Download Other binomial approaches –

Cox, John C., Ross, Stephen A., Rubinstein, Mark, (1979) “Option Pricing: A Simplified
Approach,” Journal of Financial Economics, vol 7. pp. 229-263.
Contention; the CRR binomial model contains the Black-Scholes model as a limiting
case, i.e. as the number of binomial steps in a period of calendar time tends to infinity.
Calendar time (0-t) divided into increments 0,1,2, ….,t-1,t. We will investigate the
properties of the binomial model as the number of increments (binomial steps), n.
Black-Scholes model: Given constant r, m, 
S t  S 0 exp(Wt  mt)
Bt  exp( rt )
Given the Black-Scholes assumptions, stock prices are distributed log-normal,
continuously compounded returns are distributed normal.
ln( S1 S 0 ) ~ N (m,  2 )
ln( S t S 0 ) ~ N (mt, t 2 )
S t S 0  S1 S 0 * S 2 S1 * .... * S t S t 1
t
ln( S t S 0 )   ln( S i S i 1 )
i 0
E ln( S t S 0 )  tm
VARln( S t S 0 )  t 2
Under the assumptions of the Black-Scholes model the continuously compounded rate of
return 0….t is distributed normal with mean, tu, and variance t2.
Binomial Model: n = #trials, j = 1(uptick) with probability=q, j = 0 (downtick) with
probability =(1-q).
For a sequence of n-trials, j = number of upticks is distributed binomial
E(j) = nq,
VAR(j) = nq(1-q)
S t  S 0 u j d n j
S t S 0  u j d n j
ln( S t S 0 )  j ln( u d )  n ln d
E ln( S t S 0 )  nln( u d )q  ln d   ̂ n
1
VARln( S t S 0 )  E ln( S t S 0 )  uˆ n 2  ˆ n2
ˆ n2  E j ln u  (n  j ) ln d  (ln( u d ) E ( j )  n ln d )2 
E j ln( u d )  ln( u d ) E ( j )2 
ln( u d ) 2 E  j  E ( j )
2
ˆ n2  nq(1  q) ln( u d ) 2
The first step in illustrating the contention above is to show that the first two moments of
the return distribution of the binomial process match the moments of the continuously
compounded return distribution as the number of binomial time steps increases.
lim n  
ˆ n  tm
ˆ n2  t 2
Select:
u  exp( t n)
d  exp(  t n)
q  1  1 (m  ) t n
2
2
Substituting for u, d, q in the mean and variance of the binomial return process using the
specified values for u,d,q;
ˆ n  mt
ˆ n2   2 t  m 2 t 2 n
Hence in the limit as n the first two moments of the Brownian motion and the
binomial process match. Alternative specifications of the steps/probability (u,d,q) give
rise to models of stock returns (price) with different characteristics.
The second step to illustrate the contention is to demonstrate that as n the terminal
distribution of returns produced by the proportional binomial process is normal
distributed.
It is natural to rely on the central limit theorem to make this argument. However the
Central Limit Theorem (CLT) that applies to identically and independently distributed
random variables is not appropriate in this setting.
The CLT applies to a sum of random variables as additional random variables are added
to the sum. This is essentially the circumstance for this application. However as n is
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increased, given the definition of (u,d,q), the distribution of each of the elements of the
sum is changed.
Lyapunov’s Third Formulation of the Central Limit Theorem;
Let x1,x2, . . .,xn be a sequence of independent random variables with means 1,2, . . .,n
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and variances ai = E(xi - i)2 and third absolute moments d i  E  xi   i  .


Then if in the limit as n the ratio defined below tends to zero then the distribution of
the summation of the standardized x’s tends to the standard normal distribution.
If in lim
n
(d1  d 2  ...  d n ) 2
(a1  a 2  ...  a n ) 3
 0 then


x  1  ...  xn   n
1 t
2
Pr  s  1
 t 
 exp(  x )dx
2(a1  ...  a n )
 s


Two levels of substitutions are necessary to check this condition:
xi  ln( Si Si 1 )  i  ˆ
ai  ˆ 2 and
uˆ  q ln( u d )  ln d ˆ  ln( u d ) q(1  q)
Substituting the specifications for (u,d,q), the ratio reduces with much algebra to
checking;
n
(1  q) 2  q 2
 0 which it does. Hence in the limit as n the
nq(1  q)
binomial proportional process produces normal distributed returns such that
If in lim
 ln( S t S 0 )  ˆ n

Pr 
 z   N ( z )
ˆ n


Each of the previous arguments illustrated a convergence of the terminal distribution of
the n-step binomial process with the normal distribution with mean and variance
equivalent to the mean and variance produced from the continuous return model
(arithmetic Brownian motion).
The third step to illustrate the contention is to impose the results derived from the
Arbitrage Theorem.
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In the single period – two state setting of Neftci Chapter 2 we discovered that under the
risk neutral measure the expected return on the underlying risky asset is the risk free
rate.
In the continuous environment of the Black-Scholes Model:
~
dSt  St dWt  rS t dt
then applying Ito’s Lemma
~
S t  S 0 exp(Wt  (r  12  2 )t )
Hence, under the risk neutral probability measure and the assumptions of the BlackScholes model the share price will have expected value, S 0 (r  12  2 )t at the end of
calendar period (0,t).
u  exp( t n)
d  exp(  t n)

