Subjective Distributions for Tactical Asset Allocation
Research explores how to express views on market returns through a derivative overlay.
September 2025, Academic Research, by Gerard Brunick, Ph.D., CFA, Robert Harlow, CFA, CAIA
Originally published March 2025 in The Journal of Portfolio Management.
Key findings
- The authors construct an asset return distribution by performing a risk adjustment to the risk-neutral distribution implied by option prices.
- They show how this initial distribution can be modified to reflect an investor’s subjective views on asset returns.
- The authors then solve an optimal investment problem to determine how these subjective views should be reflected in portfolio derivative positioning.
Gerard Brunick, Ph.D., CFA
Associate Director, Research
Robert Harlow, CFA, CAIA®
Associate Head, Global Multi-Asset Research
Insights Out: Options With Options
The authors consider the problem of an active manager who would like to express views on market returns through a derivative overlay on an equity portfolio. Translating these views into derivative positions can be challenging as most quantitative portfolio construction strategies require a fully specified return distribution as an input. To construct such a distribution, the authors start with the risk-neutral probability distributions implied from option prices. These distributions are time varying and provide conditional information on higher moments that is difficult to extract from historical data alone. Risk-neutral distributions do not include risk premia, however, so mean returns under the risk-neutral distributions are “too low” relative to investor expectations anchored in the “real-world” distribution. To address this, the authors perform a risk adjustment to recover an approximate real-world distribution from the risk-neutral distribution. They then show how one may construct a subjective distribution for asset returns that expresses an investor’s views using the implied real-world distribution as the starting point. The authors provide examples of such views and solve the associated optimal investment problem to show how these views should be expressed through a portfolio derivative overlay.
…When Paul Samuelson was originally studying options, he went out to the brokers to understand these contracts.
And at the time, a lot of these dealers were European refugees.
So he's asking a lot of questions. And apparently one of these dealers was quite skeptical of his efforts, and he told them that he would never understand options because it requires a sophisticated European mind. And so when Samuelson started publishing on this topic, he named the more complicated version the American version. As sort of a revenge to get back at this former slight.
Music
Hello, and welcome to Insights Out, our new video series focused on quantitative insights from the various research teams here at T. Rowe Price.
My name is Stefan Hubrich, and I'm the head of global multi-asset research at our firm. Our first episode today is titled “Options With Options.” We'll be looking at options securities and how they can be used in portfolios. That's a really complex topic, and so I'm lucky that I don't have to tackle it alone. In fact, today, I'm joined by my colleague Gerard Brunick, who recently wrote a really relevant paper in this very area.
The paper is titled “Subjective Distributions for Tactical Asset Allocation.”
Let's talk about you a little bit before we jump into the content. So, how come you know so much about options?
So I did a Ph.D. in financial mathematics, and I've done some academic work related to option pricing. I've also worked in the energy industry and now in the asset management industry, where I've managed option models and done research related to option-based investment strategies.
Okay. I think we're in good hands. Now I feel very confident. So let's start with the basics. Not everybody in our audience may know what an option is. So what's an option?
All right. So an option is a contract that gives the holder the right, but not the obligation, to buy or sell a security at a future point in time at a fixed price.
And so that fixed price is called the strike of the option. And the act of choosing to buy or sell is called exercising the option. And so, more generally, there's two dimensions in which we classify options: We have put and call options. So a put gives you the right to sell something; a call gives you the right to buy something.
And we also have American and European options. And so that relates to when you have to make the exercise decision. So with the European option, there's a fixed maturity date, and you choose whether to exercise on that maturity date. Whereas, with an American option, you can choose to exercise at any point up to and including the maturity date.
Let's start with an example first that has nothing to do with finance. Let's say I've got my old Ford F-150 truck in my garage.
Let's say I want to sell you a European call option on that.
Right.
So that would give you the right—let's say, a one-year maturity…
Right.
So that would give you the right to buy that old truck for that agreed-upon price one year from now, no matter what, then, I think it's actually worth.
Exactly.
