Stochastic Volatility and Sticky Delta

Question

"Stochastic volatility models can be thought of as sticky delta model. And Local volatility model as sticky Strike." Please help me understand how the author has reached this conclusion.

Answer 1

Intuitively, in a (log)-space homogenous diffusion model $$ S_t \propto S_0, \forall t \geq 0 $$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot level, which is precisely the definition of sticky delta.

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Mathematically, consider a (log)-space homogeneous diffusion model (be it stochastic or not) $$ \frac{dS_t}{S_t} = \mu(\cdot) dt + \sigma(\cdot) dW_t,\,\,\,S(0) = S_0 $$ where by (log)-space homogeneous we mean that the drift and diffusion coefficients on the RHS do not involve $S_t$. As such:

• a LV model is not space homogeneous since $\sigma(\cdot) = \sigma(t,S_t)$
• a SV model à la Heston is space homogeneous since $\sigma(\cdot) = \sqrt{v_t}$ with $v_t$ given by a separate SDE.

[Homogeneity relationship] Because of (log-)space homogeneity we have that the price of a European vanilla option is a homogeneous function of degree 1 i.e. $$ C(\xi S_0, \xi K, T) = \xi C(S_0, K, T), \forall \xi > 0 $$ such that by Euler's theorem (i.e. taking the derivative of the above wrt to $\xi$ and evaluating it at $\xi = 1$) we get $$ C = \frac{\partial C}{\partial S_0} S_0 + \frac{\partial C}{\partial K} K $$

[IV stickiness (1/2)] Consider a space homogeneous diffusion model with parameters $\Theta$. The corresponding implied volatility surface is the mapping \begin{align} \Sigma &: (S_0, K, T) \to \Sigma(S_0,K,T) \\ \text{such that } & C(S_0,K,T;\Theta) = C_{BS}(S_0, K, T; \Sigma(S_0,K,T)) \end{align} where $C_{BS}(.)$ denotes the Black-Scholes pricing formula for a European call option.

Given the model's space homogeneity, we just showed that: $$ S_0 \frac{\partial C}{\partial S_0}(S_0,K,T;\Theta) + K \frac{\partial C}{\partial K}(S_0,K,T;\Theta) = C(S_0,K,T;\Theta) $$ Plugging in the above implied volatility definition then allows one to write (chain rule) $$ S_0 \left[ \frac{\partial C_{BS}}{\partial S_0}(S_0, K, T; \Sigma) + \frac{\partial C_{BS}}{\partial \Sigma} \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) \right] + K \left[ \frac{\partial C_{BS}}{\partial K}(S_0, K, T; \Sigma) + \frac{\partial C_{BS}}{\partial\Sigma} \frac{\partial \Sigma}{\partial K}(S_0,K,T) \right] = C_{BS}(S_0, K, T; \Sigma) $$

Denoting the Black-Scholes Vega by $\nu$ and noting that the Black-Scholes model is itself space homogeneous, one gets \begin{gather*} S_0 \left[ \frac{\partial C_{BS}}{\partial S_0}(S_0, K, T; \Sigma) + \nu \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) \right] + K \left[ \frac{\partial C_{BS}}{\partial K}(S_0, K, T; \Sigma) + \nu \frac{\partial \Sigma}{\partial K}(S_0,K,T) \right] = C_{BS}(S_0, K, T; \Sigma) \end{gather*} Or equivalently rearranging terms: \begin{gather*} \nu \left[ S_0 \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) + K \frac{\partial \Sigma}{\partial K}(S_0,K,T) \right] + \underbrace{S_0 \frac{\partial C_{BS}}{\partial S_0}(S_0, K, T; \Sigma) + K \frac{\partial C_{BS}}{\partial K}(S_0,K,T;\Sigma) - C_{BS}(S_0, K, T; \Sigma)}_{=0 \text{ (BS space homogenenity) }} = 0 \end{gather*}

such that the following relationship holds for all space homogeneous models \begin{equation} \frac{\partial \Sigma}{\partial S_0}(S_0,K,T) = -\frac{K}{S_0} \frac{\partial \Sigma}{\partial K}(S_0,K,T) \end{equation}
which is consistent with a sticky moneyness (= sticky delta) behaviour, see below.

[IV stickiness (2/2)] A sticky moneyness (= sticky delta) implied volatility surface is such that $$ \Sigma(S_0+\delta S_0, K, T) = \Sigma(S_0, K^*, T) $$ provided, as the name indicates, that we are working iso-moneyness, meaning that $$\frac{K^*}{S_0} = \frac{K}{S_0+\delta S_0} \iff K^* = K(1 + \delta S_0/S_0)^{-1}$$

Under such circumstances, \begin{align} \frac{\partial \Sigma}{\partial S_0}(S_0, K, T) &= \lim_{\delta S_0 \to 0} \frac{\Sigma(S_0+\delta S_0, K, T) - \Sigma(S_0, K, T)}{\delta S_0} \nonumber \\ &= \lim_{\delta S_0 \to 0} \frac{\Sigma\left(S_0, K(1 + \delta S_0/S_0)^{-1}, T\right) - \Sigma(S_0, K, T)}{\delta S_0} \nonumber \\ &= \lim_{\delta S_0 \to 0} \frac{\Sigma\left(S_0, K(1 - \delta S_0/S_0), T\right) - \Sigma(S_0, K, T)}{\delta S_0} \nonumber \\ &= \lim_{\delta K \to 0} \frac{\Sigma\left(S_0, K-\delta K, T\right) - \Sigma(S_0, K, T)}{\frac{S_0}{K}\delta K} \nonumber\\ &= -\frac{K}{S_0} \frac{\partial \Sigma}{\partial K}(S_0, K, T) \end{align} which is the relationship stemming from price homogeneity we found above.