{"id":18674,"date":"2026-06-07T18:40:48","date_gmt":"2026-06-07T18:40:48","guid":{"rendered":"https:\/\/www.simtrade.fr\/blog_simtrade\/?p=18674"},"modified":"2026-06-07T19:14:17","modified_gmt":"2026-06-07T19:14:17","slug":"option-implied-risk-neutral-distribution","status":"publish","type":"post","link":"https:\/\/www.simtrade.fr\/blog_simtrade\/option-implied-risk-neutral-distribution\/","title":{"rendered":"Option Implied Risk-Neutral Distribution"},"content":{"rendered":"\n\"Saral<\/a>\n\n

In this article, Saral BINDAL<\/a> (Indian Institute of Technology Kharagpur, Metallurgical and Materials Engineering, 2024-2028 & Research assistant at ESSEC Business School) explains how option prices can be used to build an implied risk-neutral distribution.<\/p>\n\n\n

Introduction<\/h2>\n\n

Derivative markets provide a rich source of information for market expectations. For example, a futures price is the market\u2019s expectation of the future value of an asset. More interestingly, we can derive the moments of the statistical distribution of future asset values from the market prices of options, like the variance (second moment), the skewness (third moment) and the kurtosis (fourth moment). More generally, we can extract the ex-ante risk-neutral probability distribution of future asset prices at a given date from option market prices with the corresponding maturity date.<\/p>\n\n\n

Physical vs Risk-Neutral Probability Measures<\/h2>\n\n

A real-world probability<\/b> measure represents the statistical distribution of asset returns typically estimated using historical data. These measures incorporate risk premia, market frictions, and investor behaviour, and are primarily used for statistical inference and risk modelling.<\/p>\n\n

In contrast, risk-neutral probability<\/b> measure is a mathematical pricing measure used in no-arbitrage valuation of financial derivatives. Under this framework, asset prices are evaluated as discounted expected payoffs under an equivalent martingale measure. In this setting, the expected return of any risky asset is adjusted to the risk-free rate within the pricing measure, simplifying valuation by transforming uncertain future payoffs into present values computed via expectation (Hull, 2018; Shreve, 2004).<\/p>\n\n\n

Historical vs Risk-Neutral Distributions<\/h2>\n\n

Historical Distributions<\/b> are constructed from observed past returns under the physical measure (P-measure). They empirically capture the true statistical behaviour of asset prices, including fat tails, skewness, and volatility clustering driven by real market shocks and investor behaviour. These distributions exhibit higher variance and kurtosis, making them particularly valuable for stress testing, Value-at-Risk estimation, and portfolio risk management where realistic loss scenarios matter.<\/p>\n\n

Risk-Neutral Distributions<\/b> are derived from option market prices rather than historical data, under the implied measure by no-arbitrage pricing (Q-measure). They reflect market-implied expectations of future payoffs discounted at the risk-free rate resulting in smoother, less skewed densities. While highly effective for pricing derivatives and contingent claims, they tend to underestimate tail risk and do not directly represent the actual probabilities investors assign to future market outcomes.<\/p>\n\n\n

Risk-neutral distribution: the Black\u2013Scholes\u2013Merton framework<\/h2>\n\n

Having distinguished between the physical and risk-neutral probability measures, it is useful to examine the risk-neutral distribution implied by the Black\u2013Scholes\u2013Merton (BSM) model, which is a standard model in quantitative finance. The BSM framework assumes that the underlying asset follows a geometric Brownian motion and provides a simple illustration of how the transition from the physical measure to the risk-neutral measure alters the distribution of future asset prices.<\/p>\n\n

Under the BSM, the standard assumption is that the underlying asset follows a geometric Brownian motion given by the following expressions:<\/p>\n\n\n


\n\"SDE<\/p>\n\n

where:<\/p>\n