Dynamic Safety First Expected Utility Model Mei Choi Chiua,∗ , Hoi Ying Wongb,∗ , Jing Zhaoc,∗ a
Department of Mathematics and Information Technology, The Education University of Hong Kong, Tai Po, Hong Kong b Department of Statistics, The Chinese University of Hong Kong, Hong Kong c Department of Economics and Finance, La Trobe University, Melbourne, Australia
Abstract Levy and Levy (2009, The safety first expected utility model: Experimental evidence and economic implications, Journal of Banking and Finance 33, 1494– 1506.) empirically show that a combination of safety first and expected utility (SFEU) principles play a key role in human decision-making process. This paper extends the SFEU model to the optimal dynamic investment in a continuous-time economy. We derive the analytic optimal trading strategy using the martingale approach. Interestingly, the optimal trading strategy replicates a portfolio of a vanilla call, a vanilla put, a digital put option, and a cash reserve. These derivatives therefore match the objective of SFEU investors, which offers an explanation to their popularity in the market. The model also implies that investors with more awareness of crash risk demand put options with lower strike price. Using option data of US major market indices and alternative proxies for market awareness of crash risk, we empirically test the model implications and find that market awareness of crash risk can explain the dynamics of index option open interest. Key words: Finance; Dynamic Investment; Safety First; Expected Utility; Martingale Approach; Crash risk 1. Introduction Markowitz (1959, 1989)’s mean-variance theory and Roy (1952)’s safety first theory are two of the most influential portfolio theories developed in the last cen∗
Joint corresponding authors Email addresses:
[email protected] (Mei Choi Chiu),
[email protected] (Hoi Ying Wong),
[email protected] (Jing Zhao)
Preprint submitted to European Journal of Operational Research
January 23, 2018
tury. Although expected utility models are used widely in analyzing decision making, these models are inapplicable for many investors if they are principally concerned with avoiding a possible disaster. Investors in Roy’s safety first portfolio theory aim to minimize the disaster probability, i.e., the probability of the final wealth going below the disaster level. There exist abundant theoretical and empirical evidences showing the crucial role played by the safety first principle in human decision-making process, see for example Shefrin and Statman (2000) and Levy and Levy (2009). It has also led to several important concepts in financial risk management such as downside risk, value-at-risk, and expected shortfall, see Jorion (2006) for references. Although safety first investors and mean-variance investors have different principles, the two principles complement each other and lead to same optimal solutions in special circumstances. As the original safety first portfolio problem is difficult to obtain a closed-form solution, Roy (1952) in fact solves a surrogate problem that approximates the disaster probability by its upper bound derived from the Chebyshev inequality. That upper bound is a ratio between variance and mean of the portfolio. Levy and Sarnat (1971) find that when the disaster level is equal to the gross return on risk-free investment, 1 the (surrogate) safety first principle and the mean-variance (MV) principle lead to the same optimal unlevered portfolios in a single-period setting. Arzac and Bawa (1977) show that the capital asset pricing model (CAPM) is robust to safety first investors under traditional distributional assumptions in a single-period setting, although safety first valuation formula allows for different assumptions concerning the probabilistic information possessed by investors.2 Alternatively, shortfall constraints have been taken into consideration under some mean-variance portfolio selection formulations, see for example, Korn and Trautmann (1995), Korn (1997), Bielecki et al. (2005) and Mansini et al. (2014). Roy (1952) claims that minimizing the chance of disaster can be interpreted as maximizing expected utility if the utility function has 1
The present paper however considers general disaster levels. In practice, the regulatory capital based on the expected shortfall implies the shortfall threshold being the disaster level (Ramponi and Campi, 2017). 2 For an excellent review and examination of the validity of CAPM in behaviour economics and psychologists paradigms, see Levy (2010). Kahneman and Tversky (1979) based on experimental results, suggest prospect theory (PT) as an alternative to the MV paradigm. Levy and Levy (2004) point out that the CAPM equilibrium does not necessarily hold when investors have PT preferences. Levy (2010) shows that various versions of CAPM are intact in Roy (1952)’s framework conditional on the assumption that there is no riskless asset, or that the riskless asset exists but investors are constrained by having to invest some proportion of the wealth in the risky assets.
