Week 6: APT and Basic Options Theory

Introduction

• What is an arbitrage?• Definition: Arbitrage is the earning of risk-fr

ee profit by taking advantage of differential pricing for the same physical asset or security.

• Typical example: two banks with different interest rate.

• 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

• 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?

• Consider two different assets i and j with different sensitivities (bibj) 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).

• 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

• 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.

• 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. This result still holds if error terms are added to the multi-factor equation.

2. 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).

• We assume that cov(ei,rM)=0. Dividing this equation by M

2, we get iM = bi1 F1M + bi2 F2M with

F1M=cov(F1,rM)/M2 and F2M=cov(F2,rM)/M

2

• 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.

3. 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 We often use historical data to estimate the para

meters. 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. If we assume that the monthly returns are uncorr

elated with mean and variance 2, by taking expectations, we have y = 12 and y

2 =122. In other words, we can express the monthly mea

n in terms of annual means by = y /12 and 2

= y2 /12.

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 = py and 2 = py

2 . 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.

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 Let r1 ,…, rn be iid having the same mean and

variance 2. Then an estimate of the mean is i=1

n 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%.

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.

If longer periods are used, each sample is more reliable but fewer independent samples are found.

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.

How to price options? The law of one price: if two financial instruments

has the same payoff, then they will have the same price.

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.

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

Volatility of the underlying stock. More volatile, more value.

The interest rate.

What about the growth rate of the stock?

Single-period Binomial Option Theory

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.

More generally, suppose that the current price is s

1 and after one period the stock either goes up to s3 or s2. The exercise price is E. The risk-free 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 s1 x+b.

Assume that s2 <E< s3, then the hedge ratio δ=(s

3 -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=(1+r-d)/(u-d).This is the so-called Option pricing formula and q i

s the risk-neutral probability, which is the solution of

s1=[qu s1 +(1-q)d s1]/(1+r).

Two-step option pricing

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 2

j-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)