Non-traded Assets Research Articles

Robust utility maximization for a diffusion market model with misspecified coefficients

The paper studies the robust maximization of utility from terminal wealth in a diffusion financial market model. The underlying model consists of a tradable risky asset whose price is described by a diffusion process with misspecified trend and volatility coefficients, and a non-tradable asset with a known parameter. The robust functional is defined in terms of a utility function. An explicit characterization of the solution is given via the solution of the Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation.

Asset Allocation with Shadow Assets

1. Bernd Scherer 1. is a professor of finance at EDHEC Business School in London, UK and CIO at FTC Capital in Vienna, Austria. (bernd.scherer{at}edhec.edu) <!-- --> 1. To order reprints of this article, please contact Dewey Palmieri at dpalmieri{at}iijournals.com or 212-224-3675. The wealth of most investors contains both financial assets as well as nonfinancial assets. The author defines shadow assets as (mostly) nonfinancial and nontradable assets that are exogenous to the investor’s asset allocation decision, such as human capital, nonfinancial sovereign assets (e.g., underground oil reserves), the present value of future alumni contributions for university endowments, or the nonlisted family business for the client of a family office. Ignoring shadow assets is unfortunate, as it is the existence and nature of shadow assets that distinguishes private investors, university endowments, sovereign wealth funds, and family offices and hence leads to different demands for risky securities. Shadow assets influence the outcome of asset allocation decisions via their covariance with financial assets and their effect on total wealth. Adding shadow assets has two effects on investors: It makes investors more aggressive and can create demand for hedges. Investment advice in the client’s best interests needs to incorporate shadow assets as well as shadow liabilities, as optimal allocations differ considerably when shadow components are properly accounted for TOPICS: [Wealth management][1], [portfolio theory][2] [1]: https://www.pm-research.com/topic/wealth-management [2]: https://www.pm-research.com/topic/portfolio-theory

A New Method to Estimate Risk and Return of Nontraded Assets from Cash Flows: The Case of Private Equity Funds

AbstractWe develop a new methodology to estimate abnormal performance and risk exposure of nontraded assets from cash flows. Our methodology extends the standard internal rate of return approach to a dynamic setting. The small-sample properties are validated using a simulation study. We apply the method to a sample of 958 private equity funds. For venture capital funds, we find a high market beta and underperformance before and after fees. For buyout funds, we find a relatively low market beta and no evidence for outperformance. We find that self-reported net asset values significantly overstate fund values for mature and inactive funds.

Utility-Based Pricing, Timing and Hedging of an American Call Option Under an Incomplete Market with Partial Information

This paper studies the pricing, timing and hedging of an American call option written on a non-tradable asset whose mean appreciation rate is not observable but is known to be a Gaussian random variable. Our goal is to analyze the effects of the partial information on investment in the American option under an incomplete market. The objective of the option holder is to maximize the expected discounted utility of consumption over an infinite lifetime. Thanks to consumption utility-based indifference pricing principal, stochastic control and filtering theory, under CARA utility, we derive the value and the exercise time of the American call option, which are determined by a semi-closed-form solution of a free-boundary PDE problem with a finite time horizon. We provide numerical results by finite difference methods and compare the results with those under a fully observable case. Numerical calculations demonstrate that partial information leads to a significant loss of the implied value of the American call option. This loss increases with the uncertainty of the mean appreciation rate. If the option holder is risk-averse enough, a growth of the systematic/idiosyncratic risk will increase/decrease the implied option value and the option is exercised late/early. Whether a stronger positive correlation between the tradable asset and the non-tradable underlying asset increases the option value and the information value depends on the risk attitude of the option holder. But on the contrary, a stronger negative correlation will definitely make the option and the information more valuable. In addition, as a corollary of our results, explicit expressions of the utility-based pricing for a perpetual American call are presented if the tradable risky asset is perfectly correlated with the underlying asset.

Accounting for risk aversion in derivatives purchase timing

We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent’s risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors’ first paper (Leung and Ludkovski, SIAM J. Fin. Math., 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.

Convergence results for the indifference value based on the stability of BSDEs

We study the exponential utility indifference value h for a contingent claim H in an incomplete market driven by two Brownian motions. The claim H depends on a non-tradable asset variably correlated with the traded asset available for hedging. We provide an explicit sequence that converges to h, complementing the structural results for h known from the literature. Our study is based on a convergence result for quadratic backward stochastic differential equations. This convergence result, which we prove in a general continuous filtration under weak conditions, also yields that the indifference value in a setting with trading constraints enjoys a continuity property in the constraints.

The Implied Market Price of Weather Risk

Weather derivatives (WD) are end-products of a process known as securitization that transforms non-tradable risk factors (weather) into tradable financial assets. For pricing and hedging non-tradable assets, one essentially needs to incorporate the market price of risk (MPR), which is an important parameter of the associated equivalent martingale measure (EMM). The majority of papers so far has priced non-tradable assets assuming zero or constant MPR, but this assumption yields biased prices and has never been quantified earlier under the EMM framework. Given that liquid-derivative contracts based on daily temperature are traded on the Chicago Mercantile Exchange (CME), we infer the MPR from traded futures-type contracts (CAT, CDD, HDD and AAT). The results show how the MPR significantly differs from 0, how it varies in time and changes in sign. It can be parameterized, given its dependencies on time and temperature seasonal variation. We establish connections between the market risk premium (RP) and the MPR.

