Correlation and implied volatility

Question

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 ?

Answer 1

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.

Answer 2

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.

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Original answer:

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.