Advanced Finance

Objective of the course:

Kowledge and comprehension:

  • Knowledge of international finance, derivatives and credit risk, active management, efficient markets, behavioural finance, factor models and hedgefunds.

Skills

  • Be able to explain important topics and derive results in international finance, derivatives and credit risk, active management, efficient markets, behavioural finance, factor models and hedgefunds.
  • Apply theoretical concepts and ideas to new areas.

Competense:

  • The student should be able to develop independently their own competence and expertise in the field of international finance, derivatives and credit risk, active management, efficient markets, behavioural finance, factor models and hedgefunds. Moreover, he or she should be able to discuss central questions, analyses and conclusions pertaining to these topics.

Topic 1: Contemporary Research in finance

Topic 2: APT and Fama French

Topic 3: Options - Binomial trees

Topic 4: Options - The Stock price process

Topic 5: Black-Scholes-Merton option pricing

Topic 6: Options and derivatives - The greeks

Topic 7: Options and derivatives - Volatility smiles

Topic 8: Active Management - compensation

Topic 9: Behavioral Finance

Topic 10: Market efficiency - Grossman Stiglitz

Chordia et al.2015:
Evidence on the speed of convergence to market efficiency

Order book

Shows number of Stocks avaliable at different prices

Order imbalance (OIB):

  • Number of buyer initiated order and number of seller initiated orders

Buyer initiated: Buyer Accepts an order in the order book

Seller initiated: seller Accepts an order in the order book

Assuming a New big investor starts buying (accepting ask orders in the order book)

The number of buyer initiated orders increases -> increased OIB => increased price

There is a Connection between OIB and the price in the very short horizon - it takes time for the market to react

The market is not strong form efficient (Fama French)

If OIB is known, prices can be predicted (at least a couple of minutes). However only specialists have this information at NYSE

Market maker/specialist offer bid and ask quotes. If many buyers arrive, it suggests that there is New information in the market. Could the Stock be Worth more? -> the market maker adjusts bid and ask up

Negative conditional Serial correlation: - Prices are not unconditionally serially correlated
-Given OIB, there is however a significant negative serial correlation
-That is, if you know the last period order imbalance is positive (buyer initiated trades)
-Current price is expected to increase
-Next period price is expected to decrease
→ Negative conditional serial correlation


Reason:

  • Market makers adjust the quotes to much when OIB > 0. Perhaps to be safe?
  • In the Next period the price falls on average

Conclusion:

  • Prices are not strong form efficient. May suggest that marketmakers are not optimally performing? Could have exploited the systematic reversal by adjusting quotes more efficiently
  • The market is efficient With respect to other market paraticipants

Picking winners? Investment consultants recommendations of fund managers (Jenkins, Jones and Martinez 2016)

What influences reccomendations?

Factors:

  • “Soft investment factors”: clear decision-making, capable portfolio manager, and consistent investment philosophy
  • Past performance
  • Past return volatility
  • “Service factors”: capabilities of relationship professionals, usefulness of reports prepared by the fund manager, effective presentations to consultants
  • Fee
  • Assets Under Management (AUM)

Conclusions

  • Consultants'’ behavior confirms our worst fear
  • Rely on insignificant or counterproductive information
  • Highly successful in generating asset flow, changes in recommendations generates more flow
  • Fail miserably in spotting winners

Admati & Pfleiderer

Questions :

  • How to compensate managers?
  • Fraction of Portfolio or success fee?
  • Admati & Pfleiderer -> fraction of Returns

Benchmark:

  • Manager receives h2(x-b)’r where b is benchmark and x the chosen portfolio

Optimal porfolio (tegn og forstå denne)!

Optimal compensation

Benchmarking does not align the interests of the manager with those of the investor

The manager invests some in the benchmark rather than in the optimal portfolio

There exists only one optimal Portfolio - the manager has no reason for his effective porfolio to deviate from the optimal one

Solution?:

  • He therefore invests such that the benchmarking is completely canceled out
  • and only the optimal portfolio remains
  • The manager achieves a more balanced “portfolio”
  • The investor gets a suboptimal Portfolio

Conclusion:

The investor is better of giving the manager a share of the Portfolio.

The norwegian oilfunds compensation of external fund managers

More Extreme than pure benchmarking

Benchmarking needs to be symmetric - if the manager looses he pays to the investor

Success fee:

  • Only fees if all goes well
    → Incentives to take risks (in addition to a misaligned portfolio)
    Not considered by AP

Fama & French: Disagreements, tastes and assets prices

What if the uninformed only buy index funds? (viktig figur)

When is the market Portfolio "M" efficient?

When the uninformed holds the index Portfolio. Market Portfolio is only efficient when uninformed holds the market portfolio

When the prices are 100% efficient - We know from Grossman & Stiglitz that prices cannot be fully efficient!

Too many investors think they can beat the market portfolio, and hold inefficient portfolios in stead of the index

Taste and belief

If suboptimal porfolio is a result of difference in taste -> permanent inefficiency
If suboptimal Portfolio is a result of differences in belief -> temporary inefficiency

Anomalies

Is the suboptimal Portfolio far from the optimal one?

