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STAC70 Statistics & Finance I (Winter 2021)

Syllabus

Instructor: Leonard Wong (email: [email protected])

Lectures: Monday 9am-10am, Wednesday 12pm-2pm. Lectures are conducted via Bb Collaborate
on Quercus. All materials (lecture notes, assignments, etc) will be posted on Quercus.

Office hours: TBD (Bb Collaborate on Quercus)

Teaching assistant: Peng Liu (email: [email protected])

Online forum: You are encouraged to use Piazza to (anonymously) ask questions about the course
and discuss among yourselves. You can join using the following link:

piazza.com/utoronto.ca/winter2021/stac70h3

1 Outline

This course is an introduction to the basics of option pricing theory. Options, futures and other
derivatives are financial contracts that can be traded in the market for hedging and speculation, and
they play important roles in modern financial markets. Option pricing theory is about how to value
these contracts and how to manage their risks.

The main financial results in this course include:

• The Black-Scholes formula based on the ideas of no arbitrage and dynamic replication. We
will also see that the Black-Scholes(-Merton) model can be viewed as the limit of the binomial
model as the number of time periods tends to infinity.

• The martingale approach which puts option pricing on a rigorous probabilistic basis.

Along the way we will develop some theories of stochastic processes including martingale, Brow-
nian motion and stochastic integration. We try to offer a balanced treatment featuring financial
ideas, mathematical intuition and concrete examples. The R programming language will be used to
illustrate some of the concepts.

Upon completing the course, the student will have a good understanding of the basic ideas of
option pricing and some vocabularies of modern quantitative finance. The student will have a solid
background to tackle more advanced treatments on theory and implementation.

UTSC course website: https://utsc.calendar.utoronto.ca/course/STAC70H3

Textbook: We will use our own lecture notes.

Here are some books (from senior undergraduate to Master level) that may be consulted for alter-
native treatments:

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• Mathematical Finance: A Very Short Introduction by Mark Davis (2019).

• Mathematics of Finance: An Intuitive Introduction by Donald G. Saari (2019).

• Investment Science by David G. Luenberger (2013).

• Options, Futures, and Other Derivatives by John Hull (2017).

• The Concepts and Practice of Mathematical Finance by Mark S. Joshi (2008).

• Stochastic Calculus for Finance (two volumes) by Steve Shreve (2004).

• Probability Theory in Finance: A Mathematical Guide to the Black-Scholes Formula by Seán Di-
neen (2013).

• Stochastic Calculus and Financial Applications by J. Michael Steele (2010).

2 Grading scheme

• 30%: Assignments

• 25%: Midterm (2 hours) (around week 7, date to be confirmed)

• 45%: Final exam (3 hours)

Assignments (30%)

There are 5 assignments which will be announced on the course website. The assignments are to
be submitted online through Quercus. Your work may be handwritten (professionally scanned) or
typed (preferably using LATEX). Some assignment problems require R programming.

Late submission policy:

• 1 minute to 23 hours and 59 minutes: 20% penalty

• 24 hours to 1 day 23 hours 59 minutes: 40% penalty, and so on.

Midterm (25%)

The date and time will be announced later. If you are not able to attend the midterm for a valid
reason (e.g. medical), you must let me know as soon as possible. Then you will need to take a
make-up test (which may be conducted as an oral exam).

Final exam (45%)

The date and time will be announced later. It will cover all materials covered in the course (before
and after the midterm).

