Why Stochastic Volatility Models Are Incomplete Without Volatility Hedges
Summary
The document explains why common stochastic volatility models, including Hull–White, Stein–Stein, and Heston, generally describe incomplete markets when the underlying asset is the only tradable risk asset. Volatility adds a source of uncertainty that cannot necessarily be hedged by trading the underlying alone. In that setting, the market price of volatility risk is not fixed by replication.
The explanation also uses the equivalent martingale measure perspective: if the volatility process can be assigned different risk-neutral laws while the discounted asset remains a martingale, the pricing measure is not unique, indicating incompleteness. The discussion adds that another tradable asset exposed to the volatility risk, such as a suitable option, can provide an additional hedge and help identify the volatility risk premium. Completeness depends on the assets available for trading and their exposure to the model's risk sources; the document presents a general principle rather than model-specific proofs.
Key ideas
- Stochastic volatility introduces risk that may not be spanned by trading the underlying asset alone.
- A non-unique equivalent martingale measure is a criterion for market incompleteness.
- Different risk-neutral laws for volatility can preserve the discounted underlying's martingale property.
- An additional tradable asset exposed to volatility risk can help complete the market and identify its risk premium.
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# Stochastic Volatility Models - are they complete markets? # 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. ## Answer by starovoitovs (score 3) https://quant.stackexchange.com/a/54640 Any stochastic volatility model will be incomplete. 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. ## Answer by river_rat (score 2) https://quant.stackexchange.com/a/54639 You generally need one tradable asset per source of risk in your model that is someway dependent on that noise. So in a world where you can trade a single stock, but have two sources of variance your model would be incomplete as there would be no way to fix the market price of volatility risk. However, if some other asset was tradable (say some reference option) it would complete the model and let you calibrate the market price of risk.
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