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only two values, e.g., one if disaster does not occur and zero if it does. This is obviously a non-concave utility function but the classic expected utility theory assumes a concave utility function. The safety first principle has been extended in many aspects in the literature. For instance, Telser (1956) considers a hedging problem with the safety first criterion. Kataoka (1963) develops a stochastic programming model for a portfolio selection problem under the safety first principle. Li et al. (1998) use the embedding technique of Li and Ng (2000) to develop a mathematical foundation for the multi-period (surrogate) safety first problem. Milevsky (1999) obtains optimal trading strategy on a portfolio of one risk-free asset and two risky assets by optimizing a safety first utility function under the standard geometric Brownian motion assumptions. In asset-liability management problem, Chiu and Li (2009) show that the optimal trading policy under the (surrogated) safety first principle locates on the mean-variance efficient frontier. Chiu et al. (2012) investigate the effect of the mean constraint in the original safety first portfolio problem in a continuous-time economy. Dorfleitner and Utz (2012) study socially responsible investing by introducing stochastic sustainability returns into safety first models for portfolio choice. Gao et al. (2016) obtain analytical solutions to dynamic mean-variance safety first principle formulation. An alternative extension of the safety first principle incorporates an expected utility objective. Levy and Levy (2009) experimentally show that Roy’s safety first principle is an important factor in economic decisions but it is not the unique factor in the decision-making process. Their experiments show that the combination of the safety first and expected utility principles can explain human decisions, which risk-aversion, loss-aversion and prospect theories fail to do. In particular, Levy and Levy (2009) find that individuals make decisions based on a weighted average of the safety first principle and the expected utility. They empirically estimate the relative weights and discuss their economic implications. In the conclusions of their paper, they suggest future work to derive optimal dynamic investment decision using the safety first and expected utility (SFEU) model. Motivated by empirical evidences of SFEU investors and wide applications of safety first theories, this paper investigates the optimal dynamic investment in a continuous-time economy under the SFEU problem and obtains the analytical solution of the optimal trading strategy with a general utility function. We find that the SFEU optimal trading strategy replicates a portfolio of a vanilla call, a vanilla put, a digital put option and a cash reverse. In the case of CRRA utility function, we further investigate the optimal trading strategy and their theoretical implications. The discussions in the case of power utility and logarithmic utility 3
functions are also provided. Theoretically, this paper contributes to the existing literature in two aspects. On the one hand, it aligns with the recent advances in the quantitative treatment of behavior portfolio selection problems. The pioneering work of Jin and Zhou (2008) establishes the mathematical foundation for behavioral portfolio selection problems in continuous-time. He and Zhou (2011) further investigate the possibility of obtaining analytical results, which provides the ground for explaining the effects of human characteristics such as greed, fear, loss aversion, and optimism on investment decisions (see Jin and Zhou (2013) and Yao and Li (2013a)). Yao and Li (2013b) find that prospect-theory preferences can lead investors to behave endogenously as contrarian noise traders under incomplete information. As the safety first principle includes aversion of extreme losses, it is believed that investors with SFEU preferences naturally behave like contrarian traders regardless of the information quality. However, the lack of an analytical solution to the SFEU portfolio selection problem limits the possibility of conducting further empirical analysis. In this paper, we fill this gap on the quantitative treatment of the SFEU portfolio problem. On the other hand, our results are related to the literature on the role of options in financial markets. Grossman and Zhou (1996) consider a model in which portfolio insurers receive infinite disutility if their wealth level approaches a given lower bound, and thus these agents use options as optimal risk sharing contracts. Bates (2001) proposes a modified utility specification with the feature of crash aversion, under which the less crash-averse agents insure the more crash-averse agents through options. Buraschi and Jiltsov (2006) characterize the way in which differences in beliefs affect option volumes. Our paper sheds new light on the role of options in the perspective of the SFEU principles. We show that the SFEU optimal trading strategy replicates a portfolio of options and SFEU investors have a fundamental need for option exposure. In particular, the optimal SFEU investment strategy for a specific investment horizon replicates a portfolio of an index call, an index put, an index digital put option, and a cash reserve. As range accrual note (RAN) is a portfolio of digital options with different maturities, SFEU investors with multiple investment horizons are interested in index calls and puts and RANs. This explains why these three financial derivatives are popular in the market. We highlight the differences among expected utility (EU) principle, prospect theory (PT) and SFEU approach. In fact, all these three approaches can be viewed as maximization of the expected value function. Their major differences are the choice of the value functions. The classic EU principle considers concave value functions and the risk-aversion is characterized through the curvature of the con4
