## Why Black-Scholes is Better Than We Think

Thu, Dec 6, 2012 | Jared Woodard

Every options trader knows about or at least of the Black-Scholes-Merton (BSM) pricing model. Because it is the oldest formalized pricing model and only the first of many, some traders regard it as outdated and inferior. Perhaps it is a victim of the familiarity that breeds contempt. But a recent paper gives some reasons why traders should give BSM a second look.

Delta hedging is an essential component of any volatility trading strategy. When straddle buyers also buy and sell the underlying asset to “scalp” the gamma in the trade, they are hedging the deltas created by that gamma to prevent the position from becoming a simple directional bet.

Any pure volatility trade will require hedging to eliminate directional price exposure, but even traders who want a mixture of volatility and price exposure will need, at times, to alter the delta bias of positions. To hedge the delta exposure of an option position, it is necessary to have a confident estimate of that exposure. But the delta of a given option is not dictated from on high or declared by fiat. It depends upon some pricing model. Now, although many traders do not make active use of multiple complex pricing models when finding and structuring trades, that does not mean understanding the relative merits of different models is not of practical importance.

Carol Alexander, Andreas Kaeck and Leonardo Nogueira note in a 2009 *Journal of Futures Markets* article that the existing literature is unclear about whether BSM deltas are more cost-effective for hedging than deltas produced by either of the two contemporary types of models for pricing options, local or stochastic volatility models:

[O]nce stochastic volatility is modeled [e.g., Heston 1993], the inclusion of jumps leads to no discernible improvement in hedging performance. It is conjectured that this is because the likelihood of a jump during the hedging period is too small, at least when the hedge is rebalanced frequently … .Dumas et. al (1998) test several para- metric and semi-parametric forms of the local volatility function, and conclude that BSM deltas appear to be more reliable than any of the local volatility deltas that they tested.

This is already an interesting result, because ambiguity about the effectiveness of contemporary models (relative to BSM) should lead us to reduce whatever bias we might have had in their favor.

To gain some clarity about how well different models actually hedge vanilla option deltas, the authors review the effectiveness of six pricing models: BSM, BSM adjusted to allow for stochastic volatility, Heston, SABR, a lognormal mixture diffusion model and the lognormal model with four stochastic parameters. The adjusted BSM model was configured to allow for the fact that equity implied volatility tends to correlate negatively with asset prices. Both delta-hedging and delta-gamma- hedging strategies were tested, but I will only review results for the former.

One S&P 500 option at each strike between 0.8 and 1.2 moneyness was sold, and the net portfolio delta was re-hedged daily. The primary purpose of delta hedging is, again, to remove exposure to price fluctuations in the underlying asset. All of the tested models will accomplish this in a general sense. So to compare them, the authors emphasize two other metrics, the standard deviation of daily profit or loss.

*Fig. 1. Pricing model P/L standard deviation and R ^{2}. Source: Alexander et al. 2009*

The thought here is that it is intuitively better to have a smoother time series of returns, and as low a correlation as possible (R^{2}) between the portfolio P/L and the underlying asset returns. Table 1 shows these results. The results are surprising. The adjusted BSM model outperforms every other model on the R^{2}criterion and has the lowest standard deviation of returns except for the SABR model. The paper also compares model hedging error by moneyness and across several maturity buckets. BSM-adjusted performed the best at low moneyness, with SABR providing lower errors for at- and out-of-the-money options. Similar results were obtained for hedging options with different times to expiration: SABR topped other models, with adjusted BSM a very close second. Although the original 1973 BSM model has some well-known limitations, adjusting the basic model to allow for stochastic volatility improves it considerably.

Tags: black scholes, delta hedging, options, SABR, Volatility

December 6th, 2012 at 10:04 am

The SSRN link points to a different research paper, the correct link is http://carolalexander.org/publish/download/JournalArticles/PDFs/JFM2009.pdf

December 6th, 2012 at 1:03 pm

Thanks, fixed!

December 7th, 2012 at 2:33 am

Personally, I think the “well-known limitations” of Black-Scholes is one if its greatest strengths. Its both intuitive and fairly clear where the model tends to break down, and I think that’s an oft over-looked feature in models.