Book of Greeks Edition 1.0 (Preview)

The

Book

of Greeks Edition 1.0

Rahul Bhattacharya, M.Sc., MBA, CFE

© CFE School, Risk Latte Company Limited, Hong Kong

2011

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The Book of Greeks

Certificate in Financial Engineering (CFE) www.risklatte.com

Foreword

This book is a reference book, a kind of cheat sheet that will help the user navigate through

various formulas, equations and algorithms that are used in the four modules of our

Certificate in Financial Engineering (CFE) course. It will also help the student prepare

better for the CFE examination conducted by the CFE School, the learning and education

arm of Risk Latte Company Limited. However, by no means is this comprehensive or an

exhaustive list of all formulas and equations that are used in the study of financial

engineering and quantitative finance. It is a very tiny portion of the entire discipline. This

book aims to provide to the reader a glimpse of the giant canvas that is the discipline of

Financial Engineering

All formulas and equations that are used in this reference book are meant to be

implemented on an Excel™ spreadsheet. In our CFE course, we implement all these

formulas and equations in Excel/VBA and then build quantitative models of financial

derivatives and asset portfolio pricing and risk analysis.

Financial Engineering is neither just physics nor applied mathematics, and much more than

the sum total of the two. Financial Engineering is concerned with the application of all

those physics-like theories and advanced mathematical concepts and methodologies for the

development of quantitative models and algorithms with which financial derivatives and

financial asset portfolios can be traded, valued and their risks can be analyzed. However,

such is not possible unless all those complex and abstract formulas and equations are

transformed into simpler workable format and implemented on a platform. And that

platform has to be something that is very simple to understand and work on.

For the CFE course that platform is the Microsoft Excel™ spreadsheet.

This reference book is exclusively meant for supplementing the knowledge imparted in the

CFE course and helping students take the CFE examination. Other than that, this book has

no other objective or pretense. This book should not be seen as an alternative or

supplement to any of other great books on the subject of quantitative finance and financial

engineering, a partial list of which is given at the end in the reference section.

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The Book of Greeks


Contents

Page

Chapter M

Applied Mathematics and Numerical Techniques 4

Chapter S

Stochastic Processes for Asset Price Modeling

(For Monte Carlo Simulation) 24

Chapter V

Closed form Formulas for Implied and Historical Volatility 36

Chapter O

Closed form Formulas & Approximations

For Vanilla and Exotic Options 43

Chapter G

Closed Form Greeks for Vanilla & Binary Options 55

Chapter E

Exotic Option Payoffs, Structured Products & Product Engineering 59

Chapter P

Portfolio Analytics and Risk Management 69

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Chapter M

Applied Math & Numerical Techniques

For Quant Modelling

Part A

Matrices & Applications

Types Matrices and their Applications

1. Square Matrix

In a square matrix the number of rows is equal to the number of columns. It is generally

nn mn written as matrix, where, . A matrix is a square matrix with 3 rows and 3

columns. All matrices used in financial engineering and quantitative finance are square

33matrices. An example of a matrix would be:

320

841

102

P

An example of a square matrix in finance is the correlation matrix of asset returns. A

covariance matrix of asset returns is also an example of a square matrix.

2. Identity Matrix

IAn Identity Matrix, denoted by, , has one on its diagonal and zeros in the off-diagonals.

33It is the equivalent of number 0 in number theory. An example of a identity matrix

is:

100

010

001

I

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3. Diagonal Matrix

A diagonal matrix is one whose diagonals have non-zero values and the off-diagonals

have zeros. An identity matrix is a special case of a diagonal matrix. An example of a

33 diagonal matrix is:

200

060

003

D

Say, we have a three asset portfolio (1, 2, 3) with asset (return) volatilities of 15%, 10%

and 12%. Then, we can write the volatilities as a diagonal matrix:

12.000

010.00

0015.0

V

4. Triangular Matrices

A triangular matrix is one where all elements above or below the diagonal are zeros. If all

elements above the diagonal are zero then the matrix is known as lower triangular matrix

whereas if all elements below the diagonal are zero then the matrix is known as upper

33triangular. Examples of lower and upper triangular matrices are:

Lower Triangular

829

011

002

L

Upper Triangular

400

530

681

U

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5. Symmetric Matrix: Correlation & Variance-Covariance Matrices

A symmetric matrix is one where all elements above the diagonal are mirror images of

nn Athe elements below the diagonal. In other words, for any symmetric matrix, , the TAtranspose of the matrix, , is the matrix itself. The following relationship holds:

AAT

A transpose operation flips the rows of any matrix into columns and columns into rows.

33Consider a correlation matrix of asset returns, which is a symmetric matrix:

125.055.0

25.0165.0

55.065.01

M

MAll elements of below the diagonal are mirror images of the elements below the

diagonal. That is obvious in the case of asset correlation matrix. The correlation of asset 1

with that of asset 2 is the same as the correlation of the return of asset 2 with that of asset

12 211. If is the correlation of asset 1 with asset 2 then is the correlation of asset 2

2112 with asset 1 such that .

MThe transpose of the asset correlation matrix, , above is given by:

MM T

125.055.0

25.0165.0

55.065.01

If we have three assets (1, 2, 3) in a portfolio then the symmetric, 33 asset correlation

matrix is given by:

1

1

1

3231

2321

1312

M

If the volatilities of the three assets (1, 2, 3) are given by 1 , 2 and 3 respectively

then the symmetric, 33 asset covariance matrix is given by:

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2

332233113

2332

2

22112

13311221

2

1

6. Volatility, Correlation and Variance-Covariance Matrix of a Portfolio

(Three Asset Case)

1 2 3Given three assets (1, 2 and 3) with volatilities of , and respectively. Then the

volatility matrix can be expressed as a diagonal matrix:

3

2

1

00

00

00

V

MGiven a correlation matrix of these three assets as , where,

1

1

1

3331

2321

1312

M

M jiij Where, in the matrix , we have . Then the Variance-Covariance matrix of the

three asset portfolio would be given by:

MVV T

2

323321331

2332

2

21221

13311221

2

1

3

2

1

3231

2321

1312

3

2

1

00

00

00

1

1

1

00

00

00

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7. Transpose of a Matrix

As explained above a transpose operation flips the rows of any matrix into columns and

33columns into rows. Consider the following matrix.

2.030

5.021

041

B

BThe transpose of would be given by

2.05.00

324

011TB

8. Determinant of a Matrix

Determinant symbolizes the area or the volume enclosed by the row vectors of any

matrix. It is a scalar quantity (a single number). Determinants exist only for square

22 matrices. Consider a matrix below:

31

12C

C C CdetThe determinant of this matrix, , denoted by or is given by:

7)1(13231

12

C

33For a matrix the determinant is calculated as:

251

203

121

E

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EThe determinant of would be given by:

9

15182101

51

03)1(

21

232

25

201

E

Why are determinants important?

(i) They are essential for calculating the inverse of a matrix. In fact, a determinant tells

us if a matrix can be inverted or not.

(ii) They are needed for estimation of eigenvalues and eigenvectors of a matrix

(iii) They measure the area or the volume of shape defined by the row vectors of a

matrix.

9. Inverse of a Matrix

A1A A

1AThe inverse of a matrix, , is defined as where, multiplied by is equal to the

I IAA 1identity matrix, . Mathematically, we can write, .The inverse of a matrix is

calculated as:

A

AadjA 1

If the determinant of a matrix is zero then the inverse of the matrix will not exist. We say

that the matrix is non-invertible. Such matrices are known as Singular matrices.

10. Matrix Multiplication

Two matrices can be multiplied to produce a third matrix. However, two matrices can

only be multiplied if the column of the first matrix is equal to the row of the second

matrix. The dimension of the third matrix, which is the product of the first and the second

matrix, would be the rows of the first matrix and the columns of the second matrix.

22 A BIf we have two matrices, and given as follows

2221

1211

2221

1211

bb

bbB

aa

aaA

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BAThen the product of the two matrices, is given by

2222122121222121

2212121121121111

2221

1211

2221

1211

babababa

babababa

bb

bb

aa

aaBAC

Say, we have a three asset portfolio (1, 2, 3) with asset (return) volatilities of 10%, 15%

33and 12% respectively and expressed as a diagonal matrix as:

12.000

015.00

0010.0

V

33And, if the correlation matrix of asset returns is given by:

125.055.0

25.0165.0

55.065.01

M

Then, we can obtain the 33 covariance matrix of asset returns as:

0144.00045.00066.0

0045.00225.000975.0

0066.000975.00100.0

12.000

015.00

0010.0

125.055.0

25.0165.0

55.065.01

12.000

015.00

0010.0T

T MVV

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11. Power of a Matrix

One of the problems of matrices is that they cannot be raised to non-integer powers. We

can raise a square matrix to a power of 2, 3, 4….. but we cannot raise a matrix to the

power of 0.5 (half) or 0.75. Thus we cannot find the square root of a matrix. Consider the

33following matrix:

651

113

012

X

If we want to find the square of the matrix we simply multiply the matrix with itself

(same as raising the matrix to the power of 2).

