Model Reference Compendium¶

Comprehensive reference documentation for all pricing models implemented in Neutryx, with mathematical specifications, parameters, implementation details, and validation notes.

For theoretical foundations and mathematical derivations, see Pricing Models Theory.

Table of Contents¶

Equity Models
Black-Scholes-Merton
Heston Stochastic Volatility
SABR Model
Rough Bergomi
Local Volatility (Dupire)
Jump-Diffusion Models
Interest Rate Models
Vasicek
Cox-Ingersoll-Ross (CIR)
Hull-White
References

Equity Models¶

Black-Scholes-Merton¶

Overview¶

The foundational model for European option pricing, developed by Black, Scholes, and Merton in 1973.

Nobel Prize: Scholes and Merton, 1997

Key References: - [Black & Scholes, 1973] "The Pricing of Options and Corporate Liabilities" - [Merton, 1973] "Theory of Rational Option Pricing" - See Theory: Black-Scholes-Merton Model

Model Specification¶

Under risk-neutral measure \(\mathbb{Q}\):

\[ dS_t = r S_t \, dt + \sigma S_t \, dW_t^{\mathbb{Q}} \]

Parameters: - \(S_t\): Stock price at time \(t\) - \(r\): Risk-free interest rate (constant) - \(\sigma\): Volatility (constant) - \(W_t^{\mathbb{Q}\): Brownian motion under risk-neutral measure

Solution (Geometric Brownian Motion):

\[ S_t = S_0 \exp\left(\left(r - \frac{\sigma^2}{2}\right)t + \sigma W_t^{\mathbb{Q}}\right) \]

Option Pricing Formulas¶

European Call:

\[ C(S_0, K, T) = S_0 \Phi(d_1) - K e^{-rT} \Phi(d_2) \]

European Put:

\[ P(S_0, K, T) = K e^{-rT} \Phi(-d_2) - S_0 \Phi(-d_1) \]

where:

\[ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]

and \(\Phi(\cdot)\) is the standard normal CDF.

Put-Call Parity:

\[ C - P = S_0 - K e^{-rT} \]

Greeks (Analytical)¶

Greek	Formula	Description
Delta	\(\Phi(d_1)\) (call), \(-\Phi(-d_1)\) (put)	Price sensitivity to underlying
Gamma	\(\frac{\phi(d_1)}{S_0 \sigma \sqrt{T}}\)	Curvature (convexity)
Vega	\(S_0 \phi(d_1) \sqrt{T}\)	Sensitivity to volatility
Theta	\(-\frac{S_0 \phi(d_1) \sigma}{2\sqrt{T}} - r K e^{-rT} \Phi(d_2)\)	Time decay
Rho	\(K T e^{-rT} \Phi(d_2)\)	Interest rate sensitivity

where \(\phi(\cdot)\) is the standard normal PDF.

Implementation¶

File: src/neutryx/models/bs.py

Key Functions: - price_european_call(S, K, T, r, sigma): Analytical call price - price_european_put(S, K, T, r, sigma): Analytical put price - implied_volatility(price, S, K, T, r, option_type): IV solver (bisection) - delta(S, K, T, r, sigma, option_type): Delta computation - gamma(S, K, T, r, sigma): Gamma computation

Validation: - Analytical formulas verified against Hull textbook examples - Put-call parity enforced to machine precision - Greeks validated via finite difference comparison

Model Limitations¶

Constant volatility: Market exhibits volatility smile/skew
Log-normal distribution: Underestimates tail risk (fat tails empirically observed)
Continuous trading: Ignores transaction costs and discrete hedging
No jumps: Cannot capture gap risk or earnings surprises

Extensions: See Heston, SABR (stochastic volatility), Merton/Kou (jumps), Local Volatility (smile fitting)

Heston Stochastic Volatility¶

Overview¶

The Heston model (1993) is the most popular stochastic volatility model with semi-analytical solutions via characteristic functions.

