statistical methods: from physics to financev) k. ito and r. stratonovich (~1950-60): calculus with...

Statistical Methods: from Physics to Finance

Luca DoriaTRIUMF, 2011

Palazzo Mezzanotte, Milano (Italy)

The PIENU Experiment

Precise Measurement of the Branching Ratio: B= e

e

- TRIUMF Based: pion beam thanks to the 500 MeV Cyclotron - Precision Experiment- Uncover Physics beyond the Standard Model

Jun 2011 Luca Doria, TRIUMF 3

The intersections between Physics and Finance/Economy are many!A very personal (and limited) selection of topics:

I) Introduction and Short History II) Some Physics Problems and Probability Distributions - Heavy Tails: Multiple Scattering - Gaussians: Heat Flow and Diffusion

- Brownian Motion

III) Stochastic Calculus IV) Stable Distributions V) Stocks VI) Derivates: Options VII) Summary & Conclusions


Is there any logic behind Economics or the Stock Market?


Historical Overview

Louis Bachelier (1870-1946)The Theory of Speculation

I) R. Brown (1773-1858): Observation of random motion in pollen samples II) L. Bachelier (1870-1946): First attempt to model stocks movements III) A. Einstein (1905): First model of the brownian motion IV) Mandelbrot and Pareto Distributions V) K. Ito and R. Stratonovich (~1950-60): Calculus with random variables VI) Black, Merton, Scholes: Stochastic model for options (~1973) 1997 Nobel Prize in Economics

Now:“Econophysics”: tries to apply physics methods to Finance and Economy:

- Stochastic Processes- Statistical Physics- Agent-based Models- Statistical Analysis- Feynman's Path Integrals

K. Ito (1915-2008)

R. Black, M. ScholesB. Mandelbrot (1924-2010)


PART 1:

PHYSICS MODELSand

PROBABILITY DISTRIBUTIONS:EXPONENTIALS vs POWER LAWS


Exponentials and Power-Laws Distributions

Probability Distribution: p(x)

Probability:

Moments:

Paxb=∫a

bp xdx

k=∫ x k pxdx



“Tails”

Exponential (e.g. Gaussian) Power-Law (e.g. Lorentzian)


Probability:

Moments:

Paxb=∫a

bp xdx

p x= 1

2 2e− x−2

2 2

k=∫ x k pxdx

- All the moments are finite- Central Limit Theorem- Limit of many distributions



“Tails”

Exponential (e.g. Gaussian) Power-Law (e.g. Lorentzian)


Probability:

Moments:

Paxb=∫a

bp xdx

p x= 1

2 2e− x−2

2 2

k=∫ x k pxdx

p x = 1

x−22

- All the moments are finite- Central Limit Theorem- Limit of many distributions

- Undefined moments k>0- CLT cannot hold


Multiple Scattering

pdf ∝1

0=13.6MeV c p

z x /X0[10.038 ln x /X0]

Used e.g. in:- Detector Physics and Simulation- Medical Physics

Two physics mechanisms: - EM Force - Strong Force

MaterialCharged Particle

θ


Heat Transfer and Diffusion

q=−k∇ uFourier's Law

Q=c PuThermodynamics:

If a change in internal energy in a material is given only by heat flux across the boundaries in a space/time region:

c P∫x− xx x

[u , t t −u ,t− t ]d=cP∫x− xx x

∫t−ttt ∂ u

∂d d

k∫t− tt t[∂ u∂ xx x ,−

∂ u∂ xx− x ,]d =k∫x− x

x x

∫t− tt t ∂2u

∂2 d d

∂u∂ t−k

∂2u

∂ x2=0

By energy conservation:

x− xx x ; t− tt t


Solution of the Heat Equation

∂u∂ t−k

∂2u

∂ x2=0∂u∂ tx , t −k∇ 2

u x , t =0

∂c∂ tx ,t −k∇2

c x ,t =0

Green's Function (Propagator):

General Solution:

Gx ,t =1

4 kte− x2

4kt

u x , t =∫G x− y ,t f y , t=0dy

Temperature

Concentration


Why Physicists are interested in power-law distributions?

