# Stochastic Volatility Models - are they complete markets?

I'm reading about stochastic volatility models - the ones which resulted after Wiggins proposed in 1986/7 that $$\sigma$$ in Black-Scholes should be a stochastic process rather than a constant.

In particular, I am looking at 3 models:

• Hull & White
• Stein & Stein
• Heston

The book says, after introducing the last one (Heston), that it is an incomplete market because the model has 2 Brownian motions but we only have one risk asset for replication. What I don't understand is, whether this statement about the market completeness relates to just the last model or all 3. To me, they all seem to be incomplete markets.

• You are right, they are all incomplete. – user39119 Jun 3 '20 at 21:44
• They are incomplete until they are complete(d). – Frido Rolloos Jun 4 '20 at 9:09

Asset price under no arbitrage satisfies an SDE $$dS_t = r(t, S_t) dt + \sigma(t, S_t) dW_t$$, where $$r(t, S_t)$$ and $$\sigma(t, S_t)$$ can be stochastic. Second Fundamental Theorem of Asset Pricing states that a model is incomplete if and only if the associated equivalent martingale measure is not unique. In stochastic volatility model we can change the law of the volatility process $$\sigma_t$$ (which changes the law of $$S_t$$) without affecting the martingality of the discounted asset price $$\exp\left(-\int _0 ^t r_s ds\right) S_t$$. This yields another martingale measure and thus incompleteness.