19-Measuring Risk of Derivative Instruments

7/28/2019 19-Measuring Risk of Derivative Instruments

1/9

Measuring Risk Characteristics of

Derivative Instruments


2/9

Derivative Greeks

Delta

Gamma

Vega

Theta

Rho


3/9

Delta Delta measures the rate of change of option price in relation to change in the

underlying asset value or portfolio.

Delta is the first derivative of option price with respect to the underlying stock

price.

Delta Formulae/Values:

Delta of a call option is given by N(d1) in B-S Formula. It is positive.

Delta of a pall option is given by N(d1) 1 in B-S Formula. It is negative.

Delta of an underlying security is 1.

Delta of a forward instrument is 1.

Delta of a debt instrument is 0.

Delta values are used to determine hedge ratio of underling asset or portfolio

using the options since the value of the delta depends on the underlying asset

price.

Delta of a portfolio is referred to as position delta. It may be zero, positive, or

negative.

Generally, in order to shield the delta risk, the option traders try to construct

the portfolio in such a way that the portfolio delta will become approximately

zero. Such a portfolio is called as delta neutral portfolio.


4/9

Gamma

Gamma measures the sensitivity of delta with respect to change in the

underlying portfolio or asset value. Gamma is the second partial derivative of option price with respect to the

underlying stock price.

Delta Formula/Values:

Gamma is always positive for both Call and Put.

Gamma of an underlying security is 0.

Gamma of a forward instrument is 0.

Gamma of a debt instrument is 0.

The Gamma of a portfolio is termed as position Gamma. It may be zero,

positive, or negative (due to sort position).

Generally, in order to shield the gamma risk, the option traders try to

construct the portfolio in such a way that the portfolio gamma will

become approximately zero. Such a portfolio is called as gamma neutral

portfolio.

2 2

;c pC P

S S


5/9

Delta vs. Gamma Sensitivity

Delta captures linear relationship between the option price and the underlying;

where as Gamma captures non-linear relationship between the two.


6/9

Vega Vega is the sensitivity of the options price with respect to the change in the

volatility of the underlying asset price. Vega risk is also termed as kappa risk,

lambda risk or sigma risk.

Vega Measurement: In B-S Formula_

Vega Values:

Vega is always positive for both Call and Put.

Vega of an underlying equity security is 0.

Vega of a forward instrument is 0.

Vega of a debt instrument is 0.

Since Vega risk exposes the sensitivity of an option instrument to that of the

volatility of underlying portfolio, exposure of the portfolios to sudden market

movements such as market crashes and its impact on the profits can be assessed.

The Vega of a portfolio is termed as position vega. It may be zero, positive, or

negative (due to sort position).

21

0 1

2

1

2

0

1

1

2

ln 2

d

S T N d

N d e

S X r T

d T


7/9

Theta Theta measures sensitivity of an option value with respect to time to expiration.

Time value of an option instrument declines as time to expiry of the option

contract nears zero. Theta is thus a measure of time decay and is expressed as the

loss of time value per day.

Theta Measurement: In B-S Formula_

Theta Values:

Theta is positive for long calls and puts.

Theta is negative for short calls and puts. Theta of an underlying equity security is 0.

Theta of a forward instrument is negative.

Theoretically, Theta Hedging does not make any sense since you cannot control

this systematic process of time decay. However, some traders do indulge into it.

Research on this issue reveals that the gamma value happens to be closely related

to the theta value.

2

2

2

1

0 1

0 1

2

1

2

0

1

2 1

2

2

1

2

ln 2

r T N d

c

r T N d

p

d

S N drXe

T

S N drXe

T

N d e

S X r T d

T

d d T


8/9

Rho

Rho is the sensitivity of change in option price to change in risk free

interest rates. Rho Measurement: In B-S Formula_

Rho Values:

Rho is positive for calls and puts.

Rho of an underlying equity security is 0.

Rho of a forward instrument is positive. Rho of a debt instrument held is negative.

Interest rates are assumed to be constant over the entire option period,

which is a quite tenable proposition.

Unless an option instrument has a very long time to expiry, changes ininterest rates may affect option price minimally.

When the economy itself is undergoing drastic policy changes, then

interest rate sensitivities of all option instruments would be significant.

2

2

2

0

2

ln( ) 2

rTc

tT

p

XTe N d

XTe N d

S X r T d

T


9/9

Derivative Greeks

Asset Position Hedge Delta Gamma Vega Theta RhoLong share (Positive Delta, Zero

Gamma, Vega, Theta, Rho) Sell call Negative Negative Negative Positive NegativeBuy put Negative Positive Positive Negative Positive

Short share (Negative Delta, Zero

Gamma, Vega, Theta, Rho) Buy call Positive Positive Positive Negative PositiveSell put Positive Negative Negative Positive Negative

Long call (Positive Delta, Gamma,

Vega, Rho, Negative Theta) Buy put Negative Positive Positive Negative PositiveSell shareNegative Zero Zero Zero Zero

Short call (Negative Delta, Gamma,

Vega, Rho, Positive Theta) Sell put Positive Negative Negative Positive NegativeBuy sharePositive Zero Zero Zero Zero

Long put (Negative Delta, Theta,

Rho, Positive Gamma, Vega) Buy call Positive Positive Positive Negative PositiveBuy sharePositive Zero Zero Zero Zero

Short put (Positive Delta, Theta,

Rho, Negative Gamma, Vega) Sell call Negative Negative Negative Positive NegativeSell shareNegative Zero Zero Zero Zero

Documents

19-Measuring Risk of Derivative Instruments