Key References: - [Heston, 1993] "A Closed-Form Solution for Options with Stochastic Volatility" - [Gatheral, 2006] The Volatility Surface, Chapter 3 - See Theory: Heston Model

Model Specification¶

\[ \begin{align} dS_t &= r S_t \, dt + \sqrt{v_t} S_t \, dW_t^S \\ dv_t &= \kappa(\theta - v_t) \, dt + \sigma_v \sqrt{v_t} \, dW_t^v \\ \langle dW_t^S, dW_t^v \rangle &= \rho \, dt \end{align} \]

Parameters:

Parameter	Symbol	Range	Description
Initial variance	\(v_0\)	\(> 0\)	Spot variance (e.g., 0.04 = 20% vol)
Mean-reversion speed	\(\kappa\)	\(> 0\)	Speed of reversion to \(\theta\)
Long-term variance	\(\theta\)	\(> 0\)	Equilibrium variance level
Volatility of volatility	\(\sigma_v\)	\(> 0\)	Vol-of-vol parameter
Correlation	\(\rho\)	\([-1, 1]\)	Price-vol correlation (typically negative)

Feller Condition: To ensure \(v_t > 0\) almost surely:

\[ 2\kappa\theta > \sigma_v^2 \]

If violated, variance can reach zero (boundary attainable).

Characteristic Function¶

The characteristic function of \(\log S_T\) is:

\[ \phi_T(u) = \mathbb{E}^{\mathbb{Q}}[e^{i u \log S_T}] = \exp\left(C(\tau, u) + D(\tau, u) v_t + i u \log S_t\right) \]

where \(\tau = T - t\) and \(C, D\) solve Riccati ODEs with analytical solutions.

Application: Use Carr-Madan FFT or COS method for fast pricing across many strikes.

Simulation Schemes¶

Euler Scheme (simple, can produce negative variance):

\[ v_{t+\Delta t} = v_t + \kappa(\theta - v_t) \Delta t + \sigma_v \sqrt{v_t^+} \sqrt{\Delta t} Z^v \]

Quadratic-Exponential (QE) Scheme [Andersen, 2008] (recommended): - Preserves non-negativity - Maintains first two moments exactly - Adaptive: quadratic for small \(v\), exponential for large \(v\)

Implementation: QE scheme is default in simulate() method.

Calibration¶

Target: Implied volatility surface (strikes × maturities)

Loss Function (vega-weighted IV error):

\[ \mathcal{L}(\boldsymbol{\theta}) = \sum_{i,j} w_{ij} \left(\sigma_{ij}^{\text{market}} - \sigma_{ij}^{\text{Heston}}(\boldsymbol{\theta})\right)^2 \]

Typical Parameter Ranges (equity): - \(v_0 \in [0.01, 0.1]\) (10%-30% spot vol) - \(\kappa \in [0.5, 5]\) (half-life of 0.14-1.4 years) - \(\theta \in [0.01, 0.1]\) (10%-30% long-term vol) - \(\sigma_v \in [0.1, 1]\) (vol-of-vol) - \(\rho \in [-0.9, -0.5]\) (negative for equities, leverage effect)

Optimizer: Adam (via optax) with parameter transformations

Volatility Smile Features¶

Vol-of-vol (\(\sigma_v\)): Controls smile curvature (ATM convexity)
Correlation (\(\rho\)): Creates skew (negative \(\rho\) → left skew for puts)
Mean-reversion (\(\kappa\)): Flattens long-dated smile (mean-reversion dominates)

Stylized Fact: Equity markets exhibit \(\rho \approx -0.7\) (leverage effect: stock drops → volatility rises)

Implementation¶

File: src/neutryx/models/heston.py

Key Functions: - characteristic_function(u, S0, v0, r, T, kappa, theta, sigma_v, rho): CF for FFT pricing - price_european_fft(S0, K, T, r, v0, kappa, theta, sigma_v, rho): FFT pricing (Carr-Madan) - simulate(S0, v0, T, r, kappa, theta, sigma_v, rho, n_steps, scheme='qe'): Path simulation - calibrate(market_data, initial_guess): Calibration routine

Calibration File: src/neutryx/calibration/heston.py

Validation: - Characteristic function verified against [Heston, 1993] Appendix A - FFT prices match reference implementations (QuantLib) - QE scheme preserves moments (tested against exact solution for simpler cases) - Calibration tested on SPX surface

SABR Model¶

Overview¶

The SABR (Stochastic Alpha Beta Rho) model, introduced by Hagan et al. (2002), is the industry standard for interest rate smile modeling.

