How Stochastic Interest Rates Affect Black–Scholes Price Uniqueness

Summary

The document explains how introducing stochastic interest rates changes the uniqueness of a Black–Scholes call price. The key issue is whether the interest rate risk can be traded and hedged. If no claims linked to the stochastic rate are available, the market is incomplete, so more than one risk-neutral measure may be consistent with observed prices and the option need not have a unique price.

If claims on the rate are traded, the answer can be different: the market may be complete, allowing all relevant risks to be hedged and yielding a unique risk-neutral measure and price. A zero-coupon bond is given as an example of a claim on interest rates. The response cautions that complete markets are more of a theoretical ideal; practical models often simplify by reducing the number of modeled risk drivers or ignoring some sources of risk, which can mask incompleteness.

Key ideas

• Stochastic rates can make an option market incomplete when rate-linked claims are not traded.
• In an incomplete market, multiple risk-neutral measures may produce different prices.
• Trading claims on stochastic rates can permit hedging and support a unique price.
• Zero-coupon bonds are examples of claims linked to interest rates.
• Practical models may assume away risk factors and thereby understate market incompleteness.

Tags

# Does Black Scholes + Stochastic interest rates result in a unique price


# Does Black Scholes + Stochastic interest rates result in a unique price












Black-Scholes with its assumptions results in a unique price for the call. If we introduce stochastic interest rates, would this still remain the case?

## Answer by user34971 (score 2, accepted)

https://quant.stackexchange.com/a/54656

No, if there are no claims traded on the stochastic interest rate because then the market is incomplete.

Yes, if claims on stoch ir are traded because then the market is complete (all risks can be hedged) and there is a unique risk-neutral measure. Recall that in fact a zero coupon is a claim.

The latter ("yes"), in theory, is by far the more common situation. But in practice there is always some form of market incompleteness (we just make it complete in by reducing the number of variables in our models and/or ignoring other risk drivers in our models).

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