crr probability

7/27/2019 CRR probability

1/4

,(1 ).n nX X

World Applied Sciences Journal 21 (Mathematical Applications in Engineering): 84-87, 2013

ISSN 1818-4952

IDOSI Publications, 2013

DOI: 10.5829/idosi.wasj.2013.21.mae.99927

Corresponding Author: K. Bayram, Department of Computational and Theoretical Sciences,

International Islamic University Malaysia, Jalan Istana, Bandar Indera Mahkota,

25200 Kuantan, Pahang, Malaysia.

84

Random Cox-Ross-Rubinstein Model and Plain Vanilla Options

N.N. Ganikhodjaev and K. Bayram

Department of Computational and Theoretical Sciences,

Faculty of Science, International Islamic University Malaysia, Jalan Istana,

Bandar Indera Mahkota, 25200, Kuantan, Pahang, Malaysia

Abstract:In this paper we introduce and study random Cox-Ross-Rubinstein (CRR) model. The CRR model is

a natural bridge, overture to continuous models for which it is possible to derive the Black Scholes option

pricing formula. An attractive property of CRR model is that the binomial tree for geometric Brownian motion

is consistent with the standard Black-Scholes formula for European options in that no mismatch occurs if a

tree is used to price an American option where no closed form solutions exist, as well as several exotic options.

The usual CRR model is prescribed by pair of numbers (u, d), where udenotes the increase rate of the stock

over the fixed period of time and ddenotes the decrease rate, with 0< d< 1< u. We call the pair (u, d) an

environment of the CRR model. A pair (U, D ), where {U} and {D } are the sequences of independent,n n n nidentically distributed random variables with 0


2/4

[ (1 ) ]r t

u df e pf p f = +

r te d

pu d

=

World Appl. Sci. J., 21 (Mathematical Applications in Engineering): 84-87, 2013

85

Fig. 1: Stock shares of Rolls Royce for period from

6 April to 28 June 2011th th

certain probability. Let us call the pair (u, d) environment

of CRR model. Note that in this model pair (u, d) is the

same for any moment of time.

Now we define random environment and random CRR

model as follows. Let {U} and {D } be the sequences ofn nindependent, identically distributed random variables with

0


3/4

2 2 2[ 2 (1 ) (1 ) ]r t uu ud dd f e p f p p f p f = + +

1 1 2 1 22 2[ (1 ) ]r tu u u u d f e p f p f

= +

1 1 2 1 22 2[ (1 ) ]r td d u d d f e p f p f = +

1 2 1 2

1 2 1 2

21 2 1 2

1 2 1 2

( , ) [ (1 )

(1 ) (1 )(1 ) ]

r tu u u d

d u d d

f H T e p p f p p f

p p f p p f

= + +

+

2 2f= ( , ) 2 (1 )( , ) (1 ) ( , )f H H H T f T T+ +

0

0

(1 ) max( ,0)

NrT j N j j N j

j

Ne p p S u d K

j

=

1 11 1

1 2 21

0 0

1 10 2 21 1

( 1)!( (1 ) (1 )

( 1 )! !

max( ,0)

NjrT N j k k

j k

j N j k k

Ne p p p p

N j j

S u d u d K

= =


86

Then we define option price f for simplest random Then the option price ffor simplest random two-step

single-period binomial tree model as expectation of

random variable f: R, i.e.,1

f= f(H) + (1- )f(T) (4)

Example 1:Let us consider simplest CRR model with two

environment u = 1.1, d=0.9 and u = 1.4, d=0.7. If a stock1 1 2 2price is currently $50 and the risk-free interest rate is 6%

per annum with continuous compounding, find the value

of 1 year European put option with a strike price of $52

for simplest random single-period binomial tree model.

We consider the probability of turning out of the

first environment as =0.9 and of the second 1- = 0.1.

Such as the second environment has high u and low d,

this kind of situation indicates to a crisis or too good

economical state, which happens seldom. So the price of

an option calculated by using two environments,considering casual and extraordinary situations of market,

is more realistic one. Using formulas (2-4) one can

compute that

(H)=1.2585; (T)=7.7344and= (H) + (1- )(T)=1.906

Options On Two Period Binomial Tree:In this caseN=2

and ={(H,H);(H,T),(T,H);(T,T)} and random2environments (u , d) and (u , d) correspond to outcomes1 1 2 2Head and Tail respectively.

