ch09 by Salim Kasap

Finance and Derivatives: Theory and Practice Sébastien Bossu and Philippe Henrotte

Chapter 9 Models for asset prices in continuous time

1. Continuously compounded interest rate 

The annual interest rate r[m] is fractioned m times when payments are split over the year into m equal fractions.

t (years) Cash flow Capital

1 m

0 r[ m ]

— K

m K1 m

2 m r[ m ]

× K0

r[ m ]   = K 1 +  m  

m K2 m

m m

…

×K1

r[ m ]

…

r   = K 1 + [ m ]  m  

…

=1 × K m −1 m

r[ m ]   K1 = K  1 +  m  

Chapter 9 Models for asset prices in continuous time

1. Continuously compounded interest rate (2) 

The equivalent annual compound rate for r[m] is: m

 r[ m ]  r = 1 +  −1 m   

r[m] < r: a fractioned rate underestimates its equivalent compound rate.

Chapter 9 Models for asset prices in continuous time

1. Continuously compounded interest rate (3) 



The annual rate r[c] is said to be continuous or continuously compounded when it is fractioned an infinite number of times. The equivalent compound rate is: m

 r[ c ]  r[ c ] r = lim 1 + −1 = e −1  m →+∞ m  

Hence, after one year the capital amounts to: Kexp(r[c]), and after T years: Kexp(r[c]T).

Chapter 9 Models for asset prices in continuous time

2. Models for the behavior of asset prices in continuous time 



The study of the historical time series of stock prices, which is the core focus of econometrics, leads to a satisfying mathematical model for the future behavior of asset prices. The graph on the next slides shows the daily historical prices of the S&P 500 index and their moving average over 50 days.

Chapter 9 Models for asset prices in continuous time

Figure p.143: S&P 500 index 2000 1750 1500

50-day average

1250 1000

S&P 500

750

Jan-04

Sep-03

May-03

Jan-03

Sep-02

May-02

Jan-02

Sep-01

May-01

Jan-01

Sep-00

May-00

Jan-00

500

Chapter 9 Models for asset prices in continuous time

2. Models for the behavior of asset prices in continuous time (2) 

The graph of the daily prices has two components: 



A general upward or downward trend, whose cycles are best observed on the moving average curve; Random variations around the trend, which add up positively or negatively to the latter.

Following this observation and denoting Xt the asset price at time t, mt the market trend, and Zt the random variation, we can decompose the daily change in asset price as:

X t +1 day − X t = (mt +1 day − mt ) + ( Z t +1 day − Z t )

Chapter 9 Models for asset prices in continuous time

2. Models for the behavior of asset prices in continuous time (3) 

Generally, over an infinitesimal time interval dt:

X t + dt − X t = (mt + dt − mt ) + ( Z t + dt − Z t ) 

Or: dX t = dmt + dZ t From this basic decomposition, we can derive a sensible model for the future behavior of Xt: 

The general trend m(t) = mt is a reasonably smooth, differentiable function of time. Denoting its firstorder dm derivative: µ = t t

we can write: dmt = μtdt + dZt. Hence: dXt = μtdt + dZt. Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.

Chapter 9 Models for asset prices in continuous time

2. Models for the behaviour of asset prices in continuous time (4) 



On the other hand, the random variations Z(t) = Zt constitute a fairly irregular, nondifferentiable function of time. Thus, we cannot express dZt with a derivative, and we must instead model it as a random variable. The classic distribution used here is the normal distribution.

To verify that this approach is sensible, we can simulate random series on a computer and compare the results with reallife examples. 

Example: dt = 1 day, X0 = 100, µ t = 3, dZt ~ N(0, 5)

Chapter 9 Models for asset prices in continuous time

Figure p.144: Simulated asset prices 300

250

200

150

100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Chapter 9 Models for asset prices in continuous time

3. Introduction to Stochastic Processes 



We now introduce in further detail the various random processes, also called stochastic processes, which are commonly used in finance to model the behavior of asset prices in continuous time. We have the usual representation of uncertainty as a universe Ω of all possible states of nature ω which can happen in the future with a given probability.