In an efficient market there is only one risk factor - the market (CAPM). However we know that there are numbers of anomalies affecting Returns (book-to-market ratio, share issuing, net size, accruals, momentum, change in assets and profit etc.. )

Why there is no trade in a fully efficient market

Noise traders are accually needed

If market is fully efficient - all information is reflected into the price. There would be no incentive to acchieve information since it has no value at all

Who sell when others buy, and buys when other sell

Stock prices move in discrete intervals, both in time and value. The assumtion of contioneous time and contineous prices is however convenient in Mathematical terms. It is also a fairly good approximation - we therefor often model Stock price movements as contioneous process.

1. Wiener process

The change in the price divided by the Stock price itself, deltaS/S. Also adds a radom variable element, since prices are not fully predictable. The random element should be so that the variance increases proportionally With time.

The additive property of the Wiener process

2. Stock price process

In addition to the random element, it is reasonable to assume that the Stock is expected to change by a given percantage each period - including the drift term. Adding the drift term and the stochastic Wiener process toghether gives the Stock price process:

  • deltaS/S = mudeltat+sigmadeltaz

If the time between intervals is short we denote "d" instead of delta as difference operator. Which gives:

  • dS = muSdt + sigmaSdz

3. Itos lemma and a strange contract

Imagine that you buy a contract that you can Exchange for the square of the Stock price at any time. That is: G(S,t)=S^2

Note that the function does not actually depend on the t (time). However some derivatives do depend on time, so it is common to have t as an argument.

Now imagine that you would like to predict changes directly from the Stock price rather by observing the value of the contract. This would be particulary useful if S^2 is actually only paid out some terminal date/at maturity. This is not uncommon (European option)

A man name Ito found a way to calculate the change in G(S,t) many years ago (1944). His Method is called Itos lemma:

  • dG = (GsmuS + Gt + 1/2Gss sigma^2S^2)dt + GssigmaSdz

In this case it is assumed a specific Stock price process. These can both be expressed in more general terms as well.

The general case: generalized wiener process

Assume a more generalized formulation of the Stock price process. Then a generalized Wiener process can be formulated as: dx = a dt + b dz, where -> dz = epsilon*dt^0,5

The wiener process was first used in physics to describe the motion of particles. We now see that the Stock price process is a special case of the Wiener procecess where a= muS and b=sigmaS

In general the Itos lemma says that given the general wiener process, the process of G(x,t) is the wiener process:

  • dG = (Gxa + Gt + 1/2Gxxb^2)dt + Gxbdz
  • Where Gx and Gt are partial derivatives of G With respect to x and t respectively. And Gxx is the partial double derivative With respect to x.

Some definitions:

  • Markov process: A stochastic process where only the present value of the variable is relevant for predicting the future
  • Wiener process: A contineous time Markov provess where the random changes are normally Distributed. dz=epsilon*( dt^0,5). When d approaches 0, the Markov process z approaches a continous Wiener process.
  • Generalized wiener process: a Wiener process With added drift term
  • An Ito process: A generalized wiener process where the parameter are functions of the time and the variable x.

Typically options are sold by Financial institutions, who act as market makers in the option market. These institutions need to cover the risk of their option Portfolios. "The Greeks" are Properties of options that Portfolio managers use to Control risk.

4 ways to hedge a position

Naked position (not hedging):

  • This is the simplest form of hedging strategy. Holding a naked position meaning just to hold a simple position without hedging the risk. This obviously exposes the investor to a lot of risk, and is therefor not a alternative for issuers of options.

Stop loss:

  • A hedge that is very simple to use. Within this Method the issuer buys the option whenever the option is in the Money, and holds a zero position. The disadvantage With this strategy is that the issuer is still exposed to a lot of risk.

Delta hedging:

  • Is a better approach.
  • This means holding a position in the underlying asset such that the change in the option Equals the change in the assets (show it graphically).
  • The number of assets you need to hold to neutralize changes in the option is called delta. This is simply the derivative of the option price With respect to the asset price: delta = dF/dS. Where F is the option value and S is the option price.

Each time the asset increases, we need to buy and each time it declines we need to sell. There is a cost of hedging as well as a cost of holding a fixed leverage ratio - this cost is actually compensation for risk. The holder of one unit of the option or someone doing delta hedging are exposed to a lot less risk than the owner of one unit of the underlying asset.

Delta neutrality:

  • Options that you buy on Exchanges are typically sold by banks that act as market makers. These banks earn profit on the option spread (the difference between bid and ask quote of the option)
  • However in contrast to market making in the underlying asset, the bank does not have to buy and sell an Equal amount of options - instead banks tries to obtain delta neutrality by trading the underlying asset.
  1. A bank that sells options typically calculate the delta of its Portfolio as: delta = SumWi* delta i
  1. "Wi" is the amount of option "i" With delta "delta i" in a Portfolio of options With the same underlying asset. If delta > 0 (the bank holds mostly Call options), then the bank attemt to hold a short position of "delta" assets.
  1. The net position of the bank is now riskless for this asset, until the underlying asset changes significantly. Then the bank need to rebalance. How often this happens depend on the volatility of the underlying asset, and how sensitive the delta is to changes in the asset (the latter is called the option gamma)

Gamma hedging

  1. Gamma is the change in delta. It can be defined as the second partial derivative of the option value With respect to the asset price. Gamma = d2F/d2S = Fss
  1. If gamma is high, it means that delta may change a lot even though the underlying asset only changes a little. Hence, With a high gamma and/or high underlying asset volatility, rebalancing should be done frequently.
  1. By selling additional neutralized options, the Portfolio can be made gamma neutral. In particular if the bank sells the option With the highes gammas (options at the Money), the bank might get away With selling fewer options than what is in the original hedging Portfolio.

