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

Jan 06, 2016

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Evaluation and pricing of risk under stochastic volatility. Giacomo Bormetti Scuola Normale Superiore, Pisa. 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 - PowerPoint PPT Presentation
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Page 1: Evaluation and pricing of risk under stochastic volatility

Evaluation and pricing of risk under stochastic volatilityGiacomo BormettiScuola Normale Superiore, Pisa

Page 2: Evaluation and pricing of risk under stochastic volatility

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

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

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

Page 5: Evaluation and pricing of risk under stochastic volatility

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

Page 6: Evaluation and pricing of risk under stochastic volatility

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Realizing quantiles

Unexpectedly we find

with