Option Pricing and Dynamic Modeling of Stock Prices

1
Option Pricing and Dynamic Modeling of Stock
Prices

• Investments 2003

2
Motivation

• We must learn some basic skills and set up a
  general framework which can be used for option
  pricing.
• The ideas will be used for the remainder of the
  course.
• Important not to be lost in the beginning.
• Option models can be very mathematical. We/I
  shall try to also concentrate on intuition.

3
Overview/agenda

• Intuition behind Pricing by arbitrage
• Models of uncertainty
• The binomial-model. Examples and general results
• The transition from discrete to continuous time
• Pricing by arbitrage in continuous time
• The Black-Scholes model
• General principles
• Monte Carlo simulation, vol. estimation.
• Exercises along the way

4
Here is what it is all about!

• Options are contingent claims with future
  payments that depend on the development in key
  variables (contrary to e.g. fixed income
  securities).

Value(0)?
?
0
T
Værdi(T)ST-X
5
The need for model-building

• The Payoff at the maturity date is a
  well-specified function of the underlying
  variables.
• The challenge is to transform the future value(s)
  to a present value. This is straightforward for
  fixed income, but more demanding for derivatives.
• We need to specifiy a model for the uncertainty.
• Then pricing by arbitrage all the way home!

6
Pricing by arbitrage - PCP
Therefore C P S0 PV(X) ..otherwise there
is arbitrage!
7
Pricing by arbitrage

• So if we know price of
• underlying asset
• riskless borrowing/lending (the rate of interest)
• put option
• then we can uniquely determine price of otherwise
  identical call
• If we do not know the put price, then we need a
  little more structure......

8
The Worlds simplest model of undertainty the
binomial model

• Example Stockprice today is 20
• In three months it will be either 22 or 18
  (-10)

Stockprice 22
Stockprice 20
Stockprice 18
9
A call option

• Consider 3-month call option on the stock and
  with an exercise price of 21.

Stockprice 22 Option payoff 1
Stockprice 20 Option price?
Stockprice 18 Option payoff 0
10
Constructing a riskless portfolio

• Consider the portfolio long D stocks short
  1 call option
  
• The portfolio is riskless if 22D 1 18D ie.
  when D 0.25.

11
Valuing the portfolio

• Suppose the rate of interest is 12 p.a.
  (continuously comp.)
• The riskless portfolio was
• long 0.25 stocks short 1 call option
• Portfolio value in 3 months is 220.25 1
  4.50.
• So present value must be 4.5e 0.120.25
  4.3670.

12
Valuing the option

• The portfolio which was
• long 0.25 stocks short 1 option
• was worth 4.367.
• Value of stocks 5.000 ( 0.2520 ).
• Therefore option value must be 0.633 ( 5.000
  4.367 ),
• ...otherwise there are arbitrage opportunities.

13
Generalization

• A contingent claim expires at time T and payoff
  depends on stock price

14
Generalization

• Consider portfolio which is long D stocks and
  short 1 claim
• Portfolio is riskless when S0uD u S0d D d
  or
• Note ? is the hedgeratio, i.e. the number of
  stocks needed to hedge the option.

S0 uD u
S0 f
S0dD d
15
Generalization

• Portfolio value at time T is S0u D u.
  Certain!
• Present value must thus be (S0u D u )erT
• but present value is also given as S0D f
• We therefore have S0D (S0u D u )erT

16
Generalization

• Plugging in the expression for D we get
• q u (1 q )d erT
• where

17
Risk-neutral pricing

• q u (1 q )d e-rT e-rT EQfT
• The parameters q and (1 q ) can be interpreted
  as risk-neutral probabilities for up- and
  down-movements.
• Value of contingent claim is expected payoff wrt.
  q-probabilities (Q-measure) discounted with
  riskless rate of interest.

q
(1 q )
18
Back to the example
S0u 22 u 1

• We can derive q by pricing the stock
• 20e0.12 0.25 22q 18(1 q ) q 0.6523
• This result corresponds to the result from using
  the formula

q
S0
S0d 18 d 0
(1 q )
19
Pricing the option

• Value of option is
• e0.120.25 0.65231 0.34770 0.633.

