Transcript Week 6: APT and Basic Options Theory

Week 6: APT and Basic
Options Theory
Introduction
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What is an arbitrage?
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Definition: Arbitrage is the earning of riskless profit by taking advantage of differential
pricing for the same physical asset or security.
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Typical example: two banks with different
interest rate.
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Implications of a factor model: Securities
with equal factor sensitivities will behave
the same way except for non-factor risk.
Consequently, securities or portfolios
with same factor sensitivities should offer
the same expected return. This is the
logic behind APT.
Pricing Effects
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Under APT, it turns out that the mean return is
linearly related to the sensitivity of the factor. In
short, the pricing of the security would result in
equilibrium to eliminate arbitrage opportunities so
that the mean return would satisfy
i  0  1 bi,
where 0 and 1 are constants related to the parameters
ai and bi in the simplified one factor model
ri = ai + bi F.
How this comes about?
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Consider two different assets i and j with different
sensitivities (bibj) in the one factor model.
Construct a new portfolio with return:
r =wri+(1-w)rj =wai + (1-w) aj + [wbi+(1-w)bj]F
Now pick w so that the coefficient of F becomes zero,
that is, w= bj/(bi- bj). Then this portfolio will have no
sensitivity to the factor and the return of this portfolio
becomes r = wai+(1-w) aj = ai bj /(bj- bi) + aj bi /(bi- bj).
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This portfolio is risk-free so its return must
equal to the risk-free rate rf. Otherwise, there
will be arbitrage opportunities (How?). Even
if there is no risk-free rate, all portfolios
constructed this way must have the same
return with no dependence on F. Denote the
return of this portfolio by 0 (knowing that 0
= rf). Then 0 = ai bj /(bj- bi) + aj bi /(bi- bj).
0 (bj- bi) = ai bj - aj bi,
bi(aj - 0) = bj(ai - 0),
bi/(ai - 0) = bj/(aj - 0), for all i and j
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Set
(ai - 0)/ bi = c, a fixed constant.
Thus, ai = 0 + bi c for all i.
Taking expected values,
i = ai+bi F = 0+bi(c+ F) = 0 + 1bi with
1=c+F as claimed.
Once 0 and 1 are known, the expected return of
all the assets are completely determined by the
factor sensitivity bi.
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Consider a special portfolio p* with bi=1. It has
expected return p* = 0 + 1. Thus, 1 = p* - 0 =
p* - rf. This value represents the expected excess
return of a portfolio that has unit sensitivity to the
factor. Hence, the value 1 is usually known as the
factor risk premium.
Substituting this 1 into i = 0 + 1bi, we get i = rf
+ (p* - rf)bi.
There is a very nice interpretation to this equality, the
mean return of any asset is the sum of two
components. The first is the risk-free rate, the
second is the factor risk premium times the
sensitivity to that factor.
Arbitrage Pricing Theorem
Theorem: Suppose that there are n assets whose
returns are governed by m factors (m<n)
according to the multi-factor model
ri = ai + mj=1 bijFj for i =1,…,n. Then there
exist constants 0, …, m such that for
i = 1,…,n,
i = 0 + jm=1 bij j.
Remarks
1.
2.
This result still holds if error terms are added
to the multi-factor equation.
We can reconcile CAPM and APT. Using a
two factor model from APT, suppose that
CAPM holds, then
ri = ai + bi1 F1 + bi2 F2 + ei. Taking the
covariance with the return of the market, we
get
cov(ri,rM) = bi1 cov(F1,rM) + bi2 cov(F2,rM).
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We assume that cov(ei,rM)=0. Dividing this equation
by M2, we get iM = bi1 F1M + bi2 F2M with
F1M=cov(F1,rM)/M2 and F2M=cov(F2,rM)/M2
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The overall beta of the asset with the market is made
up of the betas of the underlying factor betas (that is
independent of the asset) weighted by the
corresponding factor sensitivities of the asset.
Therefore, different assets have different betas
because they have different sensitivities.
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Looking at it differently, with the two factor
model, APT gives i = rf + 1 bi1 + 2bi2.
For CAPM, we have the SML: i – rf = iM
(M – rf). Substituting iM = bi1 F1M + bi2
F2M into the SML, we get i – rf = (bi1 F1M +
bi2 F2M)(M – rf).
When both APT and CAPM hold, we have
1 = F1M (M – rf) and 2 = F2M (M– rf).
Blur of History
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We often use historical data to estimate the
parameters. But this has a drawback. Suppose that
the yearly return r is expressed as the compound
return of 12 monthly returns, 1+ry = (1+r1)…(1+r12).
For small ri, this can be written as 1+ry~1+r1+…+r12 In
other words, ry~r1+…+r12.
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If we assume that the monthly returns are
uncorrelated with mean  and variance 2, by taking
expectations, we have y = 12 and y2 =122.
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In other words, we can express the monthly mean in
terms of annual means by  = y /12 and 2 = y2 /12.
