4.1

Option Prices:numerical approach

Lecture 4

4.2

Pricing:1.Binomial Trees

Binomial Trees

• Binomial trees are frequently used to approximate the movements in the price of a stock or other asset

• In each small interval of time the stock price is assumed to move up

by a proportional amount u or to move down by a

proportional amount d

4.4

A Simple Binomial Model

• A stock price is currently $20

• In three months it will be either $22 or $18

Stock Price = $22

Stock Price = $18

Stock price = $20

probabilities can’t be 50%-50%, unless you are risk-neutral

4.5

Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A Call Option

A 3-month call option on the stock has a strike price of 21.

if you were risk-neutral (and r=0), you could say that the option is worth: 0.5=50%*1$+50%*$0

4.6

18 2220

U(18)

U(22)U(20)

Expcted U(50% prob)

• prob(22) has to be > prob (18), because otherwise Utility function is linear

• Then, in order to know prob we need to know the Utility function. But this is an impossible task, and we have to find a shortcut .... i.e. we have to find a way of “linearizing” the world

4.7

• Consider the Portfolio: long sharesshort 1 call option

• Portfolio is riskless when 22– 1 = 18 or = 0.25

22– 1

18

Setting Up a Riskless Portfolio

4.8

Valuing the Portfolio• risk-fre rate=12% p.a. ---> 3% quarterly ---> disc.

factor=exp(-0.12*0.25)=0.970446

• The riskless portfolio is:

long 0.25 sharesshort 1 call option

• The value of the portfolio in 3 months is 220.25 – 1 = 4.50

• Note that this pay-off is deterministic, so its PV is obtained by simple discounting

4.9

Valuing the Option• The value of the portfolio today is

4.5e – 0.120.25 = 4.3670

• The portfolio that is

long 0.25 shares short 1 option

is worth 4.367

• The value of the shares is 5.000 (= 0.2520 )

• The value of the option is therefore 0.633 (= 5.000 – 4.367 )

4.10

Valuing the Option• note that the value of the option has been

obtained without knowing the shape of the utility function

• but if the solution is independent of preferences functional form, then it is valid also for all utility function

• Then, it is valid also for risk-neutral preferences .....

• ... eureka !!! let’s imagine a risk-neutral world ---> derive risk-neutral probabilities

Summing up... Movements in Time t

Su

Sd

S

p

1 – p

4.12

Risk-neutral Evaluation

• Since p is a risk-neutral probability20e0.12 0.25 = 22p + 18(1 – p ); p = 0.6523

• p is called the risk-neutral probability• show simple_example.xls

Su = 22 ƒu = 1

Sd = 18 ƒd = 0

S ƒ

p

(1– p )

hyp: risk-free rate=12% p.a.; t = 3m

Tree Parameters for aNondividend Paying Stock

• We choose the tree parameters p, u, and d so that the tree gives correct values for the mean & standard deviation of the stock price changes in a risk-neutral world

er t = pu + (1– p )d

2t = pu 2 + (1– p )d 2 – [pu + (1– p )d ]2

• A further condition often imposed is u = 1/ d

Tree Parameters for aNondividend Paying Stock

• When t is small a solution to the equations is

u e

d e

pa d

u d

a e

t

t

r t

The Complete Tree

S0

S0u

S0d S0 S0

S0u 2

S0d 2

S0u 2

S0u 3 S0u 4

S0d 2

S0u

S0d

S0d 4

S0d 3

Backwards Induction

• We know the value of the option at the final nodes

• We work back through the tree using risk-neutral valuation to calculate the value of the option at each node, testing for early exercise when appropriate

4.17

Valuing the Option

The value of the option is e–

0.120.25 [0.65231 + 0.34770]

= 0.633

Su = 22 ƒu = 1

Sd = 18 ƒd = 0

Sƒ

0.6523

0.3477

4.18A Two-Step Example

• Each time step is 3 months

20

22

18

24.2

19.8

16.2

4.19Valuing a Call Option

• Value at node B = e–0.120.25(0.65233.2 + 0.34770) = 2.0257

• Value at node C =0• Value at node A

= e–0.120.25(0.65232.0257 + 0.34770)

= 1.2823

201.2823

22

18

24.23.2

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F

4.20

Pricing:2.Monte Carlo

An Ito Process for Stock Prices(See pages 225-6)

where is the expected return is the volatility.

The discrete time equivalent is

dS Sdt Sdz

S S t S t

Monte Carlo Simulation

• We can sample random paths for the stock price by sampling values for

• Suppose = 0.14, = 0.20, and t = 0.01, then

S S S 0 0014 0 02. .

see simple_example.xls

Monte Carlo Simulation – One Path (continued. See Table 10.1)

PeriodStock Price atStart of Period

RandomSample for

Change in StockPrice, S

0 20.000 0.52 0.236

1 20.236 1.44 0.611

2 20.847 -0.86 -0.329

3 20.518 1.46 0.628

4 21.146 -0.69 -0.262

Monte Carlo Simulation When used to value European stock options, this

involves the following steps:

1. Simulate 1 path for the stock price in a risk neutral world

2. Calculate the payoff from the stock option

3. Repeat steps 1 and 2 many times to get many sample payoff

4. Calculate mean payoff

5. Discount mean payoff at risk free rate to get an estimate of the value of the option

A More Accurate Approach(Equation 16.15, page 407)

Use

The discrete version of this is

or

e

d S dt dz

S S S t t

S S St t

ln /

ln ln( ) /

/

2

2

2

2

2

2

ExtensionsWhen a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative

To Obtain 2 Correlated Normal Samples

Obtain independent normal samples

and and set

A procedure known a Cholesky's

decomposition can be used when

samples are required from more than

two normal variables

1 2

1 1

2 1 221

x

x x

Standard Errors

The standard error of the estimate of the option price is the standard deviation of the discounted payoffs given by the simulation trials divided by the square root of the number of observations.

Application of Monte Carlo Simulation

• Monte Carlo simulation can deal with path dependent options, options dependent on several underlying state variables, & options with complex payoffs

• It cannot easily deal with American-style options