Accounting for Risk Aversion in Derivatives Purchase Timing

We study the problem of optimal timing to buy/sell derivatives by a risk-averse agent in incomplete markets. Adopting the exponential utility indifference valuation, we investigate this timing flexibility and the associated delayed purchase premium. This leads to a stochastic control and optimal stopping problem that combines the observed market price dynamics and the agent's risk preferences. Our results extend recent work on indifference valuation of American options, as well as the authors' first paper (Leung and Ludkovski, SIAM J. Fin. Math., 2011). In the case of Markovian models of contracts on non-traded assets, we provide analytical characterizations and numerical studies of the optimal purchase strategies, with applications to both equity and credit derivatives.

A New Method to Estimate Risk and Return of Non-Traded Assets from Cash Flows: The Case of Private Equity Funds

We develop a new methodology to estimate abnormal performance and risk exposure of non-traded assets from cash flows. Our methodology extends the standard internal rate of return approach to a dynamic setting. The small-sample properties are validated using a simulation study. We apply the method to a sample of 958 private equity funds. For venture capital funds, we find a high market beta and underperformance before and after fees. For buyout funds, we find a relatively low market beta and no evidence for outperformance. We find that self-reported net asset values significantly overstate fund values for mature and inactive funds.

Optimal Selling Time for Real Non-Traded Assets: When Should You Sell Your Mansion?

Optimal Selling Time for Real Non-Traded Assets: When Should You Sell Your Mansion?

A Multidimensional Exponential Utility Indifference Pricing Model with Applications to Counterparty Risk

This paper considers exponential utility indifference pricing for a multidimensional non-traded assets model subject to intertemporal default risk, and provides a semigroup approximation for the utility indifference price. The key tool is the splitting method, whose convergence is proved based on the Barles-Souganidis monotone scheme, and the convergence rate is derived based on Krylov’s shaking the coefficients technique. We apply our methodology to study the counterparty risk of derivatives in incomplete markets.

Utility-Based Valuation and Hedging of Basis Risk With Partial Information

We analyse the valuation and hedging of a claim on a non-traded asset using a correlated traded asset under a partial information scenario, when the asset drifts are unknown constants. Using a Kalman filter and a Gaussian prior distribution for the unknown parameters, a full information model with random drifts is obtained. This is subjected to exponential indifference valuation. An expression for the optimal hedging strategy is derived. An asymptotic expansion for small values of risk aversion is obtained via partial differentiation equation (PDE) methods, following on from payoff decompositions and a price representation equation. Analytic and semi-analytic formulae for the terms in the expansion are obtained when the minimal entropy measure coincides with the minimal martingale measure. Simulation experiments are carried out which indicate that the filtering procedure can be beneficial in hedging, but sometimes needs to be augmented with the increased option premium, which takes into account parameter uncertainty in order to be effective. Empirical examples are presented which conform to these conclusions.

Optimal static-dynamic hedges for exotic options under convex risk measures

We study the problem of optimally hedging exotic derivatives positions using a combination of dynamic trading strategies in underlying stocks and static positions in vanilla options when the performance is quantified by a convex risk measure. We establish conditions for the existence of an optimal static position for general convex risk measures, and then analyze in detail the case of shortfall risk with a power loss function. Here we find conditions for uniqueness of the static hedge. We illustrate the computational challenge of computing the market-adjusted risk measure in a simple diffusion model for an option on a non-traded asset.

Utility-based hedging and pricing with a nontraded asset for jump processes

Optimal strategies for hedging a claim on a nontraded asset X are analyzed. The claim is valued and hedged in an exponential utility maximization frame using a correlated traded asset S . The traded asset is described as a geometric jump process and the nontraded asset as a jump-diffusion process having common jump times with S . The classical dynamic programming approach leads to characterizing the value function as a solution to the Hamilton–Jacobi–Bellman equation. Closed form formulas for the value function, in terms of a new probability measure Q ∗ equivalent to the real world probability measure P , and for the optimal investment strategy are given. Admissibility for the optimal strategy is discussed and, via a duality result, an explicit expression for the density of the minimal entropy martingale measure (MEMM) is provided. Closed form formulas are also given for the writer’s indifference price in terms of Q ∗ and the MEMM.

