CSE: Financial Engineering Track

Robbin Tops, SAMrobbin.tops@sam.math.ethz.ch

CSE: Financial Engineering Track

Financial Engineering

��������� ����������������������� ������������� ����!�� �" �#!�������������� �������������� ��

$�����%%��&'��&%(�%)���!��* ��+���%%��&'�,������"���%%��&'�%�"�������#�������!���!������-��* ��!.������ /��"���"�" !�� 0���"�� ������0���� �0��#�( �������1�1��-�(��.����"� �!�!�0! " �� �!� ������2�� !" �� �!� ������������#��� ��+������"�+�3� ��+���" �+�4�������"�5 ��������-����3 ���� �!.�1��(��� !���/� ���� /��"�!�0! " ���� 6��� �0��#��1�1��-���.�1������� * "�!��(���/ �������#� 0������7�� �#���"� ����� ����!��� �����"�!��* ���6 ��������* ��!���"�" !�� 0���!�������* ��!�� �����" ������� ����� �#����� ���(���!�0! " ���� ����������3 ���� �!1�$�����* ��!� ����"�������� � ����" �#���"� �"���� �� �#�!��* ��!+�/� ��������*� ��0��� ����� �! �� !� ����"� �!� ��� ���� �*�!� �!���"� ����/���������!!������#������������!���*��0���� 0�� ��"1��(��+�������"���� ���66 � ���!�" �� ���� * "�� �*�!�������"* ��� ��#���������������������� 6��� ��!� �� �6 ���� �� ��������* ��!1�) �� �# ��������* ��!�!������ �!� �������� 66�� �#� 6�6 ���� ��� �!�������!�0���(��+����� ����� ���66 � ���!1���%%��&'+���%%��&'��&%(�%)��+���%%��&'���&5$+���%%��&'�)8+���%%��&'��)98�&+���%%��&'�$&�,�%%5+���%%��&'��%),$&�,&+���%%��&'$�:��%)+���%%��&'�&�,�%+���%%��&'��&���"���%%��&'13%���������"����7!���"�!��* ������7!� 6��(��+���,���/����� � ��"��������!� �+� �� �!�!�0! " �� �!1

�� �0��#�;3����! <� ��<

Stock prices Apple Inc. and Lehman Brothers Inc. in May 2008before the Bankruptcy of Lehman Brothers.

CSE: Financial Engineering Track

Financial Engineering

Financial Engineering combines any fields of mathematics andcomputer science:

Complicated Financial Structures: (CDOs)

Obligor 1 →Obligor 2 →

.

.

.

↓

Obligor m →

PortfolioBond 1Bond 2

.

.

.

↓

Bond n

Periodicpayments−→

←−Sp. payment

(Cash)

SPV

Periodic couponpayments−→

←−Initial cashinvestment

Super Senior TrancheLowest return/Residual loss

Senior Tranche

2nd lowest return/3rd ..% of loss

Mezzanine Tranche

2nd highest return/2nd ..% of loss

Equity TrancheHighest return/1st ..% of loss

Partial (Integro-)Differential Equations: (HeatEquation)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stochastic Processes: (Brownian Motion, Levyprocesses)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time

Pure drift

BM

Compound Poisson

Algorithms: (PSOR)

Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,

For i = 1, . . . , N ,

xk+1i :=

1

Aii

(bi −

i−1∑j=1

Aijxk+1j −

N∑j=i+1

Aijxkj

)xk+1i := max

{ci, x

ki + ω(xk+1

i − xki )}

Next i

If ‖xk+1 − xk‖2 < ε stop elseNext k

CSE: Financial Engineering Track

Financial Engineering

Financial Engineering combines any fields of mathematics andcomputer science:

Complicated Financial Structures: (CDOs)

Obligor 1 →Obligor 2 →

.

.

.

↓

Obligor m →

PortfolioBond 1Bond 2

.

.

.

↓

Bond n

Periodicpayments−→

←−Sp. payment

(Cash)

SPV

Periodic couponpayments−→

←−Initial cashinvestment

Super Senior TrancheLowest return/Residual loss

Senior Tranche

2nd lowest return/3rd ..% of loss

Mezzanine Tranche

2nd highest return/2nd ..% of loss

Equity TrancheHighest return/1st ..% of loss

Partial (Integro-)Differential Equations: (HeatEquation)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stochastic Processes: (Brownian Motion, Levyprocesses)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time

Pure drift

BM

Compound Poisson

Algorithms: (PSOR)

Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,

For i = 1, . . . , N ,

xk+1i :=

1

Aii

(bi −

i−1∑j=1

Aijxk+1j −

N∑j=i+1

Aijxkj

)xk+1i := max

{ci, x

ki + ω(xk+1

i − xki )}

Next i

If ‖xk+1 − xk‖2 < ε stop elseNext k

CSE: Financial Engineering Track

Financial Engineering

Financial Engineering combines any fields of mathematics andcomputer science:

Complicated Financial Structures: (CDOs)

Obligor 1 →Obligor 2 →

.

.

.

↓

Obligor m →

PortfolioBond 1Bond 2

.

