Lecture 6.2 - Option Greeks
1
Lecture 6.2
Option Greeks
Prof. Olivier BOSSARD
https://www.coursehero.com/file/62725998/Lecture-62-Greeks-part-2pdf/
Introduction to Derivatives
Lecture 6.2 - Option Greeks
2
Course Content
• Intro to derivatives
• Futures/Forwards Basics
• Option Basics
• Black-Scholes Model
• Volatility
• Option Greeks
Prof.
Prof.Olivier
OlivierBOSSARD
BOSSARD
https://www.coursehero.com/file/62725998/Lecture-62-Greeks-part-2pdf/
• Hedging Strategy
• Dynamic Hedging
• Limitations of Black-Scholes
• Market Making Games
• Volatility Workshop
• Conclusion and References
Intro to Derivatives Pricing
Introduction to Derivatives
Lecture 6.2 - Option Greeks
3
Theta - !
• What:
• Theta is the 1st order partial derivative of the option value with
regards to the passage of time
"# = ! ∗ "&
• It measures how sensitive the option price is to the passage of
time.
• The theta of a long call or long put is usually negative. This
means that, if time passes, the value of a long call or long put
option declines.
• The opposite implies that the theta of a short call or short put
is usually positive.
• dt is typically one trading day = 1/252 years ≈ 0.004 years.
• E.g. suppose that a call has a theta of −13.50. With all else
being equal, the passage of one trading day will cause a change
in option value of −13.50 * 0.004 = −0.054. That is, the call
value will decline by $0.054.
Prof. Olivier BOSSARD
https://www.coursehero.com/file/62725998/Lecture-62-Greeks-part-2pdf/
Introduction to Derivatives
Lecture 6.2 - Option Greeks
4
Gamma Theta Trade-off
• From the Black-Scholes Partial Differential Equation (PDE):
!" 1 ( ( ! ("
!"
+ ' )
+ *)
= *"
(
!# 2
!)
!)
• When we are delta hedged,
,,.
•
,where
,.
Prof. Olivier BOSSARD
is 4, and
/ ( ( ,0 + ( ' ) ,10
,0 ,10
,-
= −*) ,1 + *" = 0
is Γ.
Introduction to Derivatives
Lecture 6.2 - Option Greeks
5
Vega - !
• What:
• Vega is the 1st order partial derivative of the option value with
regards to change in volatility
"# = ! ∗ "&
• It measures how sensitive the option price is to volatility.
• The portfolio vega is calculated as the weighted average of the
vega of individual options included in the portfolio. It reflects the
portfolio’s weighted average sensitivity to implied volatility
• E.g. An option has vega = 9.80 and volatility falls by 1% (so dσ =
−0.01). Then the change in the option value is 9.80*(−0.01) =
−0.098. That is, the option value falls by 9.8 cents.
• Gamma vs Vega: to some extent these are “similar” measure
• Gamma: sensitivity to actual volatility.
• Vega: sensitivity to implied volatility.
Prof. Olivier BOSSARD
Introduction to Derivatives
Lecture 6.2 - Option Greeks
6
Rho - !
• What:
• Rho is the 1st order partial derivative of the option value with regards to
change in interest rate
"# = ! ∗ "&
• It measures how sensitive the option price is to change in interest rate
• From Put-call parity, C − P = S − e−r (T−t)K, we can differentiate both sides
with regards to r, then we will have the relationship of ρcall and ρput
ρcall − ρput = (T − t)e−r (T−t)K
• E.g. the initial call price is 6.889 and ρcall = 26.44. Consider an increase of
25 basis points in interest rates from 5% to 5.25%, the estimated change in
the call value is 26.44 * (+0.0025) = +0.0661.
• Thus, the estimated new call value is 6.889 + 0.0661 = 6.955. This estimate
is accurate to three decimal places: as can be checked using the BlackScholes formula, the actual new call price at an interest rate of 5.25% is
6.955.
Prof. Olivier BOSSARD
Introduction to Derivatives
Lecture 6.2 - Option Greeks
7
The Greeks Summary - 1
Prof. Olivier BOSSARD
Introduction to Derivatives
Lecture 6.2 - Option Greeks
8
The Greeks Summary - 2
Long call
Short call
Long put
Short put
Option
value
δ
Γ
Θ
γ
Prof.
Prof.Olivier
OlivierBOSSARD
BOSSARD
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Introduction to Derivatives