Key References: - [Hagan et al., 2002] "Managing Smile Risk" - [Gatheral, 2006] The Volatility Surface, Chapter 4 - See Theory: SABR Model

Model Specification¶

\[ \begin{align} dF_t &= \alpha_t F_t^\beta \, dW_t^F \\ d\alpha_t &= \nu \alpha_t \, dW_t^\alpha \\ \langle dW_t^F, dW_t^\alpha \rangle &= \rho \, dt \end{align} \]

Parameters:

Parameter	Symbol	Range	Description
Initial volatility	\(\alpha_0\)	\(> 0\)	ATM volatility level
CEV exponent	\(\beta\)	\([0, 1]\)	0=normal, 0.5=shifted lognormal, 1=lognormal
Vol-of-vol	\(\nu\)	\(> 0\)	Volatility of volatility
Correlation	\(\rho\)	\([-1, 1]\)	Forward-vol correlation

Note: \(\beta\) is typically fixed (0.5 is common for rates), leaving 3 parameters to calibrate.

Hagan's Implied Volatility Approximation¶

At-the-money (\(F = K\)):

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

Away from ATM: Full asymptotic expansion involves \(z = \frac{\nu}{\alpha} (FK)^{(1-\beta)/2} \ln(F/K)\)

Application: Direct mapping from parameters to implied volatility (no numerical pricing required)

Calibration¶

Target: Implied volatility smile at single maturity (or term structure)

Loss Function:

\[ \mathcal{L}(\alpha, \rho, \nu) = \sum_i \left(\sigma_i^{\text{market}} - \sigma_i^{\text{SABR}}(\alpha, \beta, \rho, \nu)\right)^2 \]

Typical Parameter Ranges (interest rates): - \(\alpha \in [0.001, 0.1]\) (1%-10% ATM vol) - \(\beta = 0.5\) (fixed, common choice) - \(\rho \in [-0.5, 0.5]\) (often negative for rates) - \(\nu \in [0.1, 1]\) (vol-of-vol)

Implementation¶

File: src/neutryx/models/sabr.py

Key Functions: - implied_volatility(F, K, T, alpha, beta, rho, nu): Hagan's approximation - calibrate(market_strikes, market_vols, F, T, beta_fixed): Calibration to smile

Calibration File: src/neutryx/calibration/sabr.py

Validation: - Hagan's formula implementation tested against reference (QuantLib, original paper) - Asymptotic behavior verified: \(\sigma \to \alpha/F^{1-\beta}\) as \(K \to F\) - Calibration tested on swaption market data

Rough Bergomi¶

Overview¶

The Rough Bergomi model, introduced by Bayer, Friz, and Gatheral (2016), uses fractional Brownian motion to capture "rough" volatility dynamics with Hurst exponent \(H \approx 0.1\).

Key References: - [Bayer et al., 2016] "Pricing under rough volatility" - [Gatheral et al., 2018] "Volatility is Rough" - See Theory: Rough Bergomi Model

Model Specification¶

\[ \begin{align} dS_t &= \sqrt{v_t} S_t \, dW_t^S \\ v_t &= \xi_0(t) \exp\left(\eta W_t^H - \frac{\eta^2}{2} t^{2H}\right) \end{align} \]

Parameters:

Parameter	Symbol	Range	Description
Forward variance curve	\(\xi_0(t)\)	\(> 0\)	Deterministic term structure
Vol-of-vol	\(\eta\)	\(> 0\)	Volatility of volatility
Hurst parameter	\(H\)	\((0, 1)\)	Roughness (typically \(H \approx 0.1\))
Correlation	\(\rho\)	\([-1, 1]\)	Price-vol correlation

Fractional Brownian Motion: \(W_t^H\) is not a semimartingale for \(H \neq 0.5\) (no standard Itô calculus).

Empirical Evidence¶

Stylized Fact [Gatheral et al., 2018]: Realized volatility exhibits roughness with \(H \approx 0.1\) (anti-persistent, rougher than Brownian motion).