3 Tentative schedule

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Week Topic
1 Introduction
2 Single period models
3 Conditional expectation and martingale
4 Binomial model
5 Brownian motion
6 Properties of Brownian motion
7 Stochastic integration
8 Stochastic calculus
9 Black-Scholes-Merton model
10 Black-Scholes equation and formula
11 Greeks, hedging and volatility
12 Further topics and review

4 Important information

Accessibility

Students with diverse learning styles and needs are welcome in this course. In particular, if you have
a disability/health consideration that may require accommodations, please feel free to approach the
instructor and/or the UTSC AccessAbility Service as soon as possible. Enquiries are confidential. The
UTSC AccessAbility Services staff are available by appointment to assess specific needs, provide refer-
rals and arrange appropriate accommodations, at (416) 287-7560 or [email protected]

Religious accommodations

The University has a commitment concerning accommodation for religious observances. I will make
every reasonable effort to avoid scheduling tests, examinations, or other compulsory activities on re-
ligious holy days not captured by statutory holidays. According to University Policy, if you anticipate
being absent from class or missing a major course activity (like a test, or in-class assignment) due
to a religious observance, please let me know as early in the course as possible, and with sufficient
notice (at least two to three weeks), so that we can work together to make alternate arrangements.

Academic integrity

The University treats cases of cheating and plagiarism very seriously. The University of Toronto’s Code
of Behaviour on Academic Matters (http://www.governingcouncil.utoronto.ca/policies/
behaveac.htm) outlines the behaviours that constitute academic dishonesty and the processes for
addressing academic offences. Potential offences in papers and assignments include using someone
else’s ideas or words without appropriate acknowledgement, submitting your own work in more
than one course without the permission of the instructor, making up sources or facts, obtaining or
providing unauthorized assistance on any assignment. On tests and exams cheating includes using
or possessing unauthorized aids, looking at someone else’s answers during an exam or test, mis-
representing your identity, or falsifying or altering any documentation required by the University,
including (but not limited to) doctor’s notes.

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Course materials, including lecture notes

Course materials are provided for the exclusive use of enrolled students. Do not share them with
others. I do not want to discover that a student has put any of my materials into the public domain,
has sold my materials, or has given my materials to a person or company that is using them to earn
money.The University will support me in asserting and pursuing my rights, and my copyrights, in
such matters.

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STAC70 Statistics & Finance I

Assignment 5 Solutions

These problems cover sections 5.1 to 5.6 from the textbook.

1. (Exercise 4.18 from textbook: this is from chapter 4, but really relates to risk-neutral pricing

from chapter 5 ) Let a stock price be a geometric Brownian motion ( ) ( ) ( ) ( )dS t S t dt S t dW t  

and let r denote the interest rate. We define the market price of risk to be
r



 and the state

price density process to be   
21

2
( )21

2
( ) exp ( )

W t trt
t W t r t e e

 
  

 
      . (Note: think of

the process ( )t as the discount factor combined with the Radon-Nikodým derivative of the risk-

neutral measure . When you multiply a process ( )X t with ( )t , it is like looking at the

discounted process ( ) ( )D t X t under measure .)

a. Show that ( ) ( ) ( ) ( )d t t dW t r t dt     .

SOL: Let    2 21 12 2( ) ( ) ( ) ( )Y t W t r t dY t dW t r dt          , so that
( )

( )
Y t

t e

 

 ( )f Y t , where ( ) ( ) & ( )x x xf x e f x e f x e        . By Itō’s formula, we have

   

     

1
2

2
2 21 1 1

2 2 2

21
2

( ) ( ) ( ) ( ) [ , ]( )

( ) ( ) ( ) ( )

( ) ( )

d t f Y t dY t f Y t d Y Y t

t dW t r dt t dW t r dt

t dW t r

     

  

   

       

      212dt dt  ( ) ( ) ( )t dW t r t dt   

b. Let ( )X t denote the value of an investor’s portfolio when he uses a portfolio process Δ(t). From

the self-financing portfolio dynamics: ( ) ( ) ( )( ) ( ) ( ) ( ) ( )dX t rX t dt t r S t dt t S t dW t    

Show that ( ) ( )t X t is a martingale. (Hint: Show that the differential ( ) ( )t X t has no dt

term.)