.

.

↓

Bond n

Periodicpayments−→

←−Sp. payment

(Cash)

SPV

Periodic couponpayments−→

←−Initial cashinvestment

Super Senior TrancheLowest return/Residual loss

Senior Tranche

2nd lowest return/3rd ..% of loss

Mezzanine Tranche

2nd highest return/2nd ..% of loss

Equity TrancheHighest return/1st ..% of loss

Partial (Integro-)Differential Equations: (HeatEquation)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stochastic Processes: (Brownian Motion, Levyprocesses)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time

Pure drift

BM

Compound Poisson

Algorithms: (PSOR)

Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,

For i = 1, . . . , N ,

xk+1i :=

1

Aii

(bi −

i−1∑j=1

Aijxk+1j −

N∑j=i+1

Aijxkj

)xk+1i := max

{ci, x

ki + ω(xk+1

i − xki )}

Next i

If ‖xk+1 − xk‖2 < ε stop elseNext k

CSE: Financial Engineering Track

Financial Engineering

Financial Engineering combines any fields of mathematics andcomputer science:

Complicated Financial Structures: (CDOs)

Obligor 1 →Obligor 2 →

.

.

.

↓

Obligor m →

PortfolioBond 1Bond 2

.

.

.

↓

Bond n

Periodicpayments−→

←−Sp. payment

(Cash)

SPV

Periodic couponpayments−→

←−Initial cashinvestment

Super Senior TrancheLowest return/Residual loss

Senior Tranche

2nd lowest return/3rd ..% of loss

Mezzanine Tranche

2nd highest return/2nd ..% of loss

Equity TrancheHighest return/1st ..% of loss

Partial (Integro-)Differential Equations: (HeatEquation)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Stochastic Processes: (Brownian Motion, Levyprocesses)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

Time

Pure drift

BM

Compound Poisson

Algorithms: (PSOR)

Choose an initial guess x0 ≥ c.Choose ω ∈ (0, 1] and ε > 0.For k = 0, 1, 2,

For i = 1, . . . , N ,

xk+1i :=

1

Aii

(bi −

i−1∑j=1

Aijxk+1j −

N∑j=i+1

Aijxkj

)xk+1i := max

{ci, x

ki + ω(xk+1

i − xki )}

Next i

If ‖xk+1 − xk‖2 < ε stop elseNext k

CSE: Financial Engineering Track

Different Challenges

I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices

(Stochastic Processes)

I Default scenarios and BankruptcyI Dependence structure between stock prices

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Two independent Kou Processes

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

1.8Model for Asset Price Processes

Time

X 1t

X 2t

S 1t

S 2t

Path simulation of a two dimensional Kou model.

CSE: Financial Engineering Track

Different Challenges

I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices

(Stochastic Processes)I Default scenarios and Bankruptcy

I Dependence structure between stock prices0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Two independent Kou Processes

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

1.8Model for Asset Price Processes

Time

X 1t

X 2t

S 1t

S 2t

Path simulation of a two dimensional Kou model.

CSE: Financial Engineering Track

Different Challenges

I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices

(Stochastic Processes)I Default scenarios and BankruptcyI Dependence structure between stock prices

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Two independent Kou Processes

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

1.8Model for Asset Price Processes

Time

X 1t

X 2t

S 1t

S 2t

Path simulation of a two dimensional Kou model.

CSE: Financial Engineering Track

Different Challenges

I Modelling ChallengesI Modelling uncertain evolution of the market/stocks prices

(Stochastic Processes)I Default scenarios and BankruptcyI Dependence structure between stock prices

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3Two independent Kou Processes

Time

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.4

0.6

0.8

1

1.2

1.4

1.6

1.8Model for Asset Price Processes

Time

X 1t

X 2t

S 1t

S 2t

Path simulation of a two dimensional Kou model.

CSE: Financial Engineering Track

Different Challenges

I Mathematical ChallengesI Casting the stochastic models into a mathematical framework

(Stochastic Calculus)

I Relation between stochastic stock-price processes and value offinancial contracts

I To determine the value of financial contracts non-standardmethods are necessary (Monte Carlo, Finite Elements, FourierMethods)

00.511.522.533.544.55

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot s1

Spot s2

Op

tio

n p

rice

Value of an American Basket option in the 2-dimensional CGMYLevy Model.

CSE: Financial Engineering Track

Different Challenges

I Mathematical ChallengesI Casting the stochastic models into a mathematical framework

(Stochastic Calculus)I Relation between stochastic stock-price processes and value of

financial contracts

I To determine the value of financial contracts non-standardmethods are necessary (Monte Carlo, Finite Elements, FourierMethods)

00.511.522.533.544.55

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot s1

Spot s2

Op

tio

n p

rice

Value of an American Basket option in the 2-dimensional CGMYLevy Model.

