefly touched upon.
After covering the derivation of the pricing formulas for the options we now proceed and investigate properties of the solutions, especially their robustness. Then we
discuss the use of options in order to create a portfolio insurance as well as other
financial instruments to bet on some expected behavior of the market.
6.1 The Greeks
Usually, options are used to hedge the risk of shares in a portfolio. The number
of shares that are needed to balance one option is given by the condition that the
portfolio , given by (5.3), is zero. In that case any variation in the share value S
is balanced by a corresponding opposite variation in the value of the option. The
proportionality constant
=
∂c
∂ S
(6.1)
is called—to everybodies surprise—Delta and is the first member of the quantities
to characterize the stability of a portfolio named the Greeks.
Electronic supplementary material The online version of this chapter
(https://doi.org/10.1007/978-3-030-63643-2_6) contains supplementary material, which is
available to authorized users.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_6
59