Andrey Itkin http://www.chem.ucla.edu/~itkin
Course web page http://www.chem.ucla.edu/~itkin/CompFinanceC ourse/rutgers_course.html
My email : itkin@chem.ucla.edu
Andrey Itkin, Math 612-02

• Why computational?
– New sophisticated models
– Performance issue
– Calibration
– Data issue and historical data
• Market demand for quant people
• Pre-requisites:
– Stochastic calculus and related math
– Financial models
– Numerical methods
– Programming
Result: CF - very complex subject
Andrey Itkin, Math 612-02

1.
Closed-form solutions (BS world, stochastic volatility and Heston world, interest rates and Vasichek and Hull-White world)
2.
Almost closed-form solutions – FFT, Laplace transform
3.
Traditional probabilistic solutions – binomial, trinomial and implied trees
4.
Modern solutions – finite-difference
5.
Last chance - world of Monte Carlo, stochastic integration
6.
Calibration – gradient optimizers, Levenberg-Marquardt
7.
Advanced optimization – pattern search
8.
Specificity of various financial instruments – exotics, variance products, complex payoffs.
9.
Programming issues: Design of financial software, Excel/VBA-C++ bridge, Matlab-C++ bridge
10.
Levy processes, VG, SSM.
Andrey Itkin, Math 612-02

Excerpt from Yuji Yamada’s course
Mathematics
(Basic stochastic calculus)
Engineering
(Numerical technique)
This course
Finance
(Derivative pricing
And hedging)
Andrey Itkin, Math 612-02

1. Short overview of stochastic calculus
2. All we have to know about Black-Scholes
3. Traditional approach – binomial trees
Andrey Itkin, Math 612-02

• Binomial trees are used to approximate the movements in the price of a stock or other asset
• In each small interval of time the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d
Andrey Itkin, Math 612-02

d t
S
Andrey Itkin, Math 612-02
Su
Sd

• We choose the tree parameters p , u , and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world
(from John Hull: ) – the expected value of the stock price
E(Q) = Se r d t = pSu + (1– p ) Sd
Log-normal process: var = S
2 e 2r d t (e
σ2 d t
-1 ) = E(Q 2 ) – [E(Q)] 2 s
2 d t = pu 2 + (1– p ) d 2 – [ pu + (1– p ) d ] 2
Andrey Itkin, Math 612-02

2 equations, 3 unknown. One free choice: u

1 d
Cox Ross Rubinstein
(CRR)
Andrey Itkin, Math 612-02 u
 e s d t d
 e
 s d t p a

 a
 d u
 e r d t d

• By Jarrow and Rudd u
 e
( r
 s 2
/ 2 ) d t
 s d t d
 e
( r
 s 2
/ 2 ) d t
 s p a

 a
 u
 e r d t d d

1
2 d t
Andrey Itkin, Math 612-02

• CRR – it leads to negative probabilities when
σ < |(r-q)√ d t|.
• Jarrow and Rudd – not as easy to calculate gamma and rho.
• If many time steps are chosen – low performance
Andrey Itkin, Math 612-02

An alternative exlanations. Single period binomial model
(
Excerpt from Yuji Yamada’s course ) t= 0 t= 1 uS p
(1 +r )
W
Stock S Bond
W
1 -p d< 1 +r<u dS (1 +r )
W
Portfolio
X
1
( uS ) =
D uS+ q (1 r )
W
X
0
=
D
S+ qW
X
1
( dS ) =
D dS+ q (1 r )
W
Andrey Itkin, Math 612-02

Single period binomial model
C
1
( uS )
 max( uS

K , 0 )
C
0
C
1
( dS )
 max( dS

K ,
• Compare with portfolio process
0 )
C
1
( uS )
C
1
( dS )


X
1
( uS )
 D uS
X
1
( dS )
 D dS
 q
( 1
 r )
W
 q
( 1
 r )
W
Two equations for two unknowns
Solve these equations for
D and q
Andrey Itkin, Math 612-02

C
1
( uS )
 D uS
C
1
( dS )
 D dS
 q
( 1
 r )
W
 q
( 1
 r )
W q
D 
W 
C
1
( uS )

C
1
( dS ) uS
 dS uC
1
(
( uS
1

)
 r )( u dC
1
 d
( dS
)
)
C

1
X
1 for each state
Comparison principle
C
0

X
0
 D
S
 q W
C
0

1
1
 r


1
 u r

 d d
C
1
( uS )
 u

( 1
 u
 d r )
C
1
( dS )


Andrey Itkin, Math 612-02

C
0

1
1
 r


1
 u r

 d d
C
1
( uS )
 u
 u
( 1

 d r )
C
1
( dS )


C
0

1
1
 r
 p C
1
( uS )
 q C
1
( dS )

 
1 ,

0 ,

0
It is (notationally) convenient to regard p and as probabilities
~
, : Risk neutral probability (real probability is irrelevant)
Andrey Itkin, Math 612-02