Advantages over Heston: - Better fit to short-dated skew (steeper) - Flatter vol-of-vol term structure - Fewer parameters

Simulation¶

Method: Cholesky decomposition of fBm covariance matrix

Discretize time: \(t_0, t_1, \ldots, t_N\)
Compute covariance: \(\Sigma_{ij} = \frac{1}{2}(t_i^{2H} + t_j^{2H} - |t_i - t_j|^{2H})\)
Cholesky: \(\Sigma = L L^T\)
Sample: \(\mathbf{W}^H = L \mathbf{Z}\), where \(\mathbf{Z} \sim \mathcal{N}(0, I)\)

Complexity: \(O(N^3)\) for Cholesky (can be reduced to \(O(N \log N)\) with hybrid schemes)

Implementation¶

File: src/neutryx/models/rough_vol.py

Key Functions: - simulate_fractional_brownian_motion(T, n_steps, hurst, n_paths): fBm generation - simulate_rough_bergomi(S0, xi0, eta, hurst, rho, T, n_steps, n_paths): Full path simulation - price_european(S0, K, T, r, xi0, eta, hurst, rho, n_paths): MC pricing

Validation: - fBm covariance structure verified - Reduces to standard Heston when \(H = 0.5\) - Smile behavior matches [Bayer et al., 2016]

Local Volatility (Dupire)¶

Overview¶

Dupire (1994) showed that any arbitrage-free implied volatility surface can be realized by a deterministic local volatility function \(\sigma_{\text{loc}}(S, t)\).

Key References: - [Dupire, 1994] "Pricing with a Smile" - [Gatheral, 2006] The Volatility Surface, Chapter 5 - See Theory: Local Volatility Models

Model Specification¶

\[ dS_t = r S_t \, dt + \sigma_{\text{loc}}(S_t, t) S_t \, dW_t \]

Key Property: Local volatility is a function of current stock price and time (deterministic, not stochastic).

Dupire's Formula¶

Given market option prices \(C(K, T)\) for all strikes and maturities:

\[ \sigma_{\text{loc}}^2(K, T) = \frac{\frac{\partial C}{\partial T} + rK \frac{\partial C}{\partial K}}{\frac{1}{2} K^2 \frac{\partial^2 C}{\partial K^2}} \]

Derivation: Forward Kolmogorov equation for the probability density.

In terms of implied volatility: More complex formula involving \(\frac{\partial \sigma_{\text{imp}}}{\partial K}\), \(\frac{\partial \sigma_{\text{imp}}}{\partial T}\), etc.

Calibration Procedure¶

Interpolate implied volatility surface \(\sigma_{\text{imp}}(K, T)\) (e.g., bicubic spline, SVI parameterization)
Compute derivatives: Numerical differentiation (with smoothing/regularization)
Apply Dupire's formula to obtain \(\sigma_{\text{loc}}(K, T)\)
Enforce arbitrage-free conditions:
\(\frac{\partial C}{\partial T} \geq 0\) (calendar spread)
\(\frac{\partial^2 C}{\partial K^2} \geq 0\) (butterfly, non-negative density)

Challenges: - Noisy derivatives (market data is discrete and noisy) - Extrapolation in illiquid regions - No forward smile (volatility collapses to flat for future dates)

Implementation¶

File: src/neutryx/models/local_vol.py, src/neutryx/models/dupire.py

Key Functions: - dupire_local_vol(K, T, implied_vol_surface, S0, r): Dupire formula application - simulate_local_vol(S0, T, n_steps, local_vol_func): Monte Carlo with local vol - price_european_pde(S0, K, T, r, local_vol_surface): PDE pricing

Calibration File: src/neutryx/calibration/local_vol.py

Validation: - Perfect fit to market by construction (within interpolation accuracy) - Arbitrage-free constraints checked - Comparison with stochastic volatility models

Jump-Diffusion Models¶

Merton Jump-Diffusion¶

Overview: Merton (1976) added a compound Poisson jump process to GBM.

Model:

\[ dS_t = (\mu - \lambda \bar{J}) S_t \, dt + \sigma S_t \, dW_t + S_{t-} \, dJ_t \]

where \(J_t = \sum_{i=1}^{N_t} (Y_i - 1)\), \(N_t \sim \text{Poisson}(\lambda)\), \(Y_i \sim \text{LogNormal}(\mu_J, \sigma_J^2)\).