SOL: By Itō’s product rule     ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )d t X t t dX t X t d t d t dX t      

   

  

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

t rX t dt t r S t dt t S t dW t X t t dW t r t dt

rX t dt t r S t dt t S t dW t t dW t r t dt

t rX t dt

    

   

         

        

 ( ) ( )( ) ( ) ( ) ( )t t r S t dt X t r t dt     

 
0

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( )

t S t t dt dW t

r t t S t dt dW t

r
r

 

  


 

    

      

 
   
 

( ) ( ) ( ) ( ) ( ) ( ) ( )t t S t dt dW t dW t     

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c. Let 0T  be a fixed terminal time. Show that if an investor wants to begin with some initial

capital (0)X and invest in order to have portfolio value ( )V T at time T , where ( )V T is a

given ( )T -measureable random variable, then he must begin with initial capital

 (0) ( ) ( )X T V T . In other words, the present value at time 0 of the random payment

( )V T at time T is  ( ) ( )T V T . This justifies calling ( )t the state price density process.

SOL: We have shown that any self-financing portfolio, defined by some process ( )t , has the

property that when it is multiplied by ( )t it becomes a martingale. Thus, for a particular self-

financing portfolio with value ( ) ( )X T V T at time T, we must have that (0) (0)X 

   ( ) ( ) | (0) ( ) ( )T X T T X T   . Since (0) 1  , we get that  (0) ( ) ( )X T V T ,

where (0)X is the initial capital needed for this portfolio.

2. (Exercise 5.3 from textbook)

According to the Black-Scholes formula, the value at time 0 of a European call on a stock whose

initial price is (0)S x is given by    (0, ) ( , ) ( , ) ,rTc x xN d T x Ke N d T x   where

21 1
( , ) log , ( , ) ( , )

2

x
d T x r T d T x d T x T

KT
 


  

  
      

  
. The stock is modeled as a

geometric Brownian motion with constant volatility 0  , the interest rate is constant r , the call

strike is K , and the call expiration time is T . This formula is obtained by computing the discounted

expected payoff of the call under the risk-neutral measure  (0, ) ( )rTc x e S T K
   

 

   212exp ( )rTe x W T r T K 

 
   

  
, where W is a Brownian motion under the risk-

neutral measure . In exercise 4.9(ii) the delta of this option is computed to be

 (0, ) ( , )xc x N d T x . This problem provides an alternative way to compute (0, )xc x .

a. We begin with the observation that if ( ) ( )h s s K

  , then
0 if

( )
1 if

s K
h s

s K


  


. If s K , then

( )h s is undefined, but that will not matter in what follows because ( )S T has 0 probability of

taking the value K . Using the formula for ( )h s , differentiate inside the expectation to obtain a

formula for (0, )
x

c x .

SOL:    (0, ) ( ) ( )rT rT
d d d

c x e S T K e h S T K
dx dx dx

             

     212( ) exp ( )rT S T K
rT

d
e I x W T r T K

dx

e

 

 
     

 


 ( )

exp ( )
S T K

I W T r

      
2 21 1

2 2( )
exp ( )

S T K
T I W T T  


     

  

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b. Show that the formula you obtained in a. can be rewritten as  ˆ(0, ) ( )xc x S T K  , where ˆ

is a probability measure equivalent to . Show that ˆ ( ) ( )W t W t t  is a Brownian motion

under ˆ .

SOL: From Girsanov’s theorem with    , we have that  212( ) exp ( )Z t W T T   is a

Radon-Nikodým process which defines a new measure ˆ , equivalent to , such that

ˆ ( ) ( )W t W t t  is a ˆ -BM. Thus
     

21
2( ) ( )

ˆexp ( )
S T K S T K

I W T T I 
 

     
   

 ˆ ( )S T K  . Note that ˆ ( ) ( )W t W t t  is a ˆ -BM iff ˆ( ) ( )Z t W t is a -BM. We have:

 

    

    

 

21
2

( ) exp ( ) ( ) ( ) ( )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

ˆ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

Z t W t t dZ t Z t dW t

d Z t W t Z t dW t W t dZ t dZ t dW t

Z t dW t dt W t Z t dW t Z t dW t dW t dt

dW t Z t dt

  

   

   

   

     

   ( )Z t dt   ( )dW t 

c. Rewrite ( )S T in terms of ˆ ( )W t and show that  
ˆ ( )ˆ ˆ( ) ( , )

W T
S T K d T x

T

  
     

  

 ( , )N d T x .

