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Advanced Finance (Topic 2: APT and Fama French (Factor portfolios (We can…

- - - - Shows number of Stocks avaliable at different prices
    - - 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
  - - - 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)
- - - - 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.
      - 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
    - - 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.
    - - 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.
    - - 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?
  - - - 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 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.
    - - 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:
        
        calculate "u" and "d"
        
        calculate the risk neutral probability "q"
        
        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.
    - - 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.
- - - - 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.
  - - - 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
    - - 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
        
        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
        
        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
        
        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
        
        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
    - - 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.
- - - - 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.
  - - - 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:
        
        Replace drift term "mu" With risk free interest rate r in the Stock price
        
        calculate the expected value of the derivative at maturity
        
        discount the result to the present value
- - - - 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.
        
        A bank that sells options typically calculate the delta of its Portfolio as: delta = SumWi* delta i
        
        "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.
        
        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 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
      - 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.
      - 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!
- - - - 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?
  - - - It is likely that the difference is due to different views on the assets volatility through the market.
  - - - 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
    - - 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
    - - Mainly two reasons for the fat tails and volatility smile:
        
        The volatility is not constant
        
        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.
- - - - 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.
- - - - 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
    - - If suboptimal porfolio is a result of difference in taste -> permanent inefficiency
        If suboptimal Portfolio is a result of differences in belief -> temporary inefficiency
    - - 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.. )
  - - - 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.
    - - Three assumtions must hold for a person to be rational:
        
        1. Completeness
        
        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)
        
        2. Transitivity
        
        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?
        
        3. Independence
        
        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.
  - - - 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!
- - - - Who sell when others buy, and buys when other sell
  - - - Weak form efficiency: (price history)
        
        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.
        
        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:
        
        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.
        
        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.
      - Semi-strong efficiency: (public information)
        
        There are several results in literature that suggests that Public information have a effect on asset Returns,
        
        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
      - Strong-form efficiency: (private information)
        
        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.
      - 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.
      - Anomalies - interpretated and a possible explanation
        
        Mainly two explanations:
        
        A result of risk Premiums (as suggested by Fama-French)
        
        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.