Gamma hedging is cheaper to use due to no need for buying more Stocks to rebalance!

The Greeks:

Theta:

  • It is the time decay of an option. At the beginning of the life of a option it will have a relatively high intrinsic value (see graphically). As the date of expiry draws closer the intrinsic value will fall towards the exercise value.
  • At expiry there is no intrinsic value left, and the value will Equal the exercise value for any asset price. Hence, for a given asset price, the option price will consistently fall through the options Lifetime.
  • The decay for options is simply the partial derivative of the option value With respect to time: Theta = dF/dt = Ft
  • Since the intrinsic value of an option always falls With time for a given asset price, it is generally true that Ft<0.

Rho:

  • It is the sensitivity of the option value to changes in the interest rate (effect of interest rates).
  • Rho = dF/dR = Fr
  • Usually for options the effect is Limited and the volatility of interest rate is generally small. However, for interest rate derivatives this is an important property.

Vega:

  • Vega is the sensitivity of the option value to volatility.
  • Vega = dF/Dsigma = Fsigma
  • Relevant når det omhandler volatilitetssmil

Portfolio Insurance:

  • This can be obtained by using options and hedging. The simples way is to buy a Call option and put the rest of the Investment in the bank.
  • Any point of insuring against the Equity market? Not if you want to gain a risk Premium. It does not make much sense to insure against market risk, since it will only reduce Your risk Premium by the same amount. Including fees, such Insurance will imply a loss. Trading options are in general much more costly than trading the underlying asset.

Transaction costs:

Relationship between the Greeks

Transaction costs are mainly bid and ask spreads. If these are about 0,05%, there will be a 2% hedging cost for the hedging strategy. Hedging can increase trasaction costs significantly. If rebalancing is done With higher frequency, then the transaction costs would also be higher. The more volatile the underlying Stock, the higher gamma, the more frequent the rebalancing would be. Hence options With need for frequent rebalancing should trade at high spread.

  • Estimate: Multiplying the percentage cost of 2% by the times the Portfolio is rebalanced each week.

The Black and Scholes Equation is defined as:

  • FsSr + Ft + 1/2Fsssigma^2S^2
  • Replacing the partial derivatives With the Greek
  • DeltaSr + Theta + 1/2gammasigma^2S^2 = Fr
  • We can se that when the Portfolio is gamma and delta neutral, then its sensitivity to the interest rate should be proportionally to the option value.

From preveously we know that Itos lemma represent the innovations for the contract and that Stock price is defined as dS=muS dt + sigmaSdz. We can use this to derive a condition that needs to hold for any derivative

What is a option?:

  • It is a right to buy or sell a an asset for a certain price. Options can be used for both hedging and speculation. It is a type of derivative.
  • Perhaps the most important property of a option is that one can use them to Place large bets in the market With little Capital (gearing). Of course the risk is bigger, should be carefully and not investing to big part of the portfoliio in options.

What is a derivative?:

  • A asset where the price is determined completely by other assets. Hence the value of the derivative is a function of the asset value.

Value of a call option: (x-S)

Value of a put option: (S-x)

Important features:

It is in general never advantageous to exercise an option early before expiry date. Usually one get more by selling in the rights in the market, since the option has an intrinsic value Beyond the exercise value before this date.

This means that it if it is a long time until expiry date, then an option has some value, even if the price is lower than the exercise price since it is always a possibility that the option will pay out as long as we are not at the expiration date.

Distinguishes between European and American options

American: you can choose whether you wish to exercise the option before the terminal date

European: the option can only be exercised on the terminal date.

Put-call parity

St-NPVt(X) = Ct-Pt

  • this formula applies to any time, also before the terminal date. If one knows the value of the Call option, one can use this formula to find value of a put option.

A share and the present value of a bank loan of NPT (X) will always Equal the difference between a Call and a put option. The loan however will be charged With interests, therefor the difference is the NPV of X:

  • NPV (X) = Xt / (1+r)^T-t

When the current value of the Stock is lower than the exercise value of a Call option, then the owner will not get anything if it is exercised immediately - we than say that the option is out of the money

In general:

  • A Call option is in the Money when St > X and out the Money when St < X
  • A put option is in the Money when St < X and out the Money when St > X

When St = X then the option is at the Money for both option types

Combining options

  • One can combine the purchase and sale of puts and Calls With different exercise price, to design some special cash flow at different share prices.
  • When taking into account the intrinsic value a combination will provide cash flow as long as the Stock price is between "a" and "b". Possible to use this information in order to Place a bet that the price of some Stock will be within a certain Level (Telenor example)
  • Similary one can use options to bet that the price is going to be different, positive or negative, from a specific target - imagination is the only limit to cash flows that is possible to obtain.

