# Simple Black-Scholes alternatives

I work at an accountancy firm and we use Black-Scholes to value equity in private companies that has option like features. The equity we typically value is akin to deeply out of the money European call options and we source volatility using historical share price volatilities of quoted comparable volatilities.

It's a very rudimentary approach which I'm hoping to improve given all the problems associated with Black-Scholes (my primary concern being the volatility skew). I've done a bit of research on local vol and stochastic vol models (a lot of which went over my head) and I'm not sure which would work best given this fact pattern. I understand you need to calibrate these models to market data, which we obviously do not have for private companies. Unless it would be reasonable to calibrate the model to comparable companies (though many of these do not have traded options), or are there some 'general' parameters which could be used? It would also need to be implemented in Excel.

Any ideas for what would work best given this fact pattern? Ideally something which would address the volatility skew given the equity is deeply out of the money and therefore using constant volatility is overvaluing the equity. The simpler the better really. Bonus if there's an template excel model I could download!

Note I have no quant experience, but do have a math degree from many years ago, so I'm somewhat mathematically literate. Thanks in advance.

• Sorry, so what is your question exactly? To value an OTM European (or American?) call option on an unlisted company?
– user34971
Commented Sep 16, 2022 at 15:34
• To value equity in an unlisted company which has similar features to an OTM European call option. Currently I just use Black-Scholes with a volatility input sourced from the historical share price volatilities of listed comparable companies. But is there something which can easily be implemented which (as a minimum) takes into account volatility skew. If I use a constant volatility I fear that I will be overvaluing deeply OTM equity / undervaluing deeply ITM equity. Commented Sep 16, 2022 at 17:00
• In that case, if you have a similar company which is listed and with has an options market, then as a first rough approximation copy the skew of the listed similar firm, but adjust the overall level to your own preference.
– user34971
Commented Sep 16, 2022 at 19:18
• Thanks, which model should I use? Commented Sep 17, 2022 at 17:12

Too long for a comment.

My impression is that the only information you have is that the unlisted company $$X$$ is similar to the listed company $$Y$$. Your task/aim is to value an OTM European call option on $$X$$ with strike $$K$$ and maturity date $$T$$.

1. Single name options (jargon for options on companies), in contrast to index options (eg options on S&P500), are usually American options.
2. The price of an American option is greater than or equal to the price of an European option

Correct me if I am wrong, but although you probably have the necessary maths background, you're quite new to quant finance / derivatives.So my suggestion is forget about models and all that, and simply set the value of an European call on firm $$X$$ with strike $$K$$ and maturity date $$T$$ equal to the price of an American option on $$Y$$ with the same strike and maturity date (the prices of these American options on $$Y$$ should be observable in the market). The price difference between an American and an European option on $$Y$$ is good for you as it gives you an extra `pad'.

As options on $$Y$$ are priced with skew, so will your options on $$X$$.

HTH.

Edit: Note that there is something called the Merton model in which a company's equity is regarded as an option on the firm's assets, where the strike of the option is its outstanding debt. To value the firm's equity you would then need to calculate the volatility of its assets. An option on the firm's equity is then an option on an option. This is probably a good / fundamental approach, but even more likely also a very painful approach. Just mentioning this alternative as a fyi.

• Just adding that neither local vol nor stochastic vol would add any benefit here in any case. Without an option market, you cannot calibrate. Even if you could, you would not price a vanilla option any better than Black Scholes. Commented Sep 17, 2022 at 20:28
• Thanks, that’s really helpful and I’ve been trying to read up a bit more, as you’re right I’m new to quant finance. I am somewhat familiar with the Merton approach and indeed routinely value equity as a call option on the enterprise value using Black Scholes, which involves estimating an asset volatility from historical movements in equity prices of comparable companies. But I’m still mindful that surely there has to be a better approach than Black Scholes if I can say that Company X is identical to Company Y? Commented Sep 29, 2022 at 17:44
• Important bit of context, it’s beneficial if the value is lower than Black Scholes would give (otherwise I’d probably just stick with Black-Scholes), hence the idea behind a model which takes into account the volatility skew. Commented Sep 29, 2022 at 17:58