Week 6: APT and Basic Options Theory

Introduction What is an arbitrage? What is an arbitrage? Definition: Arbitrage is the earning of risk- less profit by taking advantage of differential pricing for the same physical asset or security. Definition: Arbitrage is the earning of risk- less profit by taking advantage of differential pricing for the same physical asset or security. Typical example: two banks with different interest rate. 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. 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 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 b i,  i  0  1 b i, where 0 and 1 are constants related to the parameters a i and b i in the simplified one factor model where 0 and 1 are constants related to the parameters a i and b i in the simplified one factor model r i = a i + b i F. r i = a i + b i F.

How this comes about? Consider two different assets i and j with different sensitivities (b i  b j ) in the one factor model. Construct a new portfolio with return: Consider two different assets i and j with different sensitivities (b i  b j ) in the one factor model. Construct a new portfolio with return: r =wr i +(1-w)r j =wa i + (1-w) a j + [wb i +(1-w)b j ]F r =wr i +(1-w)r j =wa i + (1-w) a j + [wb i +(1-w)b j ]F Now pick w so that the coefficient of F becomes zero, that is, w= b j /(b i - b j ). Then this portfolio will have no sensitivity to the factor and the return of this portfolio becomes r = wa i +(1-w) a j = a i b j /(b j - b i ) + a j b i /(b i - b j ). Now pick w so that the coefficient of F becomes zero, that is, w= b j /(b i - b j ). Then this portfolio will have no sensitivity to the factor and the return of this portfolio becomes r = wa i +(1-w) a j = a i b j /(b j - b i ) + a j b i /(b i - b j ).

This portfolio is risk-free so its return must equal to the risk-free rate r f. 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 = r f ). Then 0 = a i b j /(b j - b i ) + a j b i /(b i - b j ). This portfolio is risk-free so its return must equal to the risk-free rate r f. 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 = r f ). Then 0 = a i b j /(b j - b i ) + a j b i /(b i - b j ). 0 (b j - b i ) = a i b j - a j b i, 0 (b j - b i ) = a i b j - a j b i, b i (a j - 0 ) = b j (a i - 0 ), b i (a j - 0 ) = b j (a i - 0 ), b i /(a i - 0 ) = b j /(a j - 0 ), for all i and j b i /(a i - 0 ) = b j /(a j - 0 ), for all i and j

Set (a i - 0 )/ b i = c, a fixed constant. Set (a i - 0 )/ b i = c, a fixed constant. Thus, a i = 0 + b i c for all i. Thus, a i = 0 + b i c for all i. Taking expected values, Taking expected values,  i = a i +b i  F = 0 +b i (c+  F ) = b i with 1 =c+  F as claimed.  i = a i +b i  F = 0 +b i (c+  F ) = b i 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 b i.

Consider a special portfolio p * with b i =1. It has expected return  p* = Thus, 1 =  p* - 0 =  p* - r f. 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. Consider a special portfolio p * with b i =1. It has expected return  p* = Thus, 1 =  p* - 0 =  p* - r f. 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 = b i, we get  i = r f + (  p* - r f )b i. Substituting this 1 into  i = b i, we get  i = r f + (  p* - r f )b i. 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. 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 r i = a i +  m j=1 b ij F j for i =1, …,n. Then there exist constants 0, …, m such that for r i = a i +  m j=1 b ij F j for i =1, …,n. Then there exist constants 0, …, m such that for i = 1, …,n, i = 1, …,n,  i = 0 +  j m =1 b ij j.  i = 0 +  j m =1 b ij 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 r i = a i + b i1 F 1 + b i2 F 2 + e i. Taking the covariance with the return of the market, we get r i = a i + b i1 F 1 + b i2 F 2 + e i. Taking the covariance with the return of the market, we get cov(r i,r M ) = b i1 cov(F 1,r M ) + b i2 cov(F 2,r M ). cov(r i,r M ) = b i1 cov(F 1,r M ) + b i2 cov(F 2,r M ).