Pricing of options

Why would anyone issue a option when it only involves a obligation to buy or sell at a certain price?

  • The answer is of course that the issuer will charge to do this. How to know what to charge?

Relationship between option value and Stock is actually a formula that allows to calculate the value of the right that the option represent - Black and Scholes option pricing formula:

  • Probably one of the most useful inventions in the history of Finance, the formula is in daily use in the Financial markets.

A general binomial formula

When the Stock and option can take only two values, we Call it binomial option pricing.

(Få med eksempel på notatark)

Want to find a combination of loans and Equity Investments than provide the same Return as an equivalent position in the option - this is the goal in all option pricing.

Risk neutral probability

Can be calculated as: q = ((1+r)^T- d) / (u-d)

q is really just a tool when calculating option prices, not a probability at all. q does however have in common With real probabilities that it always lies between one and zero, and we use it to calculate an expectation in the same way as we do With normal probabilities.

Called risk neutral probability due to the fact that if we use it on the terminal values of the underlying (Su and Sd), then the expected value Equal to the current Stock price + interest:

  • Su q + Sd(1-q) = S0 * (1+r)

Once we find q, the option price is simply given as the present value of the expected option value at exercise, where we use risk-neutral probability q as the probability of "up" and "Down":

  • f0 = (qfu+(1-q)fd)/(1+r)^T

Need to discount expected values of because "fu" and "fd" are not present cash flows, but represent a cash flow at the terminal time T.

The way one calculates a binomial option is as follows:

  1. calculate "u" and "d"
  2. calculate the risk neutral probability "q"
  3. calculate the present value of expected option value With probability "q"

Although we only use the risk-neutral probability to calculate simple binomial options, this is a remarkably general Method for finding prices for all types of derivatives. Generally, the price of any derivative can be found by defining risk neutral probability distribution.

How many shares we need to buy in order to replicate is also a important feature (delta):

  • delta = fu - fd / Su - Sd
  • delta is useful since it indicates how many shares you must buy if you want to replicate it. Since more shares means more risk, the delta also indicates the amount of risk inherent in the option.

Options are frequently used in employee compensation shemes, specially for middle and top management.

By issuing Stock options a Company can provide management With strong incentives to increase the value of the shareholders and thus the share price, without the need to dilute shareholderst interest to much today.

To strong insentives? Short term thinking priority over long term?

Multi-period options

Not very realistic that Stock prices can take only two values. In principle it is possible to buy shares for any cost. We can improve the binomial model!

Need to consider the Stock price as a continuous variable:

  • put togheter several simple binomial models from a binomial tree.

4. Derivation of Itos lemma

Taylor Approximation

In order to understand the derivation of Itos lemma, we need to know what a Taylor approximation is:

  • A Taylor approximation (also called Taylor expansion) is a way of approximating the value of a function nearby some point using a series of increasing derivatives.

If we know the function and its derivative at a point x, we can find a approximate value for the function nearby this point, at x+dx by using the second order Taylor approximation (see illustration).

  • f(x+dx) = f(x)+f'(x)dx + (1/2)f''(x)(dx)^2
  • this is the known function value, at a point "A" (in figure). Now, by adding the first derivative we can follow the slope of the function and get a fine estimate at point B in the figure.

However, since non linear functions are curvative we can do even better by using the information that lies in the second derivative (the second derivative is negative for the figure). Adding a second derivative term therefor brings us approximately Down to accurate point "C". In general one can keep adding terms With increasing derivatives to increase accuracy, but in this case it is sufficient With a second order expansion.

We define:

  • df(x) = f(x+dx) -f(x)
    Therfor we can rewrite Taylor approximation as:
  • df(x) = f'(x)dx + (1/2)f''(x)dx^2

How do we arrive at Itos lemma?

We want to approximate the general function:

  • G(S,t).
  • The principle for expanding functions With two arguments is almost the same, except we get one extra set of terms and we also need to add a last term to take account of any Independence between the two variables.

The Taylor expansion of dG(S,t) With respect to both S and t are therfor:

  • dG(S,t) = Gt dt + 1/2Gttdt^2 + GsdS + 1/2GssdS^2 + GtsdSdt
  1. First we look at (dS^2). Inserting from (9) we get:
  • dS^2 = (muSdt + sigmaSdz)^2
    It is the same as:
  • muS^2dt^2 + 2mu sigma S^2 dt (dtdt epsilon)^0,5 + sigma^2S^2 dt epsilon^2
  1. When time goes to zero:
  • As "dt" goes to zero, any dt term of higher Power than 1 will become insignificant compared to dt, since there are no terms With lower Power than 1 - we can delete any term raised to more than 1. Therefor:
  • dS^2 = sigma^2S^2dt*epsilon^2
  1. Predictable variance in the very short run:
  • As dt goes to 0 the number of draws of the random variable epsilon^2 even within a very short time interval will go to infinity. Hence it can be replaced With its expectation E(epsilon^2)=1 (remember it was standard normal). Therefor:
  • dS^2=sigma^2S^2dt
  1. Now substitute (23), (24) and the definition of the Stock price itself (9) into the approximation for dG(S,t) in (20) and delete the dt^2 term. After rearranging we get Itos lemma:
  • dG(S,t) = (Gs muS + Gt + 1/2Gsssigma^2S^2)dt + GssigmaS*dz

Application of Itos lemma on Stock prices:

In addition to calculate the innovations in Stock price process that depend on S, we can use Itos lemma to derive the price Level of the Stock price process (9).