313623

718

137

651

113

012

651

113

0122 SSS

S 3 2S SSimilarly we can raise the matrix to the power of by multiplying with ; and so

Son for higher powers of

SHowever, if we want to raise the matrix, , to the power of half (0.5), i.e. if we want to

S Sfind the square root of the matrix , we cannot do it. Square roots of the matrix, exists

but they are very different from the notion of a square root that we have from number

theory. In fact, a square matrix can have numerous square roots, something that we don’t

3 3see in number theory. The number, has a unique square root given by, .

12. Square Root of a Matrix

KA square matrix can have many square roots. A matrix is said to be the square root of

M MM K 22 a matrix, , if the product equals to . For example, consider a matrix B

as given by

3721

2816B

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This matrix has many square roots. One of the square roots of this matrix is a matrix given

53

42R BRR by because, .

13. Methods of Finding the Square Root of a Matrix

(Please see the Eigenvalues and Eigenvectors of a Matrix to understand this part fully)

thNThere are many methods for finding the square root or the root of a square matrix.

One of the most common and robust ways of finding the square root of a matrix is via the

diagonalization method using the eigenvalues and the eigenvectors of the matrix.

X jk WIf is a square matrix of dimension given by and is the matrix of the

X jk eigenvectors of with the same dimension of and is a diagonal matrix of the

X Xeigenvalues of then can be decomposed as:

1 WWX

thN XThen, the root of the matrix is given by:

1

1

ˆ WWX N

thNWhere, is a diagonal matrix with the diagonal elements being the root of the

k .......,,, 21 33individual eigenvalues, . For a matrix we get the following.

33 X

333231

232221

131211

xxx

xxx

xxx

XIf we have a matrix, , given by:

W And if the matrix of eigenvectors, , and the diagonal matrix of the eigenvalues, , are

given by:

333231

232221

131211

www

www

www

W

3

2

1

00

00

00

and

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14. Cholesky Matrix

(See Part B, Chapter S for more on Cholesky matrix)

Cholesky is a lower triangular matrix that is obtained using the following

M nn Atransformation. If is a correlation or a variance-covariance matrix and is

the Cholesky matrix of same dimension then the Cholesky matrix is obtained by:

MAAT

The Cholesky is an important matrix in quantitative finance. A correlation matrix (or a

variance-covariance matrix) of asset returns is valid, i.e. workable and usable, if and

only if the Cholesky of that matrix exists. This is related to the condition of positive

semi-definiteness. A correlation matrix is positive semi-definite if all its eigenvalues

are positive and the Cholesky of that matrix exists.

If the Cholesky of a correlation matrix does not exist then the correlation matrix is

commonly known as “nonsensical”.

15. Solution of a System of Linear Equations using Matrices

Matrices can be used to solve a system of linear equations. Consider the following

system of three linear equations:

22

1

03

zyx

zyx

zyx

Using matrices, we can write the above as:

2

1

0

211

111

113

z

y

x

BAX The above can be written as: where,

2

1

0

211

111

113

B

z

y

x

XA

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The solution can be obtained using matrix multiplication and using the concept of the

BAXBAX 1inverse of a matrix:

8.0

1.0

3.0

2

1

0

211

111

1131

z

y

x

X

3.0x 1.0y 8.0z Thus, , and

Applications of Matrices in Quantitative Finance

I. Hedging an Option Portfolio

An FX options trader wants to hedge a short position of 100 contracts in an OTC FX

option position that he is running that has a Gamma of 2.12, Vega of 0.1956 and a Volga

of 1.028 such that his overall portfolio (original position plus the hedge) is gamma-vega-

volga neutral. He identifies three traded options in the market with the following greeks:

Option 1 Option 2 Option 3

Gamma 2.56 1.08 2.32

Vega 0.2259 0.3596 0.1862

Volga 0.3122 -0.0024 0.876

How many of each of these three traded options should he buy or sell to make his

portfolio gamma-vega-volga neutral?

For gamma neutrality the equation would be:

12.232.208.156.2 321 www

For vega neutrality the equation would be:

1956.01862.03596.02259.0 321 www

For 14olga neutrality the equation would be:

028.1876.00024.03122.0 321 www

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The Book of Greeks


Therefore, to solve for the weights (i.e. the number of options to buy or sell) for making

the portfolio gamma-vega-volga neutral we need to solve the system of three linear

equations given by:

12.232.208.156.2 321 www

1956.01862.03596.02259.0 321 www

028.1876.00024.03122.0 321 www

The solution is given by:

028.1

1956.0

12.2

876.00024.03122.0

1862.03596.02259.0

32.208.156.2

3

2

1

w

w

w

The matrix equation is solved as: BAWBAW 1

Therefore, the number of options to buy or sell is given by:

3257.1

1251.0

4261.0

028.1

1956.0

12.2

876.00024.03122.0

1862.03596.02259.0

32.208.156.2

3

2

1

1

3

2

1

w

w

w

W

w

w

w

W

Since the trader needs to hedge 100 contracts, he needs to sell (short) 42.61 contracts of

option 1, buy (long) 12.51 contracts of option 2 and buy (long) 13.25 contracts of

option 3 to make his portfolio gamma-vega-volga neutral. Of course, this will cause his

delta to change and he needs to re-hedge his delta with FX spot contracts.

II. Estimating the Parametric Value at Risk (VaR)

A Risk Manager wants to estimate the VaR of an FX portfolio of a spot FX trader. The

spot FX positions of the trader with raw USD exposures and respective annualized

volatilities are given by:

Asset Exposure (in USD) Volatility

USD/JPY $2.50 million 8%

USD/CHF $4.56 million 9%

EUR/USD $1.85 million 11%

GBP/USD $3.18 million 7%

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The asset (return) correlation matrix of the above portfolio is given by:

112.025.035.0

12.0132.045.0

25.032.0167.0

35.045.067.01

M

What is the Value at Risk (VaR) of this portfolio?

The VaR with a 66.67% confidence limit is given by:

MEEVaR T

E M In the above formula, is the vector of net exposure, is the correlation matrix and TE Eis transpose of the vector, . The vector of net exposure is calculated by multiplying

the raw exposures given above with the respective volatilities. For the above portfolio

Ethe vector of net exposure, , in USD is given by

600,222

500,203

400,410

000,200

E

The 66.66% Value at Risk of the portfolio is calculates as:

600,222

500,203

400,410

000,200

112.025.035.0

12.0132.045.0

25.032.0167.0

35.045.067.01

600,222

500,203

400,410

000,200T

Z

272,766 ZVaR

Thus, the 66.66% VaR of the FX portfolio is $766,272. This signifies that there is a

66.66% chance that the trader will not lose more than $766,272 in one year provided

that the volatilities and the correlations did not change over this period and the markets

behaved normally.

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The Book of Greeks


III. Portfolio Optimization: Strategic Asset Allocation Problems

Linear equations also appear in portfolio optimization problems (mean-variance

optimization) that appear in strategic asset allocation models used by fund managers to

allocate funds between various asset classes and individual securities within asset

classes.

See Chapter P for more details on this.

IV. Applications in Algorithmic Trading (Minimization of MCR)

Matrices are also used to solve complex mathematical algorithms used by algorithmic

and high frequency traders to buy and sell stocks and other assets. A particular example

is the minimization of the Marginal Contribution of Risk (MCR) to determine the

number of stocks to buy and/or sell in a trade list.

See Chapter P for more details on this.

Part B

Binomial & Trinomial Trees

1. Cox-Ross-Rubenstein (CRR) Tree

The quantum of up and down moves is given by:

uedmovedown

eumoveup

t

t

1

The drift and the probabilities are given by:

updown

up

tr

pp

du

dap

ea

1

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2. Jarrow- Rudd (JR) Tree

The quantum of up and down moves is given by:

11

11

2

2

ttr

ttr

eed

eeu

The most popular choice for the quantum of up and down moves for a JR tree is:

2

2

1

r

ed

eu

tt

tt

The probabilities are given by:

2

1 du pp

3. Trinomial Tree

The quantum of up move, down move and for remaining at the same level is given by:

t

tt

tt

emsameStay

eddown

euup

3

3

The probabilities of the moves are given by:

3

2

6

1

m

du

p

pp

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4. Tian’s Equal Probability Tree

Given the following:

tqr

t

eL

eB

LLJ

2

4

3

The quantum of up and down moves and that of staying the same is given by:

2

3

22

22

BLm

mLLd

mLLu


3

1 mdu ppp

5. Kamrad and Ritchken Tree

This tree is used by many quants to value convertible bonds. There is a stretching

parameter which makes the tree recombining. This tree also posits a horizontal jump

which is given by 1m

The quantum of up move, down move and staying the same is given by:

1

m

ed

eu

t

t

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2

2

2

2

2

11

2

2

1

2

1

2

2

1

2

1

m

d

u

p

tqr

p

tqr

p

The value of has to be bigger than one and is taken by many practitioners to be

equal to 23 which makes the probability of a horizontal jump equal to 31 .

