fe570 financial markets and trading -...

Asset Returns and Distributional Properties of Returns Spread and Liquidity The Random-Walk Model of Security Prices Statistical Analysis of Price Series The Roll Model of Bid, Ask, and Transaction Prices

FE570 Financial Markets and TradingLecture 3. The Roll Model of Trade Prices

(Ref. Joel Hasbrouck - Empirical Market Microstructure )

Steve Yang

Stevens Institute of Technology

9/11/2012


Outline

1 Asset Returns and Distributional Properties of Returns

2 Spread and Liquidity

3 The Random-Walk Model of Security Prices

4 Statistical Analysis of Price Series

5 The Roll Model of Bid, Ask, and Transaction Prices


Asset Returns:Let Pt be the price of an asset at time t, and assume nodividend.One-period simple return: Gross return

1 + Rt =Pt

Pt−1orPt = Pt−1(1 + Rt) (1)

Simple return:

Rt =Pt

Pt−1− 1 =

Pt − Pt−1

Pt−1(2)

Multiperiod simple return: Gross return

1 + Rt(k) =Pt

Pt−k

=Pt

Pt−1× Pt−1

Pt−2× ...× Pt−k+1

Pt−k

= (1 + Rt)(1 + Rt−1)...(1 + Rt−k+1).


Example

Asset Returns Example: Suppose the closing prices of anasset are

Period 1 2 3 4 5

Price 37.84 38.49 37.12 37.60 36.30

What is the simple return from period 1 to period 2?

Ans: R2 = 38.49−37.8437.84 = 0.017.

What is the simple return from period 1 to period 5?

Ans: R5(4) = 36.30−37.8437.84 = −0.041.

Verify that 1 + R5(4) = (1 + R2)(1 + R3)...(1 + R5).


Continuously Compounded (or Log) Returns:Log return:

rt = ln(1 + Rt) = lnPt − Pt−1

Pt−1= pt − pt−1 (3)

where pt = ln(Pt).

Multiperiod log return

rt(k) = ln[1 + Rt(k)]

= ln[(1 + Rt)(1 + Rt−1)...(1 + Rt−k+1)]

= ln(1 + Rt) + ln(1 + Rt−1) + ...+ ln(1 + Rt−k+1)

= rt + rt−1 + ...+ rt−k+1.


Example

Log Returns Example: Suppose the closing prices of anasset are

Period 1 2 3 4 5

Price 37.84 38.49 37.12 37.60 36.30

What is the log return from period 1 to period 2?

Ans: r2 = ln(38.49)− ln(37.84) = 0.017.

What is the log return from period 1 to period 5?

Ans: r5(4) = ln(36.30)− ln(37.84) = −0.042.

Is it easy to verify that r5(4) = r2 + r3...r5?


Distributional Properties of Returns:Moments of a random variable X with density f (x) : `− thmoment

m′

` = E (X `) =

∫ ∞−∞

x`f (x)dx (4)

`− th central moment

m` = E [(X − µx)`] =

∫ ∞−∞

(x − µx)`f (x)dx (5)

First moment: mean or expectation of XSecond moment: variance of XSkewness (symmetry) and Excess Kurtosis (fat-tails)

S(x) = E [(X − µx)3

σ3x

] (6)

K (x) = E [(X − µx)4

σ4x

] (7)


Example

Properties of Returns:

Why are mean and variance of returns important?

Ans: They are concerned with long-term return and risk,respectively.

Why is symmetry of interest in financial study?

Ans: Symmetry has important implications in holding short orlong financial positions and in risk management.

Why is kurtosis important?

Ans: Related to volatility forecasting, efficiency in estimationand tests, etc. High kurtosis implies heavy (or long) tails indistribution.


Estimation for a given sample data {x1, ..., xT }:

Sample mean:

µx =1

T

T∑t=1

xt , (8)

Sample variance:

σ2x =

1

T − 1

T∑t=1

(xt − µx)2, (9)

Sample skewness:

S(x) =1

(T − 1)σ3x

T∑t=1

(xt − µx)3, (10)

Sample kurtosis:

K (x) =1

(T − 1)σ4x

T∑t=1

(xt − µx)4, (11)


Normality Test: Under the normality assumption, we have:

S(x) ∼ N(0,6

T), K (x)− 3 ∼ N(0,

24

T) (12)

Test for symmetry:

S∗ =S(x)√

6/T∼ N(0, 1), (13)

Description rule: Reject H0 of a symmetric distribution if |S∗| > Zα/2 or

p-value is less than α.

Test for tail thickness:

K ∗ =K (x)− 3√

24/T∼ N(0, 1), (14)

Description rule: Reject H0 of a symmetric distribution if |K∗| > Zα/2 or

p-value is less than α.


Figure: Simulated E-mini Market Return Distribution.


Spread: The size of the bid/ask spread is an important objectof the microstructure theory.

