> For the complete documentation index, see [llms.txt](https://docs.algenta.ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.algenta.ai/machine-learning/ml-chain-greeks.md).

# Batched chain greeks

SIMD-batched Black-Scholes price and greeks across a whole option chain in one call, matching the scalar pricing functions to 1e-9.

`option_greeks_batch` prices an entire option chain — hundreds to thousands of strikes and expiries — in a single call, vectorizing the Black-Scholes math across hardware SIMD lanes. It replaces per-contract scalar pricing loops where call overhead and scalar transcendentals dominate.

## The modules

| Module                | What it does                                                                                                                             |
| --------------------- | ---------------------------------------------------------------------------------------------------------------------------------------- |
| `option_greeks_batch` | SIMD-batched Black-Scholes price, delta, gamma, vega and theta for an entire option chain in one call                                    |
| `option_pricing`      | the scalar Black-Scholes, binomial-tree, greeks and implied-volatility functions that `option_greeks_batch` is verified to match to 1e-9 |

## Functions

| Function                                                       | What it does                                                                                                                                                           |
| -------------------------------------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `chain_greeks(spots, strikes, times, rates, sigmas, is_calls)` | Batched BS price, delta, gamma, vega and theta for mixed call/put chains (`is_calls` 1.0/0.0). Returns `ChainGreeksResult` with five parallel `List[Float64]` outputs. |

## Semantics and guarantees

* Results match the scalar `option_pricing` functions to 1e-9 on the valid domain (verified across a 144-cell hostile grid of moneyness, expiry and volatility, calls and puts).
* Put greeks derive from parity — `delta_put = delta_call - 1`, `theta_put = theta_call + r·K·e^(-rT)`; gamma and vega are shared. Theta is per year, the same convention as `theta_call`.
* Degenerate lanes (`T <= 0` or `sigma <= 0`) return discounted intrinsic value with zero greeks instead of NaN.
* The SIMD main loop and scalar tail produce identical results at every batch size (verified for n = 1..11 against single-element calls).

## Throughput

Measured on a 2-lane NEON float64 machine (wider AVX lanes scale further), with dead-code elimination defeated by checksum accumulation:

```
1,000-contract chain: batch 28.2 us | scalar loop 88.1 us | 3.1x
= 28 ns per contract for price + all four greeks
```

**Next:** feed the greeks into [statistical model comparison](/machine-learning/ml-model-comparison.md) experiments, or browse the [engine catalog](/engine-runtime/catalog.md).


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