We first define:

  • G = lnS
    Which gives the derivatives:
  • Gs = (1/S)
  • Gss = -(1/(S^2))
  • Gt = 0

Inserting the derivatives into Itos lemma (10) we get:

  • dlnS = (mu - 1/2sigma^2)dt+sigma*dz
  • hence the Natural log difference of Stock prices is normally Distributed - therefor we say Stocks are log normally distributed

Furthermore we can add these differences together from initial time 0 up to time T. Rewrite this as:

  • lnSt - lnS0 = sum dlnSt = (mu - 1/2sigma^2)dt T + sigmazt
    additive Wiener process - convinient. Get a explicit solution for Stock prices:
  • St = S0 exp((mu - 1/2sigma^2)T+sigma*zt

the exponential term is almost identical as to the proposed Stock process (9). Why -1/2sigma^2? Reason:

  • the effect of the random term is different depending the Stock moves up or Down. For example, if the Stock moves up 10% from 100, and then Down 10%, the price wil be 99. Because a 10% decrease from 110 is larger than a 10% increase from 100.
  • The term -1/2sigma^2 cansels this effect ensuring that E(St)=S0e^muT

The log normal property of Stock prices:

Since St = S0e^(mu-1/2sigma^2)T+sigmazt, and since zt is normally Distributed With zt ~ N(0,T) it follows that the log of Stock prices is normally Distributed. A random variable that becomes normal when taking log, is called a log-normal variable.

Using Itos lemma process (1) and the Stock price process (2) we can Write out the changes in the Portfolio when selling Gs short (must sell Gs Stocks in order to remove any stochastic movements):

  • dG - GsdS = (Gs + muS + Gt+ 1/2Gss sigma^2S^2)dt + GssigmaSdz - Gs(muSdt + sigmaS*dz)
  • = (Gt+1/2Gss * sigma^2^S^2)dt

This result is free from any stochastic disturbance

If a Return is risk free it has to pay the risk free interest rate or else there would be arbitrage opportunities. Hence:

  • dG - GsdS = (G-GsS)r*dt
    substituting for dG-GsS and rearranging we get Black and Scholes:
    rG=Gt + rGsS + 1/2Gsssigma^2S^2

The Black and Scholes differantial Equation gives us a condition that any derivative must satisfy if arbitrage is not allowed and trading is done approximately contineously - hence it is a very general result.

Risk neutral valuation

Assume that there exists no compensation for risk. Under this assumption we can replace mu With r (risk free rate) in the Ito Equation. Since E(dz)=0, the expected change is then:

  • E(dG,mu=r) = (GsrS+Gt+1/2Gsssigma^2S^2)dt
  • the only thing done here is remove dz since it on average will be 0, and replace mu With r.

Since there is no compensation for risk, we will expect the contract altso Return the risk free rate:

  • E(dG, mu=r)=rGdt
    Hence we get
  • rG=Gt+rGsS+1/2Gss sigma^2S^2
  • this is the Black and Scholes differential Equation

This means that if we start out by assumin there is no risk compensation, we end up at the same Place as we did when we used the no arbitrage argument.

  • the conclusion is that any contract can be priced by assuming a probability distribution where we replace drift parameter "mu" With risk free interest rate "r".

The procedure to find the value of any contract:

  1. Replace drift term "mu" With risk free interest rate r in the Stock price
  2. calculate the expected value of the derivative at maturity
  3. discount the result to the present value

Risk neutral option valuation

The general formula:

  • St(mu,epsilon)=S0e^(mu-1/2sigma^2)T+sigma(Tepsilon)^0,5

Irrationality

Market efficiency

Are markets efficient?: - unless everybody holds the same Portfolio (which we know is not the case) there will always be winners and losers. Hence, if we focus on the winner, he will always appear exceptionally talented, even if he selected Portfolio by throwing dart.

Difficult tho seperate Luck from actual ability (CEO example - pure Luck?)

Testing for market efficiency (Eugine Fama)

  1. Weak form efficiency: (price history)
  1. Semi-strong efficiency: (public information)
  1. Strong-form efficiency: (private information)

if it is easy for managers to beat the market - it is not efficient. Most scholars agree the market are not strong form efficient, the interesting tests are within the two first.

Tests for correlation. If there are any covariance/correlation between the change in price today to tomorrow and the change from yesterday to today, it means that yesterdays price carry information about today.

There are several results in literature that suggests that Public information have a effect on asset Returns,

A different approach: divide the market into winners and losers for some time period. If the winners outperform the losers in the next time period, it is evidence of momentum - some support for this in a 3-12 month period.

If we exted period to several years there may be a reversal effect, where Companies that have done it well in previous years do worse than their peers in future.