Is that the way to think about it?
Right.
Or alternatively, I could sell you an American put option on that truck. And so we would agree you'd pay me a premium right now. We’d also agree on the price that you could sell it to me. And then at any point over the next year, you could choose to force me to buy it from you.
For example, if you got in a wreck, you could then force me to buy the truck, even though it's no longer worth what we've agreed to as the price.
And so one thing is that has a bit of an insurance feel to it—the put option, right—because it protects you against a loss in the value of your truck.
And so what we're going to talk about today is the underlying is not so much a truck; it's going to be a financial security. And, in particular, we're going to think about a broad equity index.
Think the S&P 500. So say the S&P 500 is at 6,000 today. Right. That a put option struck at 5,000 would mean in, say, three months, would mean if three months from now the S&P has dipped down below 5,000, you'd still be able to sell it at 5,000. So it'd be a kind of downside protection on the value of your portfolio.
Alternatively, if you wanted upside, you could buy a call option struck above 6,000. And in the event that the market was above that value at maturity, then you'd exercise and you'd be able to buy this basket of securities for less than its price.
Now, in the event that neither of those events happened, you'd pay the premium upfront, but then you wouldn't exercise, and so you essentially lose the premium. So there's a cost to this optionality.
Gotcha. And so that premium is important for us to keep in mind because that's the other side of the equation.
Introduction
Connsider the problem of an active manager who would like to express views on market returns through a derivative overlay on an equity portfolio. Often these views are directional or qualitative, so translating these views into derivative positions can be challenging as most quantitative portfolio construction strategies require a fully specified return distribution as an input. To construct such a distribution, we start with the risk-neutral probability distributions implied from option prices and perform a risk adjustment to recover an approximate “real-world” distribution. We then show how this real-world distribution can be further modified to reflect an investor’s subjective views. Finally, we solve an optimal investment problem based on this subjective distribution to determine how the manager’s views should be expressed using derivative positions.
As has been known since the work of Breeden and Litzenberger (1978), the distribution of an asset’s price return under the risk-neutral measure can be extracted from the set of European option prices written on that asset. These distributions are time varying and provide conditional information that is difficult to extract from historical data alone, but they also reflect the aggregate risk preferences of market participants. Economic theory (Cochrane 2005) tells us that the risk-neutral distribution is connected to the real-world distribution through the marginal utility of consumption of the representation agent, so we should not expect the risk-neutral distribution to provide an unbiased estimate of forward returns without a risk correction. For example, Bliss and Panigirtzoglou (2004) reject the hypothesis that the risk-neutral distribution agrees with the realized return distribution of the S&P 500 and FTSE 100 indices but fail to reject this hypothesis once the distribution has been adjusted to reflect risk aversion.
To recover the real-world distribution from the risk-neutral distribution, we will assume that the representative agent’s utility function takes constant relative risk aversion (CRRA) form, and we will imply the coefficient of relative risk aversion from the realized equity risk premium in our historical sample. Once the representative agent’s utility function has been specified, we will show how it can be used to transform the risk-neutral distribution into an approximate real-world distribution.
Given this market-implied return distribution as a starting point, we can then express an investor’s views as perturbations to this initial distribution. For example, bullish or bearish sentiment can be expressed by translating this distribution in log-return space and a belief that volatility is underpriced can be expressed by dilating the log-return distribution. We then solve an optimal investment problem using the distribution that expresses the investor’s views to see how the portfolio should be positioned to express each view.
An analogy can be drawn to the Black–Litterman model (Black and Litterman 1992). The construction of the Black–Litterman prior distribution requires estimating the covariance of asset returns empirically and then choosing expected returns so that the optimal portfolio in the absence of investor views is the market portfolio. Investor views are then implemented as perturbations to these prior expected returns and the strength of these views can be adjusted so that the investor is comfortable with the active risk in the resulting portfolio. Similarly, we construct a utility function and an initial distribution in such a way that the solution to the associated optimal investment problem is to hold the market, and we implement investor views as perturbations to this initial distribution.
