Evaluation and pricing of risk under stochastic volatility

1. Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa

2. Agenda • P versus Q:a brief overview of two branches of quantitative finance Freely inspired by http://ssrn.com/abstract=1717163 • The Stochastic Discount Factor The link with Asset Pricing and the Consumption- Investment optimization problem • An SDF perspective over P and Q Realizing smiles and quantiles

3. Risk and portfolio management: the P world Risk and portfolio management aims at modelling the probability distribution of the market prices at a given future investment horizon The probability distribution P must be estimated from available information. A major component of this information set is the past dynamics of prices, which are monitored at discrete time intervals and stored in the form of time series Estimation represents the main quantitative challenge in the P world

4. The legacy of Basel II: the (in)famous Value-at-Risk measure

5. The legacy of Basel II: the (in)famous Value-at-Risk measure

6. Derivatives pricing: the Q world The goal of derivativespricingis to determine the fair price of a given security in terms of the underlyingsecuritieswhosepriceisdetermined by the law of supply and demand. The risk-neutralprobabilityQand the realprobabilityPassociate differentweightsto the samepossibleoutcomes for the samefinancialvariables. The transition from one set of probabilityweights to the otherdefines the so-calledrisk-premium. Calibrationisone of the mainchallenges of the Qworld. Forward-lookingmeasure.

7. Empirical comparison Physical and risk-neutral moments from 28-day options (S&P500, EGARCH, OTM options). Taken from V. Polkovnichenko, F. Zhao Journal of Financial Economics, 2013, Vol. 107 580-609

8. The Stochastic Discount Factor • We now study the consumption-investment optimization problem of an agent maximizing an intertemporal utility criterion • The optimality conditions implied by agents optimal intertemporal choices show the existence of a universal random variable, the stochastic discount factor (SDF), such that asset prices are expectations of contingent payoffs scaled by the SDF.

9. Expected utilities Consider two dates, t and t+1. A consumption plan can be interpreted as a random variable taking value in a set Agents express preferences over consumption bundles by mean of a preference relation. We are interested in preference relations that are sufficiently general to depict interesting economic behaviour. To this end, one typically introduces some behavioural axioms that permit a description of preferences by mean of some expected utility representation (for instance von Neumann, Morgenstern (1944)) We call the two-period utility function for deterministic consumption bundles.

10. Time additive utility functions Time additive multiperiod utility functions are often used for computational convenience (even though for many asset pricing applications such assumption is not only unrealistic but even undesirable) Current investor wealth Wt can be either used for current consumption or it can be invested in a set of L financial assets. The resulting budget constraint is

11. Optimal consumption and investment problem At time t+1everyfinancialassetpays a payoff xl,t+1. Allwealthavailableat time t+1will be consumed The resultingoptimproblemis

12. The marginal rate of substitution By replacing the constraints in the objective function, the first order conditions for an interior optimal portfolio allocation ω are Above formula is the key formula from our asset pricing perspective. It defines a general asset pricing equation where today’s price is obtained as a conditional expectation of the intertemporal marginal rate of substitution times the asset payoff.

13. The SDF Nowwewant to abstract from the context of expected utility maximization and wegive the general definition

14. The fundamental theorem of asset pricing The economiccontent of the existence of a positive SDF is the absence of arbitrageopportunities in the market • Under whichconditionsis the SDF unique, whenitexists? The SDF isuniquewhenmarkets are complete Whichgeneral economicassumptionsmayensure the existence of a SDF?

15. The discrete time Black-Scholes model • The investor can trade portfolios of three basic assets: a risk-free zero-coupon bond, a risky asset, and a European call option • The risky asset • The call’s payoff • The bond payoff • The payoff space is spanned by exp yt+1, (exp yt+1 - k)+, and 1, which do not span the entire space of square integrable random variables. The market is not complete.

16. The discrete time Black-Scholes model • Absolut pricingapproach (preferencebasedsetting):ifwe assume a lognormalconsumptiongrowth in a time separablepower utility frameworkwereproduce the standard B&S result • Relative pricingapproach:we assume an exp affine SDF family parametric in v0 and v1 Mt,t+1=exp( - v0 - v1 yt+1 ) • No arbitragerestrictions • Et[Mt,t+1 1] = exp (-r) • Et[Mt,t+1expyt+1] = 1 • Aboveconditionsfixunivocally the values of v0 and v1

17. Realizing smiles and quantiles An SDF perspective over Q and P Work in progress with Adam A. Majewskij and Fulvio Corsi

18. Heterogeneous AR Gamma with Leverage (HARGL) Yt+1dailyreturn RVt+1realizedvariance Ltleveragefunction rrisk-free rate gammaequityrisk premium Taken from F. Corsi, N. Fusari and D. La Vecchia, Journal of Financial Economics, 2013, vol. 107, 284-304

19. Persistent discrete time models with stochastic volatility Comparison of the out-of-sample performances of 2-week-ahead forecasts of the AR(3), ARFIMA(5, d, 0), and HAR(3) models for S&P500 Futures. Taken from F. CorsiJournal of Financial Econometrics, 2009, Vol.7 174-196

20. Exponential affine SDF • The SDF transformsexpectations from P to Q! • v2: is the equityrisk premium • v1 : combinesboth the equity and the volatilityrisk premia

21. Realizing quantiles • Musil’s imaginary bridge You begin with ordinary solid numbers, representing measures of length or weight or something else that’s quite tangible - at any rate, they’re real numbers. And at the end you have real numbers. But these two lots of real numbers are connected by something that simply doesn’t exist. Isn’t that like a bridge where the piles are there only at the beginning and at the end, with none in the middle, and yet one crosses it just as surely and safely as if the whole of it were there? That sort of operation makes me feel a bit giddy R. Musil, Young Törless

22. Realizing quantiles Unexpectedlywefind with