# How to calculate the prices of option instruments for a new underlying

Can someone with practical experience with implementing and verifying please point me in the right direction. Let's say I have 3 months of data for an underlying. I want to generate theoretical option prices of all the options strikes and expirations. How do I generate the volatility smile across the full set of option strikes and expirations? Let's say I have the historical volatility. Can I use this as the starting poing to generate the volatility surface? Is there any literature on how to do this?

• you need option prices (not underlying) to imply volatilizes. – roz Feb 11 '20 at 15:29

This is a very common real-world task in quantitative finance, because new underlyings or options series pop up frequently.

Single Volatility

At its simplest, a volatility surface can be represented by a single constant parameter, $$\sigma_C$$, which to first approximation can be taken equal to the historical return volatility of your hypothetical underlying $$H$$. With a mere 3 months of daily data, and assuming options near the money, I would put a "safety factor" on $$\sigma_C$$ (positive or negative depending on whether you will be long or short optionality) and stop there. Frankly, even a Bachelier volatility on $$H$$ (rather than returns on $$H$$) would be fine.

Volatility Mapping

Your next best method is to "map" a surface from some economically similar underlying $$U$$ for which options exist. Let's say you get a volatility surface $$\sigma_U(K/U_0, T)$$ for $$U$$. note that I have expressed it in terms of relative strike $$K/U_0$$, the proportion of strike K to current underlying value $$U_0$$. Let's also choose some canonical volatility representative of $$U$$, say

$$\sigma_A := \sigma_U(1, 3/12)$$

From this, your mapping comes from mapping strikes to the same proportions, multiplying the volatility curve by a correcting constant, and adjusting the time parameter

$$\sigma(K/H_0, T) = \sigma_C\frac{\sigma_U(K/H_0, T\frac{\sigma_A^2}{\sigma_C^2})}{\sigma_U(1, T\frac{\sigma_A^2}{\sigma_C^2})}$$

Stochastic Volatility

If you want to get a fancier surface, you will have to choose some kind of favored stochastic model, other than plain Black-Scholes, for the underlying. Any fit of such a model gives you option prices for your choice of tenors and strikes, and those option prices in turn define a volatility surface.

Given that you lack option prices, you would fit by maximum likelihood techniques, as described in Maximum likelihood estimation of stochastic volatility models by Ait-Sahalia and Kimmel. This sort of fit involves forming a power series approximation for each day's data, which in their paper comes from equation 9,

$$l_X^{(J)}(\Delta,x|x_0;\theta) = -\frac{m}2 \log(2\pi \Delta) - D_\nu(x;\theta) + \frac{C^{-1}_X(x|x_0;\theta)}{\Delta} + \sum_{k=0}^J C^{(k)}_X(x|x_0;\theta)\frac{\Delta^k}{k!}$$

and then employing a multidimensional optimization algorithm, such as BFGS, to determine the maximum likelihood coefficients.

Using maximum likelihood estimation of stochastic volatility has various practical and theoretical problems, including

• This technique locks you to a picture of stochastic behavior that is probably not realistic, say by ignoring jumps
• Past behavior does not make a good representation of the future
• You need a lot of data, not just a few dozen points, to have a hope of getting low estimation error
• The varvol or "variability of volatility" parameters are latent and always come with large error bars
• This doesn't appear to answer the question ('how to calculate a theoretical option chain') – Chris Feb 11 '20 at 19:37
• Actually, it does answer my question. Thanks to Brian B for taking the time to answer. Brian: Let's say I am using method A in your answer (Single Volatility). From there, is there any literature that shows how to build the vol surface across strikes and expirations? The center would be the ATM strike. I am trying to understand how to establish a realistic volatility surface. – JohnGa Feb 12 '20 at 4:01
• Form that starting point, you will not be estimating stochastic vol parameters so let's ignore part 3. Start by using $\sigma_C$ as your ATM vol. Find some other similar underlying with observable option prices. Convert the option prices of it to implied volatilities, and then copy over the surface shape. – Brian B Feb 12 '20 at 15:45