crrslides(2)

8/9/2019 crrslides(2)

1/19

Mathematical finance journal club

Option pricing: a simplified approach

John C. Cox, Stephen A. Ross and Mark Rubinstein

Journal of Financial Economics, 7:229-263, 1979.

From the conclusion. . .

[The] simple two-state process is really the essential

ingredient of option pricing by arbitrage methods.

. . . [It] is reassuring to find such simple economicarguments at the heart of this powerful theory.
mailto:[email protected]://www.ucalgary.ca/


2/19

Single period model

Ingredients:

Stockcurrent price S. Can move up to uS or down to dS over the period.

Bondcan be bought or sold, and increases by a factorof r over the period.

Optionthe current value (C) is to be determined. Theoption can be exercised at any point with payoff P(S)depending on the value of the stock at that time.
mailto:[email protected]://www.ucalgary.ca/


3/19

Single period model: first example

S = $50, moving to either S = $25 or S = $100.P(S

) = (S

50)

+(call option), and r = 1.25.

Now try this:

Write three call options at C each.

Buy two shares at $50 each.

Borrow $40 at 25%, to be paid back at the end.

Whatever happens youll end up with nothing. So the outlay

at the outset must be zero, requiring C = $20.


4/19

Single period model

A few things to note. . .

It is possible to reproduce the returns of the option usinga portfolio of stocks and bonds only.

We just needed to know P, S, u, d and r. We did not need to know the probabilities for the possible

movements in the stock.


5/19

Single period model: more generality

S uS prob. qdS prob. 1 q..

C

Cu = P(uS) prob. q

Cd = P(dS) prob. 1 q..

S+ B

uS+ rB prob. q

dS+ rB prob. 1

q..

Replication = Cu Cd(u d)S, B =

uCd dCu(u d)r .


6/19

Single period model: determining C

C

P(S). (The option can be exercised at any time.)

C S+ B. (The portfolio replicates the end value ofthe option.)

C max(P(S), S + B). (Must hold to avoid anarbitrage opportunity.)

Then

C = max(P(S), S+ B).


7/19

Single period model: risk-neutral probability

Set p =r d

u d, so that 1

p =

u r

u d.

Then

C = maxP(S),1

r(pCu + (1 p)Cd) .

(Notice that q is nowhere to be seen. . . )

If d < r < u then p (0, 1)a probability under which theexpected return on the stock is r:

p(uS) + (1 p)(dS) = rS.


8/19

Multi-period model

The same argument can be applied recursively.

Formula (6) applies only in the case of call options with nodividend payments, when early exercise is never optimal.

To prove this, write Cnj for the option value after n steps

and j up-steps, and similarly for Snj . Then

Cnj = maxP(Sn

j ),1

r(pCn+1j+1 + (1 p)Cn+1j ) .

If there are N steps in total, and the strike price is K,

then

Cnj

(Snj

KrnN)+.


9/19

Multi-period model

The same argument can be applied recursively.

Formula (6) applies only in the case of call options with nodividend payments, when early exercise is never optimal.

To prove this, write Cnj for the option value after n stepsand j up-steps, and similarly for S

nj . Then

Cnj = max

P(Snj ),

1

r(pCn+1j+1 + (1 p)Cn+1j )

.

If there are N steps in total, and the strike price is K,

then

Cnj (Snj KrnN)+.

M l d d l


10/19

Multi-period model

We can prove that Cnj (Snj KrnN)+, for j = 0, . . . , n,by induction. First note that it is true for n = N since

equality holds at that point. Now suppose it is true forn = k + 1. Well show that it holds for n = k.

If Skj KrkN 0 then the inequality clearly holds.Otherwise we can proceed:

rCkj pCk+1j+1 + (1 p)Ck+1j

p(Sk+1j+1

Krk+1N)+ + (1

p)(Sk+1j

Krk+1N)+

p(Sk+1j+1 Krk+1N) + (1p)(Sk+1j Krk+1N)= puSkj + (1 p)dSkj Krk+1N = r(Skj KrkN)

and since were in the case where this is positive, Ck

j (Skj KrkN)+ as required.

M l i i d d l


11/19

Multi-period model

The result is enough to show that, if r > 1,1r(pC

n+1j+1 + (1

p)Cn+1j ) > P(S

nj ) for all n < N, so that

early exercise is not rational.

Then we have (6): (for a European call in fact)

C0

0 =

1

rN

Nj=0

nj

pj

(1p)N

j

P(SN

j ).

If P(SNa ) > 0 P(SNa1), then

C00 = 1rN

Nj=a

Nj

pj(1p)NjujdNjS

1

rN

Nj=a

Nj

p

j

(1 p)N

j

K.

M l i i d d l i i i f l


12/19

Multi-period model: option pricing formula

The formula can we written:

C = S(a; N, p) KrN

(a; N, p),where

p =r du

d

, p =u

rp,

and a is the smallest integer greater than ln KSdN/ ln ud, and

(a; N, p) =

N

j=a

N

jpj(1

p)Nj.

S i 4 ki !


13/19

Section 4: making money!

If the market value for C is wrong, but the binomialmodel is right, then there is money to be made.

The safe way to do this is to trade a portfolio of options,stock and bonds, adusting the amounts of the latter two

in order to eliminate risk by replicating the option returns.

If you try to do this by adjusting the option investment,you are exposed to risk because of the wrong option

price.

S i 5 ki i h li i


14/19

Section 5: taking it to the limit

Take N steps up to time T: and write h = T /N. Keep T fixed, and vary N. Be careful about what happens to (a; N, p) as N.

This means defining u, d and r appropriately.

After N steps (using a probability q and writing U = ln uand D = ln d) we have

E[ln(S/S)] = (qU + (1 q)D)N =: Nand

V[ln(S/S)] = q(1 q)(UD)2

N =: 2

N.

S ti 5 t ki it t th li it


15/19


If we want to match these to empirical values of T and

2T (say), at least as N, we can set

U =

h, D =

h and q =1

2+

h

2.

(If we set U =

2h + 2h2 and D = U, and q = 12

+ h2U

then the correspondence is exact for all N.)

Now a central limit theorem tells us that (with provisos)

P

ln S/S N

N z

N(z).



16/19


Comparing the call option formula with B-S:

C = S(a; N, p) KrN(a; N, p)

compares with

C = SN(x) KrtN(x T),

where

x =

ln(S/Krt)

T +

T

2 .We need to show that (a; N, p) N(x)and (a; N, p) N(x T).



17/19


If j is a sum of N draws from

{1, 0

}(with probability p of

drawing 1 in each case), then

1 (a; N, p) = P[j a 1]

= P

j N pN p(1p) a 1N p

N p(1 p)

.

Butj N pN p(1p)

=ln(S/S) N

N, and

a 1 = ln K/SN D(U

D)

, (0, 1).



18/19


Then

1 (a; N, p) =

P

ln(S/S) N

N ln K/S N (UD)

N

.

Now, N (ln r 122)T and N T, and thedesired correspondence follows.

The section concludes with a brief discussion of the Poisson

distribution that results if u us bounded away from 1 as

N - corresponding to a jump process.

Section 6 adding di idends


19/19

Section 6: adding dividends

The effect of dividend payments is to reduce the future

stock value. If the payments are known functions of S,then the recursive procedure applies as before. The main

difference is that the payoff function P is evaluated, not

at the current value of S, but at a value that takes into

account future dividend payments. For example, if there are

to be payments ofS at various points between now andexpiry, then the choice whether to exercise or not should be

made with reference to P((1 )

S) rather than to P(S).

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