A Pioneer Algo Trading Training Institute Streamlined Investment Management

Manage Complex Option Portfolios:Simplifying Option Greeks – Part II

Monday, 11th September

7:30 PM IST | 2:00 PM GMT | 10:00 AM EST

Rajib manages the course segment on option derivatives and also works with exchanges,financial & educational institutions to design educational programs. He has conductedworkshops and conferences in America, Europe and Asia.

Rajib worked with leading HFT firm Optiver in Amsterdam on options derivatives marketmaking & high frequency equity arbitrage strategies across all major European & USexchanges. Before Optiver, Rajib was a management strategy consultant withPricewaterhouseCoopers where he assisted a consortium in setting up a national commodityderivatives exchange.

A national Olympiad finalist, Rajib has twice represented India at the World PuzzleChampionships. He has a post-graduate management degree from the Indian Institute ofManagement Calcutta, a bachelor’s degree in Computer Engineering from the NationalInstitute of Technology Surathkal; and has internship experiences with Bloomberg in New York(equity option derivatives research) & with Solutia’s EMEA strategy HQ in Belgium.

About the Speaker

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Rajib Ranjan BorahCo-Founder & Director - QuantInsti™

Price of option from Black Scholes formula

Delta = ∂C/∂S or ½(∂C/∂S- + ∂C/∂S+) to be more precise= N(d1)

Delta

3

)()( 21 dNXedSNC rtt

−−=

t

trXS

dt

σ

σ )2

()ln(2

1

++=

dzexN

zx 2

2

21)(

−

∞−∫=

π

Delta

4

i.e. Delta is dependent on: underlying price, time to expiry volatility

Call Delta vs Underlying Price

Gamma: Delta vs Underlying Price

5

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

80 85 90 95 100 105 110 115 120

Delta

of o

ptio

n

Underlying Price

Call 90 Strike

Call 100 Strike

Call 110 Strike

Put Delta vs Underlying Price

Gamma: Delta vs Underlying Price

6

-1.000

-0.900

-0.800

-0.700

-0.600

-0.500

-0.400

-0.300

-0.200

-0.100

0.00080 85 90 95 100 105 110 115 120

Delta

of o

ptio

n

Underlying Price

Put 90 Strike

Put 100 Strike

Put 110 Strike

Call Delta vs Time left to expiry

Charm: Delta vs Time

7

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.0001 25 50 75 100 125 150 175 200

Delta

of o

ptio

n

Days to Expiry

Call 90 Strike

Call 100 Strike

Call 110 Strike

Underlying Price = 100Volatility = 20%

Put Delta vs Time left to expiry

Charm: Delta vs Time

8

-1.000

-0.900

-0.800

-0.700

-0.600

-0.500

-0.400

-0.300

-0.200

-0.100

0.0000.0001 25 50 75 100 125 150 175 200

Delta

of o

ptio

n

Days to Expiry

Put 90 Strike

Put 100 Strike

Put 110 Strike

Underlying Price = 100Volatility = 20%

Call Delta vs Volatility

Vanna: Delta vs Volatility

9

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

1.000

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%

Delta

of o

ptio

n

Implied Volatility

Call 90 Strike

Call 100 Strike

Call 110 Strike

Put Delta vs Volatility

Vanna: Delta vs Volatility

10

-1.000

-0.900

-0.800

-0.700

-0.600

-0.500

-0.400

-0.300

-0.200

-0.100

0.0000.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%

Delta

of o

ptio

n

Implied Volatility

Put 90 Strike

Put 100 Strike

Put 110 Strike

Gamma

11

As we have seen, deltas change with underlying price (moreso towards expiry)

Gamma is the second derivative of the change of option pricewith respect to change in underlying price

= ∂2C/∂S2 = ∂Δ/∂S = N’(h)/ (Sσ√t)

Speed: Gamma vs Price of Underlying

12

Gamma vs Price of Underlying

0.000

0.010

0.020

0.030

0.040

0.050

0.060

80 85 90 95 100 105 110 115 120

Gam

ma

of o

ptio

n

Underlying Price

Call 90 Strike

Call 100 Strike

Call 110 Strike

Color: Gamma vs Time

13

Gamma vs Time

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0.0001 25 50 75 100 125 150 175 200

Delta

of o

ptio

n

Days to Expiry

Call 90 Strike

Call 100 Strike

Call 110 Strike

Zomma: Gamma vs Volatility

14

Gamma vs Volatility

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.450

0.500

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%

Delta

of o

ptio

n

Implied Volatility

Call 90 Strike

Call 100 Strike

Call 110 Strike

Vanna: Vega vs Underlying Price

15

Vega at different strikes

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.200

80 85 90 95 100 105 110 115 120

Vega

of o

ptio

n

Underlying Price

Call 90 Strike

Call 100 Strike

Call 110 Strike

Veta: Vega vs Time

16

Vega of an option with varying time left to expiry

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.0001 25 50 75 100 125 150 175 200

Vega

of o

ptio

n

Days to Expiry

Call 90 Strike

Call 100 Strike

Call 110 Strike

Vomma: Vega vs Volatility

17

Sensitivity to volatility is sensitive to volatility itself

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%

Vega

of o

ptio

n

Implied Volatility

Call 90 Strike

Call 100 Strike

Call 110 Strike

Thega: Theta v/s Time to expiration

18

Theta with changing time to expiry

-160

-140

-120

-100

-80

-60

-40

-20

00 20 40 60 80 100 120

Thet

a

Days to expiry

K=100

K=110

K=90

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Questions?