p  12  12 ((ln r  12  2 )  ) t n
For reference:
rˆ  r
t
n
Eln( St S0 )  nln( u d ) p  ln d   ̂ pn
VARln( S t S 0 )  np(1  p) ln( u d ) 2  ˆ 2pn
Substitute (u,d,p) into definitions for ̂ pn , ˆ 2pn and examine in the limit as n. The
p-subscript denotes that the probability of an uptick has been selected to be consistent
with the risk neutral measure.
lim ˆ
 lim (ln rˆ  1  2 )t  (r  1  2 )t
n   pn
n
2
2
t
lim rˆ  lim r n  e r
n
n
lim ˆ 2   2 t
n   pn
The specifications of (u,d,q) produce the desired risk-neutral results.
To illustrate the equivalence between the binomial and Black-Scholes as n start with
the binomial model (CRR pg. 239):
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C  Sa; n, p   Kr  n a; n, p 
(r  d )
p   (u ) p
r
(u  d )
ln( K n )
Sd
a
u
ln( )
d
The complementary binomial distribution, [a;n,p] = Pr[j >= a], is the probability that
the sum of n independent binomial variable with probability of up-tick, p, will be greater
than a after n-trials. Illustration of the equivalence between models takes the limit of the
complement of [a;n,p], 1 - [a;n,p], to develop the equivalence of [a;n,p] and N(d2)
of the Black-Scholes model.
p
Other binomial approaches –
Jarrow, R.A. and A. Rudd, 1983 Option Pricing, Irwin, Englewood Cliffs, NJ.
Leisen, D.P.J. amd M. Reimer, 1996, “Binomial Moels for Option Valuation –
Examining and Improving Convergence,” Applied Mathematical Finance, 3, 319-346.
Risk neutral valuation:
We have explored the equivalent martingale method for derivative pricing and also
examined the central role of the replicating portfolio. Recall that for a European call
option with strike price, k, and expiration date T, and letting t=0, the value of the
derivative is equivalent to the expectation;
V0= e  r (T ) EQ ( Max ( ST  k ,0) 0 )
We utilized methods appropriate for martingale processes (justified by the Arbitrage
Theorem) that allowed us to specify the distribution of ST under the equivalent martingale
measure and take the necessary expectation.
A slightly different approach, risk neutral valuation, which essentially carries out the
same valuation steps was developed before the equivalent martingale method was well
understood.
Recall that the equivalent martingale measure is a probability measure that changes the
drift of the underlying stochastic processes such that all non-payout assets have an
instantaneous expected rate of return equal to the risk free rate.
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Suppose that we could specify the stochastic process for the primitive asset and
conditional on this specification know the type of distribution for the terminal (time T)
asset price. This is obviously possible we already know that a stock price that follows a
geometric Brownian motion will have a log-normal terminal distribution. For other
specifications of the stochastic process the solution to the Kolmogorov backward
equation is the probability density function for the terminal price.
Ingersoll Chp. 16 – An Introduction to Stochastic Calculus
Markov: The conditional distribution of X(t) given information up until  < t depends at
most on X(). The conditional distribution of X(t) is independent of all other information
in the filtration at .
If x evolves according to the Markov diffusion process defined above;
dx(t) =u(x,t)dt + (x,t)d(t) and f(x,t; x0, t0) is the probability density function for x at
time t conditional on x(t0) = x0, then f(x,t) satisfies the partial differential equations of
motion. They are the backward Kolmogorov equation
2 f
1 2

(
x
,
t
)
0 0
2
x02
 u ( x0 , t 0 )
f
f

0
x0 t 0
and the forward Kolmogorov (Fokker-Planck) equation
1
2
2
x 2

2

( x, t ) f 

u( x, t ) f   f  0 .
x
t
The backward equation describes how the probability density function, f, changes when
the initial time, t0, is allowed to change holding x and terminal time, t, constant. The
forward equation describes how the probability density function changes when the
terminal time, t, is allowed to vary and t0, x0 are held constant.
The risk neutral valuation approach proceeds along lines analogous to the equivalent
martingale method but relies on a heuristic argument that was later validated by the more
rigorous development of the equivalent martingale method.
At the time of its heyday the risk neutral valuation method utilized the then well accepted
partial differential equation approach to pricing derivative assets as a point of departure.
As we will see the partial differential equation approach is based on the specification of
the stochastic process for the primitive assets, specification of the derivative assets
stochastic process using Ito’s Lemma and the construction of a portfolio consisting of the
derivative, the underlying primitive asset and the risk free asset. Depending on the type
of portfolio constructed zero-investment or delta-hedged, the rate of return of the
portfolio necessary to prevent arbitrage opportunities is known. The resulting partial
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differential equation that the derivative asset value must satisfy is solved subject to the
boundary conditions appropriate for the derivative contract.
Heuristic argument of the risk neutral valuation method:
Because no assumptions about risk bearing behavior are required to develop the partial
differential equation valuation method the solution obtained by the partial differential
equation method must be valid in all preference (risk bearing) environments. One
possible preference structure is risk-neutral, i.e. decision makers require no compensation
for bearing risk. In a risk-neutral environment all assets (primitive and derivative) will
have an expected rate of return equal to the risk free rate.
If it is possible to identify the terminal distribution of the primitive assets price in the
risk-neutral world, taking the expectation of the derivative contract’s payoffs with respect
to this distribution and discounting at the risk free rate will produce the value of the
derivative contract. (Sounds suspiciously like the equivalent martingale method without
the rigorous proof of the self-financing quality of the replicating portfolio).
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