Risk-Neutral Pricing and Risk Premia in Derivative Valuation
Summary
The post asks how to value a derivative using an underlying asset’s empirically estimated drift rather than risk-neutral valuation. It describes the Black–Scholes convention of modeling the asset with the risk-free drift under the natural probability measure, which permits expected payoffs to be discounted at the risk-free rate. It then questions whether a real-world expected payoff could instead be discounted using a risk-adjusted rate, perhaps informed by credit spreads or a CAPM-style premium.
The included answer argues that risk-neutral valuation is a computational consequence of no-arbitrage pricing when a delta-hedged portfolio is riskless, rather than a claim that investors are actually risk neutral. On that view, adding a discretionary risk premium to the discount rate does not replace arbitrage-consistent derivative valuation. The exchange offers no derivation or discussion of settings with market incompleteness, transaction costs, or unhedgeable risks, so its conclusion is scoped to the stated hedging framework.
Key ideas
- Black–Scholes risk-neutral valuation uses the risk-free drift for pricing expected payoffs.
- The post considers using real-world drift with a risk-adjusted discount rate.
- The answer frames risk-neutral pricing as a no-arbitrage computational method.
- A risk premium added to the discount rate is not presented as a substitute for arbitrage-consistent pricing.
Tags
Full text
# risk-premium if we leave risk-neutral world
# risk-premium if we leave risk-neutral world
Risk neutral pricing in the Black-Scholes-Model makes life easy since it solves the challenge of the choice of two parameters simultanuously: the drift-parameter $\mu$ in the underlying geometric Brownian Motion (gBM) and the risk-neutral measure. Modeling the underlying asset directly with a drift which is equal to the risk-free interest rate ($\mu=r$), allows for using the natural probability measure instead of transforming to the risk-neutral measure when it comes to taking the expectation in the valuation of the product.
Reason: Since the gBM of the underlying asset with drift equal to risk-free interest rate (considered under the under the natural probatility measure) follows the same distribution, as the underlying if it would be modeled with the 'real-world-drift' (considered under the risk-neutral measure).
Hence, choosing $\mu=r$ in the BSM, we may determine the fair price of a derivative taking the expectation (under the natural probability measure) of the corresponding payoff, where this payoff is discounted with the risk-free interest rate (in contrast to alternative pricing models, e.g. CAPM, where the discount rate is risk-adjusted).
This works perfectly fine if we stay in the risk-neutral world. However, I am interested in an appropriate modification of this approach if we leave this world.
Consider the following situation: we have an underlying which is modeled by a Brownian Motion with certain drift and we aim to price the product without applying risk-neutral valuation techniques. Modeling the BM with the empirically measured drift implies that taking the expected payoff (under natural probability measure) and discounting the resulting value with the risk-free interest rate will not be appropriate to determine the price of the product since we have to take account for the risk corresponding to the underlying assets.
The questions I would like to get some opinions on are the following:
What is an appropriate way to incorporate the risk of the underlying in this situation?
Would it be possible to discount the payoff of the product under the natural probability measure using a risk-adjusted interest rate in the form $r_{risk\space free}$ $+$ $risk$ $premium$ similar to what is done in CAPM?
How should we choose the risk premium in this case? (Credit Spreads?, CAPM-approach using market prime and underlying-specific beta factors? other ideas?)
It would be great to hear some opinions. Thanks in advance!
## Answer by Arshdeep (score 1)
https://quant.stackexchange.com/a/80238
You seem to imply there is a choice between risk neutral and risk averse valuation. There isn't, risk neutral valuation is the only correct valuation if the delta hedged portfolio has to be riskless. So any method you choose to value it any other way is wrong.
This is where jargon misleads people. Risk neutral world is not an assumption, it is a computational tool to compute the price under no arbitrage.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.