Week 6: APT and Basic Week 6: APT and Basic Options TheoryOptions Theory

IntroductionIntroduction

• What is an arbitrage?What is an arbitrage?

• Definition:Definition: Arbitrage is the earning of risk-les Arbitrage is the earning of risk-less profit by taking advantage of differential prics profit by taking advantage of differential pricing for the same physical asset or security.ing for the same physical asset or security.

• Typical example: Typical example: two banks with different intetwo banks with different interest rate.rest rate.

• Implications of a factor model: Securities Implications of a factor model: Securities with equal factor sensitivities will behave with equal factor sensitivities will behave the same way except for non-factor risk. the same way except for non-factor risk. Consequently, securities or portfolios witConsequently, securities or portfolios with same factor sensitivities should offer thh same factor sensitivities should offer the same expected return. This is the logic e same expected return. This is the logic behind APT.behind APT.

Pricing EffectsPricing Effects

• Under APT, it turns out that the mean return is linearlUnder APT, it turns out that the mean return is linearly related to the sensitivity of the factor. In short, the y related to the sensitivity of the factor. In short, the pricing of the security would result in equilibrium to pricing of the security would result in equilibrium to eliminate arbitrage opportunities so that the mean retueliminate arbitrage opportunities so that the mean return would satisfyrn would satisfy

i i 00 1 1 bbii,,

where where 00 and and 1 1 are constants related to the parameters are constants related to the parameters

aaii and b and bii in the simplified one factor model in the simplified one factor model

rri i = a= aii + b + bii F. F.

HHow this comes about?ow this comes about?

• CConsider two different assets i and j with different seonsider two different assets i and j with different sensitivities (bnsitivities (biibbjj) in the one factor model. Construct ) in the one factor model. Construct

a new portfolio with return:a new portfolio with return:

r =wrr =wrii+(1-w)r+(1-w)rj j =wa=wai i + (1-w) a+ (1-w) aj j ++ [wb[wbii+(1-w)b+(1-w)bjj]F]F

• Now pick w so that the coefficient of F becomes zero, Now pick w so that the coefficient of F becomes zero, that is, w= bthat is, w= bjj/(b/(bii- b- bjj). Then this portfolio will have no ). Then this portfolio will have no

sensitivity to the factor and the return of this portfolio sensitivity to the factor and the return of this portfolio becomes r = wabecomes r = waii+(1-w) a+(1-w) aj j = a= ai i bbj j /(b/(bjj- b- bii) + a) + aj j bbi i /(b/(bii- b- bjj).).

• This portfolio is risk-free so its return must eqThis portfolio is risk-free so its return must equal to the risk-free rate rual to the risk-free rate rff. Otherwise, there wil. Otherwise, there will be arbitrage opportunities (l be arbitrage opportunities (HowHow?). Even if th?). Even if there is no risk-free rate, all portfolios constructeere is no risk-free rate, all portfolios constructed this way must have the same return with no dd this way must have the same return with no dependence on F. Denote the return of this portependence on F. Denote the return of this portfolio by folio by 00 (knowing that (knowing that 0 0 = r= rff). Then ). Then 0 0 = a= ai i bb

j j /(b/(bjj- b- bii) + a) + aj j bbi i /(b/(bii- b- bjj). ). • 0 0 (b(bjj- b- bii) = a) = ai i bbj j - a- aj j bbii, , • bbii(a(ajj - - 00) = b) = bjj(a(aii - - 00),),• bbii/(a/(aii - - 00) = b) = bjj/(a/(ajj - - 00), for all i and j), for all i and j

• Set (aSet (aii - - 00)/ b)/ bii = c, a fixed constant. = c, a fixed constant.

• Thus, aThus, ai i = = 0 0 + + bbi i c for all i.c for all i.

• Taking expected values, Taking expected values,

i i = a= aii+b+bi i F F = = 00+b+bii(c+ (c+ FF) = ) = 0 0 + + 11bbii with with 11==

c+c+F F as claimed. as claimed.

Once Once 0 0 and and 1 1 are known, the expected return of are known, the expected return of

all the assets are completely determined by the all the assets are completely determined by the factor sensitivity bfactor sensitivity bii..

• Consider a special portfolio pConsider a special portfolio p** with b with bii=1. It has expect=1. It has expected return ed return p*p* = = 0 0 + + 11. Thus, . Thus, 11 = = p*p* - - 0 0 = = p*p* - r - rff. Th. This value represents the expected is value represents the expected excess excess return of a portfreturn of a portfolio that has unit sensitivity to the factor. Hence, the volio that has unit sensitivity to the factor. Hence, the value alue 11 is usually known as the is usually known as the factor risk premium.factor risk premium.

