Risk Neutral GBM Drift and Arbitrage Free Structured Product Pricing

Summary

The document considers whether changing the mean and variance used in a geometric Brownian motion simulation to control how often paths cross a structured product threshold conflicts with arbitrage free pricing. Its answer distinguishes the physical measure, used for forecasting, from the risk neutral measure, used to value derivatives. Volatility may be selected or modeled without violating no arbitrage, while the risk neutral drift must reflect the risk free rate adjusted for dividends.

The explanation gives the corresponding risk neutral GBM dynamics and states that they make the appropriately discounted underlying price a martingale. A physical drift can differ for real world forecasts, but it is not a substitute for the risk neutral drift in pricing. The discussion is concise and assumes a standard GBM setting; it does not address calibration, additional market frictions, or more complex structured product features.

Key ideas

• Risk neutral pricing requires the underlying's drift to equal the risk free rate less dividends in the stated GBM setting.
• The volatility can be chosen without itself creating arbitrage.
• A discounted price process is a martingale under the risk neutral measure.
• The physical measure may use a different drift for forecasting purposes.
• The explanation assumes a standard GBM framework and does not discuss calibration or market frictions.

Tags

# GBM with adjusted normal distribution


# GBM with adjusted normal distribution












To model a structured product, I thought of using a geometric Brownian motion model, where I choose a certain mean and variance for the normal distribution to make sure that a certain percentage of paths (as a result from Monte Carlo) cross a threshold value where different conditions apply. However my question is does this violate the risk-free and arbitrage free assumption? Meaning I can no longer discount using the risk-free rate?

Please let me know if i need to provide any more information.

Kind regards

## Answer by QuantCalc.net (score 0)

https://quant.stackexchange.com/a/85277

In the geometric Brownian motion model setting, choosing any specific volatility or variance does not violate no-arbitrage; the only requirement is that under the risk-neutral measure the drift equals the risk-free rate (minus dividends). The arbitrage-free dynamics must satisfy

$$ dS_t = S_t (r - q)\, dt + S_t \sigma\, dW_t^{\mathbb{Q}}, $$

which ensures that the discounted price process

$$ e^{-(r-q)t} S_t $$

is a martingale. You are free to choose any real-world drift $\mu_{\mathbb{P}}$ for forecasting and any volatility function $\sigma(t,S_t)$, but the risk-neutral drift must stay at $r-q$ to avoid arbitrage.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.