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Econophysics... Problem Sets (Exercises)

posted on 14 February 2013

I made a tutorial in Econophysics at my uni. I have three books on the subject, but none of them have end-of-chapter exercises. Because this is such a quantitative field, I feel that it would be extremely beneficial to have some sort of "homework" to do; simply reading the mathematics behind the models just doesn't seem sufficient. Anyone got anything?

Discussion

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Hello,
there are many types of problems in econophysics... I assume you are asking for beyond basic options and portfolios covered by Hull and others... that is what is a vanilla a european and an american option and what are their portfolio relationships. If so, the statistics of observables (statistical analysis) and the underlying microscopic interactions (statistical dynamics) that produce them are the physics of economics focus.
1) given a data set of prices from S&P500 or a member stock (use yahoo.com historical data sets freely) or a liquid large cap, obtain
a) obtain the Histogram of prices and or price changes
b) obtain the PDF probability density function of prices and or price changes P(X,T) here let X=x(t)-x(t') and T=t-t' ... use i) Gaussian fitting ii) use power-law distribution fitting by discrete Student's T and by generally a power-law envelope function... how is the accuracy and what works best? Read the 'stylized facts' of real market prices and compare to your results.
c) obtain the Hurst exponent of the superdiffusive trajectories of the stochastic process ...by detrended fluctuation analysis and by other methods of literature whether autocorrelators, generating functions, CDF cumulative distribution functions...
d) obtain the auto-correlation function of the superdiffusive process...extra credit, obtain the power spectrum of the superdiffusive process... extra extra credit, obtain the SR stochastic resonance Power vs. variance graph.. does your real market data exhibit a stochastic resonance? what type periodic, aperiodic, nontypical... what type would you expect? what does this say about cycles in stock markets? what are the most prominent cycles mention daily weekly monthly quarterly, seasonally and 4-6year policy cycle...can you find such effects simply by inspecting stock markets and therefore economies as measured by stock markets?
e) If the superdiffusive process is generally a nonlinear SDE Ito stochastic diffeq, dx=a(x,t)dt+b(x,t)dW(t) , what are the a(x,t), b(x,t) drift,diffusion coefficients , and do these 'jibe' with the Gaussian PDFs you obtained as (for each time ) P(x,t)=N(t) Exp[ -(x- a)^2 / 2b t]... what about Student's T ??? and power-law envelope PDFs???

... these should point out the need for apriori nonlinear statistics as made possible by nonextensive statistics my personal favourite and other methods... also the concepts of superdiffusion, the fat tails or outliers

2) given the log normal SDE dx/x= (c) dt + D dW(t) utilized inaccurately throughout finance and economics from portfolio theory to derivatives pricing,
a) derive the Black-Scholes pricing formula
b) calculate the prices of an american style call option on a real stock by such formula utilizing any method convenient
c) how accurately do you feel such pricing compares to real prices
d) research literature for generalized options formula of Black-Scholes and beyond... are they more accurate?
e) look at the Heston SV stochastic volatility model of derivative pricing... how do the coupled SDEs utilized perform as a pricing method? research literature on generalizations and improvements, are they any more precise why or why not.... when does the generalized SV models approach the nonlinear SDE dx mentioned?

... this should get them started on Black Scholes and derivatives

3) games and rules based simulations...
a) use a super spin model to simulate demand, and calculate excess demand and with market depth proportionately the price changes and or prices ... this is a 'magnetism' type of model from physics, utilize any available source from the net to your own derivation to generate a stohastic time series from such modeling
b) does the observed excess demand, proportionately prices or price changes time series go as dx=a(x,t)dt+b(x,t)dW(t) ... that is does a rule based interaction game modeled by 'spin' as bias of demand and supply describe markets as claimed by econophysicists ?
c) do similar research utilizing multi agents, minority games, cellular automata etc... same question of prices time series... simple approach, find one of the many minority game simulators and perform similar simulation of stochastic time series.

d) how would you suggest to improve the rule based simulations to better describe real markets?
e) a crash or bubble has been described as a phase transition (more accurately as a transition between scales or a frustrated phase transition...). Do your simulations exhibit such quiescent regular fluctuations followed by large fluctuations and transitions? Do you feel such an approach describes such stylized facts of real markets? If not where are the limitations to your simulations

4) networks of interactions are another point of view of markets... research the financial networks models of traders and institutions...
b) write a computer program of a network wherein N interactors can each interact with n number of the n<N interactors chosen randomly with P(n) uniform, Gaussian, power-law ...and with strength p(s) for each interactor . At each time step for the network, calculate the total strength/N ... plot this as a time series.
Does this time series with a proportionality to prices then resemble a uniform, a Gaussian (including log normal), a power-law ???
Given real markets are as dx=a(x,t)dt+b(x,t)dW(t) and PDF==power-law , what do you need to make a network model real markets? that is does a uniform or Gaussian strengther produce power law PDFs of prices? or must you start with power-law strengthers to obtain power-law PDFs of prices...
experiment with N and s and n and proportionality factors and report results as well.

... this should get them started on network models

5) nonextensive statistics, levy processes, random matrices
Recent advances in statistics have produced a generalization of the Gibbs-Boltzmann extensive entropy S(AXB)=S(A)+S(B) statistics and S(A)=lnP(A) which then derive Gaussian PDFs, and linear Fokker-Planck PDEs with nonlinearity represented by nonlinear interactions and therefore drift coefficients, at the SDE level of Ito as dx=a(x,t)dt+DdW(t) and when transformed as a mixed drift diffusion process.... the newly innovated nonextensive entropy statistics S(AXB)=S(A)+S(B)+(1-q)S(A)S(B) then are satisfied by entropy or negative to information by S(A)= (P(A)- P(A)^q )/(1-q) which then derives power-law PDFs and apriori nonlinear Fokker-Planck PDEs yet with linear interactions or drift coefficients which yet precisely model real market prices as mentioned.
extra credit... for prices of a stock or index, utilize such q-parameterized power-law PDFs to fit the histogram of data ... what are your per increment values of drift and diffusion coefficient and do they track real prices' statistics accurately.
... extra credit.. if such SDEs then are accurate, as dx=a(x,t)dt+P(x,t)^(1-q)/2 dW(t) (note Pdf shows up in SDE !!) , rederive Black-Scholes equation pricing by this supposedly 'precise' statistics of the underlying equity... is the generalized Black-Scholes with nonextensive statistics more accurate? analyze results as comparing real prices.
... this should get them started on recent statistical methods in finance. Other levy processes, RMT random matrix theory of Wigner and more recently of nonextensive q parameterized qRMT random matrix theory can be suggested as further reading unless interested.
cheers,
Fredrick

Fredrick, that was exactly what I was looking for. Thank you!

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Although I truly like this post, I think there was an spelling error near to the end of the 3rd paragraph.