20
Two-period example

• Each step represents 3 months, dt0.25

21
Pricing a call option, X21

• Value in node B e0.120.25(0.65233.2
  0.34770) 2.0257
• Value in node A e0.120.25(0.65232.0257
  0.34770)
• 1.2823

24.2 3.2
22
B
19.8 0.0
20 1.2823
2.0257
A
18
C
0.0
16.2 0.0
f e-2r?tq2fuu 2q(1-q)fud (1-q)2fdd
e-2r?t EQfT
22
General formula
23
Put option X52
u1.2, d0.8, r0.05, dt1, q0.6282

24
American put option early exercise

Node C max(52-40, exp(-0.05)(q4(1-q)20))
9.4636
25
Delta

• Delta (D) is the hedge ratio,- the change in the
  option value relative to the change in the
  underlying asset/stock price
• D changes when moving around in the binomial
  lattice
• It is an instructive exercise to determine the
  self-financing hedge portfolio everywhere in the
  lattice for a given problem.

26
How are u and d chosen?

• There are different ways. The following is the
  most common and the most simple
• where s is p.a. volatility and dt is length of
  time steps measured in years. Note u1/d. This is
  Cox, Ross, and Rubinsteins approach.

27
Few steps gt few states. A coarse model
28
Many steps gt many states. A fine model
29
Call, S100, s0.15, r0.05, T0.5, X105
30
(No Transcript)
31
Exercise
32
Alternative intertemporal models of uncertainty

• Discrete time discrete variable (binomial)
• Discrete time continuous variable
• Continuous time discrete variable
• Continuous time continuous variable
• All can be used, but we will work towards the
  last type which often possess the nicest
  analytical properties

33
The Wiener Process the key element/the basic
building block

• Consider a variable z, which takes on continuous
  values.
• The change in z is ?z over time interval of
  length ?t.
• z is a Wiener proces, if
• 1.
• 2. Realization/value of ?z for two
  non-overlapping periods are independent.

34
Properties of the Wiener process

• Mean of z (T ) z (0) is 0.
• Variance of z (T ) z (0) is T.
• Standarddeviation of z (T ) z (0) is

A continuous time model is obtained by letting ?t
approach zero. When we write dz and dt it is to
be understood as the limits of the corresponding
expressions with ?t and ?z, when ?t goes to zero.
35
The generalized Wiener-process

• The drift of the standard Wiener-process (the
  expected change per unit of time) is zero, and
  the variance rate is 1.
• The generalized Wienerprocess has arbitrary
  constant drift and diffusion coefficients, i.e.
• dxadtbdz.
• This model is of course more general but it is
  still not a good model for the dynamics of stock
  prices.

36
Ito Processes

• The drift and volatility of Ito processes are
  general functions
• dxa(x,t)dtb(x,t)dz.
• Note What we really mean is
• where we let dt go to zero.
• We will see processes of this type many times!
  (Stock prices, interest rates, temperatures etc.)

37
A good model for stock prices

• where m is the expected return and s is the
  volatility. This is the Geometric Brownian Motion
  (GBM).
• The discrete time parallel

38
The Lognormal distribution

• A consequence of the GBM specification is
• The Log of ST is normal distributed, ie. ST
  follows a log-normal distribution.

39
Lognormal-density

40
Monte Carlo Simulation

• The model is best illustrated by sampling a
  series of values of e and plugging in
• Suppose e.g., that m 0.14, s 0.20, and dt
  0.01, so that we have

41
Monte Carlo Simulation One path

42
A sample path
43
Moving further Itos Lemma

• We need to be able to analyze functions of S
  since derivates are functions of eg. a stock
  price. The tool for this is Itos lemma.
• More generally If we know the stochastic process
  for x, then Itos lemma provides the stochastic
  process for G(t, x).

