Real-world profiles can substantially deviate from risk-neutral profiles for derivatives and related structures—depending on such factors as product, sensitivities and leverage, which can often lead to inaccurate risk profile assessments.
Real-world dynamics incorporate risk premia, which has immediate consequences for the generation of economic scenarios by insurance companies. In reality, risky positions earn a premium depending on their sensitivities to market factors. For instance, a call on an equity index is seen to earn a premium, while a put is seen to pay a premium. Thus, while risk-neutral dynamics are ideal for producing valuations and hedge ratios, risk must be assessed with reference to real-world dynamics.
In the risk-neutral world, all investments grow, on average, at the risk-free rate. From a technical perspective, how do we define risk-neutral dynamics? Risk-neutral dynamics are a “special” set of dynamics, which allow us to value instruments by discounting at the risk-free rate, i.e.:
Here:
instructs us to take the expectation with respect to the risk-neutral dynamics.
Risk-neutral dynamics for any traded security must grow on average at the risk-free rate, i.e.:
Importantly, we have an expected return of:
While risk-neutral dynamics are ideal for producing valuations and hedge ratios, clearly risk premia plays an important role in ESG—as our goal is to generate as realistic dynamics as possible. In this sense, real-world dynamics are, in fact, a necessity for risk analysis and scenario generation in today’s world.
For the purposes of evaluating risk, for example, we may want to know the probability of losing more than 10%, over a 1Y horizon. Which probability would we want?
Naturally, we would seek the “actual” probability of this taking place to assess our risk, rather than a hypothetical probability. Here, we should also note that Historical VaR is implicitly a real-world risk metric—given that real-world dynamics govern historical market movements.
Real World vs. Risk Neutral Dynamics: Implications for Risk Profiles & Scenarios
In the diagram below, we can clearly observe that the real-world and risk–neutral distributions diverge quite dramatically. Therefore, again depending on which distribution is used, significantly different risk assessments will result.
Simulated Black Scholes (1973) paths using both real-world and risk-neutral dynamics, along with percentiles of projected distributions. Monthly observations for 15 years. Percentiles were computed off samples of P=30,000 paths.
Read our recently published whitepaper, “Real-World Equity & Volatility Behavior: Implications for Economic Scenario Generation” for a deeper exploration into why real-world dynamics are necessary for risk analysis and scenario generation. The paper also examines the roles of the equity premium and volatility premium in stochastic volatility models of equity markets.