Psx=asPx

Random events usually give rise to exponential tails: understood.If there are power law tails, something interesting is happening:

1) Critical State?During phase transitions, correlation functions have power-law behavior.This means e.g. that fluctuations are happening at all scales (see 2)).

2) Scale-free systemPower-law distributions are scale free:

Px =C x−

P sx=C sx−=C s− x−=a s Px

Only one candidate distribution:

Partial proof:


Brownian Motion

R. Brown (1773-1858): Observation of random motion in pollen samples

A. Einstein (1905): First quantitative explanation ofthe brownian motion

⟨ x2⟩=2Dt

D= RTN

16 k r

(For spherical particles)

AE estimate : ~6μm / minute @ 17C, N=1023

“… It is hoped that some enquirer may succeed shortly in solving the problem suggested here, which is so important in connection with the theory of heat.”Berne, May 1905 (Received, 11 May 1905).

N= 1

⟨ x2 ⟩RT3 k P

Opportunity to estimate N !

∂u∂ t−D ∂

2u

∂ x2=0


A Simple Model for the Brownian Motion: Symmetric Random Steps

t

x

t

h

p N1 t ,ih= 12p N t ,i−1h1

2p N t ,i1h

p N1 t ,ih h2

2∂x2 p N t , ihp N t ,ih∂t p N t , ih t

∂t p t , x=2

2∂x2 p t , x

⟨ x ⟩=0

Var x=⟨ x2⟩−⟨ x ⟩2= 12h2 1

2−h2=h2

t=N t⇒Nh2= th2

t

2 t⇒ h= t

If we want the variance at t being:≈ ≈

p t , x = 1

2t− se

− x−x 02

22 t−s


Physics: Summary

I) Random processes often encountered in physics (Probability/Distributions).

II) Connection to the diffusion (or heat) equation.

III) Not all random processes have gaussian tails !

IV) “Fat” tails are not always easy to explain: associated with “complex” systems, phase transitions, critical states, scale invariance, …

In general: something interesting for a physicist !

V) More formal methods needed to treat stochastic processes.


PART 2:

STOCHASTIC PROCESSES


Brownian Processes

A standard brownian process B_{t}, t>0 is a continuous stochasticprocess in t, with the following properties:

I) B_{0}=0 with probability 1

II) The increments are independent, i.e.: if 0<t0<...tk, then Bt1-Bt0, .... , Btk-Btk-1 are independent.

III) For every s,t, with 0 < s < t, the increments Bt-Bs are distributed like N(0,t-s)

In a non-standard process, N(0,σ2(t-s))


Stochastic Processes and Stochastic Calculus

X t =X t0∫t0

tF ,dt∫t 0

tG ,dW

dX t=F t , .dtG t , .dW t

It is not possible to define derivatives in a stochastic process (jumps!)It is possible to define the integral : the Ito (Stratonovich) integral:

General Stochastic Process:

Stochastic Differential:


The Ito/Stratonovich Integral

Ito Integral:

Stratonovich Integral:

Riemann-Stieltjes Integral:

∫0tf t dg t =lim n∞∑i=1

nf i g t i1−g ti

i=t i

i=12 ti1t i

If g is smooth, the limit is independent on the choice of τ.It is not true for stochastic processes!

Example:

Example:

∫0tW t dW t=

12W t

2−12W 0t

∫0tW t dW t=

12W t

2−12W 0


The Ito's Formula

dY t ={∂tU t , X t F t ∂ X U t , X t 12G2 t ∂X

2 U t , X t}dtG t ∂ X U t , X t dW t

Y t=U t , X t

dX t=F t , .dtG t , .dW tStochastic Process:

...a function of it:

The Differential:

Ito's formula tells us how to calculate derivatives of composite functions. The result is different wrt the classical calculus.


