Statistical Arbitrage Based on No-Arbitrage Models

41 Pages·2011·0.72 MB·English

by  Liuren Wu

Statistical Arbitrage Based on No-Arbitrage
Models
Liuren Wu
ZicklinSchoolofBusiness,BaruchCollege
July 1, 2011
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Review: Valuation and investment in primary securities
The securities have direct claims to future cash flows.
Valuation is based on forecasts of future cash flows and risk:
DCF (Discounted Cash Flow Method): Discount forecasted future cash
flow with a discount rate that is commensurate with the forecasted risk.
Investment: Buy if market price is lower than model value; sell
otherwise.
Both valuation and investment depend crucially on forecasts of future
cash flows (growth rates) and risks (beta, credit risk).
Key empirical skills needed: Time-series analysis.
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Compare: Derivative securities
Payoffs are linked directly to the price of an “underlying” security.
Valuation is mostly based on replication/hedging arguments.
Find a portfolio that includes the underlying security, and possibly
other related derivatives, to replicate the payoff of the target derivative
security, or to hedge away the risk in the derivative payoff.
Since the hedged portfolio is riskfree, the payoff of the portfolio can be
discounted by the riskfree rate.
No need to worry about risk premium of a zero-risk portfolio.
Models of this type are called “no-arbitrage” models.
Key: No forecasts are involved. Valuation is based on cross-sectional
comparison.
It is not about whether the underlying security price will go up or down
(given growth rate or risk forecasts), but about the relative pricing
relation between the underlying and the derivatives under all possible
scenarios.
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Readings behind the technical jargons: P v. Q
P: Actual probabilities that earnings will be high or low.
Estimated based on historical data and other insights about the
company — time-series analysis.
Valuation is all about getting the forecasts right and assigning the
appropriate price for the forecasted risk.
Q: “Risk-neutral” probabilities that we can use to aggregate expected
future payoffs and discount them back with riskfree rate, regardless of
how risky the cash flow is.
It is related to real-time scenarios, but it has nothing to do with
real-time probability.
Since the intention is to hedge away risk under all scenarios and
discount back with riskfree rate, we do not really care about the actual
probability of each scenario happening.
We just care about what all the possible scenarios are and whether our
hedging works under all scenarios.
Q is not about getting close to the actual probability, but about being
fair relative to the prices of securities that you use for hedging.
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A Micky Mouse example
Consider a non-dividend paying stock in a world with zero riskfree interest
rate. Currently, the market price for the stock is $100. What should be
the forward price for the stock with one year maturity?
The forward price is $100.
Standard forward pricing argument says that the forward price should
be equal to the cost of buying the stock and carrying it over to
maturity.
The buying cost is $100, with no storage or interest cost.
How should you value the forward differently if you have inside
information that the company will be bought tomorrow and the stock
price is going to double?
Shorting a forward at $100 is still safe for you if you can buy the stock
at $100 to hedge.
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Investing in derivative securities without insights
If you can really forecast the cashflow (with inside information), you
probably do not care much about hedging or no-arbitrage modeling.
You just lift the market and try not getting caught for inside trading.
⇒ buy stock, long forward, buy calls, sell puts of all strikes and
maturities.
But if you do not have insights on cash flows (earnings growth etc)
and still want to invest in derivatives, the focus does not need to be
on forecasting, but on cross-sectional consistency.
No-arbitrage pricing models can be useful.
From math to finance: When you write down some stochastic
process, it is important to know whether you want to use the process
to predict future movements or to use it to link the prices of different
securities together.
Different application involves different consideration and estimation.
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Estimating statistical dynamics
Construct likelihood of the L´evy return innovation based on Fourier
inversion of the characteristic function.
If the model is a L´evy process without time change, the maximum
likelihood estimation procedure is straightforward.
Given initial guesses on model parameters that control the L´evy triplet
(µ,σ,π(x)), derive the characteristic function.
Apply FFT to generate the probability density at a fine grid of possible
return realizations — Choose a large N and a large η to generate a find
grid of density values.
Interpolate to generate intensity values at the observed return values.
Take logs on the densities and sum them.
Numerically maximize the aggregate likelihood to determine the
parameter estimates.
Trick: Do as much pre-calculation and pre-processing as you can to
speed up the estimation.
Standardizing the data can also be helpful in reducing numerical issues.
Example:
CGMY,2002,TheFineStructureofAssetReturns,JournalofBusiness,75(2),305–332.
LiurenWu,DampenedPowerLaw,JournalofBusiness,2006,79(3),1445–1474.
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Estimating statistical dynamics
The same MLE method can be extended to cases where only the
innovation is driven by a L´evy process, while the conditional mean
and variance can be predicted by observables:
dS /S = µ(Z )dt +σ(Z )dX
t t t t t
where X denotes a L´evy process, and Z denotes a set of observables
t t
that can predict the mean and variance.
Perform Euler approximation:
S −S √
R = t+∆t t =µ(Z )∆t+σ(Z ) ∆t(X −X )
t+∆t S t t t+∆t t
t
From the observed return series R , derive a standardized return
t+∆t
series,
R −µ(Z )∆t
SR =(X −X )= t+∆t √ t
t+∆t t+∆t t
σ(Z ) ∆t
t
Since SR is generated by the increment of a pure L´evy process, we
t+∆t
can build the likelihood just like before.
Given the Euler approximation, the exact forms of µ(Z) and σ(Z) do
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Estimating statistical dynamics
dS /S = µ(Z )dt +σ(Z )dX
t t t t t
When Z is unobservable (such as stochastic volatility, activity rates),
the estimation becomes more difficult.
One normally needs some filtering technique to infer the hidden
variables Z from the observables.
Maximum likelihood with partial filtering: AlirezaJavaheri,InsideVolatility
Arbitrage:TheSecretsofSkewness
MCMC Bayesian estimation: Eraker,Johannes,Polson(2003,JF):TheImpactofJumps
inEquityIndexVolatilityandReturns;Li,Wells,Yu,(RFS,forthcoming):ABayesianAnalysisof
ReturnDynamicswithL´evyJumps.
GARCH: Use observables (return) to predict un-observable (volatility).
Constructing variance swap rates from options and realized variance
from high-frequency returns to make activity rates more observable.
Wu,VarianceDynamics:JointEvidencefromOptionsandHigh-FrequencyReturns.
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Estimating statistical dynamics
Wu,VarianceDynamics:JointEvidencefromOptionsandHigh-FrequencyReturns.
Use index options to replicate variance swap rates, VIX.
Under affine specifications, VIX2 = 1EQ[(cid:82)t+hv ds]=a(h)+b(h)v ,
t T t s t
where (a(h),b(h)) are functions of risk-neutral v-dynamics.
Solve for v from VIX: v =(VIX2−a(h))/b(h).
t t t
Build the likelihood on v as an observable:
t
dv =µ(v )dt+σ(v )dX
t t t t
Use Euler approximation to solve for the L´evy component X −X
t+∆t t
from v .
t
Build the likelihood on the L´evy component based on FFT inversion of
the characteristic function.
Use high-frequency returns to construct daily realized variance (RV).
Treat RV as noisy estimators of v : RV =v ∆t+error.
t t t
Given v , build quasi-likelihood function on the realized variance error.
t
Future research: Incorporate more observables.
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