Part C

Black-Scholes Diffusion Equation

& Green’s Function for Valuation of Exotics

If you believe the stock price process is GBM,

dzttSdttStdS

Then Ito’s formula and a hedging argument, leads to the Black-Scholes Equation (BSE)

for tStX , ,

rXs

XtS

s

XtrS

t

X

2

222

2

1

To get the diffusion equation, make the change of variables,

K

tSrtT

rztT

retStXU tTr ln,,,

2

2

2

2

2

2

2

2)(

2

2

2

2

2

2

Then the BSE becomes the Diffusion Equation (DE) for zU , ,

2

2

z

UU

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In 1905, Einstein showed us that the DE arises from the Brownian motion of microscopic

particles. Thus, both the BSE and the DE are based on the same underlying process. Solutions

of the DE with known initial condition zU ,0 take the form,

4

'2

4

1',,,'',0',,,

zz

ezzGdzzUzzGzU

',, zzG or Green’s Function is called the Fundamental Solution of the DE.

Option Prices

For a call option, the boundary condition at Tt is,

01

00,,

y

yyhKSKShTSTX

where )(yh is the Heaviside Step Function. Making the transformation of variables, the

boundary condition on the DE at Tt becomes,

1,02

2

2

2

2

2

rExpK

r

zhzU

z

We can put zU ,0 back into the integral, to get Black-Scholes Formula after “some”

integration

TddT

TqrK

S

ddNKedNeSX rTqT

12

20

1210 ,

ln

,

2

Alternatively, we can solve for zU , numerically and then transform back to get X

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Initial Conditions for Other Options

Kr

zhzUPutBinary

Kr

zhzUCallBinary

rExpK

r

zhzUPutVanilla

z

2

2

2

2

2

2

2

2

2

2

,0

,0

1,0

Part D

Numerical Integration Techniques

& Monte Carlo Integration Routine

1. General Numerical Integration routine

2. Rectangle Rule for Numerical Integration

3. Trapezoidal Rule for Numerical Integration

The trapezoidal rule breaks up the area under the curve traced by the function into

trapezoids and evaluates the area of all those trapezoids and sums them up.

4. Closed form solution of the Integral of Gaussian Density Function

212

21

2

1

2

1 2

2

1 zerfdzez

z

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For the Call option valuation in a Monte Carlo Integration routine using an Excel™

spreadsheet (or VBA) the limits of integration in the above definite integral would be

from 0 to 6.

5. Monte Carlo Integration Routine

If z is a random variable that is drawn from a Normal (Gaussian) distribution and if

z is the Gaussian Probability Density function given by the above closed form

solution then the value of a Call and a Put option are given by:

dzzSKePut

dzzKSeCall

T

rT

T

rT

0,max

0,max

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Chapter S

Part A

Stochastic Process for Asset Price Modelling

(For Implementing Monte Carlo Simulation on Excel™)

The following variables and constants, unless otherwise mentioned, are used in the

equations in this chapter:

Random Number (from a Normal Distribution) = 1,0~ Nt

Weiner Process (Random Walk) = tdWt

Discrete Weiner Process (Random Walk) = tWt

Poisson Process = tP with intensity and Jump size J

Stochastic Asset Price = tS

Asset Price today (time, 0t ) = 0S

Constant displacement in Asset price =

Stochastic Forward price = tF

Constant Volatility =

Stochastic Volatility = t

Stochastic Variance = tv

Constant rates = r

Stochastic rates = tr

Constant dividend yield = q

Drift = qr

Long term value of rates (constant) = r Long term value of rates (stochastic) =

tr

Long term value of variance (constant) = v

Long term value of variance (stochastic) =

tv

Long term mean of the mean reversion parameter, v = tm

Speed of mean reversion = k

Volatility of volatility (constant) = Volatility of the mean reversion parameter (constant) =

Correlation =

Long term mean of Correlation (constant) = Libor at time, ( Tt ) = TL

Forward Libor at time ( Tt ) = F

TL

Maturity value of a variable is indicated by Tt

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1. Geometric Brownian Motion with constant volatility

Stochastic Differential Equation for Equity:

t

t

t dWdtqrS

dS

Stochastic Integral Equation

tttqr

tt eSS 2

2

1

1

Stochastic Differential Equation for FX:

tfd

t

t dWdtrrS

dS

Stochastic Integral Equation

tfd ttrr

tt eSS 2

2

1

1

2. Geometric Brownian Motion for the Inverse of the Asset Price

Given the Jensen’s inequality

XEfxfE

The geometric Brownian motion for the inverse of the asset price, SY 1 , is given by

t

t

t dWdtY

dY 2

An example would be, if tS is the USD/JPY (Dollar-Yen) price then tt SY 1 , would be

the JPY/USD (Yen-Dollar) price.

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3. Arithmetic Brownian motion with constant volatility

(See Vasicek and CIR process for Rates Modelling)

tt dWdtqrdS

4. Geometric Brownian Motion for the “Square of the Asset”

tttt dWSdtSqrSd 2222 22

5. Geometric Brownian Motion for the thn Power of the Asset

t

N

t

N

t

N

t dWSNdtSNNqrNSd

212

1

6. Brownian Bridge Process

Tt

TWT

ttWtB

,0

)(*)()( TWttWtB

Tt

T WT

tW

T

t

s

S

t eSS

0

ln

0

7. Cox-Ross Square Root Process

tttt dWSdtSdS

8. Mean Reverting Vasicek Process for Interest Rates

ttt dWdtrrdr

9. Mean Reverting Cox-Ingersoll-Ross (CIR) process for Interest Rates

tttt dWrdtrrdr

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10. Hull-White Process

ttttt dWdtrrkdr

11. n Dimensional Bessel Process

t

t

t dWdtS

ndS

2

1

12. Black-Derman-Toy (BDT) Process

ttttt

t

t dWrdtrrdt

ddr

lnln

ln

13. Black-Karisinski (BK) Process

ttttttt dWrdtrrkdr lnln

14. Poisson’s Jump Diffusion Process

tt

t

t JdPdWdtrdrfS

dS

15. Kou’s Double Exponential Process

(Stochastic Integral Equation)

)(

1

2

1 )(2

1exp

tN

i

itt YtWtrSS

111 2

2

1

1

qp

2

2

1r

11 = Mean size of upward jump

21 = Mean size of downward jump

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16. Heston’s Stochastic Volatility Process

Asset Price Process with Stochastic Variance

2

1

ttt

tt

t

t

dWdtvvkdv

dWvdtS

dS

dtdWdW 21

Asset Price Process with Stochastic Volatility

dtdWdW

kwhere

dZdtkd

dWvdtS

dS

t

ttt

tt

t

t

21

2

,2

1

4,

Finite Difference Discretization of the Variance Process using Euler scheme

ztvtvvkvv tttt

1

Finite Difference Discretization of the Variance Process using Milstein’s scheme

ttvvkztvv

ztztvtvvkvv

ttt

tttt

42

14

22

1

22

1

29

The Book of Greeks


17. Double Mean Reverting Process for Variance

m

ttt

v

ttttt

v

ttttt

tt

t

t

dWmdm

dWvvmpdv

dWvdtvvkdv

dBvS

dS

In the above process the Weiner processes are given by

42

32

,

2

,

2

22

1

1

11

1

tmtm

m

t

tvvtvvvtv

v

t

tvtv

v

t

tt

WBW

WWBW

WBW

WB

18. SABR (Stochastic Alpha Beta Rho) Process

2

2

t

ttt

dWd

dWFdF

dtdWdW 21

19. Longstaff’s Double Square Root Model

This model, which was originally proposed by Longstaff in 1989 for rates and by Zhu for

the variance process in 2000, is similar to the Heston’s model for variance except that the

drift term has a mean reversion in volatility and not variance.

2

1

ttt

tt

t

t

dWdtvkdv

dWvdtS

dS

dtdWdW 21

Where, is the long term value (mean) of volatility of the asset.