Two interpretations of the medium term momentum and short term reversal effect:

  1. Either it suggest that the market overreacts slowly to News, and so over a 3-12 month time the price increase more than the index. In the subsequent years the effect is reversed back to correct Level.
  2. Alternatively momentum is compensation for risk - you can make Money betting on last years winners, high Returns often is a result of a Company that have experienced important events during the year (mergers, aquisitions or New Products). Potential gains not realized? The market requires a preimium to hold promising Companies. The reversal may be explained by a decline in risk Premium when real implications of events are better understood by the market.

Small firms tend to outperform bigger firms - goldmines or risk Associated With such firms? Types of risk:

  • More Public information on larger firms
  • Small firms often illiquid and expensive to trade (Premium cost)
  • Compensation for unobserved risk, why has the Company declined in value?

Book to market (B/M): - Firms With high book value tend to outperform other Companies. Is the Company undervalued?

  • Companies are often undervalued because they have few "Growth" prospects. These will deliver stable Return but there is no "Growth option"
  • younger Companies often have better Growth options, hence they are often assigned With a higher valuation

Post earnings anouncements drift:

  • evidence showing that there is drift after News. In the case of bad News the price declines slowly and in the case of good News it drifts slowly upwards. Clearly, one could exploit this fact by buying shares after good News and selling after bad. This is not consistent With a efficient market

A possible explanation can be that risk increases after anouncements. The Stock price is however not acting as expected if this was the case. If risk Premium was the reason one would expect gradual increase in price after the announcement in the case of bad News as well. Since this is not what is observed - it is more likely that post earnings announcement drift is evidence that the market is not perfectly semi strong efficient.

Other anomalies

Returns are affected by net size, accruals, change in asset profit etc. Difficult to explain in terms of risk premium

It would be strange if private information could not been used to generate excess Returns. Hence, the little evidence that exists here suggests insiders do have a advantage.

Anomalies - interpretated and a possible explanation

Mainly two explanations:

  1. A result of risk Premiums (as suggested by Fama-French)
  2. Or just inefficiencies ready to be exploited
    The thruth may lie in between. It is difficult to understand how well known Premiums can exist over time without beeing exploited. On the other hand, why isnt risk Premiums dissapearing after it is well known? Such as small firm effect? If the effect does not go away it is reasonable to assume some kind of compensation for risk.

Fama - French 2007:
New explanation for anomalies. (graphically - volatility diagram)

  • Uninformed hold a "irrationally" sub optimal Portfolio, rather than the index. What if all uninformed investors decided to invest in the index? Then everybody would get the optimal efficient Portfolio. But, when the market is 100% efficient, there would be no trade. The irrational traders therefor have a important role in modern Financial markets by making it worthwhile for informed traders to acchieve fundamental information.

Derivatives:

Pros and cons for structured Products/derivatives:

Pros:

  • Possible to speculate With low capital
  • Structured Products may change the distribution of the Returns. If the customer prefers some distribution of the Returns he might be willing to pay for this service

Cons:

  • High fees
  • The fact that it is not possible to increase risk/Return relaltionship show that derivatives do not affect risk adjusted returns
  • From behavoral economics we know that very few People understand probability distribution very well, so it is unlikely that the customer really understand what a preference for a distribution really means. It is therefor unlikely that there exists any willingness to pay for this. A anomali?

Implied volatility

A well known anomaly of options is the "volatility smile". In order to understand this we need to introduce implied volatility.

The volatility we need in order to get indentical prices for observed and theoretical price. Sometimes there might be a deviance in price between theoretical and observed price.

It is likely that the difference is due to different views on the assets volatility through the market.

The volatility smile

One would usually assume that the volatility is independent of strike price of a option. The strike price is a property of the contract between the seller and buyer - should not affect the riskiness of the underlying asset!

The funny thing however is that the strike price does affect the implied volatility!

It turns out that when the option is deep out the Money, or in the Money (in the endpoints), the implied volatility is much higher than at the Money.

The reason for this is probably since investors consider an option deep out of or in the Money as more risky than one where the asset price is Close to the strike price.

The most likely explanation is that Returns are not normally Distributed as assumed by Black and Scholes. That means that the actual risk might be far from the expected future value.

How is the intrinsic value affected by strike prices?

  • If Returns are not normal, but have fatter tails, it means that the actual risk in case of an Extreme event is much higher than the standard deviation would suggest.

It is well known that Stock Returns have higher peak and wider/fatter tails than normal distribution. Often called Leptokurtic distribution.

Issuers of options require compensation for this risk, therefor demanding higher prices for options into or out of the money

Determining implied and probability distribution

As we know from option pricing, the option value Equals the expected value of the option using a risk neutral probability distribution, and discounting to present value.

  • Can use this in order to infer the implicit risk neutral probability distribution.
  • With risk neutral prices, the price of a European Call can be calculated as: C=e^-rt Integral(k - infinity)* (S-K)g(S)dS

If we differentiate this twice With respect to the strike price K, we get:

  • d2c/d2K = e^-rt * g(K)
    Solving for the density function gives:
    g(K) = e^rt*d2c/d2K

Hence, by numerically calculating the partial derivative from option prices we can approximate the risk neutral distribution consistent With a the smil (volatility smile)

To obtain numerical estimate of the second derivative, we need Three prices for each point estimated:

  • Difference between the strike prices of delta, so that c1, c2 and c3 are the prices of Call option With strike prices K-delta, K and K+delta. Formula for calculation:
  • g(K)=e^rt*(c1+c2-2c2)/delta^2

Why a smile?

