1,237 research outputs found
Option-pricing in incomplete markets: the hedging portfolio plus a risk premium-based recursive approach
Consider a non-spanned security in an incomplete market. We
study the risk/return tradeoffs generated if this security is sold
for an arbitrage-free price and then hedged. We
consider recursive "one-period optimal" self-financing hedging
strategies, a simple but tractable criterion. For continuous
trading, diffusion processes, the one-period minimum variance
portfolio is optimal. Let be its price. Self-financing
implies that the residual risk is equal to the sum of the one-period
orthogonal hedging errors, . To
compensate the residual risk, a risk premium is
associated with every . Now let be the price of
the hedging portfolio, and is the total residual risk. Although not the same, the
one-period hedging errors are orthogonal to
the trading assets, and are perfectly correlated. This implies that
the spanned option payoff does not depend on y. Let
. A main result follows. Any arbitrage-free
price, , is just the price of a hedging portfolio (such
as in a complete market), , plus a premium,
. That is, is the price of the
option's payoff which can be spanned, and is
the premium associated with the option's payoff which cannot be
spanned (and yields a contingent risk premium of sum t at maturity). We study other applications of option-pricing theory as well
On the Exact Solution of the Multi-Period Portfolio Choice Problem for an Exponential Utility under Return Predictability
In this paper we derive the exact solution of the multi-period portfolio
choice problem for an exponential utility function under return predictability.
It is assumed that the asset returns depend on predictable variables and that
the joint random process of the asset returns and the predictable variables
follow a vector autoregressive process. We prove that the optimal portfolio
weights depend on the covariance matrices of the next two periods and the
conditional mean vector of the next period. The case without predictable
variables and the case of independent asset returns are partial cases of our
solution. Furthermore, we provide an empirical study where the cumulative
empirical distribution function of the investor's wealth is calculated using
the exact solution. It is compared with the investment strategy obtained under
the additional assumption that the asset returns are independently distributed.Comment: 16 pages, 2 figure
Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach
Consider a non-spanned security C_{T} in an incomplete market. We study the risk/return trade-offs generated if this security is sold for an arbitrage-free price C₀ and then hedged. We consider recursive "one-period optimal" self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let C₀(0) be its price. Self-financing implies that the residual risk is equal to the sum of the one-period orthogonal hedging errors, ∑_{t≤T}Y_{t}(0)e^{r(T-t)}. To compensate the residual risk, a risk premium y_{t}Δt is associated with every Y_{t}. Now let C₀(y) be the price of the hedging portfolio, and ∑_{t≤T}(Y_{t}(y)+y_{t}Δt)e^{r(T-t)} is the total residual risk. Although not the same, the one-period hedging errors Y_{t}(0) and Y_{t}(y) are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let C₀=C₀(y). A main result follows. Any arbitrage-free price, C₀, is just the price of a hedging portfolio (such as in a complete market), C₀(0), plus a premium, C₀-C₀(0). That is, C₀(0) is the price of the option's payoff which can be spanned, and C₀-C₀(0) is the premium associated with the option's payoff which cannot be spanned (and yields a contingent risk premium of ∑y_{t}Δte^{r(T-t)} at maturity). We study other applications of option-pricing theory as wellOption Pricing; Incomplete Markets
Option-Pricing in Incomplete Markets: The Hedging Portfolio plus a Risk Premium-Based Recursive Approach
Consider a non-spanned security CT in an incomplete market. We study the risk/return tradeoffs generated if this security is sold for an arbitrage-free price bC0 and then hedged. We consider recursive “one-period optimal” self-financing hedging strategies, a simple but tractable criterion. For continuous trading, diffusion processes, the one-period minimum variance portfolio is optimal. Let C0(0) be its price. Self-financing implies that the residual risk is equal to the sum of the oneperiod orthogonal hedging errors, Pt=T Yt(0)er(T -t). To compensate the residual risk, a risk premium yt.t is associated with every Yt. Now let C0(y) be the price of the hedging portfolio, and Pt=T (Yt(y) + yt.t) er(T -t) is the total residual risk. Although not the same, the one-period hedging errors Yt(0) and Yt(y) are orthogonal to the trading assets, and are perfectly correlated. This implies that the spanned option payoff does not depend on y. Let bC0 = C0(y). A main result follows. Any arbitrage-free price, bC0, is just the price of a hedging portfolio (such as in a complete market), C0(0), plus a premium, bC0 - C0(0). That is, C0(0) is the price of the option’s payoff which can be spanned, and bC0 - C0(0) is the premium associated with the option’s payoff which cannot be spanned (and yields a contingent risk premium of Pyt.ter(T -t) at maturity). We study other applications of option-pricing theory as well.
