Risk Neutral Densities: A Review Stephen Figlewski* * Professor of Finance Stern School of Business New York University email: [email protected] Outline of Presentation I. Overview II. Options, Probabilities, and Risk Preferences: The P and the Q-densities III. The Evolution of the Data Generating Process and the Risk Neutral Density IV. Overview of Estimation Methodology V. Implied Volatility: The Volatility Risk Premium and the VIX VI. The Volatility Surface VII. The "Pricing Kernel Puzzle"

VIII. Two Major Directions for Future Research International Risk Management Conference 2018 2018 Figlewski What is the Risk Neutral Density? The Risk Neutral Density (RND) extends the familiar concept of Implied Volatility (IV) for a single option. Given a set of options with a range of strike prices, you can extract the market's entire (risk neutral) probability distribution over the expiration day stock price ST . The RND (the "Q-distribution") combines the market's estimate of the true probabilities over ST (the "P-distribution") and the market's risk preferences (the "pricing kernel" k(ST) Q(ST)/P(ST) ). We can "observe" the Q density in the options market. A major challenge is to separate the market's true probability expectations

from risk premia. International Risk Management Conference 2018 2018 Figlewski How risk neutral probabilities are extracted from option prices Consider a call option that allows you to buy a share of some underlying stock for a price of 101 one month from now. If the stock price in one month is above 101, you will exercise the option. The market price for this option is $5.00 . There is a second call option that allows you to buy 1 share of the same stock for a price of 100 in one month. The market price for Option 2 is $5.70. For every stock price above 101, the second option pays $1 more than the first option. The market values the extra $1 that option 2 pays if the stock price is above 101 as being worth 5.70 5.00 = $0.70. So (roughly speaking) the market is saying the probability the

stock price will be above 101 is 70%. If you have a lot of options prices with strikes close together, you can get the whole density. International Risk Management Conference 2018 2018 Figlewski 4 How risk neutral probabilities are extracted from option prices Extracting the RND from the Call pricing function: Take the partial derivative w.r.t. X, rearrange and add 1 to both sides. This gives the Risk Neutral Distribution function:

Taking the partial w.r.t. X a second time gives the Risk Neutral Density (RND) The RND is the second partial derivative of call value w.r.t. the strike, futurevalued to date T. International Risk Management Conference 2018 2018 Figlewski Points to Notice The P density is what the market really is predicting. Presuming investors in aggregate are rational, expectations under the P density are the true expected values. For example, EP[ ST ] = E [ ST | ], where represents all currently available information. The pricing kernel modifies the P density to incorporate risk preferences (and speculative beliefs about mispricing). An extra dollar received in a state of the world ST that the market dislikes is

valued more highly than an extra dollar received in a state of the world in which the market is already happy. For a payoff in the bad state, the pricing kernel will have Q(ST)/P(ST) = k(ST) > 1, making that outcome seem more probable under the Q density than under the P. The more favored state will have k(ST) < 1 and lower probability under the Q than the P. International Risk Management Conference 2018 2018 Figlewski A Brief History of the RND Concept The Black-Scholes (BS) model is based on the familiar logarithmic diffusion in which a (dynamic) arbitrage can get rid of the exposure to stock price risk. A riskless delta-hedged position should pay no risk premium regardless of how

risk averse investors might be, so the drift should be equal to the riskless rate r. The RND for log(ST) comes from the above diffusion, with = r. The shape of the density for log returns is normal with the same volatility; only the mean changes. International Risk Management Conference 2018 2018 Figlewski A Brief History of the RND Concept But real world implied volatilities (IV) display a smile pattern, inconsistent with BS. The market does not behave as if volatility is a fixed constant. Ways to make options worth than BS prices more when the stock price falls:

1. CEV: 2. Use a fatter tailed process than the normal: e.g., Student-t, Normal Inverse Gaussian 3. Modify the normal distribution to fatten the tails: e.g., Gram-Charlier; Hermite polynomials; 4. GARCH: Out of the money (OTM) contracts are not so far OTM as under Black-Scholes, because volatility increases after big price changes (in either direction, but especially after a large negative return). International Risk Management Conference 2018 2018 Figlewski Points to Notice Directly modifying the RND separates the density for date T from the dynamics that produce that density. No riskless arbitrage is specified; the model

doesn't say how to delta hedge. With no arbitrage trade to force prices into alignment, option prices can be expected to reflect many things that are excluded from Black-Scholes: risk aversion and variations in risk aversion; constraints on trading, e.g., restrictions on short sales (Figlewski and Webb 1993); difficulties in hedging, e.g., high gamma or high idiosyncratic risk, that make deltahedging risky (Figlewski and Freund 1994, Cao and Han (2013)); market maker inventory positions (Garleanu, Pedersen, and Poteshman 2009, Bollen and Whaley 2004);

anything else that affects supply and demand for options. All of these will show up in empirical tests as risk premia. International Risk Management Conference 2018 2018 Figlewski A Brief History of the RND Concept, continued Another idea to connect RNDs to short run dynamics: Local volatility models A binomial or trinomial lattice model is modified to build in a different, nonstochastic, volatility at each node, such that the RND at the final time step matches the one in the market. The structure constructs short run dynamics (delta-hedging) that produce the observed RND at maturity.

