توضیح معماهای قیمت گذاری دارایی در ارتباط با سقوط بازار در سال 1987

The 1987 market crash was associated with a dramatic and permanent steepening of the implied volatility curve for equity index options, despite minimal changes in aggregate consumption. We explain these events within a general equilibrium framework in which expected endowment growth and economic uncertainty are subject to rare jumps. The arrival of a jump triggers the updating of agents' beliefs about the likelihood of future jumps, which produces a market crash and a permanent shift in option prices. Consumption and dividends remain smooth, and the model is consistent with salient features of individual stock options, equity returns, and interest rates.

The 1987 stock market crash has generated many puzzles for financial economists. In spite of little change in observable macroeconomic fundamentals, market prices fell 20–25% and interest rates dropped about 1–2%. Moreover, the crash triggered a permanent shift in index option prices: Prior to the crash, implied ‘volatility smiles’ for index options were relatively flat. Since the crash, however, the Black-Scholes formula has been significantly underpricing short-maturity, deep out-of-the-money Standard and Poor's (S&P) 500 put options (Rubinstein, 1994 and Bates, 2000). This feature, often referred to as the ‘volatility smirk,’ is demonstrated in Fig. 1, which shows the spread of both in-the-money (ITM) and out-of-the-money (OTM) implied volatilities relative to at-the-money (ATM) implied volatilities from 1985–2006. This figure clearly shows that the volatility smirk spiked upward immediately after the 1987 stock crash, and that this shift has remained ever since.Not only is this volatility smirk puzzling in its own right, but it is also difficult to explain relative to the shape of implied volatility functions (IVF) for individual stock options, which are much flatter and more symmetric (see, e.g., Bollen and Whaley, 2004; Bakshi, Kapadia, and Madan, 2003; Dennis and Mayhew, 2002). Indeed, Bollen and Whaley (2004) argue that the difference in the implied volatility functions for options on individual firms and on the S&P 500 index cannot be explained by the differences in their underlying asset return distributions. In this paper, we attempt to explain these puzzles while simultaneously capturing other salient features of asset prices. In particular, we examine a representative-agent general equilibrium endowment economy that can simultaneously explain: • The prices of deep OTM put options for both individual stocks and the S&P 500 index. • Why the slope of the implied volatility curve changed so dramatically after the crash. • Why the regime shift in the volatility smirk has persisted for more than 20 years. • How the market can crash with little change in observable macroeconomic variables. We build on the long-run risk model of Bansal and Yaron (2004, BY), who show that if agents have a preference for early resolution of uncertainty, e.g., have Kreps and Porteus (1978)/Epstein and Zin (1989), or KPEZ, preferences with elasticity of intertemporal substitution EIS>1EIS>1, then persistent shocks to the expected growth rate and volatility of aggregate consumption will be associated with large risk premiums in equilibrium. Their model is able to explain a high equity premium, low interest rates, and low interest rate volatility while matching important features of aggregate consumption and dividend time series. We extend their model in two dimensions. First, we add a jump component to the shocks driving the expected consumption growth rate and consumption volatility. These jumps (typically downward for expected growth rates and upward for volatility) are bad news for the agent with KPEZ preferences, who will seek to reduce her position in risky assets. In equilibrium, this reduction in demand leads to asset prices exhibiting a downward jump, even though aggregate consumption and dividends are smooth. That is, in our model, the level of consumption and dividends follows a continuous process; it is their expected growth rates and volatilities that jump. Since shocks to expected consumption growth rate and consumption volatility are associated with large risk premiums, jumps in asset prices can be substantial, akin to market ‘crashes.’ Our second contribution relative to BY (2004) is to allow for parameter uncertainty and learning. Specifically, we assume the jump frequency is governed by a hidden two-state continuous Markov chain, which needs to be filtered in equilibrium. This adds another source of risk to the economy, namely the posterior probability of the hidden state. We show that the risk premium associated with revisions in posterior beliefs about the hidden state can be large, as they are a source of ‘long-run risk.’ In fact, we show that it can explain the dramatic shift in the shape of the implied volatility skew observed in 1987. If, prior to 1987, agents' beliefs attribute a very low probability to high jump intensities then, prior to 1987, prices mostly correspond to a no-jump Black-Scholes type economy. However, after a jump in prices occurs as a result of the jump in expected growth rates and volatility of fundamentals, agents update their beliefs about the likelihood (i.e., intensity) of future jumps occurring, which contributes to the severity of the market crash and leads to the steep skew in implied volatilities observed in the data henceforth. Because these beliefs are very persistent, the skew is long-lived after the crash. Although the two new features, jumps and learning, dramatically impact the prices of options, we show that our model still matches salient features of U.S. economic fundamentals. Because jumps impact the expected consumption growth rate and consumption volatility, but not the level of consumption, the consumption process remains smooth in our model, consistent with the data. Further, as noted