Option Pricing and Dynamic Modeling of Stock Prices

Investments 2004

2

MotivationWe 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 arbitrageModels 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).

0 T

?Value(0)=?

Value(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

Transaction

Time 0 price

Time T flowST>X

Time T flowST<X

Long stock

-S0 ST ST

Long put(X)

-P 0 X-ST

Loan(X) PV(X) -X -X

Sum PV(X)-P-S0

ST-X 0

Long call(X)

-C ST-X 0

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 World’s simplest model of undertainty – the binomial model

Example: Stockprice today is $20In three months it will be either $22 or $18 (+-10%)

Stockprice = $22

Stockprice = $18

Stockprice = $20

9

Stockprice = $22Option payoff = $1

Stockprice = $18Option payoff = $0

Stockprice = $20Option price=?

A call option

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

10

Consider the portfolio: long stocksshort 1 call option

The portfolio is riskless if 22– 1 = 18 ie. when = 0.25.

22– 1

18

Constructing a riskless portfolio

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 optionPortfolio 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 optionThe portfolio which was

long 0.25 stocksshort 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

S0u ƒu

S0d ƒd

S0

ƒ

. where ued rT

14

Generalization

Consider portfolio which is long stocks and short 1 claim

Portfolio is riskless when S0u– ƒu = S0d – ƒd or

Note: is the hedgeratio, i.e. the number of stocks needed to hedge the option.

dSuS

ff du

00

S0 u– ƒu

S0d– ƒd

S0– f

15

Generalization

Portfolio value at time T is S0u – ƒu. Certain!

Present value must thus be (S0u – ƒu )e–rT

but present value is also given as S0– f

We therefore haveƒ = S0– (S0u – ƒu )e–rT

16

Generalization

Plugging in the expression for we get

ƒ = [ q ƒu + (1 – q )ƒd ]e–rT

where

du

deq

rT

17

Risk-neutral pricing

ƒ = [ q ƒu + (1 – q )ƒd ]e-rT = e-rT EQ{fT}

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.S0u

ƒu

S0d ƒd

S0

ƒ

q

(1 – q )

18

Back to the example

We can derive q by pricing the stock:20e0.12 0.25 = 22q + 18(1 – q ); q =

0.6523This result corresponds to the result from using the formula

6523.09.01.1

9.00.250.12

e

du

deq

rT

S0u = 22 ƒu = 1

S0d = 18 ƒd = 0

S0

ƒ

q

(1– q )

19

Pricing the option

Value of option is

e–0.120.25 [0.65231 + 0.34770] = 0.633.

S0u = 22 ƒu = 1

S0d = 18 ƒd = 0

S0

ƒ

0.6523

0.3477

20

Two-period example

Each step represents 3 months, t=0.25

20

22

18

24.2

19.8

16.2

21

Pricing a call option, X=21

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

f = e-2rt[q2fuu + 2q(1-q)fud + (1-q)2fdd] = e-2rt EQ{fT}

22

General formula

n

i

ni

initrn fqqi

nef

0

)1(

23

Put option; X=52

504.1923

60

40

720

484

3220

1.4147

9.4636

u=1.2, d=0.8, r=0.05, t=1, q=0.6282

24

American put option – early exercise

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

Node C: max(52-40, exp(-0.05)*(q*4+(1-q)*20))

9.4636

25

Delta

Delta () is the hedge ratio,- the change in the option value relative to the change in the underlying asset/stock price changes when moving around in the binomial latticeIt 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 is p.a. volatility andt is length of time steps measured in years. Note u=1/d. This is Cox, Ross, and Rubinstein’s approach.

u e

d e

t

t

27

5 skridt over et år

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

74.08182 83.52702 94.17645 106.1837 119.7217 134.9859

Aktieværdi

Sa

nd

sy

nlig

he

d

Few steps => few states. A coarse model

28

50 skridt over et år

0

0.02

0.04

0.06

0.08

0.1

0.12

74

.1

76

.8

79

.6

82

.5

85

.6

88

.7

91

.9

95

.3

98

.8

10

2

10

6

11

0

11

4

11

8

12

3

12

7

13

2

Aktieværdi

Sa

nd

sy

nlig

he

dMany steps => many states. A ”fine” model

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Konvergens i binomialmodellen

2.5

2.7

2.9

3.1

3.3

3.5

3.7

3.9

4.1

4.3

4.5

1 6 11 16 21

Antal skridt

Op

tio

ns

pri

sCall, S=100, =0.15, r=0.05, T=0.5, X=105

30

31

Alternative intertemporal models of uncertainty

Discrete time; discrete variable (binomial)Discrete time; continuous variableContinuous time; discrete variableContinuous time; continuous variable

All can be used, but we will work towards the last type which often possess the nicest analytical properties

32

The Wiener Process – the key element/the basic building blockConsider 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, if1. 2. Realization/value of z for two non-overlapping periods are independent.

(0,1). Ntz from drawn is where,

33

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)] isT

A continuous time model is obtained by letting t approach zero. When we write dz and dt it is to beunderstood as the limits of the corresponding expressions with t and z, when t goes to zero.