Stochastic Processes

dX t=F t , X t dtG t , X t dW t

dX t=2dW t

dX t= X t dt2X t dW t

dX t=−X t dt2 X t dW t

dX t=− X t dt2dW t Ornstein-Uhlenbeck (Fluctuation around zero)

“Return to the mean” process

Geometric Process

Simple Brownian Motion (L. Bachelier Model)

dX t=dt2dW t Brownian Motion with Drift


Summary

I) There is a powerful formalism for treating stochastic processes

II) Used in many scientific areas: - Condensed Matter Physics - Meteorology/Oceanography - Chemistry - Connections with Quantum Mechanics / QFT - Turbulence / Hydrodynamics - ....

III) Financial time series look like random walks: is it possible to apply these ideas to finance or economy?


PART 3:

STOCK MARKET


Financial Time Series

I) Can we describe financial time series with stochastic processes?

II) If yes, what are the properties of such a process?

III) In the market, not only stocks are exchanged: derivates. Models?

IV) Not only Finance: Economic Systems

Any relation with physics models?


Stock Market Time Chart

Goldman-Sachs Group, Inc, NYSE

0 200 400 600 800 1000 1200 1400 1600 1800

0

50

100

150

200

250

300

~today

Jan,1 2005

2007

S(t)

t(days)


From Bachelier to Mandelbrot


Observables

Nowadays: huge database of (high frequency) financial data. Opportunity to study (“data analysis”) the market and try to “understand” it. What are the important quantities to study? No reference units!

Y t

Z t =Y t t −Y t

Rt =Y t t −Y t

Y t

S t =ln Y t t −ln Y t

Price

Price Change

Return

Log-Change


Features:- Not Gaussian! It is not a Brownian Motion.

- Leptokurtic: heavy tails

- Almost no skewness

Distribution of the increments (returns)

Stock: Goldman-Sachs Group, Inc, NYSE

i=∫ x−x0i p xdxK=

4 4−3 0

S=3≈0


Correlation Functions

N-point Functions: ⟨ x t ⟩=∫ dx x f x , t

⟨ x t1 x t 2⟩=∫dx1dx2 x1 x 2 f x1 , t1 , x 2 , t 2

⟨ x t1 xt 2... xt n⟩

P [ x t ]=P [ x t]

C t 1 , t 2=⟨ x t 1 x t 2⟩−t 1 t 2=R−2 =t 2−t 1

S f =∫ℜ Re−2i f d =2 2c

12 f tc 2

Stationarity:

Autocorrelation:

Power Spectrum:S f ≈ 1

f 2

S f ≈ 1

f N 0N2

Brownian

Memory: M=∫0∞R d

Finite

Infinite

Undetermined


Correlation Functions (Single Stock)

Autocorrelation Function (Coca-Cola Company):Very fast decay (<1day).Arbitrage difficult.No memory

Spectral Density: (Coca-Cola Company):

Compatible with a random walk

S f ≈ 1

f 2

From: Stanley, Mantegna


Correlation Functions (Market Index)

Autocorrelation Function (S&P 500):Very fast decay (Corr time <4 min.).Arbitrage difficult.No memory

Spectral Density: (S&P 500):

Compatible with a random walk

S f ≈ 1

f 2

From: Stanley, Mantegna


Higher Order Correlations

- Pairwise independence does not imply independent random variables !- Look for high-order functions of the price change: e.g. the VOLATILITY.

LONG RANGE CORRELATION!Power Law with exponent ~0.3


Volatility: Summary

I) No “average” volatility existsII) Volatility is difficult to estimate (in the future)III) Clustering (Intermittency?).IV) Memory.V) ARCH / GARCH Models


Levy Stable Distributions

Definition: x is a stable random variable if for every positive A,B, exist C,D so that:

Where x1 and x2 are copies of x.

Ax1Bx2 ≈ CxD

Theorem: The sum of independent, identically distributed random variables converge to a stable distribution as the number of iid variables tends to infinity.

