# Correlation and implied volatility

Say I want to write call options on a stock, with no options written already on it. I know some asset which is highly correlated to it. How can I proceed to make use of the correlation between this asset and the first one, to recreate some implied volatility surface on this unknown option ?

This is a problem commonly faced by investment banks and buy-side firms (such as hedge funds) that deal in lots of derivatives.

There isn't much more one can do than employ a few rules of thumb, and those rules have not changed much over the decades. In this case, those tricks look something like the following:

First, let's assume you have your stock $$S$$ with no observable option prices in the market. Furthermore, you have found the asset $$A$$ which is "most similar" to $$S$$. (The rules of thumb blow can of course be generalized to set of similar assets $$\{A_1,\dots\}$$).

• Decide on a linear relationship you want to assume between the two, for example $$\sigma_S = s(\sigma_A) = c + b \sigma_A$$ where usually you force either $$b==0$$ or $$c==0$$.

• If you at least have a price history for $$S$$, then get a historical volatility $$h_S$$, as well as its analog $$h_A$$ for $$A$$. Normally you do this a 1x to 5x the tenor of the options you ultimately want to price. Find $$b$$ or $$c$$.

• If you have no price history for $$S$$, then just assume something reasonable for $$b$$ or $$c$$.

• Map at-the-money volatility $$\sigma^{(\mathrm{ATM})}$$, obtaining $$\sigma^{(\mathrm{ATM})}_S$$ from $$\sigma^{(\mathrm{ATM})}_A$$ as $$\sigma^{(\mathrm{ATM})}_S = c + b \sigma^{(\mathrm{ATM})}_A$$

• Standardize the implied volatility skew for $$A$$. First, instead of marking volatilities in $$(K, T)$$, space, mark them in moneyness terms $$(M, T)$$ where $$F$$ is the forward price and moneyness is $$M=\frac{\log(K/F)}{\sigma^{(\mathrm{ATM})}\sqrt{T}}$$

• Now, for any strike $$K_S$$ on an $$S$$-option, we have the $$S$$ ATM volatility $$\sigma^{(\mathrm{ATM})}_S$$, so we can obtain its moneyness $$m=m(K)$$.

• Take that moneyness, and find the $$A$$-option volatility for the same moneyness (usually by interpolation) $$\sigma_A(m)$$.

• Set the $$S$$-option volatility to be $$\sigma_S(K) = s(\sigma_A(m))$$

You can repeat this for all strikes and tenors, obtaining a decent guess at a full volatility surface for $$S$$.

If you are quoting options, put a nice big spread on your derived volatilities $$\sigma_S$$ before giving any quotes to the counterparty.

If you later want to get some correlation coefficient $$\rho$$ between $$S$$ and $$A$$, and then price $$S$$ options by decomposing $$S$$ into $$A+G$$, where $$G$$ is idiosyncratic, you can do that, but you will need to have the volatility surface for $$S$$ first anyway.

• Wow thanks a lot, the goal in the end is indeed to quote those 'newborns' options. I will try the method you just exposed and will come back to you If I have questions again. Commented Nov 15, 2021 at 15:45
• Does this method enable to have a reliable skew ? I obtain some realistic values for ATM prices using It but it drifts for ITM options. Do you know a method similar to obtain a better guess of the skew ? Commented Nov 18, 2021 at 15:04

EDIT:

Apologies, one more edit, but an important one:

Note, as kindly pointed out to me by an interested reader a short time ago: there is a potential issue with the simple model I proposed. Namely, as it stands the model implies that the illiquid asset$$Y_t$$ is not a martingale. But all is not lost; the model could potentially still be used if the illiquid asset is not tradable (in which case it doesn't have to be a martingale), for example the VIX Index.

I've been thinking about this question for some time. In addition to @Brian B's answer, giving here another route to constructing the skew for asset $$Y$$ given the skew for another asset $$X$$, where $$X_t$$ is a positive price process.

