# Short time to maturity behaviour of implied volatility

There are several perturbative expansions in derivatives literature on the short-time to maturity behaviour of implied volatility. When it comes to implied volatility in (local) stochastic volatility models, whose expansion is considered the "gold standard" nowadays? I.e. an expansion that is as robust / general as possible for (L)SV models.

I am interested in particular in the behaviour of $$\frac{\partial^2 \Sigma}{ \partial S \partial \sigma}$$ where $$\Sigma$$ is implied volatility, $$\sigma$$ is the instantaneous stochastic vol and $$S$$ the asset/index. Specifically I am wondering for which (L)SV models can I assume that the above cross-term is approximately zero or even exactly zero for short time to maturity. And how about for non-vanishing time to maturity?

Thanks.

• What is the instantaneous volatility for a stoch vol model? Do you get the above derivative as a function of the current instantaneous vol, spot, and time? How is that useful? – will Mar 29 '19 at 23:36
• @will The current instantaneous vol yes, which is the ATM implied volatility for time to maturity $\tau \rightarrow 0$. It is useful to know the above derivative for the problem I am working on. I am familiar with the Medvedev-Scaillet expansion to name but one. But I am just wondering if there are "better" expansions available, although "better" could be a relative concept. I.e. what would be the gold standard when it comes to IV perturbative expansion in terms of stoch vol parameters such as the current instantaneous vol, current spot/moneyness, time to maturity, vol of vol etc. – ilovevolatility Mar 30 '19 at 11:31

We need to categorise the types of models before we consider the term $$\dfrac{\partial^2 \Sigma}{\partial S \partial \sigma}$$. I will only consider local volatility models and stochastic volatility models.

Local volatility models

The local volatility function is, of course,

$$\sigma^2(K,T)=2 \dfrac{\partial_T C_{KT}}{\partial_{K}^2 C_{KT}}$$

This can be expressed as a function of the implied volatility $$\Sigma$$,

$$\sigma^2(K,T) = \dfrac{ 2 \partial_{\tau} \Sigma + \Sigma/\tau }{ K^2 \left[ \partial_K^2 \Sigma - z_1 \cdot \sqrt{\tau} \cdot \left[ \partial_K \Sigma \right]^2 + \left[ 1 / \Sigma \right] \right] \left[ 1/ \left( K \sqrt{\tau} \right) + z_1 \cdot \partial_K \Sigma \right]^2}$$

where $$\tau=T-t$$ and $$z_1 = \log(S/K)/(\Sigma \sqrt{\tau}) + (1/2) \Sigma \sqrt{\tau}$$.

As $$\tau \rightarrow 0$$, i.e., the short maturity case, we get

$$\sigma(K) = \dfrac{\Sigma}{1 + \left[ K/ \Sigma \right] \cdot \log(S/K) \cdot \dfrac{d \Sigma}{d K} }$$

To calculate the term $$\dfrac{d \Sigma}{d \sigma}$$, notice that $$\dfrac{d \Sigma}{d \sigma} = \dfrac{1}{\dfrac{d \sigma}{d \Sigma}}$$. Likewise for $$\dfrac{d \Sigma}{d S}$$ and henceforth for $$\dfrac{\partial^2 \Sigma}{\partial S \partial \sigma}$$.

You can specify a simple functional form for $$\Sigma(K)$$, i.e., something like $$\Sigma(K):=a \cdot e^{-bK+c}$$, hence allowing you to calculate the quantity you are interested in.

Stochastic volatility models

We need to be careful here because there are market models (for the stochastic volatility) and stochastic volatility models. They sound the same but they are not - a stochastic volatility model will not model the implied volatility, but will (obviously) model the stochastic volatility. A market model will model the implied volatility.

Example of stochastic volatility model: SABR model

Example of market model: Schonbucher model

There are papers (Managing Smile Risk paper for SABR model, Schonbucher's 1999 paper for Schonbucher's model) that will be (most likely) out of date - but they will give you good intuition on how to calculate the vanna term for a stochastic volatility model.

The latest papers on perturbation expansions will be proprietary and obviously not public. Both of the papers that I mentioned provide analytic expressions for the implied volatility $$\Sigma$$ as a function of all relevant parameters. Setting $$\tau=0$$ or taking the limit $$\lim_\limits{\tau \rightarrow 0} \Sigma(\tau,\ldots)$$ for the expressions given in those papers (and then calculating the vanna term) should give you the answer you are looking for.

• thanks. A parametric form as you suggest I'd rather avoid as actually I don't think it is that robust. Furthermore what you wrote above is for local vol, I am actually more interested in stochastic vol models. And I mean stochastic instantaneous vol models, not market models of implied volatility such as those by Schonbucher. As mentioned in my comment above to Will, I am familiar with the Medvedev-Scaillet expansion, I am just wondering whether there are "better" expansions available that are not proprietary, i.e. published ones. – ilovevolatility Mar 30 '19 at 11:28