Option Prices: numerical approach Lecture 4

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Option Pricesnumerical approachLecture 4
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Pricing1.Binomial Trees
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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

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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 20
Stock Price 18
probabilities cant be 50-50, unless you are
risk-neutral
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A Call Option

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

Stock Price 22 Option Price 1
Stock price 20 Option Price?
Stock Price 18 Option Price 0
if you were risk-neutral (and r0), you could say
that the option is worth 0.5501500
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• prob(22) has to be gt prob (18), because
  otherwise Utility function is linear

U(22)
U(20)
Expcted U(50 prob)
U(18)
22
18
20

• 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

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Setting Up a Riskless Portfolio

• Consider the Portfolio long D shares short
  1 call option
  
• Portfolio is riskless when 22D 1 18D or
  D 0.25

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Valuing the Portfolio

• risk-fre rate12 p.a. ---gt 3 quarterly ---gt
  disc. factorexp(-0.120.25)0.970446
• The riskless portfolio is
• long 0.25 shares short 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

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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 )

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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 !!! lets imagine a risk-neutral world
  ---gt derive risk-neutral probabilities

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Summing up... Movements in Time Dt

Su
p
S
1 p
Sd
12
Risk-neutral Evaluation
hyp risk-free rate12 p.a. t 3m
Su 22 ƒu 1

• Since p is a risk-neutral probability 20e0.12
  0.25 22p 18(1 p ) p 0.6523
• p is called the risk-neutral probability
• show simple_example.xls

p
S ƒ
Sd 18 ƒd 0
(1 p )
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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 Dt pu (1 p )d
• s2Dt pu 2 (1 p )d 2 pu (1 p )d 2
• A further condition often imposed is u 1/ d

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Tree Parameters for aNondividend Paying Stock

• When Dt is small a solution to the equations is

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The Complete Tree
S0u 4
S0u 3

S0u 2
S0u 2
S0u
S0u
S0
S0
S0
S0d
S0d
S0d 2
S0d 2
S0d 3
S0d 4
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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

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Valuing the Option

• The value of the option is e0.120.25
  0.65231 0.34770
• 0.633

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A Two-Step Example

• Each time step is 3 months

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Valuing a Call Option

• Value at node B e0.120.25(0.65233.2
  0.34770) 2.0257
• Value at node C 0
• Value at node A e0.120.25(0.65232.025
  7 0.34770)
• 1.2823

24.2 3.2
D
22
B
19.8 0.0
20 1.2823
2.0257
A
E
18
C
0.0
16.2 0.0
F
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Pricing2.Monte Carlo
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An Ito Process for Stock Prices(See pages 225-6)

• where m is the expected return s is the
  volatility.
• The discrete time equivalent is

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Monte Carlo Simulation

• We can sample random paths for the stock price by
  sampling values for e
• Suppose m 0.14, s 0.20, and Dt 0.01, then

see simple_example.xls
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Monte Carlo Simulation One Path (continued. See
Table 10.1)

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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

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A More Accurate Approach(Equation 16.15, page
407)
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Extensions

• When 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

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To Obtain 2 Correlated Normal Samples

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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.

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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