Estimating the real-world measure
We begin the construction of our approximate real-world measure by first extract- ing the risk-neutral marginal distribution of the underlying S at maturity T from the set of European call option prices. Given a sufficiently smooth call pricing function c, Breeden and Litzenberger (1978) have shown that the marginal density of the underlying at maturity T under the risk-neutral measure, PS, is given by
pST(k) = erT ∂2/∂k2 c(k)
where k denotes option strike and e−rT is the price of the zero-coupon bond with maturity T. While theoretically attractive, this relationship can be difficult to implement because it is sensitive to estimation error in c, so we instead work with a discrete approximation.
We fix a finite set of strikes K and assume that the call prices associated with these strikes are free of arbitrage. We further assume that the interest rate and dividend yield on the underlying are deterministic. We can then invoke Theorem 3.1 and the remark that follows from Davis and Hobson (2007) to construct a discrete distribution
Q(ST =si )=qi for i=1,…,n
with Σiqi = 1 that properly prices all options with strikes in K and maturity T. In practice, we start with all liquidly traded strikes and then add additional strikes with prices generated by a model that has been calibrated to current market prices. Moreover, if we let F denote the forward price of the underlying, then we can add an additional sufficiently small strike k* > 0 with associated price e−rT(F − k*) to ensure that theconstants s1, …, sn are all strictly positive.
As an example, we infer the 12-month forward risk-neutral distribution of the S&P 500 Index on June 20, 2025, as of June 18, 2024, and plot the result in Exhibit 1. Our call prices are generated using a proprietary methodology based on the SABR model (Hagan et al. 2002) that is calibrated to OptionMetrics data. We also include the normal distribution corresponding to the Black–Scholes model using at-the-money volatility for comparison. We see that the risk-neutral distribution implied from option prices is more sharply peaked than the normal distribution and has fatter tails. The distribution implied from options also evidences negative skewness.
The real-world measure, P, is equivalent to the risk-neutral measure,1 so P must take the form
P(ST = si ) = pi for i = 1,…,n
for some strictly positive constants p1, …, pn with Σipi = 1. As we are working on a discrete probability space, it is convenient to introduce the Arrow–Debreu state-price securities with payoffs E = (E )ni=1. The payoff Ei Is equal to 1 when ST = si and 0 otherwise. These elemntary securities span the space of time T payoffs in the sense that every payoff is almost surely equal to some linear combination of the payoffs in E. For example, holding one unit of each Arrow–Debreu security ensures the delivery of one unit of numeraire, so this portfolio is equivalent to holding the zero-couponbond with maturity T.
Exhibit 1: Twelve-Month Forward, Risk-Neutral SPX Density (June 18, 2024)
Source: Based on data from OptionMetrics.
To link the probabilities (pi )ni=1 and (qn )ni=1, we consider a simple two-period utility maximization problem as in Huang and Litzenberger (1988). We have an agent withinitial deterministic endowment w. This agent gains utility U0(c) from consumption level c at time 0 and utility U(c) from consumption level c at time T.2 We make the standard assumptions that U0 and U are strictly concave, continuously differentiable functions taking values in [−∞, ∞) with strictly positive first derivatives wherever they are finitely valued. For example, if we extend the natural logarithm to the whole real line by defining log(c) = −∞ when c ≤ 0, then U = log satisfies these conditions.
At time t = 0, the agent chooses to hold θi units of the ith state-price security with price ei and payoff Ei, and at time t = T, she consumes the resulting payoffs. The agent’s utility maximization problem is given by
where a · b = Σi=1aibi denotes the usual inner product. Letting θ* denote the optimal solution to this problem, the first order condition for optimality is
ei U′0 (θ*⋅e) = EP[Ei U′(θ*⋅E)] = pi U′(θi*) for i = 1,…,n
The first term in Equation (5) describes the utility of consumption at time t = 0 that is lost if the agent chooses to invest slightly more in asset i and the next two equivalent terms give the expected utility of consumption at time t = T that is gained. If these quantities do not match, the agent can increase her expected utility by adjustingher investment in asset i.
