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Transcript of Presentation Financial Time Series
Financial Time Series
Carl LindbergChalmers University of Technology
March 16, 2011
Outline
Financial models
The course
α and β
Model errors and the Financial Tsunami
Implementing investment strategies
Starting with Matlab  GARCH
Momentum
Pairs trading
Volatility
Regression and the FamaFrench model
Portfolio construction
Financial models
We know some financial models from previous courses, forexample:
I The Binomial model (Option Pricing Theory)I The BlackScholes model (Option Pricing Theory and
Portfolio Optimization)I Extreme Value Theory (Risk Management)
Financial models
You need to know that
All Models Are Wrong!
However, some are useful....
Financial models
Finance is different than Physics:
I A physical model is wrong in exactly the same way foreverI In Physics, parameters never change, everI In Finance, different models are good different timesI Parameters estimated from financial data are typically
highly unstable
Financial models
Some obvious model deficiencies not captured in the models youknow:
I Liquidity droughts at known or unknown news releasesI Bad prices at distress and at opening and closingI Intraday variation in trading activityI Releases of macro economic dataI Random BidAsk spreadsI Stochastic volatilityI Random liquidityI Corporate eventsI NonstationarityI Closed marketsI Price jumpsI ...
Financial models
I Typically, academic models explain
I It is more fun, and considerably more profitable, to predictI Further, to predict is to understand  to explain what
just happened is notI In this course, we will seek understanding through
predictionI We will probably fail!I However, even if we do, we are still better off than if we
never tried
Financial models
I Typically, academic models explainI It is more fun, and considerably more profitable, to predict
I Further, to predict is to understand  to explain whatjust happened is not
I In this course, we will seek understanding throughprediction
I We will probably fail!I However, even if we do, we are still better off than if we
never tried
Financial models
I Typically, academic models explainI It is more fun, and considerably more profitable, to predictI Further, to predict is to understand  to explain what
just happened is not
I In this course, we will seek understanding throughprediction
I We will probably fail!I However, even if we do, we are still better off than if we
never tried
Financial models
I Typically, academic models explainI It is more fun, and considerably more profitable, to predictI Further, to predict is to understand  to explain what
just happened is notI In this course, we will seek understanding through
predictionI We will probably fail!I However, even if we do, we are still better off than if we
never tried
Financial models
I Classic financial theory assumes rational, fully informedinvestors and absolute market efficiency
I In these models, prediction is impossibleI All price models you know are the offspring of these
theoriesI Modern financial theory and reality are less restrictive
The course
I This course will give some hints for how to use somemathematical tools, most of which are already in yourpossession, for implementing profitable investmentstrategies in the financial markets
I This will be done in a project, preferably in groups of two,which constitutes your examination
I You will implement at least three different strategies, andput them together in a portfolio
I I will leave class time for supervising your projects,particularly during the last weeks.
I The project is due at the end of the course
The course
I You will find your own data, clean it, and import it in asoftware program of your choosing (e. g. Matlab)
I I am fully aware that many of you have little experiencewith Matlab
I This will no longer be true in JuneI However, I will take care of you and the project is
feasible even without extensive knowledge of programming
The course
I You will find your own data, clean it, and import it in asoftware program of your choosing (e. g. Matlab)
I I am fully aware that many of you have little experiencewith Matlab
I This will no longer be true in June
I However, I will take care of you and the project isfeasible even without extensive knowledge of programming
The course
I You will find your own data, clean it, and import it in asoftware program of your choosing (e. g. Matlab)
I I am fully aware that many of you have little experiencewith Matlab
I This will no longer be true in JuneI However, I will take care of you and the project is
feasible even without extensive knowledge of programming
α and β
Hedge funds and traders aim at generating α  positive excessreturns that are ’independent’ of the direction of the underlyingmarkets, β
I α is portable to any marketI α is very hard to create, hence expensiveI Buying the market (β) is exceptionally cheapI By diversification, an investor wants to invest in as many
independent α sources as possible to offset the β riskI The existence of α is impossible in efficient markets
Why should α exist?
The ceaseless quest for alpha by active managers is the invisiblehand that pushes the financial markets toward efficiency. [...]Without active managers, the markets would be rudderless.Paradoxically, perfect market efficiency would lead to marketsbecoming inefficient
Dr. Lee R. Thomas, III, PIMCO
Why should α exist?
I Fear and greedI Different investment horizonsI Investment limitations and regulationsI Nonprofit maximizing investors such as pension funds and
central banksI Forced sellingI ...I ..I .