Pricing of Asian Temperature Risk

Weather derivatives (WD) are different from most financial derivatives because the underlying weather cannot be traded and therefore cannot be replicated by other financial instruments. The market price of risk (MPR) is an important parameter of the associated equivalent martingale measures used to price and hedge weather futures/options in the market. The majority of papers so far have priced non-tradable assets assuming zero MPR, but this assumption underestimates WD prices. We study the MPR structure as a time dependent object with concentration on emerging markets in Asia. We find that Asian Temperatures (Tokyo, Osaka, Beijing, Teipei) are normal in the sense that the driving stochastics are close to a Wiener Process. The regression residuals of the temperature show a clear seasonal variation and the volatility term structure of CAT temperature futures presents a modified Samuelson effect. In order to achieve normality in standardized residuals, the seasonal variation is calibrated with a combination of a fourier truncated series with a GARCH model and with a local linear regression. By calibrating model prices, we implied the MPR from Cumulative total of 24- hour average temperature futures (C24AT) for Japanese Cities, or by knowing the formal dependence of MPR on seasonal variation, we price derivatives for Kaohsiung, where weather derivative market does not exist. The findings support theoretical results of reverse relation between MPR and seasonal variation of temperature process.

Dynamic Hedging in Incomplete Markets: A Simple Solution

Despite much work on hedging in incomplete markets, the literature still lacks tractable dynamic hedges in plausible environments. In this article, we provide a simple solution to this problem in a general incomplete-market economy in which a hedger, guided by the traditional minimum-variance criterion, aims at reducing the risk of a non-tradable asset or a contingent claim. We derive fully analytical optimal hedges and demonstrate that they can easily be computed in various stochastic environments. Our dynamic hedges preserve the simple structure of complete-market perfect hedges and are in terms of generalized Greeks, familiar in risk management applications, as well as retaining the intuitive features of their static counterparts. We obtain our time-consistent hedges by dynamic programming, while the extant literature characterizes either static or myopic hedges, or dynamic ones that minimize the variance criterion at an initial date and from which the hedger may deviate unless she can pre-commit to follow them. We apply our results to the discrete hedging problem of derivatives when trading occurs infrequently. We determine the corresponding optimal hedge and replicating portfolio value, and show that they have structure similar to their complete-market counterparts and reduce to generalized Black-Scholes expressions when specialized to the Black-Scholes setting. We also generalize our results to richer settings to study dynamic hedging with Poisson jumps, stochastic correlation and portfolio management with benchmarking.

Dynamic Asset Allocation in the Presence of Housing and Incomplete Markets

This paper studies in continuous time and in an incomplete market setting the optimal housing, consumption, labor and portfolio choice of an agent in the presence of stochastic house prices and wages. Thus, the house prices and wage rates cannot be spanned by the financial market. In particular, the paper investigates the optimal strategies under two different preference specifications with respect to housing. The paper provides new closed-form solutions in the special case in which the market is complete. In addition, the paper also studies the optimal housing, consumption, labor, portfolio and welfare implications of frictions in the housing market. Particularly, the optimal strategies and welfare loss are analyzed if the house is a non-traded asset. This paper suggests that the consumption, labor, speculative investment and hedging of human capital is similar across preference specification and frictions in the market for housing when the economy is complete. The consumption and labor strategies are, however, dependent on frictions in the market for housing in the case where the economy is incomplete. The welfare loss from these frictions is small in magnitude but will influence the optimal consumption, labor and portfolio choice.

Black–Scholes theory for an underlying with multiple attractors

In recent years, numerous authors in the field of the macroeconomic theory of financial markets have studied multiple attractor regimes for both traded and non-traded assets, with particular emphas...

Can Exchange Traded Funds be Used to Exploit Industry Momentum?

There is overwhelming empirical evidence on the existence of country and industry momentum effects. This line of research suggests that investors who buy countries and industries with relatively high past returns and sell countries and industries with relatively low past returns will earn positive risk-adjusted returns. These studies focus on country and industry indexes that cannot be traded directly by investors. This warrants the question whether country and industry momentum effects can really be exploited by investors or are illusionary in nature because they exist only on non-tradable assets. We analyze the profitability of country and industry momentum strategies using actual price data on Exchange Traded Funds (ETFs). We find that, over the sample periods that these ETFs were traded, an investor would have been able to exploit country and industry momentum strategies with an excess return of about 5% per annum. These returns cannot be explained by unconditional exposures to the Fama-French factors. The daily average bid-ask spreads on ETFs are substantially below the implied break-even transaction costs levels. Hence, we conclude that investors that are not willing or able to trade individual stocks may use ETFs to benefit from momentum effects in country and industry portfolios.

Exponential utility indifference valuation in two Brownian settings with stochastic correlation

We study the exponential utility indifference valuation of a contingent claimBin an incomplete market driven by two Brownian motions. The claim depends on a nontradable asset stochastically correlated with the traded asset available for hedging. We use martingale arguments to provide upper and lower bounds, in terms of bounds on the correlation, for the valueVBof the exponential utility maximization problem with the claimBas random endowment. This yields an explicit formula for the indifference valuebofBat any time, even with a fairly general stochastic correlation. Earlier results with constant correlation are recovered and extended. The reason why all this works is that, after a transformation to the minimal martingale measure, the valueVBenjoys a monotonicity property in the correlation between tradable and nontradable assets.