If we also price the ith elementary security using the risk-neutral measure
ei = e-rtEQ[Ei] = ee-rt qi
then we can combine (5) and (6) to connect P and Q
Expression (7) provides some intuition about how the risk-neutral measure relates to the agent’s risk preferences. The constant erT /U′0 (θ*⋅e) ensures that Σiqi = 1. The expression U′(θi*) gives the agent’s marginal utility of consumption when ST = si. If this value is large, then ST = si represents a state of the world that is bad for the agentand the risk-neutral measure overweights this bad outcome relative to the real-world measure. This reflects the fact that a risk-averse agent is willing to pay more for claims that pay off in bad states of the world.
To make Equation (7) operational, we need to specify U and identify θ*. To thisend, we first assume that U is a CRRA utility function of the form
with coefficient of relative risk a version γ. As is standard, we take U(c) = −∞ when c < 0 and assume U is continuous from the right at zero. This class of utility functions istractable and quite natural for an asset manager because optimization with respect to CRRA utility produces solutions that are given as portfolio weights and do not depend upon the value of the fund. This is a desirable quality for an asset manager who wants to present a stable risk profile to clients as the size of the fund changes. Next, we further assume that our agent is the representative agent for the economy. This means that the agent must hold the market portfolio as aggregate debt and option positions net to zero in equilibrium. Letting y denote the deterministic dividend yield on the market, this means that θi* = (1 + y)si , and we can invert the relationship (7) to recover P from Q.
All that remains is to estimate the risk-aversion coefficient γ. To do this, we choose γ so that the average forward-looking equity risk premium in our model agrees with the realized equity risk premium in our sample. Our historical period runs from January 1, 2000, to March 31, 2024. Data are sourced from Option Metrics, S&P Global, and ICE. At each quarterly SPX option expiration date, we imply the 3-month, 6-month, and 12-month forward risk-neutral return distributions from quoted prices, transform the risk-neutral to a real-world measure using an assumed value for γ, and then compute the equity risk premium implied by this real-world distribution. We then compare the average model implied value with the realized equity risk premium in our sample, which is computed using the monthly total returns of the S&P 500 and the monthly returns on the ICE Bank of America 3-Month US Treasury Bill Index. As we see in Exhibit 2, we infer a relative risk a version of about 1.6, and the result is not particularly sensitive to the forward return period used.
Applying this transformation to the 12-month forward risk-neutral distribution that we implied as of June 18, 2024, from SPX options with expiration June 20, 2025, we obtain the distribution displayed in Exhibit 3. We see that our estimate dreal-world distribution has less negative skewness and higher expected return. The forward equity risk premium implied by this real-world distribution is about 3.8%. The estimated real-world measure also has less variance; this reflects the variance risk premium (Carr and Wu 2008).
1Two measures are equivalent if they share the same null sets. The probability measures P and Q may assign different probabilities to the same event, but they must agree on the set of events thatare possible.
2 Alternatively, we could imagine that we are in a multiperiod setting where U(w) denote the value function that returns the maximum utility of current and future consumption that is achievable given wealth level w at time t = T.
Relative risk aversion vs. implied equity risk premium
Source: Based on data from OptionMetrics, S&P Global, and ICE.
Twelve-month forward S&P 500 index return density (June 18, 2024)
Source: Based on data from OptionMetrics.
Optimal portfolio based on manager’s view on ERP
Source: Based on data from OptionMetrics.
Optimal active return based on manager’s view on ERP
SOURCE: Based on data from OptionMetrics.
Optimal portfolio based on manager’s view on risk
Source: Based on data from OptionMetrics.
Optimal active return based on manager’s view on risk
Source: Based on data from OptionMetrics.
Reflecting the investor's subjective views
We now consider the investor’s optimization problem. Let P denote the investor’s subjective view which may differ from the market-implied view P we constructed previously.The investor may invest in the bond, the market, and in puts and calls at asubset of the strikes ˆK ⊂ K that were used to construct Q. We restrict the strikes atwhich our investor can trade because our goal in this section is to produce examplesof the technique.