On the Equivalence of Quadratic Optimization Problems Commonly Used in Portfolio Theory
In the paper, we consider three quadratic optimization problems which are
frequently applied in portfolio theory, i.e, the Markowitz mean-variance
problem as well as the problems based on the mean-variance utility function and
the quadratic utility.Conditions are derived under which the solutions of these
three optimization procedures coincide and are lying on the efficient frontier,
the set of mean-variance optimal portfolios. It is shown that the solutions of
the Markowitz optimization problem and the quadratic utility problem are not
always mean-variance efficient. The conditions for the mean-variance efficiency
of the solutions depend on the unknown parameters of the asset returns. We deal
with the problem of parameter uncertainty in detail and derive the
probabilities that the estimated solutions of the Markowitz problem and the
quadratic utility problem are mean-variance efficient. Because these
probabilities deviate from one the above mentioned quadratic optimization
problems are not stochastically equivalent. The obtained results are
illustrated by an empirical study.Comment: Revised preprint. To appear in European Journal of Operational
Research. Contains 18 pages, 6 figure
A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability
We present a simulation-based method for solving discrete-time portfolio choice problems involving non-standard preferences, a large number of assets with arbitrary return distribution, and, most importantly, a large number of state variables with potentially path-dependent or non-stationary dynamics. The method is flexible enough to accommodate intermediate consumption, portfolio constraints, parameter and model uncertainty, and learning. We first establish the properties of the method for the portfolio choice between a stock index and cash when the stock returns are either iid or predictable by the dividend yield. We then explore the problem of an investor who takes into account the predictability of returns but is uncertain about the parameters of the data generating process. The investor chooses the portfolio anticipating that future data realizations will contain useful information to learn about the true parameter values.
A short note on the problematic concept of excess demand in asset pricing models with mean-variance optimization
Referring to asset pricing models where demand is proportional to excess returns and said to be derived from a mean-variance optimization problem, the note formulates what probably is common knowledge but hardly ever made an explicit subject of discussion. This is an insufficient distinction between the desired holding of the risky asset on the part of the speculative agents, which is the solution to the optimization problem and usually directly presented as excess demand, and the desired change in this holding, which is what should reasonably constitute the excess demand on the market. The note arrives at the conclusion that in models with a market maker the story of the maximization of expected wealth should be dropped
A Closed-Form Solution of the Multi-Period Portfolio Choice Problem for a Quadratic Utility Function
In the present paper, we derive a closed-form solution of the multi-period
portfolio choice problem for a quadratic utility function with and without a
riskless asset. All results are derived under weak conditions on the asset
returns. No assumption on the correlation structure between different time
points is needed and no assumption on the distribution is imposed. All
expressions are presented in terms of the conditional mean vectors and the
conditional covariance matrices. If the multivariate process of the asset
returns is independent it is shown that in the case without a riskless asset
the solution is presented as a sequence of optimal portfolio weights obtained
by solving the single-period Markowitz optimization problem. The process
dynamics are included only in the shape parameter of the utility function. If a
riskless asset is present then the multi-period optimal portfolio weights are
proportional to the single-period solutions multiplied by time-varying
constants which are depending on the process dynamics. Remarkably, in the case
of a portfolio selection with the tangency portfolio the multi-period solution
coincides with the sequence of the simple-period solutions. Finally, we compare
the suggested strategies with existing multi-period portfolio allocation
methods for real data.Comment: 38 pages, 9 figures, 3 tables, changes: VAR(1)-CCC-GARCH(1,1) process
dynamics and the analysis of increasing horizon are included in the
simulation study, under revision in Annals of Operations Researc
- …