• Substituting this Substituting this 11 into into i i = = 0 0 + + 11bbii, we get , we get i i = r= rf f + + ((p*p* - r - rff)b)bii..

• There is a very nice interpretation to this equality, the There is a very nice interpretation to this equality, the mean return of any asset is the sum of two components.mean return of any asset is the sum of two components. The first is the risk-free rate, the second is the factor r The first is the risk-free rate, the second is the factor risk premium times the sensitivity to that factor.isk premium times the sensitivity to that factor.

Arbitrage Pricing TheoremArbitrage Pricing Theorem

TheoremTheorem:: Suppose that there are n assets whose Suppose that there are n assets whose returns are governed by m factors (m<n) accorreturns are governed by m factors (m<n) according to the multi-factor model ding to the multi-factor model

rri i = a= aii + + mmj=1j=1 b bijijFFjj

for i =1,…,n. Then there exi for i =1,…,n. Then there exi

st constants st constants 00, …,, …, m m such that for such that for

i = 1,…,n,i = 1,…,n,

i i = = 00 + + jjmm

=1=1 b bijij jj..

RemarksRemarks

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

2.2. We can reconcile CAPM and APT. Using a tWe can reconcile CAPM and APT. Using a two factor model from APT, suppose that CAwo factor model from APT, suppose that CAPM holds, then PM holds, then

rri i = a= aii + b + bi1i1 F F1 1 + b+ bi2i2 F F2 2 + e+ eii. Taking the covarian. Taking the covariance with the return of the market, we get ce with the return of the market, we get

cov(rcov(rii,r,rMM) = b) = bi1i1 cov(F cov(F11,r,rMM)) + b+ bi2i2 cov(F cov(F22,r,rMM).).

• We assume that cov(eWe assume that cov(eii,r,rMM)=0. Dividing this equation b)=0. Dividing this equation b

y y MM22, we get , we get iMiM = = bbi1i1 F1M F1M + + bbi2 i2 F2M F2M withwith

F1MF1M==cov(Fcov(F11,r,rMM))//MM2 2 and and F2MF2M==cov(Fcov(F22,r,rMM))//MM

22

• The overall beta of the asset with the market is made uThe overall beta of the asset with the market is made up of the betas of the underlying factor betas (that is indp of the betas of the underlying factor betas (that is independent of the asset) weighted by the corresponding fependent of the asset) weighted by the corresponding factor sensitivities of the asset. Therefore, different assactor sensitivities of the asset. Therefore, different assets have different betas because they have different senets have different betas because they have different sensitivities. sitivities.

3.3. Looking at it differently, with the two factor modeLooking at it differently, with the two factor model, APT gives l, APT gives i i = r= rf f + + 11 b bi1 i1 + + 22bbi2i2..

For CAPM, we have the SML:For CAPM, we have the SML: i i – r– rf f = = iMiM ( (M M – r– rff). ).

Substituting Substituting iMiM = b = bi1i1 F1M F1M + b+ bi2 i2 F2M F2M into the SML, into the SML,

we getwe get i i – r– rf f = (b= (bi1i1 F1M F1M + b+ bi2 i2 F2MF2M)()(M M – r– rff). ).

When both APT and CAPM hold, we have When both APT and CAPM hold, we have

11 = = F1M F1M ((M M – r– rff) and ) and 22 = = F2M F2M ((MM– r– rff). ).

Blur of HistoryBlur of History We often use historical data to estimate the parameters. BWe often use historical data to estimate the parameters. B

ut this has a drawback. Suppose that the yearly return r is ut this has a drawback. Suppose that the yearly return r is expressed as the compound return of 12 monthly returns, 1expressed as the compound return of 12 monthly returns, 1+r+ryy = (1+r = (1+r11)…(1+r)…(1+r1212). For small r). For small rii, this can be written as 1+, this can be written as 1+rryy~1+r~1+r11+…+r+…+r12 12 In other words, rIn other words, ryy~r~r11+…+r+…+r1212..

If we assume that the monthly returns are uncorrelated witIf we assume that the monthly returns are uncorrelated with mean h mean and variance and variance 22, by taking expectations, we have , by taking expectations, we have yy = 12 = 12 and and yy

2 2 =12=1222..

In other words, we can express the monthly mean in terms In other words, we can express the monthly mean in terms of annual means by of annual means by = = yy /12 and /12 and 2 2 = = yy

2 2 /12. /12.