We assume that cov(e i,r M )=0. Dividing this equation by  M 2, we get  iM = b i1  F1M + b i2  F2M with We assume that cov(e i,r M )=0. Dividing this equation by  M 2, we get  iM = b i1  F1M + b i2  F2M with  F1M = cov(F 1,r M ) /  M 2 and  F2M = cov(F 2,r M ) /  M 2  F1M = cov(F 1,r M ) /  M 2 and  F2M = cov(F 2,r M ) /  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. 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 = r f + 1 b i1 + 2 b i2. For CAPM, we have the SML:  i – r f =  iM (  M – r f ). Substituting  iM = b i1  F1M + b i2  F2M into the SML, we get  i – r f = (b i1  F1M + b i2  F2M )(  M – r f ). For CAPM, we have the SML:  i – r f =  iM (  M – r f ). Substituting  iM = b i1  F1M + b i2  F2M into the SML, we get  i – r f = (b i1  F1M + b i2  F2M )(  M – r f ). When both APT and CAPM hold, we have When both APT and CAPM hold, we have 1 =  F1M (  M – r f ) and 2 =  F2M (  M – r f ). 1 =  F1M (  M – r f ) and 2 =  F2M (  M – r f ).

Blur of History 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+r y = (1+r 1 ) … (1+r 12 ). For small r i, this can be written as 1+r y ~1+r 1 + … +r 12 In other words, r y ~r 1 + … +r 12. 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+r y = (1+r 1 ) … (1+r 12 ). For small r i, this can be written as 1+r y ~1+r 1 + … +r 12 In other words, r y ~r 1 + … +r 12. If we assume that the monthly returns are uncorrelated with mean  and variance  2, by taking expectations, we have  y = 12  and  y 2 =12  2. If we assume that the monthly returns are uncorrelated with mean  and variance  2, by taking expectations, we have  y = 12  and  y 2 =12  2. In other words, we can express the monthly mean in terms of annual means by  =  y /12 and  2 =  y 2 /12. In other words, we can express the monthly mean in terms of annual means by  =  y /12 and  2 =  y 2 /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  = p  y and  2 = p  y 2. 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  y 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. 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. This is sometimes known as the blur of history in statistics.

For example, let  y =12% and  y = 15%. Then CV = 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 For example, let  y =12% and  y = 15%. Then CV = 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 r 1, …, r n be iid having the same mean  and variance  2. Then an estimate of the mean is  i=1 n r i /n. E( )=  and  =  /  n. Let r 1, …, r n be iid having the same mean  and variance  2. Then an estimate of the mean is  i=1 n r i /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%. 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. 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. 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. 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 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. If shorter periods are used, more samples are available but each one is worse in terms of the CV.

Basic Options Theory Introduction Introduction Option (call and put) Option (call and put) Option premium Option premium Exercise (strike) price Exercise (strike) price Option value at expiration Option value at expiration Call: (S T -E) +, Put (E-S T ) +, where S T 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. 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. 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 ? 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. Volatility of the underlying stock. More 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 Theory A simple example: A simple example:

Consider the following portfolio: Consider the following portfolio: purchase x share stock and b dollars worth of the risk-free asset. Then 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 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 s 3 or s 2. The exercise price is E. The risk- free rate of interest is r. Using the same argument as above, we have More generally, suppose that the current price is s 1 and after one period the stock either goes up to s 3 or s 2. The exercise price is E. The risk- free rate of interest is r. Using the same argument as above, we have x s 3 +(1+r)b=(s 3 – E) + and x s 2 +(1+r)b=(s 2 - E) + x s 3 +(1+r)b=(s 3 – E) + and x s 2 +(1+r)b=(s 2 - E) + As a result, we have the hedge ratio x and the amount of borrow b. The option price is x+b.

Assume that s 2 <E< s 3, then the hedge ratio δ=(s 3 - E)/(s 3 - s 2 ). The amount borrow is δs 2 /(1+r). Assume that s 2 <E< s 3, then the hedge ratio δ=(s 3 - E)/(s 3 - s 2 ). The amount borrow is δs 2 /(1+r). Write s 3 =us 1, s 2 =ds 1 and C u = (s 3 – E) +, Write s 3 =us 1, s 2 =ds 1 and C u = (s 3 – E) +, C d = (s 2 – E) + ， under no arbitrage assumption C d = (s 2 – E) + ， under no arbitrage assumption (u>1+r>d?), we have the option price C=[q C u +(1-q) C d ]/(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 s 1 =[qu s 1 +(1-q)d s 1 ]/(1+r).

Two-step option pricing A simple example: A simple example:

A general binomial tree model Assume that: At the j-th node the stock is worth s j and the option is f(j). At the j-th node the stock is worth s j 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 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. The time between the ticks is Δt.Then, f(j)=exp(-rΔt)[q j f(2j+1)+(1-q j )f(2j)], q j =[exp(r Δt) s j - s 2j ]/(s 2j+1 - s 2j ). q j =[exp(r Δt) s j - s 2j ]/(s 2j+1 - s 2j ).

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. 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)