Chapter 9 Models for asset prices in continuous time

3.1. Definitions 



A stochastic process is a sequence (Xt)t ≥ 0 of random variables indexed by time t in [0, ∞). The process (Xt)t ≥ 0 follows a particular path in each state of Nature ω which is given by the function t  Xt (ω). The process (Xt)t ≥ 0 is said to be continuous if every path t  Xt (ω) is a continuous function of time for all states of Nature ω in Ω.

Chapter 9 Models for asset prices in continuous time

3.2. Standard Brownian motion 

A standard Brownian motion, also called standard Wiener process, is a continuous stochastic process (Wt)t ≥ 0 which satisfies: 1.

W0 = 0

For all 0 ≤ t < t', the increment variable D = Wt' – Wt follows a normal distribution with zero mean and standard deviation

t′ − t

For all 0 ≤ t1 < t2 ≤ t3 < t4, the increment variables and are independent D = W −W ∆ = W −W t2

Chapter 9 Models for asset prices in continuous time

Figure p.145: Brownian motion 0,6 0,4 0,2 0 0

0,2

0,4

0, 6

0,8

-0,2 -0,4 -0,6

Chapter 9 Models for asset prices in continuous time

3.2. Standard Brownian motion (2) Remarkable properties:  Brownian paths tend to exhibit alternate cycles above or below the time axis.  Brownian paths are continuous at every point in time but nowhere differentiable.

Chapter 9 Models for asset prices in continuous time

3.3. Generalized Brownian motion 

A generalized Brownian motion or Wiener process (Xt)t ≥ 0 is defined as: dX t = a dt + b dWt where a and b are constants and (Wt) t≥0 is a standard Brownian motion.

Chapter 9 Models for asset prices in continuous time

Figure p.146: Generalized Brownian 25 20 15 10 5 0 -5 adt b dW t adt+b dW

-10

-15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Chapter 9 Models for asset prices in continuous time

3.3. Generalized Brownian motion (2) 

To understand the definition of a generalized Brownian motion, we must consider adt and bdWt separately: 



adt is a general, deterministic upward or downward trend of the process called drift bdWt corresponds to a random variation around the general trend whose amplitude is controlled by the time interval dt

Integrating both sides of the relationship, we obtain a useful analytical expression for (Xt)t ≥ 0: Xt = X0 + at + bWt.

Chapter 9 Models for asset prices in continuous time

3.4. Geometric Brownian motion 



A geometric Brownian motion or Wiener process (Xt)t ≥ is defined as: 0 dX t = (aX t )dt + (bX t )dWt The financial interpretation of this definition becomes apparent when we divide both sides by Xt and write: X t + dt − X t = adt + bdWt Xt 

For a price process (Xt)t ≥ 0, this is the rate of return over an infinitesimal time interval dt. A geometric Brownian motion is thus particularly suited to model the behavior of asset prices through their returns.

Chapter 9 Models for asset prices in continuous time

4. Introduction to stochastic calculus 



An Ito process (Xt)t ≥ 0 is an even more general form of processes seen thus far, and is defined as: dX t = a(t , X t )dt + b(t , X t )dWt where a and b are now two ‘sufficiently smooth’ functions of two variables. Note that if a and b are constant (Xt)t≥0 is a generalized Brownian motion, and if a(t, Xt) = αXt and b(t, Xt) = βXt then (Xt)t≥0 is a geometric Brownian motion.

Chapter 9 Models for asset prices in continuous time

4. Introduction to stochastic calculus (2) ItoDoeblin Theorem:  Let (X ) be an Ito process; t t≥0  

f a ‘sufficiently smooth’ function of time and X; (Yt)t≥0 the stochastic process defined as Yt = f(t, Xt).

Then  (Y ) is an Ito process which satisfies: t t≥0

∂f 1 ∂ ² f   ∂f  ∂f dYt = df =  + a + b²  dt +  b ∂X 2 ∂X ²   ∂t  ∂X

  dWt 