# How much does a rise in volatility in a short-term option affect a longer-term option

How would a rise in implied volatility on a short-term option affect the implied volatility of another short-term option with the same strike, but with slightly-longer expiry?

Assuming that the short-term volatility rise is due to an event completely contained in the time of the short-term option.

Here is my stab at this:

Let $$s$$ and $$\ell$$ be the DTE of the short-term and long-term options.

Let $$V_s$$ be the short term option implied volatility now, and $$V_\ell$$ is the longer-term option volatility.

Since variances are additive, we should be able to calculate $$V_{\ell-s}$$, the implied volatility between the expiry of the short-term option to the expiry of the longer-term option right? $$V_{\ell-s}^2 = V_{\ell}^2 - \frac{s}{\ell}V_s^2$$

Then if $$V_s$$ rises to $$V_{s*}$$, then $$V_{\ell}$$ should rise to $$V_{\ell*}$$ where

$$(V_{\ell*})^2 = \frac{s}{\ell}V_{s*}^2 + V_{\ell-s}^2$$

Giving us:

$$(V_{\ell*})^2 = \frac{s}{\ell}\left(V_{s*}^2 - V_s^2\right)+ V_{\ell}^2$$

From a more practical standpoint, I'm curious how this translates to the value of options? I've been reading about how calendar spreads are not always long-vega strategies?

• Without doing any maths, the implied vol for the slightly longer-dated option will cover the period of the shorter-dated option. If the shorter-dated option starts experiencing an increase in implied vol, the market is pricing in some stressed event (could be dividends if we're talking a stock option, or a central bank rate decision if we're talking an FX or Rates option). This stress event will affect the underlying in a way that will also affect the value of the longer-dated option. So you can definitely expect the IV to also increase for the longer-dated option. – Jan Stuller Jan 11 at 10:30
• @JanStuller Obvious. But the question is how much? – Winston Du Jan 12 at 16:57

## 1 Answer

Short answer: I believe the quantification of the impact of an increase in the shorter-dated vol on the longer-dated vol depends on what type of vol function we select to model the Volatility term-structure of the options.

Long answer (focusing on piece-wise volatility term-structure function):

Variances of log-returns in the GBM model are proportional to the time increment because of the scaling property of Brownian motion (i.e. an assumption inherent in the model):

$$ln\left(\frac{ S_{t_{i+1}}}{S_{t_{i}}}\right)=(\mu+0.5\sigma^2)(t_{i+1}-t_i)+\sigma W(t_{i+1}-t_i)\overset{d}{=}(\mu+0.5\sigma^2)(t_{i+1}-t_i)+\sigma \sqrt{t_{i+1}-t_i}Z$$

In the GBM model, each option with expiry $$T_i$$ would have its own BS volatility $$\sigma_{T_i}$$. To compute how an increase in a shorter-dated $$\sigma_{T_i}$$ affects a change in some longer-dated $$\sigma_{T_i}$$, we could map the BS vols onto some time-dependent vol function $$\tilde{\sigma}(t)$$:

$$\sigma_{T_i}T_i=\int_{h=0}^{h=T_i}\tilde{\sigma}(h)dh$$

If we assume piece-wise constant $$\tilde{\sigma}(t)$$, we will get:

$$\sigma_{T_i}T_i=\sum_{j=1}^{i}\tilde{\sigma}_j(T_{j}-T_{j-1})$$

We then get:

• $$\sigma_{T_1}^2T_1=\tilde{\sigma}^2_1T_1$$
• $$\sigma^2_{T_2}T_2-\tilde{\sigma}^2_1T_1=\tilde{\sigma}^2_2(T_2-T_1)$$
• $$\sigma^2_{T_n}T_n-\sum_{j=1}^{j=n-1}\tilde{\sigma}_j^2(T_j-T_{j-1})=\tilde{\sigma}^2_{T_n}(T_n-T_{n-1})$$

Focusing on $$T_2$$ and $$T_1$$ for simplicity, supposing $$\sigma_{T_1}$$ goes up (so that $$\tilde{\sigma}_1$$ also goes up), but the forward piece-wise volatility $$\tilde{\sigma}_2$$ between $$T_2$$ and $$T_1$$ doesn't change, the new value of the BS volatility (that I denote $$\sigma_{T_2}^*$$) for the option expiring at $$T_2$$ would be:

$$(\sigma_{T_2}^*)^2T_2=\left(\tilde{\sigma}_{1}+\delta_{\tilde{\sigma}_1}\right)^2 T_1+\tilde{\sigma}_2^2(T_2-T_1)$$

The difference between $$\sigma_{T_2}^*$$ and the old value $$\sigma_{T_2}$$ is then simply:

$$\delta_{\sigma_{T_2}}=\sqrt{(\sigma_{T_2}^*)^2-\sigma_{T_2}^2}=\sqrt{\frac{T_1}{T_2}(2\tilde{\sigma}_1\delta_{\tilde{\sigma}_1}+\delta_{\tilde{\sigma}_1}^2)}$$

To compute the impact on the value of the option expiring at $$T_2$$, we could plug the quantity $$\delta_{\sigma_{T_2}}$$ into the option Vega.

If we assume different than piece-wise constant volatility function, the quantity $$\delta_{\sigma_{T_2}}$$ would be different.

• Can you clarify all of the variables you're using? The notation is confusing. For example, what is $\tilde{\sigma}_1$ supposed to be? I thought you defined it as a function. – Winston Du Jan 15 at 8:13
• @WinstonDu: $\tilde{\sigma}_1$ is the piece-wise constant volatility between $t_0$ and $T_1$. $\tilde{\sigma}_2$ is the piece-wise constant volatility between $T_1$ and $T_2$, and so on. Together, across all indexes $j$, the function $$\sum_{j=1}^{i}\tilde{\sigma}_j(T_{j}-T_{j-1})$$ is the piece-wise volatility function $\tilde{\sigma}(t)$. I could have probably introduced a new variable for the summation, but I didn't want to have too many variable. Hope that's a bit clearer. – Jan Stuller Jan 15 at 8:19