The investor then solves the utility maximization problem
where X1, …, Xn denote the payoffs of the securities that are available to the investor,x1, …, xn denote the time t = 0 prices of these securities, and A(w) = {ˆθ ∈Rn : θ ⋅ x ≤ w} denotes the set of admissible investment strategies that can be funded from asset value w at time t = 0. If we were to allow the investor to trade all strikes in K, the solution to (9) would be available in closed form (Carr and Madan 2001). The restricted problem must be solved numerically, but it is a low-dimensional, concave optimization problem, so it is straight forward to solve with standard algorithms.
In principle, the investor’s utility could differ from the representative investor, but in practice, it is convenient to use the same utility function. If the investor uses the same utility function as the representative agent, then holding the market is the optimal solution to the investor’s optimization when the investor’s subjective view P agrees with the approximate real-world measure P we constructed in the previous section.
We will now work through a series of examples to show how this machinery can be applied. In each example, we return to our sample data from June 18, 2024, and assume that the investor may hold the bond, the market, a –20 delta put, an ATM straddle, and a 20 delta call. If we allow the investor to trade the bond, the market, and both an ATM put and an ATM call, one of these securities would be redundant, so we require the investor to hold a straddle to ensure that the solution is unique.
Example 1
We first consider the case where the investor would like to express a view onforward expected returns. In this case, the approximate real-world distribution constructedas of June 18, 2024, implies a forward-looking equity risk premium of about 3.8%. If the investor has a different view, we can translate the market-implied distribution in log-return space to produce a subjective distribution that is consistent with the investor’s view. The solutions to the associated optimal investment problem fora range of views are given in Exhibit 4.
When the investor believes that the equity risk premium is higher than the market’s view, she levers up the portfolio, and when she believes the market is over priced,she derisks as one would expect. However, the act of levering up her portfolio introduces tail risk. The investor could potentially use the out-of-the-money put option to address this, but the optimal solution is to buy convexity at-the-money and sell it in the right tail.
If we look at implied real-world distribution in Exhibit 3, we see that the mode ofthe distribution is about 11%. When the investor believes that the equity risk premium is higher than the market-implied value of 4%, her subjective distribution puts more weight on outcomes above the mode and less weight on outcomes below the mode. If we look at the active returns given in Exhibit 5, we see that the optimal portfolio clearly reflects this view.
Example 2
Rather than disagreeing with the market’s view on expected returns, the investor might instead disagree with the market’s view on risk. On our sample date, the standard deviation of log-returns under the approximate real-world distribution is 14.1%. If the investor disagrees, we can dilate the market-implied distribution in log-return space to construct an alternative distribution that reflects the investor’s view. If the investor does not wish to express a view on expected returns, we can then perform a translation in log-return space to ensure that the equity risk premium is undisturbed. If we then solve the optimal investment problem with respect to this subjective distribution, we obtain the portfolios in Exhibit 6.
When the investor expects more volatility than we see in the market-implied distribution,she derisks the portfolio and buys convexity at the money and in the right tail. In this case, she also sells back some of the risk that she removed from the left tail. If we look at the graph of the payoffs corresponding to the view that the market is underpricing volatility in Exhibit 7, we see that, despite selling some out-of-the moneyputs, the investor’s optimal payoff still dominates the market return in the left tail.
Example 3
Third, we consider the case where the investor has a view on the skew of the market return distribution. To adjust the skew of the return distribution, we define the function
gα (x) = x + sgn(α)eαx
and then use the distribution of ST = exp{gα (log ST )} to represent the investor’s view. When α > 0, gα is convex and log ST has a more positive skew than log ST. Conversely,when α < 0, gα is concave and log ST has a more negative skew than log ST. We usea root-finding algorithm to find an α that produces the target skew, and then we translate and scale the resulting distribution so that it has the same implied equityrisk premium and log-return standard deviation as the market-implied distribution.