In general, if we have yearly data and if we are interesteIn general, if we have yearly data and if we are interested in estimates at a higher frequency p (such as monthly) d in estimates at a higher frequency p (such as monthly) in each year (p=1/12 for monthly data or p=1/4 for quarin each year (p=1/12 for monthly data or p=1/4 for quarterly data), then it can be shown easily that terly data), then it can be shown easily that = p = pyy and and

2 2 = p= pyy2 2 . .

The ratio between the standard deviation and the mean iThe ratio between the standard deviation and the mean is known as the coefficient of variation (CV). It has an s known as the coefficient of variation (CV). It has an order of 1/order of 1/p, which increases as p decreases. In other p, which increases as p decreases. In other words, the more frequent we sample, the larger the relatwords, the more frequent we sample, the larger the relative error in estimation.ive error in estimation.

This is sometimes known as the blur of history in statistThis is sometimes known as the blur of history in statistics.ics.

For example, let For example, let yy =12% and =12% and yy = 15%. Then CV = = 15%. Then CV =

1.25. If we go to monthly observations, then p=1/12 1.25. If we go to monthly observations, then p=1/12 =1%, and =1%, and =4.33%, giving a CV~4. If we go furthe =4.33%, giving a CV~4. If we go further down to daily observations, p=1/250, r down to daily observations, p=1/250, =0.048%, an=0.048%, an

d d =0.95%, giving a CV~19.8. It is quite common t =0.95%, giving a CV~19.8. It is quite common t

hat stock values may easily move 3% to 5% (hat stock values may easily move 3% to 5% () withi) within each day, yet the expected change (n each day, yet the expected change () is only 0.05%.) is only 0.05%. Given the large CV (19.8), such an estimate of expe Given the large CV (19.8), such an estimate of expected change is highly inaccuratected change is highly inaccurate

Mean BlurMean Blur Let rLet r11 ,…, r ,…, rnn be iid having the same mean be iid having the same mean and varia and varia

nce nce 22. Then an estimate of the mean is . Then an estimate of the mean is i=1i=1n n rrii/n. E( )= /n. E( )= and and = = //n.n.

Using the same example, let p=1/12. Recall that the Using the same example, let p=1/12. Recall that the monthly return monthly return =1% and =1% and =4.33%. If we use 1 ye =4.33%. If we use 1 year of monthly data, we get ar of monthly data, we get = 4.33/ = 4.33/ 12 = 1.25%. T12 = 1.25%. This is pretty big since the standard error is larger than his is pretty big since the standard error is larger than the estimate itself. We may want to use more data so the estimate itself. We may want to use more data so that the standard error is of 1/10 of the mean value, i.that the standard error is of 1/10 of the mean value, i.e., 0.1%.e., 0.1%.

In other words, 4.33/ In other words, 4.33/ n = 0.1 giving n=1875, n = 0.1 giving n=1875, 156 years of data are required.156 years of data are required.

There are two drawbacks, (a) the mean value rThere are two drawbacks, (a) the mean value remains fixed over such a long period and (b) wemains fixed over such a long period and (b) where do we get 156 years of data.here do we get 156 years of data.

It is almost impossible to estimate the mean vaIt is almost impossible to estimate the mean value lue within workable accuracy using historica within workable accuracy using historical data. By increasing frequency of measuremel data. By increasing frequency of measurement cannot solve this difficulty.nt cannot solve this difficulty.

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

If shorter periods are used, more samples are aIf shorter periods are used, more samples are available but each one is worse in terms of the vailable but each one is worse in terms of the CV.CV.

Basic Options TheoryBasic Options Theory

IntroductionIntroduction Option (call and put)Option (call and put) Option premiumOption premium Exercise (strike) priceExercise (strike) price Option value at expirationOption value at expiration

Call: (SCall: (STT-E)-E)++, Put (E-S, Put (E-STT))++, where S, where ST T is the price of is the price of

the derivative at exercise date and E is its the derivative at exercise date and E is its Exercise (strike) price.Exercise (strike) price.

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

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

To valuate an option, one must find a portfolio To valuate an option, one must find a portfolio or a self-financing trading strategy with a or a self-financing trading strategy with a known price and which has the same payoffs known price and which has the same payoffs as the option. By the law of one price, it as the option. By the law of one price, it follows that the price of the options must be follows that the price of the options must be equal to that of the portfolio or self-financing equal to that of the portfolio or self-financing trading strategytrading strategy..

