GBM Dynamics and Risk Neutrality in Risk Sensitive Option Pricing

Summary

The document poses a modeling question about pricing European options with dynamic programming. The proposed setup treats option hedging as a discrete-time, continuous-state Markov decision problem and uses a risk-sensitive action-value function: instead of equating the option’s fair price with the expected hedge portfolio value, it adds a discounted variance-based risk premium.

The central uncertainty is whether assuming geometric Brownian motion for the stock price imposes risk neutrality despite that explicit risk adjustment. The question invokes Girsanov’s theorem, but supplies no answer or resolution. It also does not specify the measure under which the GBM is modeled, the precise variance penalty, or how the risk premium is calibrated. The excerpt is useful as a framing of the distinction between an asset dynamics model and a pricing criterion, but it is not a complete pricing method.

Key ideas

• The proposed option pricing method uses dynamic programming in a Markov decision framework.
• Its action-value function adds a discounted variance penalty to account for hedge risk.
• The author asks whether assuming GBM nevertheless embeds risk neutrality in the formulation.
• The document provides no answer, so the measure and risk adjustment remain unspecified.

Tags

# Does simulating price as GBM automatically implies risk neutrality?


# Does simulating price as GBM automatically implies risk neutrality?












I am using a dynamic programming approach to price European options where formulate the pricing as a discrete-time continuous-space Markov Decision process. The MDP is risk-sensitive as in I don’t take expected value of hedge portfolio to be the option fair price but rather I add a “risk-premium” which is equal to discounted variance of the hedge portfolio. By doing this I can negate the assumption of risk-neutrality.

But since Dynamic programming needs a model for the world, I have to assume the stock price dynamics as a GBM. My question is even though I incorporate risk in options in the action-value function of the MDP, does using GBM which by girsanov theorm is risk-neutral creeps in the risk-neutral assumption in the formulation?

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