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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 illiquid asset$Y_t$ cannot be 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.

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.

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.

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.

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.

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 illiquid asset$Y_t$ cannot be 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.

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.

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'.

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). So I'm curious, if you decide to use this, what results you obtain.

Hope this helps.

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.