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In general, if we have yearly data and if we are
interested in estimates at a higher frequency p (such as
monthly) in each year (p=1/12 for monthly data or
p=1/4 for quarterly data), then it can be shown easily
that  = py and 2 = py2 .
The ratio between the standard deviation and the mean
is known as the coefficient of variation (CV). It has an
order of 1/p, which increases as p decreases. In other
words, the more frequent we sample, the larger the
relative error in estimation.
This is sometimes known as the blur of history in
statistics.
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For example, let y =12% and y = 15%. Then CV =
1.25. If we go to monthly observations, then p=1/12
=1%, and  =4.33%, giving a CV~4. If we go
further down to daily observations, p=1/250,
=0.048%, and  =0.95%, giving a CV~19.8. It is
quite common that stock values may easily move 3%
to 5% () within each day, yet the expected change
() is only 0.05%. Given the large CV (19.8), such
an estimate of expected change is highly inaccurate
Mean Blur
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Let r1 ,…, rn be iid having the same mean  and
variance 2. Then an estimate of the mean is
i=1n ri/n. E( )=  and = /n.
Using the same example, let p=1/12. Recall that the
monthly return  =1% and  =4.33%. If we use 1
year of monthly data, we get  = 4.33/ 12 = 1.25%.
This is pretty big since the standard error is larger
than the estimate itself. We may want to use more
data so that the standard error is of 1/10 of the mean
value, i.e., 0.1%.
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In other words, 4.33/ n = 0.1 giving n=1875,
156 years of data are required.
There are two drawbacks, (a) the mean value
remains fixed over such a long period and (b)
where do we get 156 years of data.
It is almost impossible to estimate the mean
value  within workable accuracy using
historical data. By increasing frequency of
measurement cannot solve this difficulty.
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If longer periods are used, each sample is more
reliable but fewer independent samples are
found.
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If shorter periods are used, more samples are
available but each one is worse in terms of the
CV.
Basic Options Theory
Introduction
 Option (call and put)
 Option premium
 Exercise (strike) price
 Option value at expiration
Call: (ST-E)+, Put (E-ST)+, where ST is the price
of the derivative at exercise date and E is its
Exercise (strike) price.
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How to price options?
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The law of one price: if two financial
instruments has the same payoff, then they will
have the same price.
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To valuate an option, one must find a portfolio
or a self-financing trading strategy with a
known price and which has the same payoffs
as the option. By the law of one price, it
follows that the price of the options must be
equal to that of the portfolio or self-financing
trading strategy.
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A simple example: Suppose stock in company
A sells $100/share, the risk-free rate is 6%,
now consider a futures contract obliging one
party to sell stock of company A to the other
party one year later from now at price $P/share,
what should P be ?
Factors affecting the value of
Options
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Volatility of the underlying stock. More
volatile, more value.
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The interest rate.
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What about the growth rate of the stock?
Single-period Binomial Option
Theory
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A simple example:
Consider the following portfolio:
purchase x share stock and b dollars worth of
the risk-free asset. Then
100x+(1+r)b=20 and 60x+(1+r)b=0
This gives that x and b, where
x=volatility of the option/ volatility of the stock
is the hedge ratio.
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More generally, suppose that the current price
is s1 and after one period the stock either goes
up to s3 or s2. The exercise price is E. The riskfree rate of interest is r. Using the same
argument as above, we have
x s3 +(1+r)b=(s3 –E)+ and x s2 +(1+r)b=(s2 E)+
As a result, we have the hedge ratio x and the
amount of borrow b. The option price is x+b.
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Assume that s2 <E< s3, then the hedge ratio δ=(s3 E)/(s3 - s2). The amount borrow is δs2/(1+r).
 Write s3=us1, s2=ds1 and Cu= (s3 –E)+ ,
Cd= (s2–E)+ ， under no arbitrage assumption
(u>1+r>d?), we have the option price
C=[q Cu+(1-q) Cd]/(1+r), where q=(R-d)/(u-d).
This is the so-called Option pricing formula and q is the
risk-neutral probability, which is the solution of
s1=[qu s1 +(1-q)d s1]/(1+r).
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Two-step option pricing
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A simple example:
A general binomial tree model
Assume that:
 At the j-th node the stock is worth sj and the option is
f(j).
 The j-th node leads to either the (2j+1)-th node or the
2j-th node after the time “tick”.
 The time between the ticks is Δt.
Then,
f(j)=exp(-rΔt)[qj f(2j+1)+(1-qj)f(2j)],
qj=[exp(r Δt) sj- s2j]/(s2j+1 - s2j).
Example: Consider a stock with a volatility of its
logarithm of σ=0.2. The current price of the stock is
$62. The stock pays no dividends. A Certain call
option on this stock has an expiration date 5 months
from now and a strike price of $60. The current rate
of interest is 10%. We wish to determine the price of
this call using the binomial option approach.
(u=exp[σ(Δt)1/2], d= exp[-σ(Δt)1/2], R=1+0.1/12)
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