The quoted spread between ask At and bid Bt that is averagedover T periods equals:SQ = 1

T

∑Tt=1(At − Bt)

The average spread in terms of the asset fundamental price P∗tis defined as:S = 1

T

∑Tt=1 2qt(Pt − P∗t )

where qt is 1 for buy orders and -1 for sell orders. Since thevalue of P∗t is not observable, the effective spread in terms ofmid-price Mt = 1

2 (At + Bt) is usually used:

SE = 1T

∑Tt=1 2qt(Pt −Mt)

The realized spread is applied in post-trade analysis:SR = 1

T

∑Tt=1 2qt(Pt −Mt+1)


Spread Components: The bid/ask spread is the price ofimmediacy of trading.

The spread incorporates the dealers’ operational costs, such astrading system development and maintenance, clearing andsettlement, etc. If dealers are not compensated for theirexpenses, there is no rationale for them to stay in the business.

Dealers’ inventory costs contribute to the bid/ask spread,because they must recover their potential losses by wideningthe spread. Since deals must satisfy order flows on both sidesof the market, they maintain inventories of risky instruments(and sometimes undesirable).

Spread covers the dealers’ risk of trading with counterpartswho have superior information about true security value.Informed traders trade at one side of the market and mayprofit from trading with dealers. This component of thebid/ask spread is called the adverse-selection component sincedealers confront one-sided selection of their order flow.


Liquidity: is a notion that is widely used in finance, yet it hasno strict definition and in fact may have different meanings.Generally, the term liquid asset implies that it can be quicklyand cheaply sold for cash. A popular notion defines liquidityas the market’s breadth, depth, and resiliency.

It means that buying price and selling price of a liquidinstrument are close, that is, the bid/ask spread is small. In adeep market, there are many orders from multiple marketmakers, so that order cancellations and transactions do notaffect notably the total order inventory available for trading.

Market resiliency means that if some liquidity loss does occur,it is quickly replenished by market makers. In other words,market impact has only temporary effect.

Sometimes inverse liquidity - illiquidity based on the priceimpact caused by trading volume, is used:ILLIQ = 1

N

∑Nk=1 |rk |/Vk , where rk and Vk are the return and

trading volume at time k .


The Random-Walk Model of Security Prices1900 - Louis Bachelier The French mathematician LouisBachelier first documented the idea and provided insightsabout stock market prices in his Ph.D. dissertation titled ”TheTheory of Speculation”.1953 - Maurice Kendall A British statistician proposed theRandom-Walk hypothesis in his paper titled ”The Analytics ofEconomic Time Series, Part 1: Prices”.1964 - Paul Cootner MIT Sloan professor developed the sameidea in his book titled ”The Random Character of StockMarket Prices”.1965 - Eugene Fama Eugene Fama at University Chicagofurther developed the idea in a paper titled ”Random Walks InStock Market Prices”, and eventually he infused the idea intothe Efficient-Market hypothesis.The Random-Walk Model is no longer considered to be acomplete and valid description of a short-term price dynamics,but it nevertheless retains an important role as a model for thefundamental security value. (Martin Weber, Andrew Lo, etc.)


The Random-Walk Model - General ModelLet pt denote the transaction price at time t, where t indexesregular points of real (”calendar” or ”wall-clock”) time, forexample, end-of-day, end-of-hour, end-of-minute,end-of-second, etc. Because it is unlikely that trades occurexactly at these times, we will approximate these observationsby using the prices of the last (most recent) trade, forexample, the day’s closing price.

The Random-Walk model (with drift) is defined as:

pt = pt−1 + µ+ ut ,

- where ut , t = 0, 1, ... are independently and identicallydistributed random variables. Intuitively, they arise from newinformation that bears on the security value.

- µ is the expected price change (the drift).- The units of pt are either levels (e.g. dollars per share) or

logarithms. The log form is sometimes more convenientbecause price changes can be interpreted as continuouslycompounded returns.


The Random-Walk Model - MartingaleThe drift can be dropped in this model in most of themicrostructure analysis. When µ = 0, pt cannot be forecastbeyond its most recent value:

E [pt+1|pt , pt−1, ...] = pt ,A process with this property is generally described as amartingale.

One definition of a martingale is a discrete stochastic processxt where E |xt | <∞ for all t, and E (xt+1|xt , xt−1, ...) = xtNote that expectation in this formulation is conditioned onlagged pt or xt that is the history of the process.A more general definition of a martingale process involvesconditioning on broader information set. The process xt is amartingale with respect to another (possibly multidimensional)process zt if E |xt | <∞ for all t and E (xt+1|zt , zt−1, ...) = xtWhen the conditioning information is ”all public information”,the conditional expectation is sometimes called thefundamental value or the efficient price of the security.


The Random-Walk Model - ObservationsTo define a Random-Walk formally, take independent randomvariables Z1,Z2, ..., where each variable is either 1 or -1, with a50% probability for either value, and S0 = 0 and Sn =

∑nj=1 Zj .