30

The Book of Greeks


20. Stochastic Correlation Process

ttttt dWdtkd 11

21. Variance Gamma (VG) Process

Given v , the variance rate of the gamma process, , as the parameter that defines the

skew of the distribution, , the volatility of the asset and the following:

gg

21ln 2

T

The stochastic integral equation for the asset price in a VG process is given by

Tqr

T eSS 0

22. Displaced Diffusion Model

t

t

t dWS

Sd

23. Simplified Libor Market Model

(Single factor with Lognormal rates)

tT

F

t

F

TtT

LL

eL

LLL

2

2

1

31

The Book of Greeks


Part B

Cholesky and Eigensystem Decomposition

for Multi-asset Stochastic Processes

How to factor Correlation in a Monte Carlo Simulation

Given two or more than two stochastic processes (for two more assets in a basket) like the

following:

dtdWdWwhere

dWdtqrS

dS

dWdtqrS

dS

tt

t

t

t

t

t

t

21

2

222

,2

,2

1

111

,1

,1

,

The correlation, , between the two Weiner processes (assets) can be incorporated in the

model either using the Cholesky decomposition or the Eigensystem decomposition.

Excel™ Spreadsheet Implementation

Monte Carlo Simulation Algorithm (using Cholesky Decomposition)

1. Generate a set of uncorrelated random numbers, ε, k ......,,, 21 from a Normal

distribution with mean zero and standard deviation of one, where k denotes the

number of assets and 1,0~ Nk .

2. Given a correlation matrix, M for the asset returns, estimate the Cholesky Matrix, A ,

from this correlation matrix.

3. Generated correlated random numbers using the Cholesky matrix as shown below:

kkkk

k

k aa

aa

z

z

AZ

1

1

1111

4. If using stochastic differential equations (SDE) to model asset price paths and/or

volatility or variances, then transform these stochastic differential equation (SDE) into

Difference equations;

32

The Book of Greeks


5. Use the set of correlated random numbers, kzzz ,.....,, 21 to generate correlated asset

price paths by inputting them in the stochastic differential equation (SDE) or the

stochastic integral equation for the asset prices and/or the volatility process.

Monte Carlo Simulation Algorithm (using Eigensystem Decomposition)

1. Generate a set of uncorrelated random numbers, ε, k ......,,, 21 from a Normal

distribution with mean zero and standard deviation of one, where k denotes the

number of assets and 1,0~ Nk .

2. Given a correlation matrix, M for the asset returns, estimate the Eigenvector Matrix,

W of the correlation matrix and the Eigenvalues, , corresponding to the Eigenvector

matrix, .

3. Correlate the random numbers using the Eigensystem transformation

kkkkk

k

k ww

ww

z

z

WZ

11

1

1111

0

0

4. If using stochastic differential equations (SDE) to model asset price paths and/or

volatility or variances, then transform these stochastic differential equation (SDE) into

Difference equations;

5. Use the set of correlated random numbers, kzzz ,.....,, 21 to generate correlated asset

price paths by inputting them in the stochastic differential equation (SDE) or the

stochastic integral equation for the asset prices and/or the volatility process.

Explanation of the Special Matrices and Transformations

Given a correlation matrix, M , as follows

Correlation Matrix =

nnn

N

M

1

111

And, the corresponding Cholesky matrix and the Eigensystem of the Correlation matrix as

following:

33

The Book of Greeks


Cholesky Matrix =

nnn

n

aa

aa

A

1

111

Eigenvector Matrix =

nnn

n

ww

ww

W

1

111

Eigenvalues (scalar) = n 1

Lambda Matrix =

n

0

01

Square Root Lambda Matrix (Diagonal matrix) =

n

0

01

Uncorrelated Random Normal Numbers = n 1

Correlated Random Normal Numbers = nzzZ 1

1. Definition of Transpose

For any arbitrary, square matrix (rows equal to columns), Transpose operation flips the

rows into columns and columns into rows. For a square matrix, X , the transpose is

written as TX . As an example, the transpose of a 22 matrix is shown below

01

12

01

12TXX

2. Cholesky Transformation

MAAT

34

The Book of Greeks


3. Cholesky transformation for 2 (two) assets case

2221212

2121111

2

1

2221

1211

2

1

aaz

aaz

aa

aa

z

z

AZ

Closed Form Solution for the 22 Cholesky Matrix

2

2221

1211

1

01

aa

aaA

4. Cholesky transformation for 3 (three) assets case

3332321313

3232221212

3132121111

3

2

1

333231

232221

131211

3

2

1

aaaz

aaaz

aaaz

aaa

aaa

aaa

z

z

z

AZ

Closed Form Solution for the 33 Cholesky Matrix

35

The Book of Greeks


2

12

2

1412232

132

12

131223

13

2

1212

333231

232221

131211

11

1

01

001

aaa

aaa

aaa

A

5. Eigensystem Decomposition

MWW T

6. Eigensystem decomposition for 2 (two) assets case

2

1

222121

212111

2

1

2

1

2221

1211

2

1

0

0

ww

ww

ww

ww

z

z

WZ

Expanding as a system of linear equations

222212212

221211111

wwz

wwz

7. Eigensystem decomposition for 3 (three) assets case

3

2

1

333232131

323222121

313212111

3

2

1

3

2

1

3

2

1

333231

232221

131211

3

2

1

00

00

00

www

www

www

z

z

z

www

www

www

z

z

z

WZ

36

The Book of Greeks


Chapter V

Closed Form Formulas for

Historical and Implied Volatility

Given the following notations:

Current Price of the Asset (at 0t ) = 0S

Strike Price = K

Risk free rate = r

Dividend Yield = q

Volatility =

Time to Maturity = T

Current price of the Call = C

High of the Stock Price = HS

Low of the Stock Price = LS

Number of trading days = n

Decay Parameter =

Part A

Implied Volatility Estimation

1. Leland’s Formula for incorporating Transaction Costs in Volatility

Adjusted Implied Volatility for Long option position

2

1

81

k

t

Adjusted Implied Volatility for Short option position

2

1

81

k

t

Where, t is the frequency of rebalancing and k is the transaction cost in percentage,

taking into account the bid-ask spread. Bid-ask spread is also a measure of liquidity in

the market and one way to define a bid-ask spread is as follows:

37

The Book of Greeks


Bid-Ask Spread =

bidask

bidask

PP

PP

2

1

2. Brenner-Subrahmanyam Approximation for ATM Call and the Put

Here the rates are assume to be extremely low, theoretically equal to zero, and the spot

is assumed to equal the discounted strike ( rTKeSr ,0 ), i.e. the option is ATM.

This approximation holds only for ATM options with zero rates.

TS

C

S

C

TVol impliedATM

00

5.22

If dividends are taken into account then the above formula becomes

TeS

C

eS

C

TVol

qTqTimpliedATM

00

5.22

3. Improvement on Brenner-Subrahmanyam Approximation for ATM Option

(Steven Li’s Approximation)

Once again, rates are assumed to be zero and spot is assumed to be equal to the

discounted strike.

32

3

2

2

68

122

1

0

2

CosCosz

S

C

zz

Tz

Timplied

4. Bharadia, et al Approximation for Near the Money Option

If rates are zero (i.e. spot is equal to discounted strike) but, if the option is not strictly

at the money, but near the money, i.e. the strike does not deviate too far away from the

spot then the implied volatility can be estimated by:

38

The Book of Greeks


2

22

KSS

KSC

Timplied

5. Corrado and Miller Approximation for Near the Money option

If rates are zero and if the option is near the money, i.e. the strike is not too far away

from the spot, then the implied volatility can be estimated by a more accurate formula

given below:

22

2

22

12 KSKSC

KSC

KSTimplied

6. Implied Volatility Approximation for ITM or OTM Options

(Steven Li Approximation)

Deep in the money or deep out of the money options

For deep in or out of the money options with small volatility and short time to

expiration and zero rates the following formula is quite accurate

Timplied

2

1

14ˆˆ

2

2

Where,

1

2

1

2ˆ

0

S

C and

K

S0

Near the Money Option

When the strike does not deviate too far away from the spot and when the following

condition holds T

1

then the following formula can be used to estimate the

implied volatility

39

The Book of Greeks


12

1

2ˆ

32

ˆ3ˆ

ˆ2

ˆ6ˆ8

1ˆ

22

0

1

2

S

C

CosCosz

zz

Tz

Timplied

7. SABR Volatility

A 2-factor SABR model of stochastic volatility for the forward is given by

2

1

dWd

dWFdF

For 1 which is mostly the case for FX, the closed form solution for SABR

volatility, B is given by:

1

21ln

ln

.....3224

1

4

11

2

22

zzzz

K

Fz

Tz

zB

For 1 which is mostly the case for FX, the closed form solution for SABR

volatility, B is given by:

40

The Book of Greeks


1

21ln

ln

...24

32

241

ln

2

222

zzzz

K

FFKz

TFKz

z

KF

KFB

The SABR volatility can be input into the Black-76 formula to estimate the value of the

option. In that sense, this becomes the Black equivalent volatility.

8. CEV Volatility

A CEV (constant elasticity of variance) model for the is given by

dWFdF

The closed for solution for the CEV volatility, is given by:

2

...24

1

24

211ˆ

22

222

1

KFf

f

T

f

KF

f

This volatility can be input into the Black-76 formula for estimating the option values.

For at the money forward the CEV volatility becomes

1ˆ

F

In the above formulas, K is the strike price.