Example: the longonly equity fund investor
I Every deviation between the investor’s portfolio and themarket portfolio is an alpha bet
I Hence the investor’s portfolio is β + α, for some βI Since β exposure is ’free’, the fund’s fees are often
outrageously high relative to the alpha they supply:
Example: the longonly equity fund investor
Below are the largest portfolio weights of the SAX index (theOMX Stockholm market portfolio) and Sweden’s largest bankNordea’s Sweden fund, as of February 2009
SAX wghts Nordea Sweden Fund wghtsH&M 9.2% 8.9%Nordea 8.0% 8.7%TeliaSonera 7.2% 7.4%Ericsson 7.2% 7.4%Sandvik 2.5% 4.6%Total costs p a 0% 2.8%, 1% withdrawal fee
What are you paying for?
Example: Structured products
I Structured products typically consist of a call option and abond, both with long time to expire (35 years)
I The bond is supposed to guarantee that the investor getsher money back at expiry
I The commision is often 25%I Both the option and the bond can easily be bought at low
commisions, producing a ”homemade” product, identicalto those manufactured and sold by e.g. banks
I By exchanging the single defaultable bond for a portfolio ofbonds, or even better, a fixed income fund, bydiversification the ”moneyback guarantee” becomes muchmore certain
Risk premiums
I Risk premiums, and absolute or relative mispricings, giverise to α opportunities
I These mispricings and risk premiums can be categorized instrategies:
I MomentumI Pairs tradingI VolatilityI Value vs GrowthI ...I ..I .
Momentum
I Idea: An asset that has performed well in the past willcontinue to perform well, and vice versa
I The existence of momentum is supported by empiricalfinancial research
I Intuitively appealing
Pairs trading
I Idea: The price difference between two assets,appropriately defined, exhibit autoregressive behavior
I A pair trade is a portfolio consisting of one long (positive)and one short (negative) position
I Pairs trading is perhaps the most popular hedge fundstrategy
I A pair trade can be constructed to be market neutral
Volatility
I Empirical fact: Options often trade at an implied volatilitywhich is larger than the true (future) realized volatility
I Volatility is typically traded by taking positions in optionsI Trading options/volatility more complex than pairs trading
and momentumI An option is an interaction between time, volatility, and
directionI Volatility is important:
I You can lose money even if the market goes your wayI You can make money even if the market goes against you
Value vs Growth
I Empirical fact: Value stocks outperform growth stocks overtime
I Theoretically controversial, but widely applied principle
I Distinguished Chalmers professor: I never invest inhightech, because I work at Chalmers so I know how badthe students are
Value vs Growth
I Empirical fact: Value stocks outperform growth stocks overtime
I Theoretically controversial, but widely applied principleI Distinguished Chalmers professor: I never invest in
hightech, because I work at Chalmers so I know how badthe students are
Model errors
No strategy works always:I Momentum: When will it end? Reversals can be brutalI Pairs trading: Structural breaksI Volatility: In fast and/or distressed markets, prices will be
bad (if there are any) and volatilities have spiked. This isoften when you need to act
I Value vs Growth: At times, growth stocks brutallyoutperform value stocks (e. g. the IT stock bubble)
Model errors
I In August 2007, Goldman Sachs flagship global equity fundlost over 30% of its value in a week, on the back of thestart of the ”credit crunch”
I David Viniar, CFO Goldman Sachs commented:We are seeing things that were 25standard deviationevents, several days in a row
I The HFRX Global Hedge Fund Index was down a record15% in September and October 2008, coinciding with the”Financial Tsunami”
Hiding risk in the tails makes β look like α
Subprime
The term subprime was popularized during the credit crunchI A subprime lender has e.g. little or no assets, no job,
excessive debt, a history of missed payments, failures topay debts, and recorded bankruptcies
I In short: Subprime lenders should not get loansI NINJA – No Income, No Job, and no AssetsI When house prices rose, subprime lenders could borrow
more to finance old loans they couldn’t affordI This lending policy was encouraged by the govt, despite
what they claim today: ”House prices always rise, henceevery American should own his home”
Subprime
I Some analysts saw the problem of increasing subprimedefaults coming, but noone thought the subprimeconsumers were important enough to have any affect on thebroad economy.
I Badly designed bonus systems incited risky behavior –Portfolio managers got bonuses for taking large shortpositions in low probability events, like playing Russianroulette
I The productification of credits had caused the subprimepoison to be everywhere in the financial system
I When house prices started falling, many subprime lendersdefaulted on their loans
I The loans were tied to the real estate, not to the lenders(who could disappear like a ninja) – The banks got stuckwith the empty homes as collateral
Subprime
I The credit hedge fund industry constituted much of theliquidity in the credit markets.
