Speed and Accuracy Trade-Offs in Black-Scholes Pricing
Summary
This discussion addresses how to compute Black-Scholes-Merton option prices quickly while retaining precision. It identifies the cumulative normal distribution function, or an equivalent error function, as the key special function needed beyond the standard formula. The answer recommends using mature math-library implementations, which generally provide full precision at low computational cost, making custom approximations unnecessary for many applications.
For workloads where pricing speed is especially important, the discussion suggests precomputed lookup tables arranged to reduce CPU cache misses, or hardware-specific implementations such as GPUs and FPGAs. These are presented as possible performance approaches, not as benchmarked recommendations: the document supplies no comparative timing or accuracy measurements for them. The best choice therefore depends on workload, hardware, and acceptable approximation error. It also does not detail table construction, interpolation, or validation, so those steps would need separate analysis before using a lookup method in production.
Key ideas
- The cumulative normal distribution or an equivalent error function is the main special function used in Black-Scholes pricing.
- Standard math libraries can provide accurate function evaluations at modest computational cost.
- Lookup tables may improve throughput when designed to limit CPU cache misses.
- Specialized hardware can be considered for speed-sensitive workloads, though the discussion gives no benchmark results.
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Full text
# Black-Scholes fastest computation method
# Black-Scholes fastest computation method
What is the fastest way to numerically compute Black-Scholes-Merton option prices?
I'm trying to find fastest and still precise method. Currently I'm using numerical approximation of Normal cdf with 10-9 precision and standard formulae.
Is there another way to compute them numerically using any programming language without built-in libraries for option pricing and cdf computation? If any what is the speed of the algorithm in comparison with standard method and what is precision of an answer?
There is a question about approximations to the Black-Scholes formula, but there are aproximations for ATM options. And it is not obvious that this methods would show better results.
## Answer by Alexey Kalmykov (score 8)
https://quant.stackexchange.com/a/8336
Fastest method is a pre-generated lookup table with carefully selected in-memory structure so you don't get too many CPU cache misses (avoiding the memory latency).
If you want an absolute speed, you also can go for a hardware specific implementation (GPU, FPGA).
## Answer by q.t.f. (score 3)
https://quant.stackexchange.com/a/14561
The only special function needed for computing Black-Scholes option prices is the cumulative normal function ("N" or "Phi") or equivalently the error function ("erf"). These are very widely available with good standard library implementations. The erf function in single and double precision is part of the c99 and c++11 math standard libraries. For your favorite language, search for "error function" possibly in a "special functions" library. Generally there is no need to trade speed vs. accuracy; the library functions here give full precision with timing in the 10s or 100s of processor cycles. In almost any application the math expense is completely negligible.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.