34

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.

dx=adt+bdz.This model is of course more general but it is still not a good model for the dynamics of stock prices.

35

36

Ito ProcessesThe drift and volatility of Ito processes are general functions

dx=a(x,t)dt+b(x,t)dz.

Note: What we really mean is

where we let t go to zero.We will see processes of this type many times! (Stock prices, interest rates, temperatures etc.)

ttxbttxax ),(),(

37

A good model for stock prices

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

dzdtS

dS

σSdz μSdtdS

or

tStSS

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.

,2

lnln

or

,2

lnln

2

0

2

0

TTSNS

TTNSS

T

T

We will showthis shortly!!

39

Lognormal-density

)1()(Var

)(222

0

0

TT

T

TT

eeSS

eSSE

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 0.5 1 1.5 2 2.5 3

40

Monte Carlo Simulation

The model is best illustrated by sampling a series of values of and plugging in……Suppose e.g., that = 0.14, = 0.20, and t = 0.01, so that we have

SS

SSS

02.00014.0

01.020.001.014.0

Methods for sampling ’s…

41

Monte Carlo Simulation – One path

Period

Stock Price at Start of Period

Random Sample for

Change in Stock Price, 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

You MUST go home and try this…

42

A sample path:

Simuleret GBM

0

20

40

60

80

100

120

140

160

180

0 0.2 0.4 0.6 0.8 1

Tid

Akt

ieku

rs

43

Moving further: Ito’s 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 Ito’s lemma.More generally: If we know the stochastic process for x, then Ito’s lemma provides the stochastic process for G(t, x).

44

Ito’s lemma in brief

Let G(t,x) and dx=a(x,t)dt + b(x,t)dz

22

2

½

termextraan get we

processes stochastic with dealingBut when

xdx

Gtd

t

Gxd

x

GdG

tdt

Gxd

x

GdG

have wecescircumstan normal Under

45

Why the extra term?

Because

so

But

tbtax

orderhigher of terms)( 222 tbx

2 has expected value of 1 and variance of term is of order (t)2

So it is deterministic in the limit…..

46

Ito’s lemma

dzbx

Gdtb

x

G

t

Ga

x

GdG ½ 2

2

2

Substituting the expression for dx we get:

THIS IS ITO’S LEMMA!

The option price/the price of the contingent claim is alsoa diffusion process!

47

Application of Ito’s lemma to functions of GBM

dzSS

GdtS

S

G

t

GS

S

GdG

tSG

zdSdtSSd

½

get we and of function aFor

is process pricestock The

222

2

48

Examples

dzdtdG

SG

dzGdtGrdG

eSG

TtTr

2

ln 2.

)(

at time expirescontract -stock a of price forward The 1.

2

)(

Integrate!

49

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.

50

Pre-Nobel prize methodology

Calculate expected payoff. See note…Discount using r… or …. or something else…??

51

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.

Let’s do it! ......

52

Derivation of the Black-Scholes equation

)!( stocks :ƒ

+

option :1

of consisting portfolio aconstruct e W

ƒƒ

½ƒƒ

ƒ

222

2

S

dzSS

dtSSt

SS

d

dzSdtSdS

53

ƒ

ƒ

bygiven is

interval over time valuein this Change

ƒ

ƒ

bygiven is , portfolio, of Value

dSS

dd

dt

SS

Derivation of the Black-Scholes differential equation

The uncertainty/risk of these termscancel, cf. previous slide.

54

.. theasknown also is This

ƒƒ

½ƒƒ

:equation aldifferenti Scholes-Black the toleads

equations in these and ƒfor sexpression thengSubstituti

have therefore Weinterval. time

smallnext over the riskless is portfolio thisofreturn The

2

222

equationdiffpartiallfundamenta

rS

SS

rSt

dSd

dtrd

Derivation of the Black-Scholes differential equation

55

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 ƒ = S – K when t =T The solution to the pde is thus

ƒ = S – K e–r (T – t )

Check the pde!

56

Risk-neutral pricing

The parameter 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!

57

Risk-neutral pricing in practice

1. Assume the expected stock return is equal to the riskless rate of interest, ie. use =r in the GBM.

2. Calculate the expected risk-neutral payoff for the option.

3. Perform discounting with riskless rate of interest, i.e. )0,max( KSEec T

QrT

58

Black-Scholes formulas

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln( where

)( )(

)( )(

59

The Monte Carlo idea

General pricing relation:

For example:

These expressions are the basis of Monte Carlo simulation. The expectation is approximated by:

TQrT cEec ~

)0,max( KSEec TQrT

N

i

simiT

rT cN

ecc1

,1ˆ

60

The market price of riskThe 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 Ito’s lemma to show that derivatives will follow

where

dzdtrdzdtf

df )(

risk if pricemarket

r

61

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.

The market price of risk

62

Example of a non-priced underlying variable

63

Historical volatility

1. Observe S0, S1, . . . , Sn with interval length years.

2. Calculate continuous returns in every interval:

3. Estimate standard deviation, s , of the ui´s.

4. The historical annual volatility:

uS

Sii

i

ln1

sˆ

64

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.

65

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.