Greeks of Financial Derivatives - Mathematical finance

finance

1. Options Mastery: Theory to Trading #2
Option Greeks and P&L Analysis
Junsu Park? ? R
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

2. Table of Contents
1
Risk-Neutral Measure
2
Option Greeks
Delta
Gamma
Theta
Vega
3
P&L Analysis
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

3. Interpretation of the Black-Scholes Formula
Because the price of an option is a discounted expectation of the payoff, we
have
Vt=e
−r(T−t)
E
Q
[(ST−K)+|Ft]
=e
−r(T−t)
ˇ
E
Q
fi
ST1
{ST>K}

Ft
fl
−E
Q
fi
K1
{ST>K}

Ft
fl
ı
=e
−r(T−t)
ˇ
E
Q
fi
ST1
{ST>K}

Ft
fl
−KQ(ST>K|Ft)
ı
=e
−r(T−t)
[Φ(d1)Ft−Φ(d2)K].
Φ(d1)andΦ(d2)
(
Φ(d1)Ft=E
Q
fi
ST1
{ST>K}

Ft
fl
=Undiscounted cash exposure toST.
Φ(d2) =Q(ST>K|Ft) = Risk-neutral probability ofST>K.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

4. Implied Probability Distribution
Suppose we can observe European option prices for any positive real strike.
Then, we can calculate the risk-neutral distribution implied by the option
prices.
Implied CDF and PDF
LetP(K)be the time-tprice of a put option of strikeK. Then,



Q(ST<K|Ft) =
Model-Free
e
r(T−t)´P(K)
´K
=
BS Model
Φ(−d2)
Q(dST|Ft) =
Model-Free
e
r(T−t)´
2
P(K)
´K
2dK =
BS Model
f(d2)
Ks
√
T−t
dK
The left equalities hold as long asris constant and the right equalities hold
under the Black-Scholes assumptions.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

5. Table of Contents
1
Risk-Neutral Measure
2
Option Greeks
Delta
Gamma
Theta
Vega
3
P&L Analysis
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

6. Option Greeks
The Greeks are the quantities that represent the sensitivity of the price
of an option to parameters.
The Greeks are crucial for risk management.
The Greeks are useful for profit-and-loss (P&L) analysis.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

7. Delta (∆)
Delta,∆t, is the sensitivity of the option price toFt,
űVt
¶Ft
.
∆t=



e
−r(T−t)
Φ(d1) if Call
−e
−r(T−t)
Φ(−d1)if Put
Delta is closely related to the risk-neutral CDF,Vt=∆tFt+
´Vt
¶K
K, and
this relationship holds not only in the Black-Scholes world but also in
general assumptions.
Cash Delta=Ft∆t, is a martingale up to discounting.
Some practitioners interpret|∆t|as the probability of an option
expiring in the money which is not accurate but close enough.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

8. Delta (∆)
F=100,s=0.2, andr=0.05
Delta is stable across expires for a
fixed normalized log-moneyness
z=
log(K/Ft)
s
√
T
.
ű∆t
¶K
<0.
0<|∆t|<e
−rT
.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

9. Gamma (Γ)
Gamma,Γt, is the sensitivity of∆ttoFt,
ű
2
Vt
¶F
2
t
.
In the Black-Scholes model,Γt=e
−r(T−t)f(d1)
Fts
√
T−t
.
Cash Gamma=$Γt=ΓtF
2
tis a martingale up to discounting.
Cash Gamma is proportional to the derivative of option price w.r.t.
the total variances
2
(T−t):
1
2
$Γt=
´Vt
¶ s
2
(T−t)
.
Cash Gamma is proportional to the risk-neutral probability density
function,ΓtF
2
t=
´
2
Vt
¶K
2K
2
>0, and this relationship holds not only in
the Black-Scholes world but also in general assumptions.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

10. Gamma (Γ)
F=100,s=0.2, andr=0.05
Gamma increases as T decreases for
strikes nearF.
Gamma decreases as T decreases for
strikes far fromF.
űΓ
∂K