by BY (2004) and Shephard and Harvey (1990), it is very difficult to distinguish between a purely independent identically distributed (i.i.d.) process and one which incorporates a small persistent component. Indeed, we show that the dividend and consumption processes implied by the model fit the properties of the data well, in that we cannot reject the hypothesis that the observed data were generated from our model. Nonetheless, and as in BY (2004), the asset pricing implications of our model differ significantly from those of an economy in which dividends are i.i.d. Specifically, the calibrated model matches the typical level of the price–dividend ratio and produces reasonable levels for the equity premium, the risk-free rate, and their standard deviation. In the same calibration we show that the pre-crash implied volatility function for short-maturity index options is nearly flat, while it becomes a steep smirk immediately after the crash. Moreover, the model predicts a downward jump in the risk-free rate during the crash event, consistent with observation. Finally, the model reproduces the stylized properties of the implied volatility functions for individual stock option prices. We specify individual firm stock dynamics by first taking our model for the S&P 500 index and then adding idiosyncratic shocks, both of the diffusive and the jump types. We then calibrate the coefficients of the idiosyncratic components to match the distribution of returns for the ‘typical’ stock. In particular, we match the cross-sectional average of the high-order moments (variance, skewness, and kurtosis) for the stocks in the Bollen and Whaley (2004) sample. We simulate option prices from this model and compute Black-Scholes implied volatilities across different moneyness. Consistent with the evidence in Bollen and Whaley (2004), Bakshi, Kapadia, and Madan (2003), and Dennis and Mayhew (2002), we find an implied volatility function that is considerably flatter than that for S&P 500 options. Bakshi, Kapadia, and Madan (2003) conclude that the differential pricing of individual stock options is driven by the degree of skewness/kurtosis in the underlying return distribution in combination with the agent's high level of risk aversion. Here, we propose a plausible endowment economy that, in combination with recursive utility, yields predictions consistent with their empirical findings. Related literature: Motivated by the empirical failures of the Black-Scholes model in post-crash S&P 500 option data, prior studies have examined more general option pricing models (see, e.g., Bates, 1996; Duffie, Pan, and Singleton, 2000; Heston, 1993). A vast literature explores these extensions empirically,3 reaching the conclusion that a model with stochastic volatility and jumps significantly reduces the pricing and hedging errors of the Black-Scholes formula.4 These previous studies, however, focus on post-1987 S&P 500 option data. Further, they follow a partial equilibrium approach and let statistical evidence guide the exogenous specification of the underlying return dynamics. Reconciling the findings of this literature in a rational expectations general equilibrium setting has proven difficult. For instance, Pan (2002) notes that the compensation demanded for the ‘diffusive’ return risk is very different from that for jump risk. Consistent with Pan's finding, Jackwerth (2000) shows that the risk aversion function implied by S&P 500 index options and returns in the post-1987 crash period is partially negative and increasing in wealth; similar results are presented in Aït-Sahalia and Lo (2000) and Rosenberg and Engle (2002). This evidence is difficult to reconcile in the standard general equilibrium model with constant relative risk-aversion utility and suggests that there may be a lack of integration between the option market and the market for the underlying stocks. Several papers have investigated the ability of equilibrium models to explain post-1987 S&P 500 option prices. Liu, Pan, and Wang (2005, LPW) consider an economy in which the endowment is an i.i.d. process that is subject to jumps. They show that, in this setting, neither constant relative risk aversion nor Epstein and Zin (1989) preferences can generate a volatility smirk consistent with post-1987 evidence on S&P 500 options. They argue that in order to reconcile the prices of options and the underlying index, agents must exhibit ‘uncertainty aversion’ towards rare events that is different from the standard ‘risk-aversion’ they exhibit towards diffusive risk. This insight provides a decision-theoretic basis to the idea of crash aversion advocated by Bates (2008), who considers an extension of the standard power utility that allows for a special risk-adjustment parameter for jump risk distinct from that for diffusive risk. These prior studies assume that the dividend level is subject to jumps, while the expected dividend growth rate is constant. Thus, in these models a crash like that observed in 1987 is due to a 20–25% downward jump in the dividend level.5 Moreover, their model predicts no change in the risk-free rate during the crash event. In our setting, it is the expected endowment growth rate that is subject to jumps. Thus, in our model, dividends and consumption are smooth and the market can crash with minimal change in observable macroeconomic fundamentals. Further, the risk-free rate drops around crash events, consistent with empirical evidence. Other studies explore the option pricing implications of models with state dependence in preferences and/or fundamentals; see, e.g., Bansal, Gallant, and Tauchen (2007), Bondarenko (2003), Brown and Jackwerth (2004), Buraschi and Jiltsov (2006), Chabi-Yo, Garcia, and Renault (2008), David and Veronesi, 2002 and David and Veronesi, 2009, and Garcia et al., 2001 and Garcia et al., 2003. These papers do not study the determinants of stock market crashes, the permanent shift in