option pricing and dynamic modeling of stock prices investments 2004
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Option Pricing and Dynamic Modeling of Stock Prices
Investments 2004
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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.
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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
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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]+
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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!
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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!
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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......
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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
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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.
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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
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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.
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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.
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Generalization
A contingent claim expires at time T and payoff depends on stock price
S0u ƒu
S0d ƒd
S0
ƒ
. where ued rT
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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
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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
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Generalization
Plugging in the expression for we get
ƒ = [ q ƒu + (1 – q )ƒd ]e–rT
where
du
deq
rT
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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.S0u
ƒu
S0d ƒd
S0
ƒ
q
(1 – q )
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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 )
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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
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Two-period example
Each step represents 3 months, t=0.25
20
22
18
24.2
19.8
16.2
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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}
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General formula
n
i
ni
initrn fqqi
nef
0
)1(
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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
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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
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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.
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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
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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
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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
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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
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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,
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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)] 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.
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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.
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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 ),(),(
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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
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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!!
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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
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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…
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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…
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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
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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).
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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
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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…..
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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!
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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
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Examples
dzdtdG
SG
dzGdtGrdG
eSG
TtTr
2
ln 2.
)(
at time expirescontract -stock a of price forward The 1.
2
)(
Integrate!
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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.
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Pre-Nobel prize methodology
Calculate expected payoff. See note…Discount using r… or …. or something else…??
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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! ......
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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
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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.
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.. theasknown also is This
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:equation aldifferenti Scholes-Black the toleads
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have therefore Weinterval. time
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equationdiffpartiallfundamenta
rS
SS
rSt
dSd
dtrd
Derivation of the Black-Scholes differential equation
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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!
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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!
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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
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Black-Scholes formulas
TdT
TrKSd
T
TrKSd
dNSdNeKp
dNeKdNScrT
rT
10
2
01
102
210
)2/2()/ln(
)2/2()/ln( where
)( )(
)( )(
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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
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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
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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
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Example of a non-priced underlying variable
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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
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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 stead of price.
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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.