I They were typically highly leveraged due to very low creditspread levels – The risk was thought of as the volatility ofthe position, not in what the position had you to promise
I They all got margin calls, they all needed to sell, and theywere all marktomarket – Armageddon
I Weavering Credit Opportunities 2007, margin from 1 to10Me overnight
Subprime
I In September 2008, Lehmann Brothers was allowed to fail –The end of the world as we know it!?!
I ”9 bottles of water, one with poison” – Which banks havetrouble? Which will be saved? How many CAN be saved?Interbank lending dried up, credit markets froze
I Bernanke decided to bring risk of inflation into play toavoid depression
I Inflation maybe even desirable? Relieves burden of debt,but has the cost of lost trust from US creditors
Stoploss
I ’All’ successful fund managers use some sort of stop loss tolimit the risk, but
I Stoplosses only work when the market is open and thereare prices to trade
I Most important: Liquidity is critical to any active strategy,and liquidity vanishes when markets crash
I Have control of your tails  do not use strategies which,when they go wrong, go terribly wrong (rememberSubprime)!
I Useful cliche: Buy your umbrella before it starts raining
Slippage
When we implement a strategy we need to think through thatour evaluation is realistic and ”correct”. We need to consider:
I Transaction costs  fixed fees, bid/ask spreads, etcI Data tradability  Can we trade at closing prices? Is our
data the actual prices?I Liquidity issues, in particular at our stoplossesI Time lags for data, e. g. European and American closing
prices differ in time, a quarterly report for Q1 is notavailable the first day of Q2, etc
I ...
In sample and Out of sample
I When we look for a good investment strategy, we”optimize” models to a data set
I In doing so, we explain what would have happened if wewould have applied the strategy (which is optimized to thesample data) starting at the beginning of the sampleperiod (when we did not have the information we based ourstrategy on)
I This is called testing in sample
In sample and Out of sample
I Estimating a model from one set of data, and applying intothe (at least from the model’s perspective) unknown futureis called out of sample testing
I Out of sample testing is considerably more reliable than insample testing
I However, the only way to get investors to really believethat what you have done is correct is to show a trackrecord of the strategy with real money belonging to otherpeople than yourself  this we will not do in this course!
Overparameterization
I In finance, one can not hope for stationary data (i.e. datafrom the same distribution with the same parameters) forextensive periods of time. One year is a long time
I Therefore, it is important to not make models toocomplicated, because you will not have time to obtainenough data to get stable parameter estimates
I Further, too complicated models will have you seerelationships that are not there in reality, e.g. the modelmight show that if it rains more than 3 mm in westernUganda, S&P500 will go up two days later
I Beware of datamining!
Overparameterization
I These models will not make money in the future eventhough they might appear to have been excellent in thepast
I Einstein: Make everything as simple as possible, but notsimpler
I In essence, You need to understand why your modelworks  the intuition is important!
I Recall that the shorter the time horizon the weaker thetrading signal that you are looking for
Statistics
Look at your data!
I It is Not sufficient to just plug in your data in someoptimization algorithm
I You always need to look at your data!I This is the only way to find certain types of model errors
and modeling flawsI The Matlab command plotmatrix is usefulI Data is frequently wrong even if you download it from
Google or Yahoo (look in particular for very large pricejumps)
Important things to remember at all times
I DO NOT use data from too long time period to estimateyour parameters  one year is a very long time!
I Markets are not stationary for long! You need a good fitwith little data
I This is not the same as evaluating the performance of yourstrategy over short time horizons
Suggested investment implementation
wealth = [ ]; wealth(1) = 1; in = 0;for i = 1:trading days
strength(i) = [something];if strength(i) > a && in == 0
in = 1; %Buywealth(i+1) = wealth(i)*(1+returns(i+1)  cost);
elseif strength(i) > b && in == 1wealth(i+1) = wealth(i)*(1+returns(i+1)); % Keep
elseif strength(i) < a && in == 0wealth(i+1) = wealth(i); % Do nothing
elsein = 0; %Sellwealth(i+1) = wealth(i)*(1cost);
endend
Starting with Matlab  GARCH
As a way to get started with Matlab we will now introduce theGARCH process
I GARCH is a ”Nobel prize” awarded volatility estimationmethod
I In essence, it says that if the recent history of a stock hasbeen volatile, than the near future is likely to be volatile,too (if the market crashed yesterday, people are likely to bemore nervous than usual today)
I GARCH is not a great volatility estimate, but it is sofamous that you need to know it
I GARCH can not be used to trade on, but you might needvolatility estimates in your trading strategies
Starting with Matlab  GARCH
GARCH is a stochastic process for the variance σ2t
(volatility =√variance). We assume that the returns
R(t) = X(t)−X(t−1)X(t−1) of some process X(t) (e.g. a stock price) can
be modelled byR(t) = σtεt,
where εt are i.i.d N(0, 1) random variables, and
σ2t = µ+ σ2
t−1(α+ βε2t−1) = µ+ ασ2t−1 + βR(t− 1)2),
for µ, α, β > 0, t = 0, . . . ,∞. Hence
GARCH states that the volatility of tomorrow dependsONLY on what has happened up until today.