Parameters: - \(\sigma\): Diffusion volatility - \(\lambda\): Jump intensity (jumps per year) - \(\mu_J\): Mean of log-jump size - \(\sigma_J\): Volatility of log-jump size

Pricing: Analytical series expansion (weighted sum of Black-Scholes prices) or FFT (Carr-Madan).

Key References: - [Merton, 1976] "Option Pricing when Underlying Stock Returns are Discontinuous" - [Cont & Tankov, 2004] Financial Modelling with Jump Processes, Chapter 9 - See Theory: Merton Jump-Diffusion

Implementation: src/neutryx/models/jump_diffusion.py

Kou Double Exponential¶

Overview: Kou (2002) used asymmetric exponential jump distributions.

Jump Distribution:

\[ p_J(y) = p \cdot \eta_u e^{-\eta_u y} \mathbf{1}_{y > 0} + (1-p) \cdot \eta_d e^{\eta_d y} \mathbf{1}_{y < 0} \]

Parameters: - \(p\): Probability of upward jump - \(\eta_u > 1\): Decay rate of upward jumps - \(\eta_d > 0\): Decay rate of downward jumps

Advantage: Captures asymmetry (negative jumps larger), analytically tractable for barriers/lookbacks.

Key References: - [Kou, 2002] "A Jump-Diffusion Model for Option Pricing" - See Theory: Kou Double Exponential

Implementation: src/neutryx/models/kou.py

Variance Gamma¶

Overview: Variance Gamma (Madan et al., 1998) is a pure jump Lévy process (infinite activity, no Brownian component).

Model: \(S_t = S_0 \exp((r + \omega)t + X_t)\) where \(X_t = \theta G_t + \sigma W_{G_t}\), \(G_t \sim \text{Gamma}(\nu t, \nu)\).

Parameters: - \(\theta\): Drift of Brownian motion - \(\sigma\): Volatility of Brownian motion - \(\nu\): Variance rate of gamma time change

Characteristic Function:

\[ \phi_T(u) = \left(1 - i u \theta \nu + \frac{\sigma^2 \nu u^2}{2}\right)^{-T/\nu} \]

Advantages: Fits smile with fewer parameters than stochastic vol, captures excess kurtosis (fat tails).

Key References: - [Madan et al., 1998] "The Variance Gamma Process and Option Pricing" - [Cont & Tankov, 2004], Section 8.4 - See Theory: Variance Gamma

Implementation: src/neutryx/models/variance_gamma.py

Interest Rate Models¶

Vasicek¶

Overview¶

Vasicek (1977) introduced the first mean-reverting short rate model.

Key References: - [Vasicek, 1977] "An Equilibrium Characterization of the Term Structure" - [Brigo & Mercurio, 2006] Interest Rate Models, Section 3.2 - See Theory: Vasicek Model

Model Specification¶

\[ dr_t = \kappa(\theta - r_t) \, dt + \sigma \, dW_t \]

Parameters:

Parameter	Symbol	Range	Description
Mean-reversion speed	\(\kappa\)	\(> 0\)	Speed of reversion (half-life = \(\ln(2)/\kappa\))
Long-term mean	\(\theta\)	\(\mathbb{R}\)	Equilibrium rate level
Volatility	\(\sigma\)	\(> 0\)	Interest rate volatility

Properties: - Ornstein-Uhlenbeck process (Gaussian) - Mean-reverting: \(\mathbb{E}[r_t | r_0] = \theta + (r_0 - \theta) e^{-\kappa t}\) - Limitation: \(r_t\) can be negative (though \(\mathbb{P}(r_t < 0)\) can be small if parameters chosen carefully)

Zero-Coupon Bond Pricing¶

Analytical formula:

\[ P(t, T) = A(t, T) e^{-B(t, T) r_t} \]

where:

\[ B(t, T) = \frac{1 - e^{-\kappa(T-t)}}{\kappa} \]

\[ A(t, T) = \exp\left(\left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(t, T) - (T-t)) - \frac{\sigma^2 B(t, T)^2}{4\kappa}\right) \]

Yield curve: \(y(t, T) = -\frac{\ln P(t, T)}{T - t}\)

Implementation¶

File: src/neutryx/models/vasicek.py

Key Functions: - bond_price(r, t, T, kappa, theta, sigma): Zero-coupon bond pricing - yield_curve(r, t, maturities, kappa, theta, sigma): Yield curve construction - simulate(r0, T, n_steps, kappa, theta, sigma, scheme='exact'): Path simulation (exact or Euler) - bond_option_price(...): European option on zero-coupon bond (Jamshidian formula)

Validation: - Bond pricing formula verified against [Brigo & Mercurio, 2006] - Exact simulation matches analytical distribution (Gaussian) - Convergence to long-term mean verified

Cox-Ingersoll-Ross (CIR)¶

Overview¶

CIR (1985) ensured non-negative interest rates via square-root diffusion.