A simple example: Suppose stock in company A simple example: Suppose stock in company A sells $100/share, the risk-free rate is 6%, A sells $100/share, the risk-free rate is 6%, now consider a futures contract obliging one now consider a futures contract obliging one party to sell stock of company A to the other party to sell stock of company A to the other party one year later from now at price party one year later from now at price $P/share, what should P be ?$P/share, what should P be ?

Factors affecting the value of Factors affecting the value of OptionsOptions

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

The interest rate.The interest rate.

What about the growth rate of the stock?What about the growth rate of the stock?

Single-period Binomial Option Single-period Binomial Option TheoryTheory

A simple example:A simple example:

Consider the following portfolio:Consider the following portfolio:

purchase x share stock and b dollars worth of purchase x share stock and b dollars worth of the risk-free asset. Thenthe risk-free asset. Then

100x+(1+r)b=20 and 60x+(1+r)b=0100x+(1+r)b=20 and 60x+(1+r)b=0

This gives that x and b, where This gives that x and b, where

x=volatility of the option/ volatility of the stock x=volatility of the option/ volatility of the stock is the hedge ratio.is the hedge ratio.

More generally, suppose that the current price More generally, suppose that the current price is sis s11 and after one period the stock either goes and after one period the stock either goes

up to sup to s33 or s or s22. The exercise price is E. The risk-. The exercise price is E. The risk-

free rate of interest is r. Using the same argumfree rate of interest is r. Using the same argument as above, we haveent as above, we have

x sx s33 +(1+r)b=(s +(1+r)b=(s33 –E) –E)++ and x s and x s2 2 +(1+r)b=(s+(1+r)b=(s22 -E) -E)++

As a result, we have the hedge ratio x and the amAs a result, we have the hedge ratio x and the amount of borrow b. The option price is x+b.ount of borrow b. The option price is x+b.

Assume that sAssume that s22 <E< s <E< s33, then the hedge ratio , then the hedge ratio δδ=(=(ss33 -E)/ -E)/

((ss33 - - ss22). The amount borrow is ). The amount borrow is δδss22/(1+r)./(1+r).

Write sWrite s33=us=us11, s, s22=ds=ds11 and C and Cuu= (s= (s33 –E) –E)++ , ,

CCdd= (s= (s22–E)–E)++ ， ， under no arbitrage assumptionunder no arbitrage assumption

(u>1+r>d?), we have the option price(u>1+r>d?), we have the option price

C=[q CC=[q Cuu+(1-q)+(1-q) CCdd]/(1+r), where q=(R-d)/(u-d).]/(1+r), where q=(R-d)/(u-d).

This is the so-called Option pricing formula and q is the This is the so-called Option pricing formula and q is the risk-neutral probability, which is the solution of risk-neutral probability, which is the solution of

ss11=[qu s=[qu s11 +(1-q)d s +(1-q)d s11]/(1+r).]/(1+r).

Two-step option pricingTwo-step option pricing

A simple example:A simple example:

A general binomial tree modelA general binomial tree model

Assume that: Assume that: At the j-th node the stock is worth sAt the j-th node the stock is worth s jj and the option is and the option is

f(j).f(j). The j-th node leads to either the (2j+1)-th node or the The j-th node leads to either the (2j+1)-th node or the

2j-th node after the time “tick”.2j-th node after the time “tick”. The time between the ticks is The time between the ticks is ΔΔt.t.

Then,Then,

f(j)=exp(-rf(j)=exp(-rΔΔt)[qt)[qjj f(2j+1)+(1-q f(2j+1)+(1-qjj)f(2j)],)f(2j)],

qqjj=[exp(r =[exp(r ΔΔt) t) ssjj-- ss2j2j]/(s]/(s2j+1 2j+1 -- ss2j2j).).

Example: Consider a stock with a volatility of its logaExample: Consider a stock with a volatility of its logarithm of rithm of σσ=0.2. The current price of the stock is $62. =0.2. The current price of the stock is $62. The stock pays no dividends. A Certain call option on The stock pays no dividends. A Certain call option on this stock has an expiration date 5 months from now athis stock has an expiration date 5 months from now and a strike price of $60. The current rate of interest is nd a strike price of $60. The current rate of interest is 10%. We wish to determine the price of this call usin10%. We wish to determine the price of this call using the binomial option approach. g the binomial option approach.

(u=exp[(u=exp[σσ((ΔΔt)t)1/21/2], d= exp[-], d= exp[-σσ((ΔΔt)t)1/21/2], R=1+0.1/12)], R=1+0.1/12)