The skew of the market’s log-return distribution is –1.7 in our sample data. Given the investor’s view on skew, we can apply the algorithm described to produce a subjective distribution that reflects the investor’s view on the skew of the log returns. We constructed a number of these distribution and solved the related optimal investment problems. The resulting portfolios are described in Exhibit 8.
We see that the behavior of these portfolios in the tails reflects the investor’s view on skew. When the investor believes that returns are more negatively skewed than the market, she buys out-of-the-money puts and sells out-of-the-money calls as one would expect. Introducing additional negative skew without changing the implied equity risk premium or the standard deviation of log-returns increases the median of the return distribution. If we look at the optimal active payoff in Exhibit 9 that corresponds to an investor who believes that the return distribution is more negatively skewed than the market-implied distribution, we see that the portfolio positioning around the money is constructed to reflect the fact that the median of the investor’s distribution is shifted to the right relative to the market-implied distribution.
Optimal portfolio based on manager’s view on skew
Source: Based on data from OptionMetrics.
Optimal portfolio based on manager’s view on skew
Source: Based on data from OptionMetrics.
Conclusions
In this article, we constructed a discrete approximation of the risk-neutral market return distribution from call prices. We then fit a CRRA utility function to the realize dequity risk premium in our historical sample and show how this calibrated utility function can be used to construct an approximate real-world return distribution. Finally, we give a number of examples showing how one can perturb this real-world distribution to reflect an investor’s views, and we solve an optimal investment problem to determine how these views should be expressed through option positioning.
These tools can used by active managers to size option positions. We imagine a manager whose strategic allocation is 100% long an active equity portfolio with some tracking error constraint to an equity index. That manager may then have a tactical view on the underlying equity market. By generating a range of potential subjective distributions and examining the resulting impact on portfolio position, the manager can quantitatively calibrate the strength of their view. Once they are happy with the result, the overlay would be implemented using derivatives where the option positions are given explicitly, and a futures position in the underlying equity index would be used to adjust the market exposure.
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References
Breeden, D. T., and R. H. Litzenberger. 1978. “Prices of State-Contingent Claims Implicit in OptionPrices.” Journal of Business 51 (4): 621–651.
Black, F., and R. Litterman. 1992. “Global Portfolio Optimization.” Financial Analysts Journal48 (5): 28–43.
Bliss, R. R., and N. Panigirtzoglou. 2004. “Option Implied Risk Aversion Estimates.” The Journalof Finance 59 (1): 407–446.
Cochrane, J. H. 2005. Asset Pricing: Revised Edition. Princeton, NJ: Princeton University Press.
Carr, P., and D. Madan. 2001. “Optimal Positioning in Derivative Securities.” Quantitative Finance1: 19–37
Carr, P., and L. Wu. 2008. “Variance Risk Premiums.” Review of Financial Studies 22: 1311–1341.
Davis, M. H. A., and D. G. Hobson. 2007. “The Range of Traded Option Prices.” MathematicalFinance 17 (1): 1–14.EXHIBIT 9Optimal Active Return Based on Manager’s View on SkewSOURCE: Based on data from OptionMetrics.−40 −20 0 20 40−50510SPX ReturnActive ReturnView on Skew−2.1−1.7−1.5
Hagan, P. S., D. Kumar, A. S. Lesniewski, and D. E. Woodward. 2002. “Managing Smile Risk.”Wilmott Magazine (September): 84–108.
Huang, C.-F., and R. H. Litzenberger. 1988. Foundations for Financial Economics. New York:North-Holland.
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Data shown is in USD currency, where relevant.
Data source information for Exhibits 1 and 3-9 – T. Rowe Price analysis, based on Option Metrics prices for the SPX Index options with expiration date of 6/20/2025 and market pricing date of 6/18/2024.
Data source information for Exhibit 2 – T. Rowe Price analysis , based on Option Metrics data for SPX Index options prices and GPAR returns for SP500 and MLM3T between 1/1/2000 to 3/31/2024.
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