For outcomes (H,H) and (T,T) the value if

corresponding options one can compute directly using

well known formula for the case of two time steps of usual

binomial tree model

(5)

wheref the value of the stock after two up movements,uuf after one up and one down and f after two downud dd movements. For outcome (H, T) repeated application of

equation (2) gives

(6)

(7)

Substituting from equations (6) and (7) into (2),

we get

(8)

and it is evident thatf(H,T)=f(T,H).

binomial tree modelis defined as expectation of random

variable f: R,2

Example 2:Now let us look at example1 in a two steps

model. Using formulas (5) and (8) one can compute that

f(H,H)=4.1791; (H,T) = (T,H) =6.3569; (T,T)=6.4002

and using formula (9) the value of our option will be

= (H,H) + 2 (1- )(H,T) + (1- ) (T,T)=5.16932 2

Options On N-Period Binomial Tree:Now suppose that

a tree withN- time steps is used to value a European call

option with strike priceKand life T. Each step is of length

T/N. For usual binomial tree model, if there have been

j upward movements and N-j downward movements on

the tree, the final stock price is S u d , where u is the0j N-j

proportional up movement, d is the proportional down

movement and S is the initial stock price. Then option0pricefis computed as follows [3, 4].

(9)

Now consider simplest CRR model where the

randomness of environment is defined by tossing a coin(non necessarily fair)Ntimes. Let be a sample spaceNwith outcome = ( , ,, ), where {H,T}, i =1 2 N i1,2,,N, observing Heads or Tails. Let Pr{H}= and

Pr{T}= 1- , where 0 1. For any outcome letNM( ) be the number of tails in tossing of coin N times.

IfM( ) =0 orM( )=N, then one can compute option price

for European call using previous formula. Now we show

how to compute option price for 0


4/4

1

11

0 0

12 2 0 2 21 1

( )!( (1 )

( )! !

(1 ) max( ,0),

N mjrT N m j

j k

j N jk m k k m k

N me p p

N m j j

p p S u d u d K

= =

Pr( ) = (1 )N m mmN

Am

0= (1 )

N N m mmm

Nf f

m

=


87

Since in the case of random binomial tree models we have

(10) and Peteritis [9]. Also it is interesting problem to derive

where p and p can be computed using formula (3) for coming papers.1 2corresponding environments (u , d) and (u , d)1 1 2 2respectively. ACKNOWLEDGMENTS

LetA = { ;M( )=m This work was supported by the IIUM Grant EDW Bm NIt is well known that 12-403-0881.

REFERENCES

Now option price ffor simplest random N-period Economics, 7: 229-263.

CRR tree model we define as expectation of random

variable f: R, i.e.,N

(11)

CONCLUSION

One way of deriving the famous Black-Scholes-

Merton result for valuing a European option on a

non-dividend-paying stock is by allowing the number

of time steps in a binomial tree to approach infinity.Above we have introduced random binomial tree model

and derived formula for computing European call options

for such models. Similarly one can define and study other

options.

To derive formula for the valuation of a

European call option for random binomial tree models that

similar the Black-Scholes-Merton formula for usual

binomial tree models, we have to investigate this model by

allowing the number of time steps to approach infinity.

doubled randomness we need to apply and develop

methods suggested by Solomon [8] and Menshikov

the Black-Scholes-Merton differential equation withrandom coefficients. All these problems will be subject of

1. Cox, J.C., S.A. Ross and M. Rubinstein, 1979. Option

Pricing: A Simplified Approach. Journal of Financial

2. Rendleman, R.J. and B.J. Bartter, 1979. Two-State

Option Pricing. Journal of Finance, 24: 1093-1110.

3. Hull, J.C., 2011. Options, Futures and Other

Derivatives. 8 Edn., Pearson.th

4. Pascucci, A., 2011. PDE and Martingale Methods in

Option Pricing. Bocconi Iniversity Press, Springer.

5. Black, F. and M. Scholes, 1973. The Pricing of

Options and Corporate Liabilities. J. Political

Economy, 81: 637-654.

6. Georgiadis, E.,2011. Binomial Options Pricing Has No

Closed-Form Solution. Forthcoming in Algorithmic

Finance, pp. 1-6, http://algorithmicfinance.org.7. Merton, R.C., XXXX. Theory of rational option

pricing. Bell J. Econom. and Management Sci.,

4: 141-183.

8. Solomon, F., 1975. Random walks in a random

environment. The Annals of Probability, 3(1): 1-31.

9. Menshikov, N. and D. Peteritis, XXXX. Random

walks in random environment on trees and

multiplicativechaos. arXiv:math/0112103v1 [math.PR]

pp: 1-30.

crr probability

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