amamef 2013 & banach center, warszawa *1cm an application...

AMaMeF 2013 & Banach Center, Warszawa

An application of the method of moments to range-basedvolatility estimation using daily high, low, opening, and

closing prices

Cristin BuescuKing’s College, London

Joint work with M. Taksar and F.J.Kone

Cristin Buescu Range-based vol estimation using daily HLOC prices 1 / 27

Disclaimer

The following expresses the views of its author(s) only, and should not betaken to represent views of institutions with which the authors are or havebeen based. In particular, the information provided is personal opinion andshould not be relied upon for financial advice. If you require financialadvice or guidance please contact an appropriate professional.

Dedication

In memoriam Michael Taksar (1949 - 2012)

Outline

1. Motivation (THE STORY)The Volatility Estimation Problem

2. Our approach (THE MATHEMATICS)1. Expectation of range of arithmetic Brownian motion2. Method of moments using HLOC data3. Solving implicit equation and including opening jumps

3. Comparison results (SIMULATED DATA)

4. Application (REAL DATA)

5. Summary

Outline

5. Summary

Notation (HLOC: high, low, close, open stock price)

−Oi+1

trading day after−hours

Volatility Estimation Problem in Black-Scholes I

• Black-Scholes setting: stock price dSt = µStdt + σStWt (GBM)

• Volatility estimation σ (equivalently, variance estimation σ2)

• Literature on this:assumes security price has no drift⇒ OVERESTIMATION

assumes no price jump at opening⇒ UNDERESTIMATION

• Garman and Klass (1980): variance estimator VGK assumes no drift

• Parkinson (1980): variance estimator VP uses only high-low data

Volatility Estimation Problem in Black-Scholes II

• Rogers and Satchell (1991) and Rogers et al (1994): VRS assumes noopening jumps

VRS =1

n∑i=1

Oi− log

)+ log

Oi− log

• JUMP (Ci → Oi+1): the sample variance V0 of log(Oi+1/Ci )

V0 :=1

n − 1

n∑i=1

Ci− 1

n∑j=1

logOj+1

)2. (2)

• Incorporate opening jumps in VRS by using V0 + VRS?

Volatility Estimation Problem in Black-Scholes III

Yang and Zhang (2000)

• caution against easy fix (adding V0 to VRS), not min variance

• propose estimator assuming stock has drift and opening jumps:

VYZ = V0 + kVC + (1− k)VRS , (3)

V0 sample variance of log(Oi+1/Ci )VC is the sample variance of log(Ci/Oi )k is constant chosen to minimize estimator variance for fixed n

Volatility Estimation Problem in Black-Scholes IV

In contrast, our approach is based in part on Kone (1996):

1. derive expectation of range of arithmetic Brownian motion

2. estimate range using method of moments and HLOC daily data

3. solve implicit equation for intra-day volatility and include openingjumps

We propose the variance estimator:

VZ := V0 + Vi , (4)

where Vi is solution of an implicit equation.

Compare to VYZ = V0 + kVC + (1− k)VRS .

Outline

5. Summary

Range of arithmetic Brownian motion

In a filtered probability space (Ω,F , Ftt ,P) consider:

• Arithmetic Brownian motion: dXt = µdt + σdWt , X0 = 0

• running max: Mt := sup0≤s≤t

• running min: mt := inf0≤s≤t

• Range: Rt := Mt −mt

Joint density of aBM and max: (Xt ,Mt)

P(Xt ∈ da,Mt ∈ db) =2(2b − a)√

2πt3σ3exp

−(2b − a)2

2tσ2+

σ2a− 1

da db.

• Derived directly.

• Can be obtained from equation (1.8.8) of Harrison (1985)

• For σ = 1 this was used in Example E5 of Karatzas and Shreve(1998) in relationship to Clark’s formula to obtain explicitly thehedging portfolio.