cave utility. For a smooth concave utility function U, the risk-aversion is related to −U /U > 0. PT advocates S-shaped value functions and probability distortion. In other words, the PT value function is convex for loss but concave for gain. Although the SFEU value functions are locally concave except for the point at the disaster level, they are non-concave functions as a whole. The introduction of the disaster level aims to emphasize on the crash aversion or safety-first behavior. The EU principle reflects a certain level of crash aversion through risk aversion too. However, Levy and Levy (2009) show experimentally that SFEU objective better captures human-decisions against extreme losses than the EU principle does. Based on the analytical solution of the optimal trading strategy under the SFEU model, we further obtain empirically testable model implications. When an SFEU investor is concerned more with crash risk, he assigns a higher relative weight to the safety first principle in his decision making. Our results imply that SFEU investors with more awareness of crash risk demand put options with lower strike price. Using option data of four US major market indices, we empirically test the model implications. In order to proxy for market awareness of crash risk, we consider three alternative measures: (1) the Chicago Board Options Exchange (CBOE) VVIX index, which contains information on the crash risk priced by the option markets as demonstrated in Park (2015); (2) the Financial and Economic Attitudes Revealed by Search (FEARS) index proposed by Da et al. (2015), which uses daily Internet search volume from millions of households to reveal marketlevel sentiment; and (3) the revised investor sentiment index proposed by Huang et al. (2015). Our analyses show that market awareness of crash risk is positively related to open interest of out-of-the-money (OTM) put options and negatively related to average strike price weighted by put option open interest, consistent with our model implications. Empirically, this paper demonstrates that the SFEU principle helps to explain the dynamics of index option open interest. The rest of the paper is organized as follows. In Section 2, we formulate the SFEU portfolio problem and derive the optimal trading strategy under SFEU in general. In Section 3, we further investigate the SFEU optimal trading strategy under the CRRA utility function and its theoretical implications, and formally discuss the power utility and logarithmic utility. In Section 4, we empirically test the model implications by examining the relationship between market awareness of crash risk and open interest of put options. Section 5 concludes the paper.
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2. Problem Formulation Consider a financial market in which n + 1 assets are traded continuously within the time horizon [0, T ]. We label these assets S i , where i = 0, 1, 2 . . . , n, with S0 being the risk-free asset. The risk-free asset S0 (t) satisfies the following differential equation: dS0 (t) = S0 (t)r(t)dt, S0 (0) = R0 > 0,
(1)
where r(t) is the risk-free rate. The remaining n assets are risky and their price processes S1 (t), . . . , Sn (t) satisfy the stochastic differential equations (SDEs): dSi (t) = Si (t) αi (t)dt + nj=1 σij (t)dWtj , t ∈ [0, T ], Si (0) = si > 0,
i = 1, 2, . . . , n,
(2)
where Wt = (Wt1 , Wt2, Wt3 , . . . , Wtn ) is the standard n-dimensional Wiener process defined on a filtered complete probability space (Ω, F , P, Ft≥0 ), Wti and Wtj are mutually independent for all i = j, αi (t) is the appreciation rate of asset i, and the volatility matrix σ(t) = [σ ij (t)] is an Rn×n -valued continuous function on [0, T ]. Although Wti and Wtj are mutually independent for all i = j, the assets are correlated when the volatility matrix is not a diagonal matrix. Let L2FT ∧τ (Ω, Rd ) be the set of all Rd -valued FT ∧τ -adapted η such that E[η2 ] < +∞ for any stopping time τ . As widely conducted in the literature, we assume that the non-degeneracy condition of σ(t)σ(t) := Σ(t) ≥ δIn1 holds for all t ∈ [0, T ] and for some δ > 0. In other words, the variance-covariance matrix Σ(t) is positively definite. Consider an investor with initial wealth of Y 0 in the specified financial market. The investor seeks an admissible portfolio strategy so that the disaster probability, P{Y (T ) < D}, is minimized or, equivalently, the safety probability P{Y (T ) ≥ D} is maximized, where D is the disaster level and Y (T ) is the final wealth. In addition, the investor wants to maximize the expected utility E[U(Y (T ))], where U(·) is an increasing and strictly concave utility function such that lim U (x) = +∞, x↓0
lim U (x) = 0.