Stable (Levy) Distributions

lnk =i k− ⌈ k ⌉ [1−i k

⌈k ⌈tan /2] ≠1

i k− ⌈k ⌉[1i k⌈ k ⌈2 / tan ⌉k ⌈ ] =1

S , ,

02 0 ∈ℜ ∈−1,1Stability Index Scale Factor Shift Factor Asymmetry Factor


Power Law (Fat) Tails

limx∞ P Xx≈1sin /2

⌈ x ⌈1≈ 1

⌈ x ⌉1

Stable Distributions have fat tails:

Some known Analytic Forms:

=2

=1 =1

=12

=1

Gaussian

Landau

Levy

Lorentz=1 =0


The “stylized” Facts

I) Fat TailsLarge returns follow a fat-tailed distribution P(r) ~ r-α . Generally, α>2, therefore the variance is finite. This questions the use of Levy stable distributions. Truncated Levy pdfs have been proposed, among the many other proposals. The issue is still debated.

II) Asymptotic NormalityWhen the time scale of the returns is increased, the distribution tends to the normal one. Conclusion: the shape of the distribution changes with the time scale and therefore the underlying stochastic process has a non trivial temporal structure.

III) No linear autocorrelations (efficient market, no arbitrage)The autocorrelation falls quickly to 0: after ~15min it is practically =0. There is no memory effect: support of the “efficient market hypothesis”. Over longer timescales, (months) there are some (non conclusive) hints of autocorrelation.

IV) Clustered VolatilityIt is evident that the volatility of the market is not constant in time and there is autocorrelation: high volatility periods cluster in time and large changes are followed by large changes.


Stochastic Processes in Finance: Conclusions

Data:- Today, thanks to computers, we have a very large dataset of financial data to study.- It was not possible at the times of Bachelier!

Stochastic Process:- It is not a simple Brownian Motion (Bachelier Model)- Exhibits heavy (fat) tails- Very short correlation time (No memory)- Volatility has longer correlation (Memory?)- …

Models:- Brownian Motion (Not accurate, possibility of negative prices)- Lognormal Process (Better, but not fully accurate)- Ornstein-Uhlenbeck Process- Levy, Truncated Levy, Tsallis, … (Better heavy tails, but debated)- ARCH / GARCH , .... (Powerful, simple: worthed the 2003 Nobel Price!)


The Black-Merton-Scholes Model

Black, F. och M. Scholes, 1973, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy, Vol. 81, pp. 637-654.

Black, F., 1989, "How We came Up with the Option Formula", The Journal of Portfolio Management, Vol. 15, pp. 4-8

Hull, J.C., 1997, Options, Futures and Other Derivates, 3rd edition, Prentice Hall

Merton, R.C., 1973, "Theory of Rational Option Pricing", Bell Journal of Economics and Management Science, Vol. 4, pp. 141-183.

Robert C. Merton Myron S. Scholes

The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997 was awarded jointly to Robert C. Merton and Myron S. Scholes "for a new method to determine the value of derivatives "


Derivates: The Options

Definition:

In finance, an option is a derivate financial instrument that establishes a contract between two parties concerning the buying or selling of an asset at a reference price.

“Put” (“Call”) European Options:

The owner of the option has the right (but no obligation) to sell (buy) a certain

good at a certain time for a certain price.

Other jargon terms:

Underlying (Asset): A stock for a simple option (can be any good).

Strike Price: Price for buying or selling the underlying.

Expiration Date: When the option might be used.

American Options: The option can be used at any time before expiration.

Exotic Options: All other kind of options (e.g. asian options, ...).


The Black-Merton-Scholes Option Pricing ModelHypoteses: I) The short-term interest rate r is known and constant and it is possible to ask for money at that rate.

II) The strike price X is known and constant

III) The stock price S follows a geometric stochastic process

Where W is a Wiener process, μ is the expected return, σ the volatility.