44
Itos lemma in brief

• Let G(t,x) and dxa(x,t)dt b(x,t)dz

45
Itos lemma
Substituting the expression for dx we get
THIS IS ITOS LEMMA! The option price/the price
of the contingent claim is also a diffusion
process!
46
Application of Itos lemma to functions of GBM
47
Examples
Integrate!
48
The Black-Scholes model

• We consider a stock price which evolves as a GBM,
  ie.
• dS ?Sdt ?Sdz.
• For the sake of simplicity there are no
  dividends.
• The goal is to determine option prices in this
  setup.

49
The idea behind the Black-Scholes derivation

• The option and the stock is affected by the same
  uncertainty generating factor.
• By constructing a clever portfolio we can get rid
  of this uncertainty.
• When the portfolio is riskless the return must
  equal the riskless rate of interest.
• This leads to the Black-Scholes differential
  equation which we will then find a solution to.
• Lets do it! ......

50
Derivation of the Black-Scholes equation
51
Derivation of the Black-Scholes differential
equation
The uncertainty/risk of these terms cancel, cf.
previous slide.
52
Derivation of the Black-Scholes differential
equation
53
The differential equation

• Any asset the value of which depends on the stock
  price must satisfy the BS-differential equation.
• There are therefore many solutions.
• To determine the pricing functional of a
  particular derivative we must impose specific
  conditions. Boundary/terminal conditions.
• Eg For a forward contrakt the boundary condition
  is S K when t T
• The solution to the pde is thus
• S K er (T t )

Check the pde!
54
Risk-neutral pricing

• The parameter m does not appear in the
  BS-differential equation!
• The equation contains no parameters with relation
  to the investors preferences for risk.
• The solution to the equation is therefore the
  same in the real World as in a World where all
  investors are risk-neutral.
• This observation leads to the concept of
  risk-neutral pricing!

55
Risk-neutral pricing in practice

• Assume the expected stock return is equal to the
  riskless rate of interest, ie. use mr in the
  GBM.
• Calculate the expected risk-neutral payoff for
  the option.
• Perform discounting with riskless rate of
  interest, i.e.

56
Black-Scholes formulas
57
The Monte Carlo idea

• General pricing relation
• For example
• These expressions are the basis of Monte Carlo
  simulation. The expectation is approximated by

58
The market price of risk

• The fundamental pde. holds for all derivatives
  written on a GBM-stock.
• If the underlying is not traded (eg. a rate of
  interest, a temperature, a snow depth, a
  Richter-number etc.) we can derive a similar pde,
  but there will be a term for the market price of
  risk of this factor.
• For example we can use Itos lemma to show that
  derivatives will follow
• where

59
The market price of risk

• The market price of risk can not be determined
  from arbitrage arguments alone. It must be
  estimated using market data.
• When simulating the risk neutralized underlying
  variable the drift must be adjusted with a term
  which includes the market price of risk.

60
Example of a non-priced underlying variable
61
Historical volatility

• Observe S0, S1, . . . , Sn with interval length t
  years.
• Calculate continuous returns in every interval
• Estimate standard deviation, s , of the uis.
• The historical annual volatility

62
Implied volatility

• The implied volatility is the volatility which
  when plugged into the BS-formula creates
  correspondence between model- and market price of
  the option.
• The BS-formula is inverted. This is done
  numerically.
• In the market volatility is often quoted in stead
  of price.

63
Exercises/homework!

• Simulate a GBM and show the result graphically
  using a spread sheet.
• Compare the Black-Scholes price with the price of
  options found using the binomial approximation.
  How big must N be in order to obtain a good
  result?
• Try to estimate the volatility using a series of
  stock prices which you have simulated (so that
  you know the true volatility).
• Try to determine some implied volatilities by
  inverting the BS formula.
• Try to determine a call price using Monte Carlo
  simulation and compare your result with the exact
  price obtained from the BS formula.