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SABR Volatility Model

Overview

cookbook/numerics/sabr.eta implements the Hagan et al. (2002) SABR implied volatility approximation and computes all model sensitivities using Eta’s built-in tape-based reverse-mode AD.

Key ideas demonstrated:

• SABR Hagan implied vol formula (ATM + general-K branches)
• Vol surface generation across a strike × expiry grid
• First-order Greeks (∂σ/∂α, ∂σ/∂β, ∂σ/∂ρ, ∂σ/∂ν) in a single backward pass
• Plain arithmetic (+, -, *, /, exp, log, sqrt) with transparent tape recording

etai cookbook/numerics/sabr.eta

Note

The SABR formula uses plain arithmetic — no lifted nd+, nd*, ndlog wrappers. The VM transparently records onto a tape when TapeRef operands are present, making the source code identical to a non-AD version.

The SABR Model

The SABR (Stochastic Alpha Beta Rho) model describes the dynamics of a forward rate F and its instantaneous volatility σ:

$$dF_t = \sigma_t F_t^\beta , dW_1$$

$$d\sigma_t = \nu \sigma_t , dW_2$$

$$dW_1 \cdot dW_2 = \rho , dt$$

Hagan Approximation

General Case (F ≠ K)

$$\sigma_{\text{impl}} = \frac{\alpha}{(FK)^{(1-\beta)/2} \left[1 + \frac{(1-\beta)^2}{24}\ln^2\frac{F}{K} + \frac{(1-\beta)^4}{1920}\ln^4\frac{F}{K}\right]} \cdot \frac{z}{x(z)} \cdot \left[1 + \epsilon , T\right]$$

where:

$$z = \frac{\nu}{\alpha}(FK)^{(1-\beta)/2}\ln\frac{F}{K}$$

$$x(z) = \ln\frac{\sqrt{1-2\rho z+z^2}+z-\rho}{1-\rho}$$

$$\epsilon = \frac{(1-\beta)^2\alpha^2}{24(FK)^{1-\beta}} + \frac{\rho\beta\nu\alpha}{4(FK)^{(1-\beta)/2}} + \frac{(2-3\rho^2)\nu^2}{24}$$

ATM Case (F ≈ K)

$$\sigma_{\text{ATM}} = \frac{\alpha}{F^{1-\beta}} \left[1 + \left(\frac{(1-\beta)^2\alpha^2}{24 F^{2-2\beta}} + \frac{\rho\beta\nu\alpha}{4 F^{1-\beta}} + \frac{(2-3\rho^2)\nu^2}{24}\right) T\right]$$

(defun sabr-atm-vol (F T alpha beta rho nu)
  (let ((one-minus-beta (- 1 beta)))
    (let ((F-pow (ndpow F one-minus-beta)))
      (let ((base-vol (/ alpha F-pow)))
        (let ((term1 (/ (* (* one-minus-beta one-minus-beta)
                            (* alpha alpha))
                         (* 24 (ndpow F (* 2 one-minus-beta)))))
              (term2 (/ (* rho (* beta (* nu alpha)))
                         (* 4 F-pow)))
              (term3 (/ (* (- 2 (* 3 (* rho rho)))
                            (* nu nu))
                         24)))
          (* base-vol
             (+ 1 (* T (+ term1 (+ term2 term3))))))))))

The |F − K| < 10⁻⁷ tolerance switches between the ATM and general formulas. The branch is a plain if on numeric values and is not differentiated — correct because the SABR approximation is smooth across the ATM boundary.

Market Parameters

Vol Surface

The example generates an implied vol surface across strikes (80 %–120 % of forward) and expiries (0.25 Y–5 Y):

With ρ < 0, low strikes have higher implied vol — the characteristic negative skew of interest-rate markets.

First-Order Greeks

A single grad call with inputs (α, β, ρ, ν) produces the implied vol and all four partial derivatives in one backward pass:

(grad (lambda (alpha beta rho nu)
         (sabr-implied-vol F K T alpha beta rho nu))
       (list alpha-val beta-val rho-val nu-val))
;; => (sigma_impl  #(∂σ/∂α  ∂σ/∂β  ∂σ/∂ρ  ∂σ/∂ν))

Tip

In practice β is typically fixed (0.5 for rates, 1.0 for FX). Pass it as a plain number instead of a TapeRef to exclude it from differentiation.

Tape-Based AD Architecture

SABR’s ~50 elementary operations on 4 scalar parameters work naturally with Eta’s tape-based AD. When grad creates TapeRef variables and activates the tape, the VM’s +, -, *, /, exp, log, sqrt transparently record each operation (~32 bytes per entry, zero closure allocations).

Key Benefits

Comparison with External Frameworks

Note

The tape approach is optimal for scalar models with few parameters (SABR, Black-Scholes, xVA with ≤ 50 risk factors). For tensor workloads (matrix multiply, batched convolutions), libtorch is faster due to BLAS/LAPACK kernels. The two approaches are complementary.

Summary

Example Output

Tip

etac -O compiles with optimisations; etai runs .eta or .etac files directly.

$ etai sabr.eta
==================================================
 SABR Volatility Surface with Tape-Based AD
==================================================

SABR parameters:
  F (forward)  = 3.00%
  alpha        = 0.035   (vol level)
  beta         = 0.50    (CEV exponent)
  rho          = -0.25   (skew)
  nu           = 0.40    (vol-of-vol)


-- Implied Vol Surface (%) --

  K/F(%)   0.25Y   0.5Y   1Y   2Y   5Y
   80%   23.0051  23.0618  23.1753  23.4022  24.0829
   85%   22.1764  22.2312  22.3409  22.5602  23.2181
   90%   21.4442  21.4974  21.6037  21.8163  22.4541
   95%   20.8057  20.8573  20.9607  21.1674  21.7876
  100%   20.2577  20.3081  20.409  20.6107  21.2159
  105%   19.7969  19.8463  19.945  20.1426  20.7352
  110%   19.4188  19.4674  19.5644  19.7586  20.341
  115%   19.1179  19.1658  19.2615  19.453  20.0275
  120%   18.8876  18.935  19.0297  19.2192  19.7877


-- First-Order Greeks (single backward pass) --

  At ATM point: F = K = 3%, T = 1Y

  sigma_impl = 20.409%

  Sensitivities:
    dsigma/dalpha = 5.82147
    dsigma/dbeta  = -0.71583
    dsigma/drho   = 0.00406239
    dsigma/dnu    = 0.0109325

  Interpretation:
    dalpha -> smile level shift
    dbeta  -> backbone / CEV curvature
    drho   -> skew sensitivity
    dnu    -> wing / kurtosis sensitivity


-- OTM Greeks: K = 2.4% (80% moneyness), T = 1Y --

  sigma_impl = 23.1753%

  Sensitivities:
    dsigma/dalpha = 6.03833
    dsigma/dbeta  = -0.765843
    dsigma/drho   = -0.0340376
    dsigma/dnu    = 0.0623731


-- Tape-Based AD Architecture --

  The SABR Hagan formula involves ~50 elementary operations
  on 4 scalar parameters — ideal for tape-based AD.
  When grad activates the tape, each operation (`+`, `-`, `*`, `/`,
  `exp`, `log`, `sqrt`) transparently records onto the tape
  (32 bytes per entry, zero closure allocations).

  Key benefits:
    - Single reverse sweep for the backward pass
    - No closure allocations — efficient memory usage
    - Transparent recording with plain arithmetic