9. Forward Interpolation of Implied Volatility

If 1 is the implied volatility for a maturity 1,0 T and 2 is the implied volatility of

the maturity 2,0 T such that 12 TT then the forward volatility between 1T and 2T is

given by:

41

The Book of Greeks


12

1

2

12

2

212 ,

TT

TTTT

Part B

Historical Volatility Estimation

1. Historical Volatility using Close to Close prices

1

lnt

t

impliedHistoricalS

SVol

2. Historical Volatility using High and Low of Closing Prices

n

i L

HHistoricalHistorical

S

S

nVol

1

2

ln361.0

3. Parkinson’s Number for Calculating Historical Volatility using High and Low

n

t L

HHistoricalHistorical

S

S

nVol

1

2

ln)2log(4

250

4. Garman-Klass Estimator

n

t C

O

L

HHistorical

S

S

S

S

n 1

22

log1)2log(2log2

1250

42

The Book of Greeks


5. Rogers-Satchell Estimator

n

t O

L

C

L

O

H

C

HHistorical

S

S

S

S

S

S

S

S

n 1

loglogloglog250

6. Historical Volatility using a Kalman Filter

n

t L

Ht

n

t

t

n

t L

Ht

HistoricalHistoricalS

SS

S

Vol1

2

1

1

2

ln)1(

ln

7. Exponentially Weighted Moving Average

11

2

1, ln)1(t

t

tEWMAEWMAHistoricalS

SVol

43

The Book of Greeks


Chapter O

Closed form Formulas & Approximations

Vanilla & Exotic Options

Asset price today ( 0t ) = 0S

Forward = f

Strike = K Alternative Strike = K

Volatility =

Maturity = T

Discrete Barrier = H

Barrier = H

Risk free rate = r

Dividend Yield = q

1. Vanilla Call Option

(Black-Scholes Formula)

Tdd

T

TqrK

S

d

dNKedNeSCall rTqT

12

20

1

210

2

1ln

2. Vanilla Put Option

Tdd

T

TqrK

S

d

dNeSdNKePut qTrT

12

20

1

102

2

1ln

3. Put-Call Parity Relationship

rTqT KeeSPutCall 0

44

The Book of Greeks


4. Put-Call Symmetry

2

1

K

fLog

f

KLog

21

2

21

KKf

fKK

1

2

K

K

Call

Put

5. Put-Call Super-symmetry

PutCall

TrKSPTrKSC

,,,,,,,, 00

6. Brenner-Subrahmanyam Approximation for a Call

(“Black-Scholes in your head”)

ATM call and put option price, with rates equal to zero ( 0r ), and dividend yield

equal to zero ( 0q ), thus making the spot is equal to the discounted strike, are given

by

20

TSPutCall

TSPutCall 040.0

7. Approximation for a Call and a Put Price

When the rates and the dividend yield are not equal to zero, i.e. 0r and 0q , the

Call and Put option prices are approximated by the following formulas

0022

12

STqrTqrT

SCall

45

The Book of Greeks


00

221

2S

TqrTqrTSPut

8. Call Option price in a Displaced Diffusion Model

T

TK

S

d

T

TK

S

d

dNKdNSC

20

2

20

1

210

2

1ln

2

1ln

0

0

S

S

A more accurate approximation for the Displaced diffusion volatility is given by

TS

S

TS

S

2

0

0

2

0

0

24

11

24

11

46

The Book of Greeks


9. Power Call Option

(Closed form Solution)

If the Call payoff is 0,max 2 KST , the closed form value of the call is given by:

Tdd

T

TqrK

S

d

dNKedNeSCallPower rTqT

2

2

22ln

12

220

1

210

10. Exchange Option

22211121 dNeSndNeSnCallExchange

TqTq

2,121

2

2

2

1

12

2

12

22

11

1

2ˆ

ˆ

ˆ

2

ˆln

Tdd

T

TqqSn

Sn

d

11. Binary (Digital) Call

Tdd

T

TqrK

S

d

dNeBC rT

12

20

1

2

2

1ln

47

The Book of Greeks


12. Binary (Digital) Put

Tdd

T

TqrK

S

d

dNeBP rT

12

20

1

2

2

1ln

13. Put-Call Parity for Binary Options

rTeBPBC

14. Range Accrual Option

BPBCAccrualRange 100

15. Pay Later Option (Contingent Premium Option)

A Pay Later option is a vanilla call or a put option with the difference that the buyer of

the option will pay the premium to the seller at maturity and only if the option is in the

money. Otherwise, the buyer pays nothing.

put

call

rtTKSVanilladN

eP

rtTKSDigital

rtTKSVanillaP

rT

1

1

,,,0,,,*

,,,0,,,

,,,0,,,

2

48

The Book of Greeks


16. Barrier options

Given the following identities pertaining to the barrier

2

2

21

0

21

0

qr

qr

S

Hh

S

Hg

T

TqrH

S

d

T

TqrK

S

d

T

TqrK

S

d

20

3

20

2

20

1

2

1ln

2

1ln

2

1ln

T

TqrH

S

d

T

TqrH

S

d

20

5

20

4

2

1ln

2

1ln

T

TqrH

KS

d

T

TqrH

KS

d

T

TqrH

S

d

2

2

0

8

2

2

0

7

20

6

2

1ln

2

1ln

2

1ln

49

The Book of Greeks


Up and Out Call

7542

86310

dNdNgdNdNKe

dNdNhdNdNeSCUO

rT

qT

Up and In Call

754

8630

dNdNgdNKe

dNdNhdNeSCUI

rT

qT

Down and Out Call

54

630

72

810

1

1

,

1

1

,

dNgdNKe

dNhdNeSCDO

thenHKIf

dNgdNKe

dNhdNeSCDO

thenHKIf

rT

qT

rT

qT

Down and In Call

542

6310

7

80

1

1

,

1

1

,

dNgdNdNKe

dNhdNdNeSCDO

thenHKIf

dNgKe

dNheSCDO

thenHKIf

rT

qT

rT

qT

Down and Out Put

5724

68130

dNdNgdNdNKe

dNdNhdNdNeSPDO

rT

qT

50

The Book of Greeks


Down and In Put

574

6830

1

1

dNdNgdNKe

dNdNhdNeSPDI

rT

qT

Up and Out Put

72

810

54

630

1

1

,

1

1

,

dNgdNKe

dNhdNeSPUO

thenHKIf

dNgdNKe

dNhdNeSPUO

thenHKIf

rT

qT

rT

qT

Up and In Put

7

80

524

6130

,

,

dNgKe

dNheSPUI

thenHKIf

dNgdNdNKe

dNhdNdNeSPUI

thenHKIf

rT

qT

rT

qT

17. Adjustment for Monitoring Discrete Barrier

Approximate Adjustment for up barrier

tHeH 8.0ˆ

tHefHf 8.0ˆ

51

The Book of Greeks


More Accurate Adjustment

(“Plus” sign for up barrier and “Minus” sign for down barrier)

mTHefHf

5826.0ˆ

18. Fixed Strike Lookback Call

If the Lookback is priced at 0t then minmax0 SSS

Tdd

T

TqrK

S

d

dNeTqr

dNK

S

qreS

dNKedNeSCall

thenSKIf

Tqr

qr

rT

rTqT

12

20

1

11

22

0

210

max

2

1ln

2

2

,

2

Thh

T

TqrS

S

h

hNeTqr

hNS

S

qreS

hNeShNeSKSeCall

SKIf

Tqr

qr

rT

rTqTrT

12

2

max

0

1

11

2

max

2

0

2max10max

max

2

1ln

2

2

2

52

The Book of Greeks


19. Fixed Strike Lookback Put

If the Lookback is priced at 0t then minmax0 SSS

Tdd

T

TqrK

S

d

dNeTqr

dNK

S

qreS

dNeSdNKePut

thenSKIf

Tqr

qr

rT

qTrT

12

20

1

11

2

0

2

0

102

min

2

1ln

2

2

,

2

Thh

T

TqrS

S

h

hNeTqr

hNS

S

qreS

hNeShNeSSKePut

SKIf

Tqr

qr

rT

rTqTrT

12

2

min

0

1

11

2

min

2

0

2min10min

min

2

1ln

2

2

2

53

The Book of Greeks


20. Floating Strike Lookback Call

Tll

T

TqrS

S

l

lNeTqr

lNS

S

qreS

lNeSlNeSCall

Tqr

qr

rT

rTqT

12

2

min

0

1

11

2

min

0

2

0

2min10

2

1ln

2

2

2

21. Floating Strike Lookback Put

Tll

T

TqrS

S

l

lNeTqr

lNS

S

qreS

lNeSlNeSPut

Tqr

qr

rT

qTrT

12

2

max

1

11

2

min

2

0

102max

2

1ln

2

2

2

22. Arithmetic Average (Asian) Call

(Turnbull & Wakeman Formula)