Local volatility models: Rubinstein (1994), Derman and Kani (1994), Dupire (1993), Jackwerth and Rubinstein (1996), Derman, Kani and Chriss (1996) International Risk Management Conference 2018 2018 Figlewski Local Volatility Models title Source: Derman, Kani and Chriss. Journal of Derivatives, 1996. Standard fixed volatility Binomial tree

Local volatility implied Binomial tree Notice that volatility varies over time but it is not stochastic it is known at each node. A Brief History of the RND Concept, continued Local volatility models Unfortunately: Local volatility models were called into serious question by Dumas, Fleming and Whaley (1998) who found that they were so unstable that they could be beaten both in matching market prices and in delta-hedging by an ad hoc polynomial formulation with just maturity and moneyness. International Risk Management Conference 2018

2018 Figlewski A Brief History of the RND Concept, continued Single-factor models of stock returns were largely abandoned in favor of stochastic volatility (SV) specifications. Heston model: K is the speed of reversion to long-term mean variance V* and is the volatility of volatility. The correlation between dz and dw is . With two sources of risk, dz and dw, and the stock as the only hedge instrument, there is no longer any possibility of riskless arbitrage between options and stock. The new parameters, , and are all "latent". Since volatility is not observable, both equations need to be estimated from the series of returns. How aggressive

should we expect investors to be in correcting "mispricing" (according to this model) in the market? International Risk Management Conference 2018 2018 Figlewski A Brief History of the RND Concept, continued And yet, even with several new parameters... SV models are not adequate to explain option prices. They can't generate enough weight in the tails to match short maturity option market prices. Jumps seem to be neededSVJ models. Typical jump term: where Jt is jump size, often defined as lognormal: log(Jt) ~ N(J*, J); (another common choice is the exponential distribution); Nt is a Poisson counting process with arrival intensity

Jumps are "doubly stochastic" because both jump occurrence and jump size are random. Jumps add a lot of parameters. Do jumps affect the returns equation? The variance equation? Both? Should there be different processes for up and down jumps? Should jump size and/or intensity depend on diffusive volatility? International Risk Management Conference 2018 2018 Figlewski A Brief History of the RND Concept, continued Current models for the stock returns process include:

stochastic diffusive volatility one or more doubly stochastic jumps in the returns equation possibly jumps in the variance equation also possible extra machinery to capture still-unexplained behavior of the left tail There are no riskless arbitrage trades here. Such models need risk premia. (And, of course, the risk premia need not be fixed parameters. They may well vary stochastically over time.) The Bottom Line: With no clear connection between the returns generating process at the short run (hedging) horizon level and the RND, there is no reason for the RND to take any particular mathematically convenient form, or for its shape to remain the same from day to day. International Risk Management Conference 2018

2018 Figlewski Extracting an RND from Option Prices Two general approaches: parametric and nonparametric. Parametric RNDs assume a specific family of densities, e.g., lognormal, Student-t, Normal Inverse Gaussian and fit them to the options market prices. There are a lot of choices, but keep in mind the "Bottom Line" from the previous slide. Nonparametric RNDs are extracted by numerical approximation of the second derivative of the option pricing function in the market w.r.t. strike. This gives the portion of the RND that lies within the range spanned by the option strikes. Some way to fill in the tails is needed to get the complete density. (Parametric approaches do this by assumption.)

International Risk Management Conference 2018 2018 Figlewski Extracting an RND from Option Prices Nonparametrically Obtaining a well-behaved risk neutral density from market option prices is a nontrivial exercise. Here are the main steps. 1. Simultaneous observations are crucial. Use bid and ask quotes, rather than transactions prices. 2. Use out of the money calls, out of the money puts, and a blend of the two at the money. 3. Convert prices to Black-Scholes implied volatilities 4. Interpolate the IVs using a 4th degree smoothing spline or similar method 5. Convert the interpolated IV curve back to option prices. Use a numerical approximation to compute the 2nd partial derivative at each strike X. This produces the middle portion of the risk neutral density

6. Append tails to the Risk Neutral Density from a Generalized Pareto Distribution (GPD) International Risk Management Conference 2018 2018 Figlewski stuff International Risk Management Conference 2018 2018 Figlewski Extracting an RND from Option Prices Bakshi, Kapadia, and Madan (Review of Financial Studies, 2003) derived formulas for the moments of an approximated RND in terms of the underlying

option prices. Given the first 4 moments, a parametric RND can be easily fitted to the observed option prices. The moments may be of interest by themselves, in particular the volatility. The formula became the basis for the new VIX index. International Risk Management Conference 2018 2018 Figlewski The Formula for the VIX Index International Risk Management Conference 2018 2018 Figlewski

Volatility Risk Premium Discretely rebalanced delta-hedges are exposed to risk that increases with volatility. We should expect a volatility risk premium in option prices. There is another type of volatility risk, because volatility itself is stochastic. This risk is what is hedged with a volatility swap. Some articles say the volatility risk premium is negative, some say it is positive. They are all saying the same thing: Investors dislike volatility and will pay extra for options that hedge large price moves. The volatility risk premium measured in terms of Option price is positive; Option expected return is negative; Option implied volatility is positive