(This can obviously not be true in reality!)
Starting with Matlab  GARCH
Since εt is not required to calculate σ2t , εt and σ2
t areindependent
We see from the definition of σ2t that
E[σ2t ] = µ+ E[σ2
t−1](α+ βE[ε2t−1]).
Since E[ε2t−1] = 1 by definition, we get that
α+ β < 1,
since otherwise σ2t explodes.
Starting with Matlab  GARCH
I According to GARCH, R(t)/σt = εt, so we can calculatethe εt
I Since the εt are i.i.d N(0, 1), the loglikelihood function is
L(µ, α, β, ε1, . . . , εn) = −t∑i=1
ε2i
I The parameters can be found by maximizing this function,which is done in the Matlab function ugarch
I Choose σ20 = V ar(R(·))
Starting with Matlab  GARCH
I GARCH has three model parameters, which is a quite a lotI If we choose µ = 0, this corresponds to removing the
(nonintuitive) lower bound on the volatility so that
σ2t = σ2
t−1(α+ βε2t−1)
Starting with Matlab  GARCH
I But for this model specification, we need that
E[σ2t ] = E[σ2
t−1]
since otherwise the volatility would either explode ordisappear.
I Hence, we get the condition that
α+ β = 1.
I Either α or β is now redundant
Starting with Matlab  GARCH
I We get in this way an alternative to GARCH, namely theEWMA (Exponentially Weighted Moving Average) model:
σ2t = σ2
t−1(α+ (1− α)ε2t−1)
I This model has only one parameterI Here, too, R(t)/σt = εt, and the parameters can be fitted in
the same way as GARCHI Matlab has no function for EWMA, but it is only five lines
in Matlab to do the job yourselfI Here, choose σ2
0 < V ar(R(·)). This is needed for numericalstability of the estimation procedure
Starting with Matlab  GARCH
I In the project, you will find a data set, e.g. onfinance.yahoo.com, and fit GARCH and EWMA models toit
I You will analyze the observed residuals R(t)/σt = εt andsee how they conform to theory
I Further, you will also compare GARCH to differentvariance estimators, and analyze parameter stability andoutofsample performance
Momentum
I Idea: An asset that has performed well in the past willcontinue to perform well, and vice versa
I The existence of momentum is supported by empiricalfinancial research
I Momentum can be either absolute (e.g. you buy astock) or relative (e.g. you buy a stock and financethis by shortselling a stock index)
I Intuitively appealingI We will now go through some common momentum and
reversal strategies
Momentum
In a set of stocks, buy at each rebalancing time the topx% performing stocks over some time period
I The time between rebalancing varies, e.g. each friday, thefirst day in each month,...
I The evaluation period varies, too, everything from a coupleof minutes (high frequency trading) up to years
I The most common evaluation periods are one week up tosix months
I Frequently, the exit strategies are e.g.I Hold the stocks until they are not among the top x%I Hold the stocks until they are not among the top y%I Hold the stocks a prespecified time period no matter what
happens
Momentum
If some ”strength function” for an asset is historicallyhigh, then buy that asset
I Moving Average: buy if three day mean is larger than onemonth mean
I Autoregressive: buy if∑∞
i=1 ∆X(t− i) ∗ λi is large,λ ∈ (0, 1)
I Moving averages and autoregressive processes may begeneralized considerably  this is Time Series Analysis (lastyear’s course)
I The machinery of Time Series Analysis adds little to theunderstanding of finance  if it does not work for the simplemodels it will not work for the more general ones
I Time Series Analysis is more important in related fieldssuch as macro economics and econometrics
Momentum
There are many other ideas to try:I Relative strength/weakness  consider prices relative other
pricesI Absolute strength/weakness  consider only absolute pricesI Best/worst in group  invest in the best and/or worst
performers in a set of assetsI What entry and exit strategies should be used?I Optimal Filter theory (related to Time Series Analysis)I Technical Analysis  this field is often considered to be
unserious by mainstream academia and financial industryI To say the least, Technical Analysis investment strategies
are hard to quantify objectively, hence they are hard toevaluate
Momentum
I Recall that you can finance a long stock position (i.e. toown a stock) by shortselling another asset
I Hence, such a trade costs nothing to set upI All strategies above can be either net long, net short, or
neutralI This is up to you to decide!