CSE: Financial Engineering Track

Different Challenges

I Mathematical ChallengesI Casting the stochastic models into a mathematical framework

(Stochastic Calculus)I Relation between stochastic stock-price processes and value of

financial contractsI To determine the value of financial contracts non-standard

methods are necessary (Monte Carlo, Finite Elements, FourierMethods)

00.511.522.533.544.55

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot s1

Spot s2

Op

tio

n p

rice

Value of an American Basket option in the 2-dimensional CGMYLevy Model.

CSE: Financial Engineering Track

Different Challenges

I Mathematical ChallengesI Casting the stochastic models into a mathematical framework

(Stochastic Calculus)I Relation between stochastic stock-price processes and value of

financial contractsI To determine the value of financial contracts non-standard

methods are necessary (Monte Carlo, Finite Elements, FourierMethods)

00.511.522.533.544.55

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Spot s1

Spot s2

Op

tio

n p

rice

Value of an American Basket option in the 2-dimensional CGMYLevy Model.

CSE: Financial Engineering Track

Different Challenges

I Computational ChallengesI Computation in high dimension (up to 500 dim.) can be tricky

I Fast algorithms are crucial in financial markets (advancedcomputer languages are key)

00.5

11.5

22.5

3

0

0.5

1

1.5

2

2.5

3

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Spot s1

Delta of S1

Spot s2

00.5

11.5

22.5

3

0

0.5

1

1.5

2

2.5

3

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Spot s1

Delta of S2

Spot s2

Sensitivity (derivative wrt s1 and s2) of an American Basket optionin the 2-dimensional CGMY Levy Model.

CSE: Financial Engineering Track

Different Challenges

I Computational ChallengesI Computation in high dimension (up to 500 dim.) can be trickyI Fast algorithms are crucial in financial markets (advanced

computer languages are key)

00.5

11.5

22.5

3

0

0.5

1

1.5

2

2.5

3

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Spot s1

Delta of S1

Spot s2

00.5

11.5

22.5

3

0

0.5

1

1.5

2

2.5

3

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Spot s1

Delta of S2

Spot s2

Sensitivity (derivative wrt s1 and s2) of an American Basket optionin the 2-dimensional CGMY Levy Model.

CSE: Financial Engineering Track

Different Challenges

I Computational ChallengesI Computation in high dimension (up to 500 dim.) can be trickyI Fast algorithms are crucial in financial markets (advanced

computer languages are key)

00.5

11.5

22.5

3

0

0.5

1

1.5

2

2.5

3

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0

Spot s1

Delta of S1

Spot s2

00.5

11.5

22.5

3

0

0.5

1

1.5

2

2.5

3

−0.35

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Spot s1

Delta of S2

Spot s2

Sensitivity (derivative wrt s1 and s2) of an American Basket optionin the 2-dimensional CGMY Levy Model.

CSE: Financial Engineering Track

Prospects

Jobs/Career

I Academics (Universities and research institutes, PhD,Post-doc, etc.)

I Banking (Private banking, asset and risk management,financial product development, etc.)

I Investment services (Hedge funds, pension funds, privateequity, etc.)

I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)

I Other financial service industries (high frequency trading, etc)

I Consulting (Model validation, Regulation, etc.)

CSE: Financial Engineering Track

Prospects

Jobs/Career

I Academics (Universities and research institutes, PhD,Post-doc, etc.)

I Banking (Private banking, asset and risk management,financial product development, etc.)

I Investment services (Hedge funds, pension funds, privateequity, etc.)

I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)

I Other financial service industries (high frequency trading, etc)

I Consulting (Model validation, Regulation, etc.)

CSE: Financial Engineering Track

Prospects

Jobs/Career

I Academics (Universities and research institutes, PhD,Post-doc, etc.)

I Banking (Private banking, asset and risk management,financial product development, etc.)

I Investment services (Hedge funds, pension funds, privateequity, etc.)

I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)

I Other financial service industries (high frequency trading, etc)

I Consulting (Model validation, Regulation, etc.)

CSE: Financial Engineering Track

Prospects

Jobs/Career

I Academics (Universities and research institutes, PhD,Post-doc, etc.)

I Banking (Private banking, asset and risk management,financial product development, etc.)

I Investment services (Hedge funds, pension funds, privateequity, etc.)

I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)

I Other financial service industries (high frequency trading, etc)

I Consulting (Model validation, Regulation, etc.)

CSE: Financial Engineering Track

Prospects

Jobs/Career

I Academics (Universities and research institutes, PhD,Post-doc, etc.)

I Banking (Private banking, asset and risk management,financial product development, etc.)

I Investment services (Hedge funds, pension funds, privateequity, etc.)

I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)

I Other financial service industries (high frequency trading, etc)

I Consulting (Model validation, Regulation, etc.)

CSE: Financial Engineering Track

Prospects

Jobs/Career

I Academics (Universities and research institutes, PhD,Post-doc, etc.)

I Banking (Private banking, asset and risk management,financial product development, etc.)

I Investment services (Hedge funds, pension funds, privateequity, etc.)

I Insurance (life-insurance, non-life insurance, Re-insurance,etc.)

I Other financial service industries (high frequency trading, etc)

I Consulting (Model validation, Regulation, etc.)

CSE: Financial Engineering Track