Transcript 4028-10-Syllabus

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28-Sep-04
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Books –
Econophysics
• Statistical Mechanics of Financial Markets
– J Voit Springer ISBN 3 540 41409 6
• Patterns of Speculation; A study in Observational
Econophysics
– BM Roehner Cambridge
• Introduction to Econophysics
– HE Stanley and R Mantegna Cambridge
– Cambridge ISBN 0 521 62008 2
• Theory of Financial Risk: From Statistical Physics to
Risk Management
– JP Bouchaud & M Potters Cambridge
• Financial Market Complexity
– Johnson, Jefferies & Minh Hui Oxford
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Econophysics SS-4013
Syllabus
• What is a stock?
– Reading the press; fundamental and noise traders
– Price formation; market and limit orders; order book
– Historical data; indices and price time series; price returns; volatility; fat tails
• Distribution functions;
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joint distributions;
Bayes’ theorem; conditional and unconditional distributions
Characteristic functions; normal distributions; Levy distributions; kurtosis;
combining random variables;central limit theorem; stable distributions;
Markov process; Chapman Kolmogorov;
Bachelier’s approach to price fluctuations;
Additive noise, Gaussian and Wiener random walks; multiplicative walks
Stochastic differential equations
– Langevin Equations
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Breakdown of Bachelier and Gaussian
– Empirical or Stylized facts
• Distribution function for Returns
• Scaling of financial data (price returns)
– Mandelbrot; Stable Levy distributions
– ‘Stanley’ data analysis for high frequency returns
• Autocorrelation functions
– Price returns
– Clustered volatility
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Stock portfolios
• Risk
– Distribution functions; cumulative distributions
– Gaussian v power laws
• Minimizing risk
– Markowicz’ theory and efficient market theory
– Correlations
– Stock taxonomy
– Minimal spanning trees
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Options
• Futures; calls and puts
• Black Scholes
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Minority Game
• El Farol problem
• Bounded rationality
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Simple agent models
– Model of Bouchaud Cont
• Noise traders
• Fundamental traders
• Non linear effects: crashes and bubbles
– Peer pressure and Lotka Volterra models
– Can agents be modelled as molecules?
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2002 Exam 1
• A) How is a 1st order Markov process defined? [1]
– Consider random variable x that takes values x1, x2, x3…..etc at times t1, t2,
t3….etc
– The probability for x1,t1 given the earlier sequence is
– p(x1,t1|x2,t2, x3,t3…etc).
– A first order Markov process is one where this probability distribution only
depends on the previous value: ie p = p(x1,t1|x2,t2)
– Generally just called a Markov process
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2
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B) The Chapman Kolmogorov equation is an integral equation for the
conditional probability
– p(x1,t1|x2,t2) = - p(x1,t1|x3,t3) p(x3,t3|x2,t2)dx3
Explain how Bachelier used this equation to obtain a Gaussian distribution for
stock price returns stating the assumptions used. You may ignore any
complications due to Ito corrections.
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Bachelier assumed
p(x1,t1|x2,t2)  p(x1-x2,t1-t2|x2,t2)
He further assumed the fluctuations were independent. I.e. p(x1,t1|x2,t2)  p(x1,t1)
CK equation now reduces to
p(x1,t1) = - p(x1-x3,t1-t3) p(x3,t3)dx3
And can be solved with ansatz p(x,t) = p0(t)exp[-p02(t)x2)
This yields p02(t1+t2) = p02(t1) p02(t2)/ [ p02(t1) + p02(t2)]
 p0(t) = H/t
Substitute 2 = t/2H2
Obtain P(x,t) = exp[-x2/22(t)] / / 2(t)
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2 continued
• Bachelier then identifies the random variable with the log of
the asset price, S.
• If this follows a random walk we have
• S –S0 ~ (rt+)S
ie lnS/S0 ~ rt +
• Thus in expression for Gaussian
– x = ln S/S0 – rt and (t) = t
• This ignores the correction of Ito. With this correction
included the correct expression is
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X = ln S/S0 – {r -2/2}t
• Hence distribution function for stock market returns of time
horizon, t follows Gaussian distribution for all t.
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• How does distribution
observed for stock price
returns deviate from
Gaussian?
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– Stock returns exhibit fat
tailed distribution function.
Modelled by Mandelbrot
as Levy distribution with
tail exponent of ~1.7.
More recently Stanley et al
have shown high
frequency returns follow
an exponent of ~4
(cumulative distribution
~3)
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• Bachelier assumes
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volatility, , to be
constant. Sketch out how
the volatility looks in
practise
• Volatility is |return| or an
average of |return| over
time.
• Illustrative sketch for
annual volatilities for FTA
index over 19th and 20th
centuries. Red line is
constant assumed by
Bachelier.
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FTA Annual volatility
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1800
1850
1900
1950
2000
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• What is a market order? [1]
– Market orders are executed immediately when a matching order(s) arrives
irrespective of the stock price
– The price may change during the waiting time
• What is a limit order? [1]
– A limit order is triggered when the market price reaches a predetermined
threshold
– Used to protect against unlimited losses or buying at too high a price
– No guarantee exists that the order will be executed at or even close to the
threshold
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6
• The pictures below are part of a market maker’s order book. Annotate
the pictures explaining their meaning and show how the order is
completed
Cum ulative Order Volum es
1600
1400
Cumulative volumes
1200
1000
800
600
400
200
0
159
160
161
162
163
164
165
Price
162.2
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All sell orders
up to 162 executed
Buy order at 162 executed
only in part ie 200 sold
Remainder lapses or new
price is negotiated
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7
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How is the order book modified by the presence of a limit order?
Now all demand
with limit of 163
executed
162.5
2000
1800
Order Volumes
1600
1400
Demand
1200
Supply
1000
800
Demand including
market order
600
400
200
0
158
160
162
164
166
100 out of 300 sold
Order completed
in part
Price
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8
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Suppose orders arrive sequentially at random each with a mean waiting time of 3
minutes and standard deviation of 2 minutes. Consider the waiting time for 100
orders to arrive. What is the approximate probability that this will be greater
than 400 minutes?
– Assume events are independent.
– For large number of events, use central limit theorem to obtain m and .
– Thus
• Mean waiting time, m, for 100 events is ~ 100*3 = 300 minutes
• Average standard deviation,  ~ 2/100 = 0.2 minutes
– Model distribution by Gaussian, p(x) = 1/[(2)½] exp(-[x-m]2/22)
– Answer required is
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P(x>400) = 400 dx p(x) ~ 400 dx 1/((2)½) exp(-x2/22)
= 1/()½ z dy exp(-y2)
where z = 400/0.04*2 ~ 7*10+3
=1/2{ Erfc (7.103)} = ½ {1 – Erf (7.103)}
– Information given: 2/ * z dy exp(-y2) = 1-Erf (x)
– and tables of functions containing values for Erf(x) and or Erfc(x)
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