Example of investment implementation
wealth=[ ]; wealth(1) = 1; in = 0; lag = 3; longlag = 30;a = 1.03; b = 0.99;for i = longlag +1:trading days
strength(i) = mean(X(ilag:lag))/mean(X(i30:i));if strength(i) > a && in == 0
in = 1; %Buywealth(i+1) = wealth(i)*(1+returns(i+1)  cost);
elseif strength(i) > b && in == 1wealth(i+1) = wealth(i)*(1+returns(i+1)); % Keep
elseif strength(i) < a && in == 0wealth(i+1) = wealth(i); % Do nothing
elsein = 0; %Sellwealth(i+1) = wealth(i)*(1cost)
endendfigure; plot(wealth)
Reversal
I Momentum is closely related to ReversalI Reversal is the notion that an asset which has performed
too well too long will perform poorly going forward
Reversal
If an asset has increased its price by more than x% in aweek, then ”go short” the following week
I Remember that what is a large price movement in one weekis not necessarily large on a monthly perspective, and so on
I To ”short” is to own a negative positionI In practice, one borrows a stock and sells it, with the hope
that one will be able to buy it back at a lower priceI By shorting a stock, the investor has a debt consisting of a
number of stocks instead of a number of eurosI It is harder to find short strategies that work satisfactory
than long onesI This has the effect that most hedge funds are implicitly
long the market, often unknowinglyI Tip: use the Suggested Investment Implementation slide to
begin with
Reversal
If an asset has increased its price by more than x% in aweek, then ”go short” the following week
I It is harder to find short strategies that work well thanlong ones
I This has the effect that most hedge funds are implicitlylong the market, often unknowingly
Pairs trading
I Idea: The price difference between two assets,appropriately defined, exhibit autoregressive behavior
I A pair trade is a portfolio consisting of one long (positive)and one short (negative) position
I Pairs trading is perhaps the most popular hedge fundstrategy
I A pair trade can be constructed to be market neutral
Pairs trading
I If S1 = V olvo and S2 = Scania, then the portfolio
X = S1 − πS2,
where π > 0, is a pair tradeI A natural model candidate is to model X as an
autoregressive process (AR(1)):
∆X(t) = α(X(t− 1)− λ) + βεt,
where εi are i.i.d. N(0,1)
Pairs trading
I If X(t) is far from λ, then the drift is large  this keeps theprocess around λ
I We want to trade on this drift!I Note that
E[∆X(t)] = E[α(X(t− 1)− λ)],
I The AR(1) process is the discrete time analogue to theOrnsteinUhlenbeck process in stochastic calculus
I Understand why X(t) is meanreverting!!
Pairs trading
How to estimate an AR(1) model?I We recognize X(t) as a linear regression with X(t− 1)
being the explanatory variable
X(t) = (1 + α)X(t− 1)− αλ+ βεt
I This is done in Matlab with the function regress, whichminimizes the least squares of the residual errors
I Alternatively, you can use Matlab’s function ar fordifferent estimation procedures
I If you do this, you may have to do a parallel shift of X (byremoving its estimated mean), corresponding to theassumption λ = 0
Pairs trading
How do we choose which stocks to analyze?