International Risk Management Conference 2018 2018 Figlewski The Volatility Surface As the RND extraction procedure makes clear, there is a one-to-one relation between the volatility smile and the RND. Calculating the VIX is the single most important real world use of the RND. The volatility surface has been covered at great length elsewhere. I will offer only a few comments on it. General gripe: Black-Scholes is not the way the market prices options, yet we extract BS IVs and model their dynamics. What is the scientific basis for taking an incorrect model, extracting a set of fudge factors that artificially set its outputs for a single date equal to market option prices and then trying to apply elaborate statistical procedures to capture and predict the behavior of the wrong model's fudge factors?

International Risk Management Conference 2018 2018 Figlewski The Volatility Surface Numerous models of the dynamics of the volatility surface have been proposed. Articles: Dumas, Fleming and Whaley (1998) Cont, da Fonseca, and Durrleman (2001) Andersen, Fusari and Todorov (2013) Carr and Wu (2016) Israelov and Kelly (2017) Books: Gatheral (2006)

Derman and Miller (2016) International Risk Management Conference 2018 2018 Figlewski The Volatility Surface Israelov and Kelly (2017) is my current favorite volatility surface model. They don't just compute a prediction of IVt+1 given IVt. They get an entire probability distribution for each option's next day IV. 1. Map traded options onto a fixed moneyness and maturity grid. 2. Conceptually separate the effect of the change in stock price from the change in the IV at a given grid point (i.e., the change in the risk premium at that moneyness/maturity point). 3. Fit a model at each grid point to the past history of IVs at that point and extract principal components.

4. Estimate the dynamics of the principal components to project the probability density of next period's IV at each grid point. 5. Map back from the standardized grid to the specific option contracts. This recognizes that the IV surface inherently involves dynamic risk preferences, and that the returns process models we use do not explain them. International Risk Management Conference 2018 2018 Figlewski The Pricing Kernel Puzzle The pricing kernel captures the effects of market risk aversion (and speculative beliefs). When the option underlying is on the aggregate stock market portfolio, we expect that the marginal value of $1 will go down if investors become more wealthy, i.e., when stock prices go up. The pricing kernel should be

monotonically downward sloping in return on the stock index. But the options market doesn't show that happening. Here's a typical example. International Risk Management Conference 2018 2018 Figlewski Probability Density The Pricing Kernel Puzzle Standard deviations International Risk Management Conference 2018

2018 Figlewski The Pricing Kernel Puzzle What's happening here? There have been a variety of possible theoretical explanations that did not work empirically (e.g., investors love risk). Two that could work are 1. Investors care about things in addition to expiration day returns. If they like high returns, hate low returns, and on top of that they dislike volatility for its own sake, a non-monotonic kernel can be generated. (Unclear how easy it is to get the full shape in the picture.) 2. Investors are not homogeneous and their expectations cannot be compressed into those of a single representative agent. International Risk Management Conference 2018

2018 Figlewski Three Different Exchange-Traded Funds on the S&P 500 Investor heterogeneity can be explored by comparing the Risk Neutral Densities extracted from options on three different ETFs, all tied to the same underlying S&P 500 Index. SPY: The original SPDR (S&P Depository Receipt) index-tracking fund. SSO: A leveraged ETF with twice the exposure to the underlying index. SDS: A leveraged inverse ETF with negative two times the exposure to the index. International Risk Management Conference 2018 2018 Figlewski 28

May 2, 2008 Risk Neutral Densities Before Transformation International Risk Management Conference 2018 2018 Figlewski RNDs on May 2, 2008 after Two Transformations All three are left-skewed SSO (double long): q(x) is highest for gains lowest for small losses higher than SPY for large losses SPY tracks the

S&P500 1 to 1 SDS (double short): q(x) lower for gains, higher for losses 02-May-2008; T = 50; SPX = 1413.90; dSPX = 0.32%; VIX = 18.18; dVIX = -0.70; p-dist'n vol'y = 15.75 International Risk Management Conference 2018 2018 Figlewski 30 More Research is Needed! Two major areas in which theories about Risk Neutral Densities are plainly inadequate, and more research is needed: Understanding the risk neutralization process: We have gone about as far as

we can with models that focus only on the dynamics of the returns process. Option prices, and RNDs, reflect market risk aversion/tolerance, that we have so far made little effort to model in any systematic way. Allowing for heterogeneous investors: The "representative investor" simplification cannot work for zero-sum contracts like options, futures, and other derivatives. These markets exist in order to transfer risk from one counterparty to another. If investors are identical, they won't trade. They will agree on the option price, but it will be the level where no one wants to take a long or short position. In short, none of our models really capture what any investor in options is actually doing. There is plenty of room for further research on Risk Neutral Densities! International Risk Management Conference 2018 2018 Figlewski

And when you do great new papers on RNDs, send them to the Journal of Derivatives! THANKS! International Risk Management Conference 2018 2018 Figlewski