Tdd

T

TK

S

d

dNKedNeSCall

A

A

AA

rTrA

12

20

1

210

2

1ln

54

The Book of Greeks


Where, the averaging parameters are given by:

T

M

T

M

A

AA

1

2

ln

ln

If is the time to the averaging period then the first two moments, 1M and 2M are

given by:

222

2

2222

1

2

12

2

2

2

22

qr

e

qrTqr

e

Tqrqr

eM

Tqr

eeM

Tqrqr

Tqr

qrTqr

23. Arithmetic Average (Asian) Put

With all the above parameters in place for the Arithmetic average call, the value of

Arithmetic average put option is given by:

102 dNeSdNKePutTrrT A

55

The Book of Greeks


Chapter G

Closed form Greeks for Vanilla

& Binary Options

Current Price of the Asset (at 0t ) = 0S

Risk free rate = r

Dividend Yield = q

Volatility =

Time to Maturity = T

Given the following parameters and identities

Tdd

T

TqrK

S

d

12

20

1

2

1ln

21

2

2

1

1

2

1

2

1

2

1

d

x

edN

exN

1. Call Delta

1dNeS

CDeltaCall qT

Approximating Call Delta of an ATM option

11 4.02

1

2

1

2

1ddCallATM

56

The Book of Greeks


2. Put Delta

11 1 dNedNeS

PDeltaPut qTqT

3. Gamma of a Call and a Put

TS

dNe

S

P

S

CGamma

qT

0

1

2

2

2

2

Approximating the Gamma for an ATM option

TT

ATM

4.0

2

1

4. Vega of a Call and a Put

TdNeSPC

Vega

TdNeSPC

Vega

qT

qT

10

10

101010%

Approximating the Vega of the ATM option

TST

SVATM 04.02

5. Vanna of a Call and a Put

12 dNde

DdeltaDvolVannaqT

6. Volga of a Call and a Put

Volga =

212110 dd

VegadddNeTS

DvegaDvolqT

57

The Book of Greeks


7. Theta of a Call

210

10

2dNrKedNeqS

T

dNeS

T

CThetaCall rTqT

qT

8. Theta of a Put

210

10

2dNrKedNeqS

T

dNeS

T

PThetaPut rTqT

qT

9. Rho of a Call

(with respect to the rate)

2dNTKe

r

CCallRho rT

10. Rho of a Put

(with respect to the rate)

2dNTKer

PPutRho rT

11. Binary Call Delta

TS

dNe rT

CallBinary

2

12. Binary Put Delta

TS

dNe rT

PutBinary

2

13. Binary Call Gamma

TS

dNde rT

CallBinary 22

21

58

The Book of Greeks


14. Binary Put Gamma

TS

dNde rT

PutBinary 22

21

15. Binary Call Vega

21 dNde

VrT

CallBinary

16. Binary Put Vega

21 dNde

VrT

PutBinary

59

The Book of Greeks


Chapter E

Exotic Option Payoffs, Structured Products

& Product Engineering

1. Call Payoff

0,max KSCall T

2. Put Payoff

0,max KSPut T

Binary Call

0,

1,

KSif

KSifBC

T

T

3. Binary Put

1,

0,

KSif

KSifBP

T

T

4. Knock-out (Barrier) Call

0,

0,max,

thenTtforHtSif

KSthenTtforHtSifKO

T

HtS means that the asset never touches (hits) the barrier level, H , at any point in

time before the maturity of the option.

5. Knock-out (Barrier) Put

0,

0,max,

thenTtforHtSif

SKthenTtforHtSifKO

T

HtS means that the asset never touches (hits) the barrier level, H , at any point

in time before the maturity of the option.

60

The Book of Greeks


6. Knock-in (Barrier) Call

0,max,

0,

KSthenTtforHtSif

thenTtforHtSifKI

T



7. Knock-in (Barrier) Put

0,max,

0,

TSKthenTtforHtSif

thenTtforHtSifKI



8. Fixed Strike Lookback Call

0,max max KSCallLookback

9. Fixed Strike Lookback Put

0,max minSKPutLookback

10. Floating Strike Lookback Call

0,max minSSCallLookback T

11. Floating Strike Lookback Put

0,max max TSSPutLookback

12. Floating Strike Ladder Call

0,,.....,,,minmax 21 TnT SLLLSCallLadder

13. Floating Strike Ladder Put

0,,.....,,,maxmax 21 TTn SSLLLPutLadder

61

The Book of Greeks


14. Arithmetic Average Call

0,max KSAC Average

N

i

iAverage SN

S1

1

15. Arithmetic Average Put

0,max AverageSKAP

N

i

iAverage SN

S1

1

16. Simple Chooser Option

ttTKPtTKCChooser ;,,,max

Where, Tt and t is the time to choose between a call and a put.

17. Asymmetric Power Option (Power of 2)

0,max 22 KSPayoff T

18. Symmetric Power Option (Power of 2)

20,max KSPayoff T

This can be decomposed, for valuation and hedging purposes, as:

KSKKSKS TTT 2222

Thus, we get the relationship that

VanillaKShortPowerAsymmetricLongPowerSymmetric 2

19. Cliquet Option

0,max1

KS

SCliquet

t

t

62

The Book of Greeks


20. A capped and floored Cliquet Option

CFKS

SCliquet

t

t

FloorCap ,,maxmin1

,

21. Locally Capped, Globally Floored Cliquet

min

24

1

,02.0,02.0,maxminmax CRCliqueti

t

Where, 1

1

t

tt

tS

SSR and minC is a constant and denotes minimum coupon payable.

22. Digital Cliquet

1 tt SSCCliquetDigital

Where, C is the coupon and is the Heaviside function

23. Reverse Cliquets

12

1

max ,0min,0maxi

treverse RCCliquet

Where, maxC is a constant and denotes maximum coupon payable and 1

1

t

tt

tS

SSR

24. Napoleon Option

tRCNapoleon ˆ,0max max

N

tttt RRRR ....,,,minˆ 21

Where, maxC is the maximum coupon and tR denotes the worst monthly (or any other

periodic) return of a basket of stocks or stock indices

63

The Book of Greeks


25. Amortizing Call Option

T

TAmortizing

S

KSCall

0,max

26. Two Asset Rainbow Option

0,,max 2,21,1 KSKSCallRaibow TT

27. Quanto Option

0,max foreignforeign

T

fixed

o KSFXCallQuanto

Where, foreign

TS is the stock price at maturity denominated in a foreign currency and foreignK is the strike price denominated in the same foreign currency. fixedFX 0 is the

fixed FX rate at time, 0t .

28. Two Asset Pyramid Option

0,max 2,21,1 KKSKSCallPyramid TT

29. Two Asset Madonna Option Call Option

0,max

2

2,2

2

1,1 KKSKSCallMadonna TT

30. Exchange Option

0,max 2211 SnSnOptionExchange

31. Two Asset Best of Call Option

0,,max

0,1

0,1,1

0,2

0,2,2

S

SS

S

SSCallofBest

TT

64

The Book of Greeks


32. Single Asset Best of Call Option

(for Hedge Fund Managers)

If a hedge fund manager wants to invest in a product that gives him the minimum

return on a stock index (asset) or 4% in a certain period maturing at time T then the

payoff of this product will be:

04.0,15.0max

0S

SPayoff T

This product can be broken up as:

0,08.1max2

1%4

04.05.0,0max%4

04.0,5.0max04.004.0

0

0

0

0

0

0

SSS

P

S

SSP

S

SSPPayoff

T

T

T

Thus, the payoff of the best of product represents a 4% coupon and a call option with

a strike of 008.1 S and leverage factor of 02

1

S. If the notional of the product is N

then the leverage of the call option would be 02SN . This best of product

represents a long zero coupon bond with a coupon of %4 and a leveraged call option.

33. “Best of” Options with CMS Floor

y

TT R

S

SPayoff 5

0

,1max75.0

Where, y

TR5 is the 5 year swap rate.

34. “Best of” Option with Inflation Floor

1,175.0max

00 I

I

S

SPayoff TT

Where, TI is the retail price index at maturity.

65

The Book of Greeks


35. FX Power Reverse Dual Currency (PRDC) Option

FX PRDC call options are a series of options embedded in a long term note (typical

maturity would be 20 or 30 years) each maturing at regular intervals.

1.......,,3,2,1;,,maxmin0

TtFFr

S

SrPayoff ULd

t

f

0

0 ,

0,max

S

rJ

r

rSK

KSJPayoff

f

f

d

t

In the above, dr is the domestic interest rate and fr is the foreign interest rate.