I To avoid exposing yourself to relative performancesbetween different branches of industry, compare stocksfrom the same branch
I For the same reason as above, choose stocks that haveabout the same market value
I If you do not comply to these suggestions, you are tradingbranches, or small cap vs large cap rather than stocks
I This is fine, as long as you know what you are doing  pairstrading in this way is more risky and more exposed tomarket bursts and irrationalities than intrabranch,intrasize pairs trading
Pairs trading
How to construct a pair trade?I If you comply to the suggestions on the previous slide, you
can often safely take 1SEK in each leg of the pair trade(one2one)
I The reason is that the volatilities will then be similar, andboth stocks are likely to behave similarly regardless ofwhether the markets go through the roof or crash
I Your analyzes should in this case analyze the portfoliowhich, in the beginning of the evaluation period, wasweighted in such a way which would have resulted in aone2one portfolio today
I This is done by normalizing each daily stock price by thestock price at the last day of the evaluation period
Pairs trading
I An alternative is to normalize each daily stock price by thestock price at the first day of the evaulation period
I You can then not apply one2one, since the portfolioweights of your evaluated portfolio might have drifted faraway from one2one during the evaluation period (ifmarkets are up or down a lot)
I One can also model the ratio Y = S1S2
as an AR(1) processI Of course, one still has to trade the actual pair X, not YI If you blend stocks from different branches of industry, and
of different sizes, you need to consider more advancedportfolio constructions involving e.g. the volatilities of thestocks
Pairs tradingExit strategies and stop losses
I It is natural, and easy to implement, exit and stoploss setswhich are of the type ”first time crossing some barrier”
I Not only is it natural, it can also be shown to be optimal ifthe investor can hold on to the position as long as he/shewants to
I For example, one can take a trade in the interval (σ, 2σ),where σ is the standard deviation of the stationarydistribution of X(t)
I We have that
σ2 = V ar(X(t)) = α2V ar(X(t− 1)) + β2V ar(εt)
= α4V ar(X(t− 2)) + α2β2V ar(εt−1) + β2V ar(εt)
= . . . = [α2n → 0]
= β2∞∑i=0
α2i = [Geometric sum] =β2
1− α2
Pairs trading
Exit strategies and stop lossesI Your stoploss might then be 3σ (remember 6σ in
statistical process control)I Try what works yourselves !I In practice, one wants to let the trade pass an eye
inspection  however, in your evaluations your trading rulesneed to be specified quantitatively
I It is useful to simulate some data so you know what youare looking for
I Tip: use the Suggested Investment Implementation slide tobegin with
I It is particularly easy to implement pairs trading when youtrade only once a week and have no stop losses
Volatility
I Empirical fact: Options often trade at an implied volatilitywhich is larger than the true (future) realized volatility
I Volatility is typically traded by taking positions in optionsI Trading options/volatility more complex than pairs trading
and momentumI An option is an interaction between time, volatility, and
directionI Volatility is important:
I You can lose money even if the market goes your wayI You can make money even if the market goes against you
Volatility
I Volatility is not only conceptually harder to comprehendthan linear instruments (like stocks, futures, etc)
I It is also harder to evaluate option strategies due to lack ofdata
I A stock is a stock, but options have different strikes andmaturities, which causes one stock to have hundreds ofoptions associated to it
I Typically, option price data is not possible/cheap to getI Even if you do, liquidity is worse than for stocks and there
are wider bid/ask spreadsI The upside is that there is plenty in the field for us
mathematicians to do!I Volatility trading will be more challenging to implement
than the other strategies, highfrequency trading excluded
Volatility
I We will have access to implied volatility data on the coursehome page
I The data is either ”1st Month”  meaning the contract’sthat are up for expiry  or ”1 Month”  a constructedvolatility which is supposed to always have 1 Month/20trading days to expiry
I For the former, evaluations are more exact, for the latteryou can check robustness by changing the date of expiry
I The data is also quoted either as percentage of”moneyness”  105% moneyness means that S(t) = 1.05K or in terms of the BlackScholes delta  implied volatility at∆ = 0.40 means that the strike is where ∆ = 0.40 inBlackScholes model
Volatility
I When you evaluate your strategies, you want to take actiononly on the points where you have data, otherwise youcan’t trust your prices
I It is possible to interpolate between data points, but thishas to be done with a high level of sophistication to avoidmisleading results
I It will affect your results a lot of you consistantly get theimplied volatility wrong even by 1%
Volatility
I You cannot be sure that you can change position when youwant to
I Transaction costs are high, so strategies with high turnoverwill have a hard time working
I Do not wait too long to take your stoploss  thenonlinearity of options strikes hard against you!