Density and expectation of half-range: Mt − Xt

Density of half-range via Jacobian (2-dim transformation):

fMt−Xt (c) = 2µ

σ2Φ(µt − c

(− 2

2tπexp

−(µt + c)2

Expectation of half-range:

E (Mt − Xt) =σ2

2µΦ(µσ

√t)−(µt +

)(1− Φ

√(t)))

+σ√

t√2π

exp(− tµ2

Expectation of range of arithmetic Brownian motion

Rt = Mt −mt = (Mt − Xt) + (Xt −mt)

Symmetry between half-ranges gives:

E [Rt ] =(µt +

)(1− 2Φ

(−√

t√2π

exp(− tµ2

Equivalently:

E [Rt ] = h( µt

)=: ER(µ, σ, t), (6)

where the function h is defined by:

h(x , y) :=

(x2 + 1)(2Φ(x)− 1) +

2x√2π

exp(− x2

)y . (7)

Corollaries

• Derive also joint density of (Mt ,mt) (see also Borodin and Salminen(1996))

• and the density of the range Rt of an arithmetic Brownian motion

1 day = intra-day + after-hours

−Oi+1

trading day after−hours

1 = (1− f ) + f

Price St (GBM):dSt

St= µs dt + σ dWt , t ≥ 0. (8)

Log-price Xt = log St (aBM):

dXt = µdt + σdWt , µ = µs −σ2

Assume after-hours= fraction f of 1 day.

Intra-day data (fraction 1− f of 1 day):

Oi = Si−1, Ci = Si−f , Hi = supt∈[i−1,i−f ]

St , Li = inft∈[i−1,i−f ]

1 = (1− f ) + f

Price St (GBM):dSt

St= µs dt + σ dWt , t ≥ 0. (8)

Log-price Xt = log St (aBM):

dXt = µdt + σdWt , µ = µs −σ2

Assume after-hours= fraction f of 1 day.

Intra-day data (fraction 1− f of 1 day):

Oi = Si−1, Ci = Si−f , Hi = supt∈[i−1,i−f ]

St , Li = inft∈[i−1,i−f ]

Application of the method of moments

Use intra-day range data to derive the implicit equation in x = σ(1− f ):

R1−f = M1−f −m1−f = log H1 − log L1 = logH1

L1⇒ k1 :=

n∑i=1

Recall E (R1−f ) = ER(µ, σ, 1− f ) = ER(µ(1− f ), σ√

1− f , 1).

Compute E[

Si−fSi−1

]= µ (1− f )⇒ k2 :=

n∑i=1

Implicit equation k1 = h(k2

). Solution: Vi = x2. (9)

Full one-day variance

Compute:

]= VAR

Si−fSi−1

]= σ2 (1− f ),

logOi+1

]= VAR

Si−f

]= σ2 f .

Thus, we can write heuristically:

σ2 = VAR[

]= VAR(intra-day)+VAR(after hours)

leading toVZ = Vi + V0 (annualised σ2a := 252VZ ).

Outline

5. Summary

Monte Carlo simulation for known vol

Eg 1: Strikingly similar pattern for the two estimated vols when trueσa = 0.2 for varying number of data points (µs = 0.015, f = 0.25,250 trading days, 50k time points)

0 20 40 60 80 100 120

σZ σ

Compare the two vol estimators: σZ vs σYZ

Goal: investigate MVUEDue to the implicit nature of the equation we achieve this by simulation.Repeat previous simulation 5k times and look at:

• unbiased• percentage of times σZ is closer to true σ than σYZ• Mean Absolute Error (in L1-norm) comparison• average σZ vs average σYZ (averaged over the number of scenarios)

• min variance• efficiency

Unbiased I

Percentage of scenarios where our estimator σZ is closer than σYZ to thetrue value. For more than 37 data points in the estimation σZ is closer tothe true value more than half the time.

Unbiased II

The Mean Absolute Error (MAE) for our estimator σZ (continuous line)follows closely that of σYZ (dotted line) and both stabilize for n ≥ 55.

