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The
Book
of Greeks Edition 1.0
Rahul Bhattacharya, M.Sc., MBA, CFE
© CFE School, Risk Latte Company Limited, Hong Kong
2011
2
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.
3
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Certificate in Financial Engineering (CFE) www.risklatte.com
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
4
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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
5
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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
6
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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:
7
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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
8
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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
9
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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
10
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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
11
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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
12
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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
13
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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
14
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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
15
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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%
16
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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.
17
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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
18
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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
19
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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
The probabilities are given by:
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
20
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The probabilities are given by:
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
21
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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
22
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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
23
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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
24
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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
25
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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.
26
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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
27
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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
28
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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6. Knock-in (Barrier) Call
0,max,
0,
KSthenTtforHtSif
thenTtforHtSifKI
T
HtS means that the asset never touches (hits) the barrier level, H , at any point in
time before the maturity of the option.
7. Knock-in (Barrier) Put
0,max,
0,
TSKthenTtforHtSif
thenTtforHtSifKI
HtS means that the asset never touches (hits) the barrier level, H , at any point in
time before the maturity of the option.
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
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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
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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
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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
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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.
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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
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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 .
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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
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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.
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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
Given the following:
E = Last Period’s Earnings
g = Growth rate of Earnings
pd = Dividend Payout Ratio
k = Equilibrium Price/Earnings multiple
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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
Given the following:
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
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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.
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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
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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
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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
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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
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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:
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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 .
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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:
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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
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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.