Volatility
I One volatility strategy is to sell one call option and one putoption, both atthemoney and front month, on eachtrading day, and hold these until expiration
I In this case, you have no strength function and the returnsdepends on how the value of your options progress
I You can still use the Suggested Implementation Slide,properly adjusted
Diversification
The power of diversification, and the difficulties of tradingoptions
I Diversification is the key to every good investment strategyI Ordinarily, one can diversify into, for example, many
different stocksI There are not even close to as many liquid underlying
option marketsI Hence, diversification in time becomes more important
Diversification
The power of diversification, and the difficulties of tradingoptions
I Example: Your initial wealth X0 = 1. Flip a very unfaircoin once  p = 0.9 of success  or a less unfair coin p = 0.51  n = 100 times? Find wealth X
I Ex. 1. E[X] = 1 + 0.9 ∗ 1 + 0.1 ∗ (−1) = 1.8I Ex. 2. E[X] = 1 + 100 ∗ (0.51 ∗ 1 + 0.49 ∗ (−1) = 3I With options, you have one coin, and have to take many
bets ”on the same coin”
Regression
I In recent years, it has become increasingly popular to useregression to estimate expected returns of stocks
I The explanatory variables can be just about anythingI The idea is that explanatory variables that are not prices
can contain information about the future direction of themarket
Regression
I This is slow trading  transaction costs, data tradability,and liquidity are not issues
I However, one needs to take extra care to make sure oneuses the explanatory variables in a correct way withregards to time
I Like we have discussed before, European and Americanclosing prices differ in time, a quarterly report for Q1 is notavailable the first day of Q2, etc
Regression
I Beware of datamining!I Using too many explanatory variables will make you see
things that are not there, they are a consequence ofrandomness
I These models will not make money going forward eventhough they might have looked great in the past
I Remember ”If it rains in Uganda, ...”
FamaFrench
I The most famous regression model is the FamaFrenchThree Factor Model
I Finance people like to think and talk like it is a naturallaw, like Newton’s laws of motion
I You know that despite its fancy name, this is not trueI Financial models always fail eventually
FamaFrench
I The model says that the excess return (Rm−Rf) (marketreturn minus risk free interest rate) of a stock dependslinearly on the stock’s correlation to the market, whetherthe underlying company is small or big in terms of marketcapitalization (SMB) (Small Cap or Large Cap), andwhether it has high or low book to market ratio (HML)(i.e. Value or Growth)
I The regression estimates the return R of a stock as
R−Rf = α1 ∗ (Rm−Rf) + α2 ∗ SMB + α3 ∗HML+ ε,
where ε is i.i.d. N(0, σ) for some σ
FamaFrench
I You can find definitions of market capitalization and thebook to market ratio, and how to calculate them, by asimple internet search
I If you want to use the more famous price to earnings ratio(P/E) instead, or anything else you might think of, this isup to you
I However, you then need to motivate this choice a littleI You can find data at Prof. Ken French homepage at
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html
FamaFrench
I One easy implementation is to just categorize a set ofstocks according to whether they are Large or Small Cap,and Value or Growth
I You now go long the stocks that are Small Cap and Value,and short the stocks that are Large Cap and Growth
I Invest for example equal amounts of money in each stock ina category
I In Matlab, this is exceptionally easy!
FamaFrench
I In this portfolio, we go long two stocks, financed byshortselling one stock:
portfolio = 1/2/S1(1)∗S1 + 1/2/S2(1)∗S2−1/S3(1)∗S3;
I This is it!I You can hold this portfolio for a long time, since the
categorization of the stocks that we use is changing veryslowly
I Do you believe that this model will always be profitable inthe future?
Markov theory
Markov theory allows us to answer simple, but natural,questions such as
I What is the probability of the market going down aftergoing up for five days in a row?
I What is the probability of the market closing lower than itopened after going down three days in a row?
I What is [your own idea]?I ...
High frequency trading
We will now briefly discuss the application of advanced Markovtheory to high frequency trading
I This is a very modern approach to modellingI The high frequency hedge funds have the most impressive
track recordsI Often, one uses high frequency data on both order book
and prices
High frequency trading
I By modelling the order book, one can get probablities for,for example, upward market moves conditional on a stateof the order book
I The idea is that the order book contains additionalinformation on the future development of prices which isnot available in the price data itself
I Trading is lightning fast, with extremely high turnover
High frequency trading
I This topic requires more advanced programming than theother strategies
I Come see me if you are interested in doing this strategy forthe project
I An interesting paper is http://ssrn.com/abstract=1273160I You need not use real data, simulated data is fine
Portfolio construction
Look at your data! Once again:I It is Not sufficient to just plug in your data in some
optimization algorithmI You always need to look at your data!I This is the only way to find certain types of model errors
and modeling flawsI The Matlab command plotmatrix is useful
Portfolio construction
Once we have our αengines, how do we put them together?I MeanVarianceI Sharpe ratio?I 1/N?I Tail behavior?I ...
