No-Arbitrage Principle and Risk-Neutral Valuation › Limitations and Practical Considerations

The no-arbitrage framework provides powerful pricing tools, but you must understand its limitations. Real markets differ from the idealized conditions assumed in our derivations.

While no-arbitrage restricts prices significantly, it does not uniquely determine them for most derivatives. The risk-neutral approach requires specifying how underlying assets evolve under the risk-neutral measure. Different model choices, such as geometric Brownian motion versus jump-diffusion versus stochastic volatility, lead to different prices for the same derivative.

For simple European options on liquid underlyings, this model dependence is manageable because we calibrate to market prices. If we use the implied volatility from the market, we will match the market price by construction. For exotic derivatives or illiquid underlyings, model risk becomes a significant concern. Two equally plausible models might produce substantially different prices, and there's no way to know which is "correct" until the derivative expires.

We address model risk through several strategies: using multiple models and examining the range of prices, stress testing against alternative specifications, and adding model uncertainty premiums to quoted prices. The more exotic the derivative, the wider the uncertainty band around its "fair" price.

The no-arbitrage framework assumes idealized conditions:

• No transaction costs
• Unlimited borrowing and lending at the risk-free rate
• Perfect divisibility of assets
• No restrictions on short selling
• Continuous trading

Real markets violate all these assumptions. Transaction costs create bid-ask spreads that allow a range of prices to coexist without arbitrage. Funding costs differ from the risk-free rate and vary across counterparties. Short-selling restrictions can prevent the execution of arbitrage strategies. Assets cannot be traded in infinitesimal quantities.

These frictions don't invalidate the framework, but they do create no-arbitrage bounds, rather than unique prices. A derivative might have any price within these bounds without triggering arbitrage. The bounds are typically tight for liquid instruments but can be wide for exotic or illiquid products.

The analysis implicitly assumes all parties fulfill their obligations. In practice, counterparty credit risk affects derivative pricing. An option purchased from a risky counterparty is worth less than one from a risk-free entity because there's a chance the counterparty defaults before paying the option's value.

Modern derivative valuation incorporates credit valuation adjustments (CVA) and other XVA adjustments that modify the idealized risk-neutral price to account for credit, funding, and capital costs. These adjustments have become increasingly important since the 2008 financial crisis highlighted the significance of counterparty risk.