36. Capped Bull Note

Payoff and Reverse Engineering

0

0,min1S

SSCNPayoff T

Where, TS is the value of the asset (stock index) at maturity and 0S is the value of

the asset (stock index) today. In the above payoff, N is the notional amount of the

note, C is a constant denoting the cap and is also a constant denoting the

participation rate. The payoff of the above capped bull note can be decomposed as:

CSSS

SCNP

CS

SSCCCNP

S

SSCCCNPayoff

T

T

T

00

0

0

0

0

0

,0min1

,min1

,min1

66

The Book of Greeks


T

T

T

SC

SS

NCNP

SC

SS

CNP

CSS

SCNP

1,0max1

1,0max1

1,0min1

0

0

0

0

0

0

Thus, the capped bull note can be decomposed as long a coupon bond with a coupon

of C and short a leveraged put option on the asset with a strike price of

CS 10

.

In the above algebraic manipulation we have made use of the following

mathematical identity: yxyx ,max,min .

37. Principal Protected Bull Note

RateionParticipat

FloorF

S

SSFNPayoff T

0

0,max1

The above note can be reverse engineered as:

0,1max1

,max1

,max1

0

0

0

0

0

0

FSS

S

NFNP

FS

SSFFFNP

S

SSFFFNPayoff

T

T

T

Thus, the above principal protected bull note decomposes as long a coupon bond

with a notional of N and long a leveraged call option on the asset with a strike

price of

FS 10 .

67

The Book of Greeks


38. Principal Protected Bear Note

0

0,max1S

SSFNPayoff T

39. Principal Protected Mixed Note

0

0

0

0 ,max,max1S

SS

S

SSFNPayoff TT

40. Principal Protected Neutral Note

0

0

0

0max ,max,max1S

SS

S

SSRFNPayoff TT

In the above payoff, maxR is a constant and denotes the maximum return that the note

will pay.

41. Note with a Short Put option embedded

KSif

K

SN

KSifN

PayoffT

T

T,

The above payoff can be decomposed as:

K

SNP

K

SNP

K

SNNPPayoff

T

T

T

,1min11

,1min

,min

68

The Book of Greeks


KSK

NNP

KSK

NP

S

KSNP

K

SNP

K

SNP

T

T

T

T

T

T

,0max

,0max1

1

,0min1

1,11min1

,1min11

Thus, the note decomposes as a long zero coupon bond on a notional of N and short

a leveraged put with strike of K . The leverage factor is KN .

42. Snowball Option

In a typical snowball option, the first two coupons are fixed and relatively high.

Subsequent coupons are given by a payoff function that is tied to the previous

coupon. A possible snowball option payoff function could be:

0,%225.0max 1 Ttt LCPayoff

Where, 1tC is the previous coupon and TL is 3 month USD Libor.

69

The Book of Greeks


Chapter P

Portfolio Analytics, Algorithmic Trading

& Risk Management

1. Sharpe Ratio

P

fP rR

2. Treynor’s Ratio

P

fp rRT

3. Jensen’s Measure (Alpha)

fmPfP rRrR

4. Portfolio Volatility

(Two Asset Case)

122121

2

2

2

2

2

1

2

1 2 wwwwP

5. Multi-asset Portfolio Volatility

For 3 (three) or more assets the portfolio volatility formula in algebraic form becomes

very cumbersome and tedious to handle. Matrix notations are used in those cases. Please

see the Part A of Chapter M above for more on this.

6. Expected Return for Stocks


E = Last Period’s Earnings

g = Growth rate of Earnings

pd = Dividend Payout Ratio

k = Equilibrium Price/Earnings multiple

70

The Book of Greeks


P = Current Stock price

Then, the expected return for stocks, tR for the period t,0 is given by:

P

PkgEdgER

p

t

11

7. Expected Return for Bonds


C Coupon payment

y = Equilibrium yield to maturity

F = Face value of the bond

P = Current bond price

T = Term to maturity (investment horizon)

Then, the expected return for the bond is given by:

P

Py

y

CF

y

CC

R

T

T

1

8. Volatility (Standard Deviation) of Spread in Stock and Bond Return

bsbsbsSpread ,

22 2

9. Probability of Stocks Outperforming Bonds

If sR is the expected return of the stock and bR is the expected return of the bond and

Spread is the volatility of the spread (of returns), then assuming that the stock and bond

returns are normally distributed, the probability that the stock will outperform the bond is

given by:

Spread

bs

bs

RRNRR

Pr

71

The Book of Greeks


10. Mean-Variance Optimization with Total Return

Constrained Optimization (Risk minimization) Problem

(Strategic Asset Allocation Model for Hedge Fund Managers)

A hedge fund manager wants to minimize the risk of his portfolio subject to his expected

(desired) portfolio return.

Here, a two asset problem in strategic asset allocation model entails allocating funds

between two assets based on minimization of risk (volatility or variance) given a certain

expectation of the return. Say, a hedge fund manager has to allocate funds between two

assets, 1 and 2, such that the risk of the portfolio – comprising asset 1 and 2 – as

measured by the variance or the volatility is minimized given a certain level of portfolio

return (fund manager’s desired return) that is expected of the portfolio. The problem

differs from the classical strategic asset allocation model in the sense that in a classical

model a long only fund manager (mutual fund manager) minimizes risk, subject to certain

constraints, such as no short selling, etc., but has not desired return expectation. A hedge

fund manager, like a venture capitalist or a private equity investor, on the other hand has

a firm expectation of how much return he or she wants over a certain investment horizon.

Mathematically, this problem reduces to:

)2(..............................1

)1(..................:

2:

21

2211

122121

2

2

2

2

2

1

2

1

xx

RRxRxtoSubject

xxxxVMinimize

P

In the above optimization problems, we have used the following notations:

V = Variance of the portfolio

1R = Return of asset 1

2R = Return of asset 2

PR = Overall portfolio return that the fund manager wants to achieve

1x = Weight of (funds invested) in asset 1

2x = Weight of (funds invested) in asset 2

1 = Volatility of asset 1’s return

2 = Volatility of asset 2’s return

12 = Correlation between the returns of asset 1 and 2.

We employ Lagrangian multiplier method to solve the above optimization problem. Let’s

say that 1 and 2 are two variables (Lagrangian multipliers) that are introduced in the

problem but we are not interested in solving for these variables; they are immaterial to

the problem. It’s just a mathematical trick to make the problem tractable.

72

The Book of Greeks


Modified Objective Function and the Optimization problem becomes:

1

2:

212

22111

122121

2

2

2

2

2

1

2

1

xx

RRxRx

xxxxZMinimize

P

Essentially, the modified objective function, Z is exactly equal to the original objective

function (variance), V because we have incorporated the two constraints (1) and (2) in the

objective function by using the Lagrangian multipliers, 1 and 2 .

We need to solve for 1x and 2x for the given level of PR . The hedge fund manager’s

objective is to find out how much he should invest in asset 1 and asset 2 such that given

his desired return, PR and the volatility and correlation of assets 1 and 2, the allocation

will minimize his portfolio variance.

Differentiating the modified objective function with respect to various variables, we get:

01

0

022

022

21

2

2211

1

22121121

2

22

2

21121122

2

11

1

xxZ

RRxRxZ

Rxxx

Z

Rxxx

Z

P

Using matrix notation we can write the above equation as:

1

0

0

0011

00

122

122

2

1

2

1

21

2

2

22112

12112

2

1

pR

x

x

RR

R

R

73

The Book of Greeks


The Solution of the above matrix equation is given by:

1

0

0

0011

00

122

1221

21

2

2

22112

12112

2

1

2

1

2

1

pRRR

R

R

x

x

For a three asset problem (asset 1, 2 and 3) the above solution will become:

1

0

0

0

00

00

1222

1222

12221

321

321

3

2

332233113

23223

2

22112

131132112

2

1

2

1

3

2

1

PR

RRR

RRR

R

R

R

x

x

x

Similarly, we can extend the solution for 4, 5,….., N assets.

Let’s consider a two asset example. A fund manager wants to invest in two assets, asset 1

and asset 2, such that his desired return is 14%. Asset 1 has an expected return of 11%

and volatility of 19% and asset 2 has an expected return of 12% and volatility of 22%.

The correlation between the two asset returns is 0.25. How much should the fund

manager invest in asset 1 and asset 2?

Solution:

715.3

03.33

0.3

0.2

1

14.0

0

0

0011

0012.011.0

112.00968.00209.0

111.00209.00722.0

2

1

2

1

1

2

1

2

1

x

x

x

x

74

The Book of Greeks


Therefore, the fund manager needs to invest -200% in asset 1, i.e. he needs to borrow

money and go short on asset 1 to the extent of 200% and he needs to invest 300% in asset

2, i.e. he needs to go long on asset 2 to the extent of 300%. His total investment is 100%

of his funds.

This example illustrates the use of leverage by hedge fund managers.

11. Mean-Variance Optimization

Maximization of Sharpe Ratio

(Strategic Asset Allocation Model with Short Sales for Mutual Fund Managers)

The mutual fund manager wants to find out how much he should invest in different assets

such that Sharpe ratio of his portfolio is maximized. There are no constraints other than

the one that his total investment in various assets should add up to 100%.