     

     

   

 

2 21 1
2 2

21
2

21
2

( ) ( )2

21
2( )

21
2

ˆ ˆSOL : ( ) exp ( )

ˆ ˆexp ( ) exp ( )

ˆ ˆexp ( ) ( ) log

log l

r T r T

r T

S T K x W T r T K

xe W T T K xe W T K

K K
W T W T r T

xxe

K
r T

x
Z Z

T

 

 

  

  

 

     

     

    
          

   

  
   

     
 
 
 

 
 

21
2

og

( , )

x
r T

K
N d T x

T


  
   

   
 
 
 

3. (Exercise 5.6 from textbook) Use the two-dimensional Lévy theorem (Theorem 4.6.5 in textbook) to

prove the two-dimensional Girsanov theorem (Theorem 5.4.1 in textbook, for d=2).

SOL: For 2d  dimensions, we have  12
0 0

( ) exp ( ) ( ) ( )
t t

Z t u d u u du     Θ W Θ

 2 21 11 1 2 2 1 22 2
0 0 0 0

exp ( ) ( ) ( ) ( ) ( ) ( )
t t t t

u dW u u dW u u du u du            

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   
1 2

2 21 1
1 1 1 2 2 2 1 22 2

0 0 0 0

( ) ( )

exp ( ) ( ) ( ) exp ( ) ( ) ( ) ( ) ( )
t t t t

Z t Z t

u dW u u du u du u dW u Z t Z t

              ,

where ( ) ( ) ( ) ( ), 1, 2
i i i i

dZ t t Z t dW t i   and  1 2( ) ( ) ( )dZ t d Z t Z t 

  1 2 2 1 1 2

2 1 2 2 1 1 2 1 1 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Z t dZ t Z t dZ t dZ t dZ t

t Z t Z t dW t t Z t Z t dW t dW t dW t

   

    

0

1 1 2 2
( ) ( ) ( ) ( ) ( ) ( )t Z t dW t t Z t dW t

  

We need to show that
0

( ) ( ) ( ) , 1, 2
t

i i i
W t W t u du i    are independent BM under . It is obvious

that [ , ]( ) [ , ]( ) , 1, 2
i i i i

d W W t d W W t dt i   and
1 2 1 2

[ , ]( ) [ , ]( ) 0d W W t d W W t  . By the multivariate

Lévy’s theorem, we only need to show that ( ), 1, 2
i

W t i  are individual -martingales, or equivalently,

that ( ) ( ), 1, 2
i

Z t W t i  are individual -martingales. For 1i  , we have  1( ) ( )d Z t W t 

  
   

  

1 1 1

1 1 1 1 1 2 2

1 1 1 1 2 2

1 2 1

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

Z t dW t W t dZ t dZ t W t

Z t dW t t dt W t t Z t dW t t Z t dW t

dW t t dt t Z t dW t t Z t dW t

dW t dW t t Z t dt

   

    

    

     
1
( ) ( )t Z t dt 

1 2
( ) ( ) ( ) ( )dW t dW t   

and similarly for 2i  .

4. (Exercise 5.10 from textbook: Chooser Option) Consider a model with a unique risk-neutral

measure and constant interest rate r . According to the risk-neutral pricing formula, for 0 t T  ,

the price at time t of a European call expiring at time T is  ( )( ) ( ) | ( )r T tC t e S T K t
   

 
,

where ( )S T is the underlying asset price at time T and K is the strike price of the call. Similarly,

the price at time t of a European put expiring at time T is  ( )( ) ( ) | ( )r T tP t e K S T t
   

 
.