Mainly two reasons for the fat tails and volatility smile:

  1. The volatility is not constant
  2. The price changes are jumpy

These are interconnected/related. If prices are jumpy then volatility is not constant!

With constant volatility, deviations from the mean will always be normally Distributed by the Law of large numbers. When it is not, it suggests that the Returns are drawn from a different distribution With different variances.

Stock prices have normal distribution With 3 different variance regimes (higher peak and fatter tails)

Volatility smile of Equity:

For Equity the volatility smile is often skewed, With lower implied volatility for high strike prices. The explanation has to do With leverage. All Equity is leveraged, since Companies are financed With debt as well as Equity.

  • when the price of the Equity is low, the leverage (debt to Equity ratio) increases. Hence the issuers of options find the options find the more risky and charge a higher price.
  • opposite if price is high and leverage low, then the risk is low and so will the option price be.

A way to explain observed anomalies is to allow for irrational investors. Behavioral Finance has identifyed several ways in which humans make systematic mistakes in decicion making.

  • If a substantial number of investors are not rational, that would not itself lead to innefficient market.
  • The mistakes made by irrational traders would lead to arbitrage opportunities for the rational.
  • The result would be that the rational would gain over time, and the irrational lose - after a while this would lead to the irrational beeing driven out of the market!

If the irrational investors are a relatively small Group, these constrains will not eliminate the markets ability to exlopit these traders. However, however rationality is a widespread and the number of rational traders i small, then the rational traders woud need to take large positions to fully exploit the arbitrage possibility.

  • Hence, - contrains on credit and short trading limits the opportunities for arbitrage

The debate - how widespread is irrationality?

Active investors cannot outperform the markets as a Whole.

  • Active investors holds the market Portfolio, but have higher costs than those just holding the index. In that sense the market is efficient.
  • It might be the case that some of the active investors are irrational and lose consitently, but that is hard to imagine that the majority of traders can be this over time, since even With limits on arbitrage these traders would lose consistently over time.

Rationality

Three assumtions must hold for a person to be rational:

1. Completeness

2. Transitivity

3. Independence

All alternatives/actions can be ranked in order of preference (indifference between two or more is possible).

  • It is not uncommon that humans have hard time to deciding between the alternatives, but that often is the case of indifference, and not the ability to rank.
  • Some rare exeptions, like deciding who should die (not subject to rationallity and therefor not economics)

If A is prefered over B, and B is prefered over C, then A is prefered over C.

  • It is sometimes observed that that People either change their preferences over time or just dont understand Choice - preference reversal? Is Limited information the cause to this?

Independence of irrelevant alternatives - if the set of options are (A,B,C) and A is prefered, then A would still be prefered if the alternatives only was (A,B).

  • Highly Associated With transitivity.
  • a real world example is the sunk cost fallacy - old car, repairs?. The purchase amount is a sunk cost that is irrelevant, but many would think that if they did not repair, they would "lose" the initial Investment from buying the car

Likelihood of irrationality in fiance

For irrationality to explain the anomalies in Finance, those anomalies need to be a result of the stated assumptions (completeness, trasitivity, independene).

Distinction between irrationality and mistakes. Mistakes can be corrected over time.

  • the idea of "irrational" traders causing market anomalies is probably misleading. A more likely explanation is that People make mistakes.

The behavioral pitfalls

Technical analysis

Misjudging probabilities - pople are terrible at it:

  • Overconfidence - once a person has made a decicion, he tends to overestimate the probability that he is right. People also overestimate their own abilities.
  • Small samples - People put to much weight on small sample evidence. Healing and technical analysis example.
  • Forecasting - put to much weight on recent experience compared to the full history/data.
  • Conservativism - People tend to slowly update their beliefs

Might be helpful if market participants are not rational. Not helpful when rational market since price history are weak form efficiency.

Risk and Return are also quite abstract terms, and most People understand it bad. Risk, variance and volatility are particular difficult concepts to understand.

A few common mistakes:

  • Framing - a decicion may rely on how the answer is formed.
  • Mental accounting - it is cognitive easier to asses different decicions seperately rather than a Whole. Different Investment Portfolios for pension, childre - the optimal would be to manage them as one.
  • Prospect theory - assumes that the utility function has a S shape and depend on current wealth. People are loss averse, a loss is felt twice as hard as a winning is positive. Loss aversion mean that you are willing to impose risk to avoid a loss. A interesting property: risk averse to gains and risk seeking on losses!

Techical analysis might be possible, but only when teqhniques are not well known.

No academic evidence that it produces excess Return. Increases risk, which again increases expected Return. A explanation why it Works for someone. Impossible to test it statistically.

Some technical trading Methods:

  • Dow Theory: Three different trends (the primary, the secondary and tertiary trend.
  • Moving averages: A sliding moving average of Stock prices to find signals for when to buy and sell.
  • Volume: Number of Stocks in the market. Both the volume in order books as well as volume of transactions. If a rise happens With little volume it is likely due to New information.