S
Multi-period binomial lattice model u
3
S
Stock price u
4
S
Call price u
4
S

K u
2
S u
3 dS u
3 dS

K uS u
2 dS udS u
2 d
2
S u
2 d
2
S

K ud
2
S dS ud
3
S 0 d
2
S d
3
S d
4
S
Andrey Itkin, Math 612-02
0
Finite number of one step models

• We know the value of the option at the final nodes
• We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate
Andrey Itkin, Math 612-02

Stock price uS
3
Call price
C
4
( uS
3
)
 max( uS
3

K , 0 )
S
3
C
3
( S
3
)

1
1
 r dS
3

C
3
( S
3
)
C
4
( uS
3
)

C
4
( dS
3
) ,

C
4
( dS
3
)
 max( dS
3

K ,

1
 u r

 d d
,

1

Apply one step pricing formula at each step, and solve backward until initial price is obtained.
0 )
C k

1
( S k

1
)

1
1
 r
 p C k
( uS k

1
)
 q C k
( dS k

1
)
 D k

1

C k
( uS k

1
) uS k

1

C k
 dS
( k dS

1 k

1
)
Andrey Itkin, Math 612-02

Multi-period binomial lattice model
• Perfect replication is possible
Market is complete
• Real probability is irrelevant
• Risk neutral probability dominates the pricing formula
Andrey Itkin, Math 612-02

Binomial Trees and Option Pricing
(Two Fundamental Theorem of Asset Pricing (FTAP))
•
1 st : T he no-arbitrage assumption implies there exists (at least) a probability measure Q called risk-neutral, or risk-adjusted, or equivalent martingale measure, under which the discounted prices are martingales
•
2 nd : Assuming complete market and no-arbitrage: there exists a unique risk-adjusted probability measure Q; any contingent claim has a unique price that is the discounted Q-expectation of its final pay-off
Andrey Itkin, Math 612-02

Binomial Trees and Option Pricing
(Cox-Ross-Rubinstein Formula)
• Cox-Ross-
Rubinstein
Formula :
• J is the set of integers between 0 and N:
• Risk-neutral probability:
Andrey Itkin, Math 612-02

Binomial Trees and Option Pricing
(Summary)
Andrey Itkin, Math 612-02

S
0
= 50; X = 50; r =10%; s
= 40%;
T = 5 months = 0.4167; d t = 1 month = 0.0833
The parameters imply u = 1.1224; d = 0.8909; a = 1.0084; p = 0.5076
Andrey Itkin, Math 612-02

Figure 18.3
79.35
0.00
70.70
0.00
62.99
0.64
62.99
0.00
56.12
2.16
56.12
1.30
50.00
4.49
50.00
3.77
50.00
2.66
44.55
6.96
44.55
6.38
39.69
10.36
39.69
10.31
35.36
14.64
31.50
18.50
Early Ex : 50 - 39 .
69

10 .
31
Continuati on Value : e

0 .
1

0 .
0833 (0.5073

6.38

0.4927

14.64)

10.36
Andrey Itkin, Math 612-02
44.55
5.45
35.36
14.64
28.07
21.93
89.07
0.00
70.70
0.00
56.12
0.00

Delta is calculated from the nodes at time d t
D 
2 .
16

6 .
96
56 .
12

44 .
55
 
0 .
41
Andrey Itkin, Math 612-02

Gamma is calculated from the nodes at time
2 d t
D
1

0
62 .
99
Gamma
.
64
=

D
3

1
.
77

50
 D
2
11 .
65
11 .
65

0 .
5 ( 62 .
99


0 .
24 ;
D
2

39
0
.
.
03
69 )

3 .
77

10 .
36
50

39 .
69
 
0 .
64
Andrey Itkin, Math 612-02

Theta is calculated from the central nodes at times 0 and 2 d t
3 .
77

4 .
49
 
4 .
3
0 .
1667
0
012
Andrey Itkin, Math 612-02

• We can proceed as follows
• Construct a new tree with a volatility of
41% instead of 40%.
• Value of option is 4.62
• Vega is
4 62

.

.
per 1% change in volatility
Andrey Itkin, Math 612-02

(Hull P.409)
Again we want to match the mean and standard deviation of price changes. Terms of higher order than d t are ignored Su u
 e s p u

3 d t d

1 / u d t
12 s 2 

 r
 s
2
2




1
6 p m

2
3 p d
 
S p p u m
S p d d t
12 s 2 

 r
 s
2
2




1
6
Sd
Equivalent to explicit FD of 1 st order
Andrey Itkin, Math 612-02

• Combine two steps of CRR: u
 e s
2 d t
, d
 e
 s
2 d t p p u d p m
 

 e e r d t
2 s d t
2

 e
 s e
 s d t
2 d t
2

 e s

 e
1
 s p d t
2 u d t
2


 p d e e
 s r d t
2 d t
2
Andrey Itkin, Math 612-02


2



2