# Optionsprisfasts忙ttelse Introduktion _amp; repetetion by yurtgc548

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```									Option Pricing and Dynamic
Modeling of Stock Prices
Investments 2004
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.             2
Overview/agenda
Intuition behind Pricing by arbitrage
Models of uncertainty
n   The binomial-model. Examples and general results
n   The transition from discrete to continuous time
Pricing by arbitrage in continuous time
n   The Black-Scholes model
n   General principles
n   Monte Carlo simulation, vol. estimation.
Exercises along the way

3
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
Value(T)=[ST-X]+
4
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!

5
Pricing by arbitrage - PCP
Transaction Time 0 price Time T flow Time T flow
ST>X         ST<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!
6
Pricing by arbitrage
So if we know price of
n   underlying asset
n   riskless borrowing/lending (the rate of interest)
n   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......

7
The World’s 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

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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

9
Constructing a riskless portfolio
Consider the portfolio:   long D stocks
short 1 call option

22D – 1

18D

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

10
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
22´0.25 – 1 = 4.50.
So present value must be
4.5e – 0.12´0.25 = 4.3670.
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Valuing the option
The portfolio which was
long 0.25 stocks
short 1 option
was worth 4.367.
Value of stocks
5.000 (= 0.25´20 ).
Therefore option value must be
0.633 (= 5.000 – 4.367 ),
...otherwise there are arbitrage
opportunities.
12
Generalization

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

S0u
ƒu
S0
ƒ              S0d
ƒd

13
Generalization
Consider portfolio which is long D stocks and short 1
claim
S0 uD – ƒu
D S0–   f
S0dD – ƒd
Portfolio is riskless when S0uD – ƒu = S0d D – ƒd or

Note: D is the hedgeratio, i.e. the number of stocks
needed to hedge the option.
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Generalization
Portfolio value at time T is
S0u D – ƒu. Certain!
Present value must thus be
(S0u D – ƒu )e–rT
but present value is also given
as S0D – f
We therefore have
ƒ = S0D – (S0u D – ƒu )e–rT
15
Generalization

Plugging in the expression for D we get

ƒ = [ q ƒu + (1 – q )ƒd ]e–rT

where

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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.
q       S0u
ƒu
S0
ƒ                   S0d
(1 –
q    )    ƒd
17
Back to the example
S0u = 22
q    ƒu = 1
S0
ƒ
(1 –       S0d = 18
)          q    ƒd = 0

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

18
Pricing the option

S0u = 22
0.6 523    ƒu = 1
S0
ƒ
0.34      S0d = 18
77
ƒd = 0
Value of option is

e–0.12´0.25 [0.6523´1 + 0.3477´0] = 0.633.

19
Two-period example
24.2
22

20                       19.8

18
16.2
Each step represents 3
months, dt=0.25

20
Pricing a call option, X=21
24.2
3.2
22
B
20             2.0257              19.8
A
1.2823                                  0.0
18
C
0.0              16.2
Value in node B                                  0.0
= e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257
Value in node A
= e–0.12´0.25(0.6523´2.0257 + 0.3477´0)
= 1.2823

f = e-2rdt[q2fuu + 2q(1-q)fud + (1-q)2fdd] = e-2rdt EQ{fT} 21
General formula

22
Put option; X=52
u=1.2, d=0.8, r=0.05, dt=1, q=0.6282
72
0
60
50          1.4147             48
4.1923                              4
40
9.4636            32
20

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American put option – early
exercise

72
0
60
B
50         1.4147                48
A
5.0894                                4
40
C
12.0                32
20

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

9.4636         24
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.

25
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
u=1/d. This is Cox, Ross, and
Rubinstein’s approach.                    26
Few steps => few states.
A coarse model

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

28
Call, S=100, s=0.15, r=0.05, T=0.5, X=105

29
30
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

31
The Wiener Process – the key
element/the basic building block
Consider a variable z, which takes on
continuous values.
The change in z is dz over time interval
of length dt.
z is a Wiener proces, if
1.
2. Realization/value of dz for two non-
overlapping periods are independent.
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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 dt
approach zero. When we write dz and dt it is to be
understood as the limits of the corresponding
expressions with dt and dz, when dt goes to zero.

33
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.
This model is of course more general but it
is still not a good model for the dynamics of
stock prices.
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35
Ito Processes
The 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 dt go to zero.
We will see processes of this type many
times! (Stock prices, interest rates,
temperatures etc.)                    36
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:

37
The Lognormal distribution
A consequence of the GBM specification is

We will show
this shortly!!

The Log of ST is normal distributed, ie. ST
follows a log-normal distribution.

38
Lognormal-density

39
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

Methods for sampling e’s…
40
Monte Carlo Simulation – One path

You MUST go home and try this…
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A sample path:

42
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).

43
Ito’s lemma in brief
Let G(t,x) and dx=a(x,t)dt + b(x,t)dz

44
Why the extra term?
Because

so

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

So it is deterministic in the limit…..

45
Ito’s lemma
Substituting the expression for dx we get:

THIS IS ITO’S LEMMA!

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

46
Application of Ito’s 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 = mSdt + sSdz.

For the sake of simplicity there are no
dividends.
The goal is to determine option prices in this
setup.

49
Pre-Nobel prize methodology
Calculate expected payoff. See note…
Discount using r… or m…. or something
else…??

50
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! ......

51
Derivation of the Black-Scholes equation

52
Derivation of the Black-Scholes
differential equation

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

54
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
–r (T – t )
Check the pde!   55
ƒ=S –Ke
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!

56
Risk-neutral pricing in practice

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

57
Black-Scholes formulas

58
The Monte Carlo idea
General pricing relation:

For example:

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

59
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 Ito’s lemma to show that
derivatives will follow

where

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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.
61
Example of a non-priced
underlying variable

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Historical volatility
• Observe S0, S1, . . . , Sn with interval
length t years.
• Calculate continuous returns in every
interval:

3. Estimate standard deviation, s , of the
ui´s.
4. The historical annual volatility:

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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

64
Exercises/homework!
Simulate a GBM and show the result graphically using
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.
65

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