IV) The stock does not pay dividends

V) No transaction costs and no limits to the short-selling

VI) There is no arbitrage

dS t= S t dt2S t dW t


The Black-Merton-Scholes Option Pricing Model

dV=dS− 1dc

dc=[∂ c∂ t 12 ∂2c∂ S 2 2S 2 ]dt ∂ c∂ S S dW

dV=−1 [ ∂ c∂ t 12 ∂2c∂ S 2 2 S2]dt

−1 [ ∂c∂ t 12 ∂2c∂ S 2 2S 2]dt=S− 1 crdt

∂c∂ t=rc−rS ∂c

∂ S− 12 ∂

2c

∂ S 2 2 S 2

Build a portfolio made of stocks and options: V =S− 1c

Consider its variation:

Apply Ito's Lemma to c(t,S):

Substituting:

Equivalence to a risk-free portfolio:(no arbitrage)

After some algebra:


Solution of the BMS Equation

c S , t =S N d 2−X e−r T− tN d 2

d 1=ln S / X r 1

2 2T−t

T−t

d 2=d1−T−t

N d i=1,2=∫−∞d i 1

2e−x2 /2dx

The solution of the BSM equation is:

Boundary condition at t=T:(for a call option)

c=max S−K ,0


Physics Interpretation of the BMS Equation

∂t c12 2S2∂SS crS ∂S c−rc=0

Diffusion Term

Convection Term

Reaction Term

Example :Dispersion of a pollutant in a river:

- Dispersion : Diffusion Term- Water Flow: Convection Term- Absorption (e.g. by sand) : Reaction Term


The BMS equation IS the heat equation !

cS , t ∂t c12 2S2∂SS crS ∂S c−rc=0

u x , ∂u=∂xx u

S=ex = 12 2T−t k= 2r

2

Substitution:

is solution of the BMS equation

is solution of the heat equation

cS , t =exp [−12 k−1 x− 12 k12]u x ,


BMS and Montecarlo (N=10): The “Greeks”

= ∂ c∂T

=∂ c∂ S

=∂ c∂ S

=∂ c∂ S

V = ∂ c∂

c=c S


BMS and Montecarlo (N-=100): The “Greeks”

c=c S

=∂ c∂ S

V = ∂ c∂

= ∂ c∂T


BMS and Montecarlo (N=1000): The “Greeks”

c=c S

=∂ c∂ S

V = ∂ c∂

= ∂ c∂T


Summary

I) Nowadays, statistical analysis of financial time series possible:- Probability Distributions (Returns, Rates, …) : Heavy-tailed distributions

- Time correlations, cross-correlations. - Spectral densities - …

II) Modeling of the time series with stochastic models: - Brownian motion (Log-normal process) - (Truncated) Levy processes - Mixed processes (jump-diffusion,...)

III) Derivates: - BMS Model and all the possible generalizations, Delta-Hedging - PDE methods (also numerical) - Montecarlo methods - Feynman Path Integrals !!


Conclusion

Do really a physics theory of finance (economics) exists?

(My) answer is: NO:

Often there are only some analogies and the economic system do not resemble closely any physics system (yet).There is no “econophysics” theory, like quantum mechanics or general relativity: it is more like a collection of methods, many of them borrowed from or used by physicists.

BUT:Physicists have skills often very useful if applied in this field:

I) Strong mathematical backgroundII) Strong computational backgroundIII) Construction of modelsIV) Statistical analysis of (huge) datasetsV) ....


Physics Models

Stochastic CalculusStatistics

Financial Time SeriesDerivatesEconomic Systems

Brownian MotionDiffusionTurbulenceStatistical Mechanics.....

(Auto)-correlation functionsLognormal ProcessIto's Formula....

Heat Equation“Fat” TailsMC methods.....


Ettore Majorana (1906 – 1938 - ?)

Il valore delle Leggi Statistiche nella Fisica e nelle Scienze Sociali(Unpublished)

The Value of the Statistical Laws in Physics and in the Social Sciences

“It is therefore important that the principles of quantum mechanics brought us to recognize the the statistical character of the ultimate laws of the elementary processes.”

“The study of the relationships, true or supposed, between physics and the other sciences, had always been of big interest because of the special influence physics had in modern times on the scientific thinking.”

My “free” translation!


Grazie!Tank Yo!

Merci!

statistical methods: from physics to financev) k. ito and r. stratonovich (~1950-60): calculus with...

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