Finally, because ( )
rt

e S t

is a martingale under , the price at time t of a forward contract for

delivery of one share of stock at time T in exchange for a payment of K at time T is

 ( ) ( ) ( )( ) ( ) | ( ) ( ) | ( ) ( )r T t rt rT r T t r T tF t e S T K t e e S T t e K S t e K                  .

Because    ( ) ( ) ( )S T K K S T S T K
 

     , we have the put-call parity relationship

    
 

( )

( )

( ) ( ) ( ) ( ) | ( )

( ) | ( ) ( )

r T t

r T t

C t P t e S T K K S T t

e S T K t F t

  

 

      
 

    

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Now, consider a date
0

t between 0 and T, and consider a chooser option, which gives the right at

time
0

t to choose to own either the call or the put.

a. Show that at time
0

t the value of the chooser option is

   0( )0 0 0 0( ) max 0, ( ) ( ) ( )
r T t

C t F t C t e K S t

 
     .

SOL: Since we can choose either the call or the put option at time
0

t , we would pick the one with the

highest value, i.e.      0 0 0 0 0 0 0max ( ), ( ) max ( ), ( ) ( ) ( ) max 0, ( )C t P t C t C t F t C t F t     

 0( )0 0( ) ( )
r T t

C t e K S t

 
   .

b. Show that the value of the chooser option at time 0 is the sum of the value of a call expiring at

time T with strike price K and the value of a put expiring at time
0

t with strike price 0
( )r T t

e K
 

.

SOL: By risk-neutral pricing 0
0 0

(0) ( ) ( ) | ( ) ( )
rtrT rT

V e V T e V T t e V t
                

    0 0 0 0 0
0

( ) ( )

0 0 0 0

( )

0

exp
exp

( ) ( ) ( ) ( )

(0; , ) (0; , )

rt r T t rt rt r T t

r T t

strike iry strike iry

e C t e K S t e C t e e K S t

C K T P e K t

 
      

 

                 

 

5. (Exercise 5.11 from textbook: Hedging a cash flow). Let ( ), 0 ,W t t T  be a Brownian motion

on a probability space  , , , and let ( ), 0 ,t t T  be the filtration generated by this

Brownian motion. Let the mean rate of return ( )t , the interest rate ( )R t , and the volatility ( )t be

adapted processes, and assume that ( )t is never zero. Consider a stock price process whose

differential is given by ( ) ( ) ( ) ( ) ( ) ( ), 0dS t t S t dt t S t dW t t T     . Suppose an agent must pay

a cash flow at rate ( )C t at each time t , where ( ), 0C t t T  is an adapted process. If the agent

holds ( )t shares of stock at each time t , then the differential of her portfolio value will be

 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )dX t t dS t R t X t t S t dt C t dt     . Show that there is a nonrandom value of

(0)X and a portfolio process ( ), 0 ,t t T   such that ( ) 0X T  almost surely.

(Hint: Define the risk-neutral measure and apply Corollary 5.3.2 of the Martingale Representation

Theorem to the process
0

( ) ( ) ( ) | ( ) , 0 ,
T

M t D u C u du t t T   
  

where ( )D t is the discount

process  
0

( ) exp ( )
t

D t R u du  )

SOL: The discounted total amount that the agent must pay by time T is
0

( ) ( )
T

D u C u du , which is an

( )T -measurable random variable. We need a portfolio ( )V t that generates exactly this discounted

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amount by time T, i.e. we need
0

( ) ( ) ( ) ( )
T

D T V T D u C u du  . Because every discounted self-

financing portfolio is a martingale under the risk-neutral measure , our target portfolio ( ) ( )D t V t is

also a martingale and we can use the martingale representation theorem (combined with its terminal

value) to get
0 0

( ) ( ) ( ) ( ) | ( ) (0) (0) ( ) ( )
T t

D t V t D u C u du t D V u dW u    
   

.

