option pricing using binomial trees

7/28/2019 Option Pricing Using Binomial Trees

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Call Option Binomial Trees

Shahed Ahmmed20041

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Example

A stock price is currently Tk20

In 3 months it will be either Tk22 or Tk18

Stock Price = Tk18

Stock Price = Tk22

Stock price = Tk20


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Stock Price = Tk18

Option Price = Tk0

Stock Price = Tk22

Option Price = Tk1

Stock price = Tk20

Option Price=?

Call Option

A 3-month call option on the stock has a strike

price of 21.


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In the Portfolio: long D sharesshort 1 call option

Portfolio is riskless when

22D 1 = 18D or

D = 0.25

22D 1

18D

How to make a Riskless Portfolio


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Delta

Delta (D) is the ratio of the change in the

price of a stock option to the change in

the price of the underlying stock

The value ofD varies from node to node


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Choosing u and d

One way of matching the volatility is to set

where s is the volatility and Dtis the length

of the time step. This is the approach used

by Cox, Ross, and Rubinstein

t

t

eud

eu

Ds

Ds

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Valuing the Portfolio

The riskless portfolio is:

long 0.25 shares

short 1 call option The value of the portfolio in 3 months is

22 0.25 1 = 4.50

The value of the portfolio today is4.5e 0.120.25 = 4.3670

(Risk-Free Rate is 12%)


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Valuing the Option

The portfolio that is

long 0.25 shares

short 1 option

is worth 4.367

The value of the shares is

5.000 (= 0.25 20 )

The value of the option is therefore0.633 (= 5.000 4.367 )


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Generalization

A derivative lasts for time Tand is dependent on a

stock

S0uu

S0dd

S0


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Generalization (continued)

Consider the portfolio that is long D shares and short 1derivative

The portfolio is riskless when

S0uDu = S0dDd or

dSuS

fdu

00

D

S0uDu

S0dDd


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Generalization (continued)

Value of the portfolio at time Tis

S0uDu

Value of the portfolio today is

(S0uDu)erT

Another expression for the portfolio

value today is S0Df

Hence = S0D (S0uDu)e

rT


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Generalization(continued)

Substituting forD we obtain

= [pu + (1p)d]erT

where

p e du d

rT


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p = Probability

It is natural to interpretp and 1-p as probabilities of up

and down movements

The value of a derivative is then its expected payoff in

a risk-neutral world discounted at the risk-free rate

S0u

u

S0d

d

S0


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Risk-Neutral Valuation

When the probability of an up and downmovements arep and 1-p the expected stock priceat time Tis S0e

rT

This shows that the stock price earns the risk-free

rate Binomial trees illustrate the general result that to

value a derivative we can assume that theexpected return on the underlying asset is the risk-free rate and discount at the risk-free rate

This is known as using risk-neutral valuation


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Example

Sincep is the probability that gives a return on thestock equal to the risk-free rate. We can find it from

20e0.12 0.25

= 22p + 18(1p )which givesp = 0.6523

Alternatively, we can use the formula

6523.09.01.1

9.00.250.12

e

du

dep

rT

S0u = 22

u = 1

S0d= 18d= 0

S0


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Valuing the Option Using

Risk-Neutral Valuation

The value of the option is

e0.120.25 (0.65231 + 0.34770)

= 0.633

S0u = 22

u = 1

S0d= 18

d= 0

S0


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Stocks Expected Return is not a factor

When we are valuing an option in terms

of the price of the underlying asset, the

probability of up and down movements inthe real world are irrelevant

This is an example of a more general

result stating that the expected return onthe underlying asset in the real world is

irrelevant


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Two step valuation of a Call Option

Value at node B is

e0.120.25(0.65233.2 + 0.34770) = 2.0257

Value at node A is

e0.120.25

(0.65232.0257 + 0.3477

0) = 1.2823

20

1.2823

22

18

24.2

3.2

19.8

0.0

16.2

0.0

2.0257

0.0

A

B

C

D

E

F


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Thank You.

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option pricing using binomial trees

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