The Affine Structure and Sign Issue in the Schöbel–Zhu Model

Summary

The document examines why the Schöbel–Zhu stochastic volatility model can be treated as affine when deriving the stock price characteristic function. It describes augmenting the state with squared volatility and applies Itô’s lemma to obtain the dynamics of that additional state variable. The resulting drift is affine in volatility and squared volatility, which appears to support the affine representation.

The central caveat is that rewriting the diffusion term using the square root of squared volatility yields an absolute value. Since the volatility process in this model can become negative, that expression may not match the original diffusion. The document presents this as an unresolved mathematical question and supplies no answer or proof of why the published characteristic function derivation remains valid.

Key ideas

• The Schöbel–Zhu model uses a mean-reverting stochastic volatility process that can take negative values.
• Applying Itô’s lemma gives dynamics for squared volatility that involve both volatility and its square.
• The diffusion term for squared volatility depends on the absolute value of volatility when expressed through its square.
• The document questions whether this transformation justifies the affine characteristic function derivation.

Tags

# Why is the Schöbel-Zhu model affine?


# Why is the Schöbel-Zhu model affine?












In the Schöbel-Zhu model, the stochastic volatility process is $dv_t=\kappa(\theta-v_t)dt+\sigma dW_t$.

The characteristic function of the stock process can be found by arguing that the model is affine if we expand the set of variables to include $v^2_t$.

The argument is that applying Ito's lemma we find $dv^2_t=[2\kappa(\theta v_t - v^2_t)+\sigma^2]dt+2\sigma v_tdW_t$, which is rewritten as $dv^2_t=[2\kappa(\theta v_t - v^2_t)+\sigma^2]dt+2\sigma \sqrt{v^2_t}dW_t$.

The second equation for $dv^2_t$ is indeed affine in $v_t$ and $v^2_t$. However it seems to me that it is really equal to $dv^2_t=[2\kappa(\theta v_t - v^2_t)+\sigma^2]dt+2\sigma |v_t|dW_t$, which is not equal to the first equation since $v_t$ can become negative.

Therefore, why is the formula derived by Schöbel and Zhu correct?

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