evaluation and pricing of risk under stochastic volatility giacomo bormetti scuola normale...
TRANSCRIPT
![Page 1: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/1.jpg)
Evaluation and pricing of risk under stochastic volatilityGiacomo BormettiScuola Normale Superiore, Pisa
![Page 2: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/2.jpg)
Agenda
① P versus Q: a brief overview of two branches of quantitative finance
Freely inspired by http://ssrn.com/abstract=1717163
② The Stochastic Discount Factor
The link with Asset Pricing and the Consumption-Investment optimization problem
① An SDF perspective over P and Q
Realizing smiles and quantiles
![Page 3: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/3.jpg)
Risk and portfolio management: the P world
a. Risk and portfolio management aims at modelling the probability distribution of the market prices at a given future investment horizon
b. 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
c. Estimation represents the main quantitative challenge in the P world
![Page 4: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/4.jpg)
The legacy of Basel II: the (in)famous Value-at-Risk measure
![Page 5: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/5.jpg)
The legacy of Basel II: the (in)famous Value-at-Risk measure
![Page 6: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/6.jpg)
Derivatives pricing: the Q world
a. The goal of derivatives pricing is to determine the fair price of a given security in terms of the underlying securities whose price is determined by the law of supply and demand.
b. The risk-neutral probability Q and the real probability P associate different weights to the same possible outcomes for the same financial variables. The transition from one set of probability weights to the other defines the so-called risk-premium.
c. Calibration is one of the main challenges of the Q world.
d. Forward-looking measure.
![Page 7: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/7.jpg)
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
![Page 8: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/8.jpg)
The Stochastic Discount Factor
① We now study the consumption-investment optimization problem of an agent maximizing an intertemporal utility criterion
② 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.
![Page 9: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/9.jpg)
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.
![Page 10: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/10.jpg)
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
![Page 11: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/11.jpg)
Optimal consumption and investment problem
At time t+1 every financial asset pays a payoff xl,t+1. All wealth available at time t+1 will be consumed
The resulting optim problem is
![Page 12: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/12.jpg)
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.
![Page 13: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/13.jpg)
The SDF
Now we want to abstract from the context of expected utility maximization and we give the general definition
![Page 14: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/14.jpg)
The fundamental theorem of asset pricing
① Which general economic assumptions may ensure the existence of a SDF?
② Under which conditions is the SDF unique, when it exists?
The economic content of the existence of a positive SDF is the absence of arbitrage opportunities in the market
The SDF is unique when markets are complete
![Page 15: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/15.jpg)
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.
![Page 16: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/16.jpg)
The discrete time Black-Scholes model Absolut pricing approach (preference based setting): if we assume a lognormal consumption growth in a time separable power utility framework we reproduce the standard B&S result
Relative pricing approach: we assume an exp affine SDF family parametric in v0 and v1
Mt,t+1=exp( - v0 - v1 yt+1 ) No arbitrage restrictions
¡ Et[Mt,t+1 1] = exp (-r)
¡ Et[Mt,t+1 exp yt+1] = 1
Above conditions fix univocally the values of v0 and v1
![Page 17: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/17.jpg)
Realizing smiles and quantiles
An SDF perspective over Q and P
Work in progress with Adam A. Majewskij and Fulvio Corsi
![Page 18: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/18.jpg)
Heterogeneous AR Gamma with Leverage (HARGL)
① Yt+1 daily return② RVt+1 realized variance③ Lt leverage function④ r risk-free rate⑤ gamma equity risk premium
Taken from F. Corsi, N. Fusari and D. La Vecchia, Journal of Financial Economics, 2013, vol. 107, 284-304
![Page 19: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/19.jpg)
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. Corsi Journal of Financial Econometrics, 2009, Vol.7 174-196
![Page 20: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/20.jpg)
Exponential affine SDF
① The SDF transforms expectations from P to Q!
② v2 : is the equity risk premium
③ v1 : combines both the equity and the volatility risk premia
![Page 21: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/21.jpg)
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
![Page 22: Evaluation and pricing of risk under stochastic volatility Giacomo Bormetti Scuola Normale Superiore, Pisa](https://reader035.vdocument.in/reader035/viewer/2022062321/56649e6c5503460f94b6a5fd/html5/thumbnails/22.jpg)
Realizing quantiles
Unexpectedly we find
with