I'll state the assumptions first:

1. $$d\ln (Y_t/Y_0) = \beta d\ln (X_t/X_0) + d\ln (Z_t/Z_0)$$, and $$d \ln X_t\, d\ln Z_t = 0$$
2. $$\beta$$ is constant (maybe can be extended to it being deterministic) and can be regarded as the regression coefficient of logreturns
3. $$Z_t$$ is also a positive process that drives the error term $$d\ln Z_t$$ of the regression and has a known distribution $$q(z)$$

From assumptions (1) and (2) it follows that $$\frac{Y_T}{Y_t} = \left(\frac{X_T}{X_t}\right)^\beta \frac{Z_T}{Z_t}$$

The price of a vanilla option on $$Y$$ is then $$E_t \left[ \left(Y_T - K\right)_+ \right] = E_t \left[ \left(\frac{Y_t}{X_t^\beta Z_t}X_T^\beta Z_T - K\right)_+ \right]$$ Since by assumption (1) $$Z$$ is independent of $$X$$, and by assumption (3) the distribution of $$Z$$ is known, we can write $$E_t \left[ \left(\frac{Y_t}{X_t^\beta Z_t}X_T^\beta Z_T - K\right)_+ \right] = \int_0^\infty E_t \left[ \left(\frac{Y_t}{X_t^\beta Z_t}z X_T^\beta - K\right)_+ \right] q(z) dz$$ The expectation in the integrand is a claim on $$X^\beta_T$$ and can be synthesised using plain vanilla options on $$X_T$$ by making use of the Carr and Madan formula. Hence, since $$q(z)$$ is assumed to be known (for example the ubiquitous lognormal distribution), you can calculate options on $$Y_T$$ and infer the corresponding implied volatilities.

Remarks:

1. Typically $$\beta$$ and $$q(z)$$ are inferred from historical data since the regression and error is based on historical data. For pricing purposesyou'd therefore have to make an educated guess about the risk-neutral values.

2. Although $$\beta$$ was assumed to be constant, you could still use this in an 'uncertain beta' framework. For instance, suppose you are comfortable with $$\beta \in [\beta_1,\beta_2]$$. Then calculate the skew of $$Y$$ for both $$\beta_1$$ and $$\beta_2$$ and based on that decide what works best for your risk appetite.

3. Other than the assumptions, no approximations are used, i.e. the computation of the skew of $$Y$$ is exact' (whatever that means in practice).

To the best of my knowledge the approach outlined above has not been treated in derivatives pricing papers about this topic (but happy to be corrected here if someone has come across it), even though it is actually similar to how one would go about pricing a geometric basket. So I'm curious, if you decide to use this, what results you obtain.

Hope this helps.

• I've worked on related application, basket correlation skew. In your assumption 1, Y would be one of the stocks in the basket and X the market asset. So we express each stock as beta x mkt + idiosyncratic term Z. If the distributions of X and Z have closed form characteristic functions (eg merton jump or heston), then you can calculate options on the geometric basket of Y_i using Fourier pricing methods. This can develop realistic basket correl skew in an intuitive manner. Think you have a nice idea here too Frido. Commented Nov 21, 2021 at 10:11
• @JamesSpencer-Lavan thanks for the encouragement. Happy to work on this / exchange thoughts with you, I suppose relative value is the applicaton you have in mind. Knowing the distribution of $X$ helps but not strictly necessary given Carr-Madan can be applied. Yes, I am starting to think this simple' framework might work.
– user34971
Commented Nov 21, 2021 at 10:31
• If one takes, say, $\beta=2$ and $Z$ to be a constant, then this turns into the formula for a "power quanto option" with power 2, since the payoff is now basically $(c \cdot X^2 - K)^+$. I would be a bit nervous about imputing skews from that. Commented Nov 21, 2021 at 15:48
• @BrianB Fair point - I have no answer to that at this point. The level of the Y volatility will certainly be higher then (which makes sense if the beta is 2) and I think that's OK, but I need to think about the impact of this on the skew.
– user34971
Commented Nov 21, 2021 at 16:13
• @BrianB See my recent edit. Not sure if this was implicitly what you were referring to as well.
– user34971
Commented Nov 21, 2021 at 20:55