K=Fe
1
2
s
2
T
=0.
ű$Γ
∂F



F=Ke
1
2
s
2
T
=0.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

11. Theta (Θ)
Theta,Θt, is the sensitivity of the option price tot.
Theta can be decomposed into interest rate and volatility Theta:
Θ=re
−r(T−t)
Xt
|{z}
=Θr
+e
−r(T−t)
´Xt
¶t
|{z}
=Θσ
whereXt=e
r(T−t)
Vtis the undiscounted
price of an option.
Theta is a martingale up to discounting.
Volatility Theta is closely related to Cash Gamma:
1
2
$Γσ
2
+Θσ=0.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

12. Theta (Θ)
F=100,s=0.2, andr=0.05
Theta decreases as T decreases for
strikes nearF.
Theta increases as T decreases for
strikes far fromF.
űΘσ
∂K



K=Fe
1
2
s
2
T
=0.
űΘσ
∂F



F=Ke
1
2
s
2
T
=0.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

13. Vega (n)
Vega,n, is the sensitivity of the option price to volatility,
űVt
¶ s
.
In the Black-Scholes model,n=e
−r(T−t)
Ftf(d1)
√
T−t.
Vega is closely related to$Γ:n=$Γσ(T−t).
Useful Relationship
Ff(d1) =Kf(d2) =
√
FKf(z)e
−
1
8
s
2
(T−t)
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

14. Vega (n)
F=100,s=0.2, andr=0.05
Vega increases as T increases.
űn
űK



K=Fe
1
2
s
2
T
=0.
űn
űF



F=Ke
1
2
s
2
T
=0.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

15. Table of Contents
1
Risk-Neutral Measure
2
Option Greeks
Delta
Gamma
Theta
Vega
3
P&L Analysis
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

16. Delta-Hedged Portfolio
Consider a∆-hedged and self-financing option portfolio,Π, whereΠ0=0.
dΠt=dVt−∆tdFt−rVtdt
=Θtdt+
1
2
Γ(dFt)
2
−rVtdt∵Itô’s Lemma
=Θtdt+
1
2
$Γt(dlog(Ft))
2
| {z }
=Gamma P&L
−rVtdt
SinceΓis positive,Πgenerates positive Gamma P&L asFmoves.
Since money is borrowed to buy the option, interest is paid.
Θσ<0compensates for Gamma P&L andΘr>0compensates for the
interest.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

17. Delta-Hedged Portfolio
Loosely speaking,Fmoves±F0s
√
dton average overdt.
Gamma P&L,
1
2
Γ(dFt)
2
, is offset by volatility Theta P&L, resulting in
an average break-even for the option’s P&L.
If Delta-hedging is done continuously, instantaneous P&L is always
break-even.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

18. Revisiting the Black-Scholes PDE
Let us rewrite the Black-Scholes PDE using the Greeks.
Black-Scholes PDE
0=
´Vt
¶t
+
1
2
¶
2
Vt
¶F
2
F
2
s
2
−rVt
=Θt+
1
2
$Γts
2
−rVt
=Θr−rVt
|{z}
=0
+Θσ+
1
2
$Γts
2
|{z}
=0
This PDE elegantly describes the break-even nature of P&L.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

19. Reference
Haug, E.G. (2007). The Complete Guide to Option Pricing Formulas.
The Complete Guide to Option Pricing Formulas
Wilmott, P. (2013). Paul Wilmott on Quantitative Finance. John
Wiley & Sons.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024

20. Disclaimer
The views expressed in this presentation/article are solely those of the author and do not
necessarily reflect the views of Optiver or its affiliates. This content is intended for
educational and informational purposes only and should not be construed as investment
advice or a recommendation to engage in any specific trading strategies or transactions.
Options trading involves substantial risk and is not suitable for all investors. Individuals
should carefully consider their own financial situation and risk tolerance before engaging
in options trading or implementing any trading strategies discussed herein. The author
makes no representations or warranties as to the accuracy or completeness of any
information provided in this presentation/article and shall not be liable for any errors or
omissions in the content or for any actions taken based on the information provided
herein. Readers are encouraged to consult with a qualified financial advisor or
investment professional before making any investment decisions. By accessing this
content, you agree to release the author and Optiver from any liability arising from your
use of or reliance on the information provided herein.
Junsu Park? ? R Option Greeks and P&L Analysis May 5, 2024