Portfolio construction
I Most strategies we consider do not require an initial capitalI However, your prime broker/bank/internet brokerage
requires you to keep a certain amount, called a margin, onyour account at all times
I This is to ensure that you will be able to honor yourobligations
I The margin is ”random”  its calculations sometimesappear irrational
Portfolio construction
I You have to keep the required margin at all times!I If you don’t, your broker will sell for you at the best bidI You DO NOT want this, as it will cost a lot and put
someone else in control of your processI Hence, for each strategy you want to set aside enough cash
to make sure you always stay within your marginI It is this cash requirement that you use to calculate the
returns of your αengines
Portfolio construction
I You have to decide on a reasonable cash requirement levelyourself
I It could be e.g. two times your maximum drawdown duringthe evaluation period
I This might seem ad hoc, and it isI However, the financial markets only regulates your risks
through taking margin, and this allows you to take Largerisks
I Hence, It is up to you to have good judgment!
Portfolio construction
MeanVariance portfolio
I ”Nobel prize” awarded, founding father is Harry MarkowitzI MeanVariance is to minimize the variance (risk) of the
overall portfolio while keeping the mean above apredetermined fixed level
I Mathematically, the problem is to find portfolio weightsπ = {πi : i = 1, . . . , n} such that
minπ{V ar(W π(T ))} while E[W π(T )] ≥ C
where W π(T ) is the wealth at terminal time T given thatthe strategy π is used, and C is the required rate of return.
Portfolio construction
You find easily the MeanVariance portfolio using Matlab:I Calculate the covariance matrix of your strategies (or
stocks): ” H = cov(prismatris); ”I The input vector f is chosen to be zero: ” f =
zeros(length(H),1); ”I After having sorted out the linear constraints A, b, etc ...
(you can do this yourselves!) the MeanVariance portfolioweights are given by:” x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0); ”
I You may want to redo this optimization at the start ofeach time period  a time period can be everything fromone day to once a year, it is your choice!
Portfolio construction
Sharpe ratio
I When funds or fund managers are ranked, people typicallyuse the so called Sharpe ratio
I Mathematically, the Sharpe ratio of a portfolio process isdefined as
S =µ− rσ
where r is the short riskfree rate, and µ and σ are theexpected rate of return and volatility of the portfolioprocess, respectively.
Portfolio construction
Sharpe ratio
I Loosely speaking, the Sharpe ratio measures theperformance on a risk adjusted level
I Hence, in principle the investor can adjust the leverage toobtain the expected return, or risk, that he wants.
I However, tail risk strategies can get great Sharperatios until they blow up!
I Example: Subprime.I Note that optimizing the Sharpe ratio is essentially the
same as finding the MeanVariance portfolio for differentrequired rates of return C.
Portfolio construction
1/N
I If we can assume that our αengines are independent ofeach other, and that they all produce approximatelynormally distributed returns with the same Sharpe ratio,then a natural strategy is to put equally much risk in eachstrategy  a so called 1/N strategy
I If the αengines are mutually independent and normallydistributed with (approximately) the same parameters,then the 1/N strategy amounts to investing the sameamount of money in each αengine.
I The 1/N strategy is actually meanvariance optimal in thetwo settings desribed above
I This can be seen immediately from Jensen’s inequality
Portfolio construction
Tail behavior
I Let’s face it, despite every good intention and seriousefforts, you will most likely NOT get return time series ofyour αengines that are i.i.d. N(µ, σ)
I Your strategies will probably have a tendency to breakdown at the same time
I Yet another way to construct your portfolio is to base theportfolio weights on an analysis of the tail behavior of theportfolio
I This may be done on a totally ad hoc basisI For example, one might want to choose the portfolio which
has the lowest maximum monthly lossI However, you can also try to apply some Extreme Value
Theory (taught at the course Financial Risk)
Portfolio construction
Maximum drawdown
I Another alternative is to minimize the portfolio maximumdrawdown
I The maximum drawdown is the maximum distancebetween the value of a portfolio at any time point, and themaximum portfolio value prior to that time point Understand this graphically!
I The maximum drawdown is easily overseen when one looksat a plot of an investment strategy in retrospective
I However, it is a Very real concern for both fund managersand investors when you are in a time period when yourmodels do not work, and you do not know if they ever willagain.
Portfolio construction
Maximum drawdown
I A year ago, the Brummer flagship hedge fund Lynx hadnot made money for almost a year and a half. Do youthink they were nervous at Lynx?
I To have low maximum drawdown is important  hedgefunds typically charge performance fees only forperformance above the so called high water mark
I The high water mark is calculated individually for eachinvestor in the fund, and it is defined as the maximumvalue that the investor’s holdings in the fund has ever had
Now you work...
This is it!
I You have now plenty to think about and work withI I am here to help you for the rest of the class lectures
Now you work...
Thank you!