Evaluation and pricing of risk
under stochastic volatility
Giacomo Bormetti
Scuola Normale Superiore, Pisa
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 ConsumptionInvestment optimization problem
③ An SDF perspective over P and Q
Realizing smiles and quantiles
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
The legacy of Basel II:
the (in)famous Value-at-Risk
measure
The legacy of Basel II:
the (in)famous Value-at-Risk
measure
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.
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
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.
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.
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
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
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.
The SDF
 Now we want to abstract from the context of expected utility
maximization and we give the general definition
The fundamental theorem of asset
pricing
① Which general economic assumptions may ensure the
existence of a SDF?
The economic content of the existence of a
positive SDF is the absence of arbitrage
opportunities in the market
② Under which conditions is the SDF unique, when it exists?
The SDF is
complete
unique
when
markets
are
The discrete time Black-Scholes
model
 The investor can trade portfolios of three basic assets: a riskfree zero-coupon bond, a risky asset, and a European call
option
 The risky asset
 The call’s payoff
 The bond payoff
①The 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.
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
① Et[Mt,t+1 1] = exp (-r)
② Et[Mt,t+1 exp yt+1] = 1
 Above conditions fix univocally the values of v0 and v1
Realizing smiles and quantiles
 An SDF perspective over Q and P
 Work in progress with Adam A. Majewskij and Fulvio Corsi
Heterogeneous AR Gamma with
Leverage (HARGL)
①
②
③
④
⑤
Yt+1 daily return
RVt+1 realized variance
Lt
leverage function
r
risk-free rate
gamma equity risk premium
Taken from F. Corsi, N. Fusari and D. La Vecchia, Journal of Financial
Economics, 2013, vol. 107, 284-304
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
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
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
Realizing quantiles
 Unexpectedly we find
with