x↑+∞
(3)
Let ui (t) be the amount invested in asset i and N i (t) be the number of the ith asset in the portfolio of the investor. The wealth of the investor at time t is 6
n n then defined as Y (t) = i=0 ui (t) = i=0 Ni (t)Si (t). The portfolio u(t) = (u1 (t), u2(t), . . . , un (t)) is said to be admissible if u(t) is a non-anticipating and T 2 Ft -adapted process such that E 0 |u(τ )| dτ < ∞. In other words, the integrated investment amount has a bounded second moment. Applying Itˆo’s lemma to Y (t) with respect to the asset price dynamics (2), the wealth process is given by dY (t) = [r(t)Y (t) + u(t) β(t)] dt + u(t) σ(t)dWt , Y (0) = Y0 ,
(4)
β(t) = α(t) − r(t)1, α(t) = (α1 (t), · · · , αn (t)) ,
(5)
where
and 1 is the column vector with all elements being 1. The determination of an optimal control with respect to an objective function is a portfolio selection problem in finance. In particular, the SFEU portfolio problem is formulated as max u(·)
s.t.
P{Y (T ) ≥ D} + λE [U(Y (T ))] u(·) ∈ L2FT ([0, T ], Rn ), Y (t) follows (4).
(6)
The space L2FT ([0, T ], Rn ) collects all Rn -valued FT -measurable stochastic processes with finite second moments. The constant λ ≥ 0 indicates the weight of the expected utility in the SFEU objective function. When λ = 0, the SFEU reduces to the safety first portfolio problem without a mean constraint on the final wealth. This investment objective is equivalent to a target-reach problem as shown by Chiu et al. (2012). When λ > 0, the investor maximizes the expected utility and the safety probability simultaneously. Levy and Levy (2009) use several different 1 experimental methods to estimate α = 1+λ and obtain α 0.1, which implies λ 9 in our case. Following Levy and Levy (2009), we postulate a positive constant λ in this paper. When λ > 0, the SFEU portfolio problem (6) can be alternatively viewed as the Lagrangian formulation of the original safety first problem subject to a constraint on the final wealth through the expected utility. Specifically, it is equivalent
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to max u(·)
s.t.
P{Y (T ) ≥ D} u(·) ∈ L2FT ([0, T ], Rn), Y (t) follows (4) and E [U(Y (T ))] ≥ U ,
(7)
for some constant U related to λ. Chiu et al. (2012) point out that the original Roy’s safety first principle takes the same form as (7) with U(x) = x, a riskneutral utility. However, this constraint makes the original safety first problem an ill-posted problem such that the optimal solution cannot be derived. They propose to use U(x) = x1{x U (D). In other words, g(x) is an unbounded increasing function for x > U (D) and lim
x→U (D)
g(x) = 0.
¯ = This ensures that there exists a unique ξ¯ > U (D) such that g(ξ) 9
1 λ
> 0.
(12)
Lemma 2.2. For any positive Y0 , there exists μ > 0 which solves the equation ¯ = 1. (12) where ξ¯ > U (D) is the solution of g(ξ) λ Proof. By the definition of I(·), it is clear that μρ(T ) μρ(T ) = +∞ and lim I = 0+ . lim I μ→+∞ μ→0+ λ λ Solving the SDE of ρ gives
T −1 1 ρ(T ) = e 0 (−r(s)− 2 β(s) Σ (s)β(s))ds−
T 0
β(s) Σ−1 (s)σ(s)dWs
.
By Girsanov’s theorem,
μρ(T ) 1 − 1{U (D)≤ μρ(T )