We can now define the portfolio ( )X t as the replicating, self-financing portfolio of the cashflow

minus the cashflow, i.e.    ( ) ( ) ( ) ( ) ( ) ( )d D t X t d D t V t D t C t dt  

( ) ( ) (0)D t X t D 

1

(0) ( ) ( ) (0)X D t V t D

 

1

0
(0) ( ) ( ) , 0

t

V D u C u du t T

    .

From the terminal condition ( ) ( )D T X T

0

(0) ( ) ( )X D T V T

 

0
( ) ( )

0
(0) ( ) ( )

t

D u C u du

t

V D u C u du



   

0

(0) (0) ( ) ( )
T

X V D u C u du   
  

. Moreover, we have that ( )X t satisfies

 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )dX t t dS t R t X t t S t dt C t dt     . To see verify this, note that

 
0

1 ( )
exp ( )

( ) ( )

t R t
d d R u du dt

D t D t

   
    

  

1
( ) ( ) ( )

( )
dX t d D t X t

D t

 
  

 

   
1 1 1

( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )

D t X t d d D t X t d D t X t d
D t D t D t

   
     

   

( )D t


( )

( )
( )

R t
X t

D t
  

 
( ) ( )

( ) ( )

( )

( )

1
( ) ( ) ( ) ( )

( )

1
( ) ( ) ( ) ( ) ( )

( )

( )
( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )[ ( ) ( )

t dW t

dW t t dt

t

t

dt d D t V t D t C t dt
D t

X t R t dt d D t V t C t dt
D t

t
X t R t dt t S t dW t C t dt

D t t S t

X t R t dt t t S t dW t t




 




  

   


   

    

 

 

 

( )

( )

( )

] ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

R t

t

dS t

dt C t dt

X t R t dt t t S t dW t t S t t R t dt C t dt

t S t t dt t S t dW t R t X t dt R t t S t dt C t dt

t dS t R t X t t S t dt C t dt

 

 

 

       

        

     

6. (Black-Scholes formula with continuous dividends) Consider an asset with constant drift ( ) ,

constant volatility ( ) , and constant continuous dividend yield ( )q , whose price dynamics are

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( ) ( ) ( ) ( ) ( )dS t q S t dt S t dW t    . Let the interest rate ( )r also be constant. Use risk-neutral

pricing to show that the price of a European call with strike price K and maturity T is given by

   0 0( , ) ( , ) ( , ) ,
qT rT

C S T S e N d T x Ke N d T x
 

 
  where

21 1
( , ) log , ( , ) ( , )

2

x
d T x r q T d T x d T x T

KT
 


  

  
       

  

(Hint: the asset price dynamics under the risk-neutral measure become

( ) ( ) ( ) ( ) ( )dS t r q S t dt S t dW t   , where ( )W t is a -Brownian motion)

SOL: From risk-neutral pricing, we know that  0( , )
rT

T
C S T e S K

   
 

  ( ) ( ) ( )T T
rT rT rT

T S K T S K T
e S K I e S I e K S K
  

 
          

Since the asset price follows ( ) ( ) ( ) ( ) ( )dS t r q S t dt S t dW t   under the risk neutral measure,

we can use the results from the usual Black-Scholes formula, with interest rate r r q   :

   

   

0 ( )

0

0

( , ) ( )

( , ) ( , )

( , ) ( , ) ,

T

rT rT

T S K T

rT r T rT

qT rT

C S T e S I e K S K

e S e N d T x Ke N d T x

S e N d T x Ke N d T x

 

 

 

 

     

  

 

where
2 21 1 1 1

( , ) log log
2 2

x x
d T x r T r q T

K KT T
 

 

      
            

      
and

( , ) ( , )d T x d T x T
 

  .

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