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blivy
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Best Black-Scholes alternative

September 16th, 2022, 1:26 pm

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.
 
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Alan
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Re: Best Black-Scholes alternative

September 16th, 2022, 7:00 pm

The simplest improvement might be a "mixing model". Option values become a weighted sum over Black-Scholes (BS) values, where the weights reflect a probability distribution for the total variance over the life of the option contract.

See Ch. 4 in "Option Valuation under Stochastic Volatility", my year 2000 book, for formulas and derivations.

You may actually be "undervaluing" the equity. While broad-based index options, like the SPX, have a pronounced downward-sloping skew pattern, single-name equities have more "U-shaped" smiles. A basic (zero-correlation) mixing model will account/lead to that.

The bottom line is that you will need to develop volatility outlook distributions. This could be done with simple block-sampling of historical returns of your comparable companies. (A more sophisticated approach is called GARCH modelling). Once you have a probability density of future (total) variance estimates, you just do an integral with the BS formula. The math is not too hard.    

You can perhaps check your work by taking the same approach with one of your traded comparable companies. How far are you from the market stock price?  Of course, maybe +/- 50% would be 'good'.  :D
 
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Alan
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Re: Best Black-Scholes alternative

September 16th, 2022, 7:35 pm

p.s. I will add that U-shaped smiles for vanilla options on single-name equities does not imply U-shaped smiles for "equity as some kind of option" which is apparently your application. I don't really have a feeling for smile shapes for the latter. Nevertheless, a basic mixing model is still a good place to start IMO.
 
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Marsden
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Re: Best Black-Scholes alternative

September 16th, 2022, 11:57 pm

Why do you say the equity has option-like features, and why is it akin to deeply out of the money European options?
 
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bearish
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Re: Best Black-Scholes alternative

September 17th, 2022, 2:15 am

Why do you say the equity has option-like features, and why is it akin to deeply out of the money European options?
Merton (‘74) points out that owning equity in a firm with a single (zero coupon) bond outstanding is essentially like having a European call option on the assets of the firm, where the face value of the bond plays the role of the strike. Many refinements and approximations have since been proffered, not the least by KMV, who successfully sold their implementation of this idea (along with a lot of data scrubbing) for a nine figure sum to Moody’s a couple of decades ago.
 
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DavidJN
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Re: Best Black-Scholes alternative

September 17th, 2022, 3:30 pm

And using put-call parity, one can derive a similarly interesting characterization of risky debt in terms of equity, assets, and options. 
 
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bearish
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Re: Best Black-Scholes alternative

September 18th, 2022, 1:53 am

And then add in some vaguely specified notion of debt covenants and we can introduce barrier options into the game (Black & Cox, 1976). Much later, Duffie & Lando (2001 - although Econometrica glacial editorial process…) solved the problem of near term debt being risk free by adding imperfect information.
 
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Gamal
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Re: Best Black-Scholes alternative

September 20th, 2022, 8:18 am

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.
If I understand you correctly, your problem is not the valuation of derivative securities with relatively liquid underlyings but marking completely illiquid instruments with no clue how to price them. BS formula with historic volatility borrowed from something similar is quite good for that purpose as long you don't change parameters too often.