Choosing GBM Drift for Derivative Valuation and Exposure Simulation
Summary
The document distinguishes the drift used to price derivatives from the drift used to simulate future underlying prices. Under the risk-neutral measure, derivative valuation uses the relevant risk-free rate; for a real-world simulation of potential future exposure, the stock is instead evolved using an expected return, often estimated from historical data. Derivatives are then valued at future dates using risk-neutral pricing assumptions.
It also recommends using the overnight indexed swap curve in the currency of the derivative to determine the risk-free rate, giving a USD option as an example. The explanation is conceptual and does not derive the measure change or discuss complications such as dividends, funding conventions, or non-flat curves. Its central practical distinction is between a pricing measure and a real-world forecasting measure.
Key ideas
- Use a risk-free drift under the risk-neutral measure when valuing derivatives.
- Use an expected return under the real-world measure when simulating potential future underlying prices.
- A potential exposure workflow can simulate the underlying under the real-world measure and revalue derivatives under the risk-neutral measure.
- The suggested risk-free input is the OIS curve in the derivative's currency.
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# Drift rate in Geometric Brownian Motion # Drift rate in Geometric Brownian Motion I have two questions regarding the drift term in the geometric Brownian motion that I cannot find any clear answers to online. - When would we use risk-free rate as drift and when would we use the expected rate of return of the stock as drift rate? - If we are in the risk-neutral framework, what is the appropriate risk-free rate to use in the drift term? Is it the risk-free rate in the country of the stock or is it the risk-free rate used in the discount rate (assuming that these differ)? Please share your insights or articles on the topic. ## Answer by Jan Stuller (score 1) https://quant.stackexchange.com/a/68870 These are fairly basic concepts, although I do acknowledge that it's often not discussed explicitly in text books in great depth. 1. When to use risk-free rate? You use the risk-free rate only when you want to value derivatives (forwards, futures, options... on the stock under consideration). On the other hand, if you (for example) want to estimate Potential Future Exposure (PFE) on a derivative portfolio against a counterparty, you need to run the Monte-Carlo simulation that computes the PFE under the real-world historical measure, where you would set the stock drift to the "expected rate of return" (often calibrated to historical data). In other words, in the PFE simulation, you first evolve the underlying stock under the real-world measure, and then you'd value the derivatives at discrete future points in time under the risk-neutral measure (using the risk-free rate). The above might sound complicated, but the crux of it is: - risk-free rate for valuation of derivatives - expected rate of return for evolving the underlying to get a distribution of "potential" future prices The key concept here is that derivative prices are independent of the underlying's future "potential" price distribution (because every market participant has a different, subjective view of these); rather, the derivative prices are only dependent on the underlying's volatility & the cost of borrowing money (where this cost is reflected in the risk-free rate). 2. What risk-free rate to use? In most cases, we would use the local currency OIS curve to get the corresponding risk-free rate. So for example in USD currency, to value an option that expires in 1 year on some stock, you'd want to get the 1-year SOFR OIS rate.
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