Deriving a Markov Transition Model from Option-Implied Risk-Neutral Densities

Abstract

This project derives a discrete-time Markov transition model for an asset price using option-implied risk-neutral densities across maturities. Risk-neutral marginals are recovered from option prices (Breeden–Litzenberger), discretized onto a state grid, and linked via transition matrices that satisfy probability constraints and the risk-neutral martingale condition. Full derivations are provided in the attached note; implementation and validation are in progress.

Methods

• Option surface → risk-neutral density: recover marginal f ^Q(S_T) per maturity from call prices via Breeden–Litzenberger (using a smoothed, no-arbitrage fit).
• Discretization: map each marginal onto a fixed state grid {s₁, …, s_M} to obtain distributions π⁽ⁱ⁾.
• Markov construction: build time-inhomogeneous transition matrices P⁽ⁱ⁾ such that rows are nonnegative and sum to 1.
• Martingale constraint: enforce E[S_ti+1 | S_ti = s_j] = s_j e^(r−q)Δt under Q.
• Locality (tri-diagonal): restrict transitions to nearby states for stability and interpretability, with feasibility bounds per row.
• Calibration objective (in progress): choose free parameters to match π⁽ⁱ⁾P⁽ⁱ⁾ ≈ π⁽ⁱ⁺¹⁾ across maturities.

Output

• A sequence of risk-neutral marginal distributions {π⁽ⁱ⁾} extracted from the option market.
• A sequence of transition matrices {P⁽ⁱ⁾} defining a discrete-time, risk-neutral Markov chain consistent with those marginals.
• Applications: risk-neutral path simulation, scenario transition probabilities, and pricing of discretely monitored/path-dependent payoffs.