0 20 40 60 80 100 120

MAE σZ MAE σ

Unbiased III

The difference of the Mean Absolute Errors of σZ and σYZ goes below 0for n ≥ 37, suggesting that our σZ is better than σYZ in this range.

-0.001

-0.0005

0.0005

0.0015

0.0025

0.0035

20 40 60 80 100 120

Unbiased IV

The volatility σZ averaged over the number of scenarios (continuous line)is closer to the true value σ = 0.2 than the averaged σYZ (dotted line)when the number of data points is n ≥ 21.

0 20 40 60 80 100 120

Mean σZ Mean σYZ

Min variance I

• Following Garman and Klass (1980) we define the efficiency of ourestimator with respect to VYZ as:

Eff :=VAR(VYZ )

VAR(VZ ). (10)

• Numerical approximation shows that the efficiency is higher than 1with fewer data points, while overall it doesn’t drop by more than 1%.

Min variance II

Efficiency vs number of data points:

0 20 40 60 80 100 120

Impact of choice of f : none

• Yang and Zhang (2000): range for f is [0.18, 0.3] (typically 0.25).

• For this reason we have considered not only the case f = 0.25, butalso f = 0, f = 0.18 and f = 0.3, but the results were similar.

• Even in the driftless case (µ = 0) the values f = 0, f = 0.18, f = 0.25and f = 0.3 resulted in findings that were qualitatively similar.

Conclude: satisfactory performance on simulated data.

Outline

5. Summary

Application to algorithmic trading

Eg 2: IBM estimated volatility σa (continuous line) is similar to, butbelow, the corresponding volatility σYZ (dotted line) for each trading daybetween May 26 and June 18, 2010. Can be used in algo trading.

2010-05-26 2010-05-29 2010-06-01 2010-06-04 2010-06-07 2010-06-10 2010-06-13 2010-06-16

σa σYZ

Outline

5. Summary

Summary

• Derived expectation of range of Brownian motion(and density, see also Borodin and Salminen (1996))

• Proposed a volatility estimator σZ based on daily range data, whichincludes opening jumps and does not assume zero drift for stock price

• Advantages over estimator of Yang and Zhang (2000):• no need to estimate the constant k that achieves min variance• captures volatility using range data (fewer measurements than

normalized highs and lows)

• Disadvantages:• implicit equation gives only numerical approximation of variance• may lose up to 1% of efficiency (could have slightly larger variance)

• Works well and can be used in algorithmic trading.

Summary

Bibliography I

A.N. Borodin and P. Salminen.Handbook of Brownian motion - facts and formulae.Birkhauser, Basel, 1996.

C. Buescu, M. Taksar and F.J.KoneAn application of the method of moments to range-based volatilityestimation using daily high, low, opening, and closing (HLOC) prices.(forthcoming in Int. J. Theor. Appl. Finance)

M. Garman and M. KlassOn the estimation of security price volatilities from historical data.Journal of Business 53(1), 67–78, 1980.

Bibliography II

F.J. KoneEstimation of the volatility of stocks using the high, low and closingprices.Ph.D. thesis, State University of New York at Stony Brook, 1996.

M. ParkinsonThe extreme value method for estimating the variance of the rate ofreturn.Journal of Business 53(1), 61–65, 1980.

L.C.G. Rogers and S. SatchellEstimating variance from high, low and closing prices.Annals of Applied Probability 1(4), 504–512, 1991.

Bibliography III

L.C.G. Rogers, S. Satchell and Y. YoonEstimating the volatility of stock prices: a comparison of methodsthat use high and low prices.Applied Financial Economics 4, 241–247, 1994.

K. Sutrick, J. Teall, A. Tucker and J. WeiThe range of Brownian motion processes: density functions andderivative pricing applications.The Journal of Financial Engineering 6(1), 31–46, 1997.

D. Yang D. and Q. ZhangDrift-independent volatility estimation based on high, low, open, andclose prices.Journal of Business 73, 477–491, 2000.

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