The series Sn is called the simple random walk on ZThis series (the sum of the sequence of -1s and 1s) gives thedistance walked, if each part of the walk is of length one. Theexpectation E(Sn) of Sn is zero.This follows by the finite additivity property of expectation:E(Sn) =

∑nj=1 E(Zj) = 0.

A Random-Walk is a process constructed as the sum ofindependently and identically distributed (i.i.d) zero-meanrandom variables - a special case of martingale.

In microstructure analysis, transaction prices are usually notmartingales, but by imposing economic and statisticalstructure it is often possible to identify a martingalecomponent of the price.


Figure: Simulated E-mini Market Prices and Returns.

Suppose we have a sample {p1, p2, ..., pT} generated from arandom walk process.


2D Random Walk(for fun!!!).

http://upload.wikimedia.org/wikipedia/commons/a/a9/2D_Random_Walk_400x400.ogv

http://upload.wikimedia.org/wikipedia/commons/a/a9/2D_Random_Walk_400x400.ogv


Statistical Analysis of Price Series - Empirical EvidenceBy the Random-Walk model (with drift):pt = pt−1 + µ+ ut ,Price changes ∆pt = pt − pt−1 should be i.i.d with meanE (ut) = µ and variance Var(ut) = σ2

u, for which we cancalculate the usual estimates. However the empirical evidenceshows the following features:

Near-Zero Mean Returns. In microstructure data samples µis usually small relative to the estimation error of its usualestimate, the arithmetic mean. Zero is, of course, a biasedestimate of µ, but its estimation error will generally be lowerthan that of the arithmetic mean.Extreme Dispersion. The convenient assumption that pricechanges are normally distributed is routinely violated.Statistical analysis of price changes at all horizons generallyencounter sample distribution with fat tails.Dependence of Successive Observations. The increments(changes) in random walk should be uncorrelated. In actualsamples, the first order autocorrelations of short-run pricechanges are usually negative.


The Roll Model of Bid, Ask, and Transaction PricesKeep the random-walk assumption, and apply it to the efficientprice instead of the actual transaction price.

Denote efficient price by mt , and we assume mt = mt−1 + ut ,where ut are i.i.d. zero-mean random variables.

All trades are conducted through dealers - the best dealerquotes are bid and ask prices, bt and at , i.e. if a customerwants to buy, he/she must pay the dealer’s ask price (thereforelifting the ask); if a customer wants to sell, he/she receives thedealer’s bid price (hitting the bid).

Dealer incur a cost of c per trade. This charge reflects costslike clearing fees and per trade allocations of fixed costs.

If dealers compete to the point where the costs are justcovered, the bid and ask are mt − c and mt + c , respectively.


The Roll Model - Two ParametersThe bid-ask spread is at − bt = 2c , a constant.

At time t, there is a trade at transaction price pt , which maybe expressed as: pt = mt + qtcwhere qt is a trade direction indicator set to +1 if thecustomer is buying and -1 if the customer is selling.

We also assume that buys and sells are equally likely, seriallyindependent (a buy this period does not change the probabilityof a buy next period), and that agents buy or sellindependently of ut (a customer buy or sell is unrelated to theevolution of mt).

The Roll model has two parameters, c and σ2u. These are most

conveniently estimated from the variance and first-orderautocovariance of the price changes, ∆pt :

The variance γ0 ≡ F(c, σ2u)

The first order autocovariance γ1 ≡ G(c, σ2u)


The Roll Model - Variance and Auto-covarianceThe variance:

γ0 ≡ Var(∆pt) = E [(∆pt)2]− (E [∆pt ])

2

= E [q2t−1c2 + q2

t c2 − 2qt−1qtc2

−2qt−1utc + 2qtutc + u2t ]

= 2c2 + σ2u.

In the expectation of this equation all of the cross-productsvanish except for those involving q2

t , q2t−1, and u2

t .The first-order auto-covariance:

γ1 ≡ Cov(∆pt−1,∆pt)2 = E (∆pt−1∆pt)

= E [c2(qt−2qt−1 − q2t−1 − qt−2qt + qt−1qt)

+c(qtut−1 − qt−1ut−1 + utqt−1 − utqt−2)] = −c2.

It can be easily verified that all autocovariances of order 2 orhigher are zero.


The Roll Model - Estimate of Model ParametersFrom the equations obtained earlier, we can compute themodel parameters as:

c =√−γ1 (15)

σ2u = γ0 + 2γ1 (16)

Given a sample data, it is sensible to estimate γ0 and γ1 andapply these transformations to obtain estimate of the modelparameters.

Example: Based on all trades for PCO in October 2003, theestimated first-order autocovariance of the price changes isγ1 = −0.0000294. This implies c = $0.017 and a spread of2c = $0.034.The Roll model is often used in situations where we don’tpossess bid and ask data. In this example, we do know thatthe time-weighted average NYSE spread in the sample is$0.032, so the Roll estimate is fairly close.

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