# Bachelier 1900 vs Black 1976

As the price of crude oil futures plummeted to -$40 a barrel, the Black 1976 options pricing model - which cannot handle negative prices - was switched to the Bachelier model. But what are the implications of this switch?

After oil futures turned negative for the first time on April 20th this year, both ICE Clear Europe and CME switched out of the Black 1976 to the Bachelier model for the pricing of their Crude Oil options.

Although not as widely recognised as the Black 1976 model, the Bachelier model has been used for a long time on interest rate, electricity and crack spread contracts.

The two models are the same except that the Black 1976 uses a Geometric Brownian Motion to model the underlying price while Bachelier a Standard Brownian Motion. This difference is what makes the Bachelier model suitable at negative prices, and it also distinguishes the two models in an analogous way of linear from compound interests. While it is easy to prove that the two models are asymptotically equal on options that are close at-the-money and with a short time to maturity, it is to be expected for these models to diverge spectacularly on long-dated options.

Transaction volume in option contracts concentrates in short-maturity at-the-money contracts, which is why the Bachelier formula is a standard among central counterparty clearing houses (CCPs). A CCP uses option pricing formulas to calculate clearing members' intraday P&L, stress tests and margin requirements. Therefore, one can calculate the impact of switching from the Black 1976 to the Bachelier model by replicating the algorithm of SPAN and comparing the results under the two models.

What follows is a SPAN margin analysis under the two models at different moneyness levels of all the CME's Crude Oil Put and Call June-2029 contracts on May 21st, 2020.

The moneyness is calculated as the underlying price minus the strike price for a call option and vice versa for a put option. Therefore, zero indicates the at-the-money (ATM) option, a negative value an out-of-the-money (OTM) option and a positive value an in-the-money (ITM) option.

We note that the two models converge on long-dated deep ITM puts but diverge elsewhere with Bachelier returning a SPAN margin of $4,300 on the ATM option while the Black 1976 returns a SPAN margin of $2,773 for the same contract.

Under the Black 1976 assumption, the underlying price can never be negative, and therefore, the price of a put option is limited to its strike price. In other words, under the Black 1976 model, a put option is riskless to issue once a premium equal to the strike price is paid.

As shown in the following table, the market is willing to pay above the strike price for any put option with a strike price below $18. That is, the market believes in negative prices, and by using the Black 1976 model, the SPAN algorithm produces zero margins for these put options.

CME's Crude Oil Put June 2029 Options - May 21st, 2020 | ||||||
---|---|---|---|---|---|---|

Strike Price | Option Price | Underlying Price | Implied Volatility Bachelier | Implied Volatility Black 1976 | SPAN Margin Bachelier | SPAN Margin Black 1976 |

$20.5 | $18.9 | $53 | $27.39 | 131% | $3,462 | $744 |

$20.0 | $18.7 | $53 | $27.40 | 137% | $3,448 | $648 |

$19.5 | $18.6 | $53 | $27.40 | 145% | $3,433 | $538 |

$19.0 | $18.4 | $53 | $27.40 | 156% | $3,420 | $404 |

$18.5 | $18.2 | $53 | $27.40 | 174% | $3,405 | $238 |

$18.0 | $18.1 | $53 | $27.40 | +∞ | $3,391 | $0 |

$17.5 | $17.9 | $53 | $27.41 | +∞ | $3,377 | $0 |

$17.0 | $17.7 | $53 | $27.41 | +∞ | $3,362 | $0 |

On consideration of the put-call parity principle, we can expect the Black 1976 model to also fail in margining any call option having a strike price below $18 or costing more than the underlying price.

The above chart is showing that as expected, the Black 1976 model cannot estimate the margin on call options with a strike price lower of $18, as the implied volatility overshoot below this point with the SPAN margin locking into the future scanning margin of $3,275.

To avoid the catastrophic under margining of some portfolios, CCPs should always switch promptly to the Bachelier model as soon as the market remotely believes in negative prices and should not wait to do so until the listing of negative strike prices is required. As shown above, it can be easily inferred when the market perceives that prices can become negative.

## A Brief History of Option Pricing

In 1900, Louis Bachelier thesis, Theory of Speculation, pioneered the idea of using Brownian motion to price European-style options. The nature of Bachelier's work was not recognised for decades because of the scepticism around the use of mathematics to model the stock market at the time.

In 1973, The Pricing of Options and Corporate Liabilities by Fisher Black and Myron Scholes, introduced the Black-Scholes (BS) model. Almost at the same time, the Theory of Rational Option Pricing by Robert Merton presented an extension to the BS model accounting for dividends, this widely used extension is known as the Black-Scholes-Merton (BSM) model.

In 1976, Fisher Black proposed a way to apply the Black-Scholes model to European-style options on forwards and futures in The Pricing of Commodity Contracts. In essence, the Black 1976 (B76) model is the same as the BSM model applied on a stock that pays dividends at a rate equal to the free risk rate (note that forward or future contracts have an initial value of zero because no money changes hands with the initial agreement).

For the past 40 years, numbers of financial mathematicians have been expanding and improving this field of quantitative finance. However, from this point in time, much of the effort has concentrated on the development of newer and better numerical methods for the pricing of exotic options.

## CCPs Quantitative Option Pricing

This GitHub repository gfiocco/option-pricing provides a library in different programming languages of the option pricing formulas currently being used by most of the global derivatives CCPs. Feel free to fork it, create a pull request or, and especially, leave a star.

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