the implied volatility smirk that followed the 1987 events, and the difference between implied volatility functions for individual and index stock options. To our knowledge, our paper is the first to focus on these issues. Also related is a growing literature that investigates the effect of changes in investors' sentiment (e.g., Han, 2008), market structure, and net buying pressure (e.g., Bollen and Whaley, 2004 and Dennis and Mayhew, 2002; Gârleanu, Pedersen, and Poteshman, 2009) on the shape of the implied volatility smile. This literature argues that due to the existence of limits to arbitrage, market makers cannot always fully hedge their positions (see, e.g., Green and Figlewski, 1999 and Figlewski, 1989; Hugonnier, Kramkov, and Schachermayer, 2005; Liu and Longstaff, 2004 and Longstaff, 1995; Shleifer and Vishny, 1997). As a result, they are likely to charge higher prices when asked to absorb large positions in certain option contracts. These papers, however, do not address why end users buy these options at high prices relative to the Black-Scholes value or why the 1987 crash changed the shape of the volatility smile so dramatically and permanently. Our paper offers one possible explanation. Finally, the large impact that learning can have on asset price dynamics has been shown previously (e.g., David, 1997, Veronesi, 1999 and Veronesi, 2000). One important difference between these papers and ours is that our agent learns from jumps rather than diffusions, as in Benzoni, Collin-Dufresne, Goldstein, and Helwege (2010), leading to different updating dynamics. The main contribution of our paper is to explain pre- and post-1987 crash asset prices in a rational-expectation framework that is consistent with underlying fundamentals. However, to our knowledge this is also the first article to examine the effect of jumps in the Bansal and Yaron (2004) economy. This has proven to be a fruitful extension of the long-run risk framework and has been further explored by, e.g., Drechsler and Yaron (2011), Eraker (2008), and Eraker and Shaliastovich (2008). The rest of the paper proceeds as follows. In Section 2, we present the model and discuss our solution approach. Section 3 shows that the model matches the relevant asset pricing facts while being consistent with underlying fundamentals. Section 4 concludes the paper.

The 1987 stock market crash is associated with many asset pricing puzzles. Examples include: (i) Stocks fell 20–25%, interest rates fell approximately 1–2%, yet there was minimal impact on observable economic variables (e.g., consumption), (ii) the slope of the implied volatility curve on index options changed dramatically after the crash, and this change has persisted for more than 20 years, (iii) the magnitude of this post-crash slope is difficult to explain, especially in relation to the implied volatility slope on individual firms. We propose a general equilibrium model that can explain these puzzles while capturing many other salient features of the U.S. economy. We accomplish this by extending the model of Bansal and Yaron (2004) to account for jumps and learning. In particular, we specify the representative agent to be endowed with KPEZ preferences and assume that the aggregate dividend and consumption processes are driven by a persistent stochastic growth variable that can jump. Economic uncertainty fluctuates and is also subject to jumps. Jumps are rare and driven by a hidden state the agent filters from past data. In such an economy, there are three sources of long-run risk: expected consumption growth, volatility of consumption growth, and posterior probability of the jump intensity in expected growth rates and volatility. Jumps in fundamentals, even small, can lead to substantial jumps in prices of long-lived assets because of the updating of beliefs about the likelihood of future such jumps. In that sense, learning acts as an amplifier of long-run risk premiums associated with small persistent jumps in growth rates and their volatility. Indeed, we identify a realistic calibration of the model that matches the prices of short-maturity at-the-money and deep out-of-the-money S&P 500 put options, as well as the prices of individual stock options. Further, the model, calibrated to the stock market crash of 1987, generates the steep shift in the implied volatility ‘smirk’ for S&P 500 options observed around the 1987 crash. This ‘regime shift’ occurs in spite of a minimal change in observable macroeconomic fundamentals. In sum, our model points to a simple mechanism, based on learning about the riskiness of the economy, that explains why market prices suddenly crashed with little change in fundamentals, and why buyers of OTM put options were willing to pay a much higher price for these securities after the crash. Of course, we acknowledge that other mechanisms probably also contributed to the crash. For example, portfolio insurance and its implementation via dynamic hedging strategies is often cited as a major culprit. Let us just point out that, while not directly a ‘shock to fundamentals,’ the failure of portfolio insurance could well have contributed to deteriorating prospects for economic fundamentals through a ‘financial accelerator’ mechanism. It is a common belief that the growth rate of consumption and consumption volatility are tied to the strength of the financial system. Thus, if the crash revealed that risk-sharing was not as effective as previously thought, then this could have negatively affected investors' expectations about the future prospects of the economy. In this respect, further learning about economic fundamentals occurs through the experience of a crash in prices and might result in a further drop in prices via the mechanism we describe. Explicitly modeling this feedback mechanism between prices and economic fundamentals is outside the scope of the present paper, but seems an interesting avenue for future research.