Given the expected return of each asset (security) as NRRR ....,,, 21 and their respective

return volatilities as N .......,,, 21 . The correlation between asset i and j is given as

ij with an NN correlation matrix for the portfolio with N assets. The covariance of

asset i and j is given by jiijij .

Maximize the objective function:

P

fP rR

Subject to:

11

N

i

ix

Modified Objective function becomes:

2

1

1 11

22

1

N

i

N

ijj

ijji

N

i

ii

N

i

fii

xxx

rRx

75

The Book of Greeks


Taking the first mathematical derivative of with ix and equating them to zero:

0......,..........,0,021

ndx

d

dx

d

dx

d

Substituting, kk xhz , where h is an arbitrary constant, we get the following system of

linear equations:

2

2211

2

2

221212

1122

2

111

................

................

................

NNNNfN

NNf

NNf

zzzrR

zzzrR

zzzrR

Using matrix notation, we can solve the above system of linear equations quite easily. For

a three asset case (asset 1, 2 and 3) we can express the above system of linear equations

as:

3

2

1

2

332233113

3223

2

22112

31132112

2

1

3

2

1

z

z

z

rR

rR

rR

f

f

f

The solution is:

f

f

f

rR

rR

rR

z

z

z

3

2

1

1

2

332233113

3223

2

22112

31132112

2

1

3

2

1

After estimating the kz (the z values), we can find out the corresponding kx (the x

values), i.e. the proportion to invest in each of the asset such that the Sharpe ratio is

maximized using the following formula:

N

j

j

kk

z

zx

1

76

The Book of Greeks


An example: a mutual fund manager wants to invest in three assets (securities), asset 1, 2

and 3. Asset 1 has an expected return of 12% and volatility of 15%. Asset 2 has an

expected return of 9% and volatility of 10%; and asset 3 has an expected return of 20%

and volatility of 28%. The risk free rate is 2% and the correlation matrix of asset returns

for the three assets is given below:

102.025.0

02.0165.0

25.065.01

M

Therefore, the algorithm to maximize the Sharpe ratio of the portfolio and find out the

amounts to invest in each of the assets is given by:

14519.2

09973.6

80014.0

18.0

07.0

1.0

0784.000056.00105.0

00056.001.000975.0

0105.000975.00225.0

3

2

1

1

3

2

1

z

z

z

z

z

z

Therefore, the proportions to invest in each of the three assets are given by:

%85.8045.9

8001.01 x , %44.67

045.9

099.62 x , %72.23

045.9

145.23 x

12. Sharpe’s Algorithm for Estimating Efficient Frontier

(Constrained Optimization: Three Asset Case)

If V is the variance of the portfolio and PR is the return of the portfolio then the

objective function if given by:

PRRxRxRxtoSubject

xx

xxxxxxxVMinimize

332211

233232

133131122121

2

3

2

3

2

2

2

2

2

1

2

1

:

2

22:

77

The Book of Greeks


We modify the objective function using a Lagrangian multiplier as:

ZMinimize : , where,

233232

133131122121

2

3

2

3

2

2

2

2

2

1

2

1

3322112

22

xx

xxxxxxxRxRxRxZ

Taking the first mathematical derivative of Z with respect to asset weights

23223131133

2

33

2

33223121122

2

22

2

33113221121

2

11

1

222

222

222

xxxRx

Z

xxxRx

Z

xxxRx

Z

Sharpe’s algorithm follows an iterative procedure whereby we increase the allocation to

the asset that has the highest value for the first mathematical derivative and reduce by

the same amount the allocation to the asset that has the lowest value for the first

mathematical derivative. And we follow this iterative procedure subject to any

constraint that we may wish to impose on the portfolio. When all mathematical

derivatives (with respect to the weights of asset 1, 2 and 3) are equal to each other,

subject to our constraints, the portfolio becomes efficient.

13. Minimization of Risk and MCR Algorithm

Marginal Contribution of Risk (MCR) is widely used by asset managers to determine a

portfolio’s overall risk sensitivity to a particular asset and the MCR is used by many

algorithmic traders to determine the number of shares in a trade list to buy and sell at a

particular point in time.

We will use the terms “Risk” and “MCR” interchangeably.

YYRisk T

In the above formula, Y is the vector of exposures (the number of stocks in a trade list),

is the variance-covariance matrix and TY represents the transpose of Y .

78

The Book of Greeks


As shown by Kissell and Glantz (see Reference below), the minimization of risk (or the

MCR) entails that:

0

Y

Risk and 0

2

2

Y

Risk

Therefore, we have

0

YY

Y

Y

Risk

T

T

The solution of the above in matrix form is given by:

NMYY

Where,

1

1

AM

DN

ofDiagonalD

In terms of the elements of N and M matrix, ijn and ijm respectively, we can write

jiif

jiifn

iiij

0

1

jiif

jiifm

ii

ij

0

One has to remember that the matrix solution NMXX can only be valid for the

execution in a single stock (share). Therefore, the number of shares in a single stock k

that minimizes the residual risk (MCR) is calculated as:

NMXkIy T

k min,

Where, kI T is the thk of the Identity Matrix. Thus, the total number of shares of stock

k to trade to minimize residual risk is given by:

79

The Book of Greeks


min,kkk yYw

But since there is a restriction around the number of shares to trade, given the fund

manager / trader’s holding in the trade list, such that, kk Yw 0 the following

adjustments need to be made for a buy order and a sell order.

Buy Order:

00

0

k

kkk

kk

buy

k

w

YwY

Yww

w

Sell Order:

00

0

k

kkk

kk

buy

k

w

YwY

Yww

w

80

The Book of Greeks


References:

1. Notes on Solving the Black-Scholes Equation, EOLA Investments, LLC, Oct 2009

2. Solution to the Black-Scholes Equation, S. Karim, MIT, May 2009

3. The Diffusion Equation – A Multidimensional Tutorial, T.S. Ursell, Caltech, 2007

4. Mathematical Methods in Earth Sciences, Lectures by Prof. Francis Nimmo, Department

of Earth and Planetary Sciences, University of California, Santa Cruz

5. Equity Hybrid Derivatives, Marcus Overhaus, Ana Bermudez, Hans Buehler, Andrew

Ferraris, Christopher Jordinson and Aziz Lamnouar (John Wiley & Sons, Inc. 2007)

6. The Volatility Surface, Jim Gatheral (John Wiley & Sons, Inc. 2006)

7. Exotic Options and Hybrids, Mohamed Bouzoubaa, Adel Osseiran (John Wiley & Sons,

Ltd. 2010)

8. Dynamic Hedging, Nassim Taleb (John Wiley & Sons, Inc. 1997)

9. The Complete Guide to Option Pricing Formulas, Espen Gaarder Haug (McGraw-Hill,

2007)

10. Volatility and Correlation, Riccardo Rebonato (John Wiley & Sons, Ltd. 2004)

11. A New Formula for Computing Implied Volatility, Steven Li, School of Economics and

Finance, Queensland University of Technology, Brisbane, Australia

12. A Primer for the Mathematics of Financial Engineering, Dan Stefanica (FE Press, NY)

13. Frequently Asked Questions in Quantitative Finance, 2nd

Edition, Paul Wilmott (John

Wiley & Sons, Limited)

14. Asset Allocation for Institutional Portfolios, Mark P. Kritzman, Richard D.Irwin, 1990.

15. Modern Portfolio Theory and Investment Analysis, 4th

Edition, Edwin J.Elton and Martin

J.Gruber, John Wiley & Sons, 1994.

16. The Handbook of Exotic Options, Instruments, Analysis and Applications, Edited by

Israel Nelken, Irwin Professional Publishing, 1996.

17. The Handbook of Convertible Bonds, Pricing, Strategies and Risk Management, Jan De

Spiegeleer and Wim Schoutens, John Wiley & Sons, Ltd. 2011.

18. FX Options and Structured Products, Uwe Wystup, John Wiley & Sons, Ltd. 2006.

19. Demystifying Exotic Products, Interest Rates, Equities and Foreign Exchange, Chia

Chiang Tan, John Wiley & Sons, Ltd. 2010.

20. Modeling Derivatives in C++, Justin London, John Wiley & Sons, Inc. 2005.

21. Optimal Trading Strategies, Robert Kissell and Morton Glantz, AMACOM, 2003.

22. Foreign Exchange Option Pricing, A Practitioner’s Guide, Iain J.Clark, John Wiley &

Sons Ltd., 2011

23. Structured Equity Derivatives, Harry M.Kat, John Wiley & Sons, Ltd. 2001.

24. Paul Wilmott on Quantitative Finance, Volume 2, Second Edition, John Wiley & Sons,

Ltd., 2006.

Book of Greeks Edition 1.0 (Preview)

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