Important assumptions under APT:

  1. Risky Returns can be explained by a factor model.
  2. There are sufficient assets to diversify any idiosyncratic risk
  3. Well functioning markets do not allow for the presistence of arbitrage

APT was Introduced due to the shorcompings of CAPM.

  • CAPM = Eri = RF + beta(rm-rf)
  • The biggest assumption under CAPM is that in equilibrium everyone holds indentical Portfolios. We know this is not true.
  • Second CAPM is only based on one factor, there is evidence suggesting that expected Returns depend on more than only the market factor.

Difference between APT and CAPM:

  • Main difference between the two is that in APT we consider difference to the mean rather than difference to the risk free Return. APT is more flexible since it does not require everybody to hold the same Portfolios. Under APT one can add any factor as explanatory variable in addition to the market index. The most imortant assumption of APT is the one regarding no arbitrage.

The one-factor model

ri = mui + BetaiF + ei

The difference is that the left term is not an expection (E) as in CAPM.

Idiosyncratic risk
ei represents the undexplained asset Return. The idiosyncratic Return is not correlated With any known factot that affects the other assets. It is therefor uncorrelated With other assets. This meaning:

  • Total varaiance of this idiosyncratic risk is simply the sum of the idiosyncratic contribution from each asset in the Portfolio: var(1/n ei)=1/n^2*sigma^2e
  • As the number of assets n increasesm the total idiosyncratic variance will deminish (well diversified Portfolio). For large enough n the variance is nedligible.

Arbitrage opportunities

Assume we have two large well diversified Portfolios, A and B With zero idiosyncratic risk. Lets assume both have factor betas Equal to one:

  • ra = muA + F
  • rb = muB + F
    now consider buying A and selling B. Return would then be:
  • ra - rb = mu A - mu B, since the diversifiaction removes all risk, this implies an arbitrage opportunity if the net Return is positive. The investors would neutralize this effect by buying one and selling the other, creating equilibrium again.
  • With only one factor, all variation in Return should in equilibrium be explained by different exposure to that factor.

Factor portfolios

We can add factors, such as maro-economic factors (GDP, gross Investment, % change in inflation), but it can also be Properties that can sort Stocks in different categories (Return of small relative to big firms, HML, excess Returns of corporate bonds over government bonds). Assume Portfolios A and B each depending on a distinct factor, With factor loadings (beta) Equal to one;

  • ra = mu a + 1*F1
  • rb = mu b + F*2

A Portfolio that Depends on only one factor With factor loadings Equal to one, in a universe of several factors, is called a factor Portfolio.

In order to construct independent factor Portfolios (factor Portfolios that depend on only one factor each), we need to decompose the covariance Matrix by finding its "eigen vectors".
Want to construct Portfolios that the covariance of each factor is zero. Since we want A and B to be independent factors With zero correlation/covariance. The independent factor Portfolios are just a tool to explain why differences in Return have to be explained by differences in factor loadings.

Arbitrage opportunities With two factors:
Assume a investor holds a Portfolio that Depends on both factors With loadings Beta1 and Beta 2

  • rp = mu p + Beta1F1 + Beta2F2
    Say we sell Beta1 of A and Beta2 of B. The net Return is then
  • rp - Beta1ra - Beta2rb = mu p - Beta1mu a + Beta2mu b
    The Return is independent of the factors, because the terms Beta1F1 and Beta2F2 are canceled out when we subtract B1ra og B2rb.
  • In equilibrium all variation in expected Returns is explained by different exposure to factors

The arbitrage pricing theory

The arbitrage pricing theory tells us that Portfolios are in general dependent on various factors:

  • rp = mu p + Beta 1F1 +Beta2F2 +...+ BnFn
  • if all investors hold the same Portfolio, there can be only one factor; the market index. But investors do not hold identical Portfolios!
  • The factors do not have to be independent as discussed earlier.
  • An additional strenght og ATP is that it does not require many strong assumptions.

Fama and French - 3 factor model

Introduced two factors:

  • SMB: the smallest Companies in the index minus the largest
  • HML: the Companies With the highest book-to-market value minus the Companies With lowest such ratio

ri = alfa + Bm(rm-rf) + BsmSMB + BhmlHML + ei

  • turns out that it is very sucsessful in explaining excess returns
  • gains from betting on HML or SMB are potentially quite big
  • the effect of transaction costs might lower this return
  • the strategy of betting on long term factors (SMB and HML) just barely beats a Equal weightet Portfolio, that is before transaction costs

Grossman - Stiglitz Paradox:

  1. Information value "c" will be zero. If there are no noise traders there is no incentive to be informed.
  2. If information is costly, no investor will aquire it if it has no value.
  3. All investors in the market will then face a undertainty Equal to the unconditional variane: var(u) = sigma^2theta + sigma^2Epsilon
  4. In that case, it will be an advantage for some investor to lower the variance they face to: var(u,theta)=sigma^2epsilon, by aquring information.
  5. If however someone aquire information "theta", this will be observed from the change in prices, and information value is again at zero Level.
    The conlusion is that without noise traders there is no unique equilibrium. That beeing said, the market efficiency Grossman & Stiglitz are talking about is strong form efficiencym which are dubtly. Whith strong efficiency all private information is reflected to the priec