Key References: - [Cox et al., 1985] "A Theory of the Term Structure of Interest Rates" - [Brigo & Mercurio, 2006] Interest Rate Models, Section 3.1 - See Theory: CIR Model

Model Specification¶

\[ dr_t = \kappa(\theta - r_t) \, dt + \sigma \sqrt{r_t} \, dW_t \]

Parameters: Same as Vasicek, but volatility now \(\propto \sqrt{r_t}\).

Feller Condition: To ensure \(r_t > 0\) always:

\[ 2\kappa\theta > \sigma^2 \]

If satisfied, \(r_t\) never reaches zero. If violated, \(r_t\) can reach zero (absorbing boundary).

Zero-Coupon Bond Pricing¶

Analytical formula:

\[ P(t, T) = A(t, T) e^{-B(t, T) r_t} \]

where:

\[ B(t, T) = \frac{2(e^{\gamma(T-t)} - 1)}{(\gamma + \kappa)(e^{\gamma(T-t)} - 1) + 2\gamma} \]

\[ A(t, T) = \left(\frac{2\gamma e^{(\kappa+\gamma)(T-t)/2}}{(\gamma + \kappa)(e^{\gamma(T-t)} - 1) + 2\gamma}\right)^{2\kappa\theta/\sigma^2} \]

where \(\gamma = \sqrt{\kappa^2 + 2\sigma^2}\).

Simulation¶

Exact: \(r_t | r_0\) follows a scaled non-central chi-squared distribution.

Euler: Simple but may produce negative rates (use \(\max(r_t, 0)\) or absorbing).

Milstein: Higher-order accuracy.

Implementation¶

File: src/neutryx/models/cir.py

Key Functions: - bond_price(r, t, T, kappa, theta, sigma): Zero-coupon bond pricing - simulate(r0, T, n_steps, kappa, theta, sigma, scheme='exact'): Exact or Euler simulation - check_feller_condition(kappa, theta, sigma): Feller condition verification

Validation: - Bond pricing verified against [Brigo & Mercurio, 2006] - Exact simulation uses non-central chi-squared (scipy) - Non-negativity enforced and tested

Hull-White¶

Overview¶

Hull-White (1990) extended Vasicek to fit the initial term structure \(P^{\text{market}}(0, T)\).

Key References: - [Hull & White, 1990] "Pricing Interest-Rate-Derivative Securities" - [Brigo & Mercurio, 2006] Interest Rate Models, Section 3.4 - See Theory: Hull-White Model

Model Specification¶

\[ dr_t = (\theta(t) - \kappa r_t) \, dt + \sigma \, dW_t \]

where \(\theta(t)\) is time-dependent, chosen to fit the initial yield curve.

Calibration Formula:

\[ \theta(t) = \frac{\partial f^{\text{market}}(0, t)}{\partial t} + \kappa f^{\text{market}}(0, t) + \frac{\sigma^2}{2\kappa} (1 - e^{-2\kappa t}) \]

where \(f(0, t)\) is the instantaneous forward rate at time 0 for maturity \(t\).

Bond Pricing¶

Analytical formula (same affine structure as Vasicek):

\[ P(t, T) = A(t, T) e^{-B(t, T) r_t} \]

where \(B(t, T) = \frac{1 - e^{-\kappa(T-t)}}{\kappa}\) and:

\[ \ln A(t, T) = \ln\frac{P^{\text{market}}(0, T)}{P^{\text{market}}(0, t)} - B(t, T) f^{\text{market}}(0, t) - \frac{\sigma^2}{4\kappa} B(t, T)^2 (1 - e^{-2\kappa t}) \]

Caplet and Cap Pricing¶

Caplet (closed-form):

\[ \text{Caplet}(t, T, K) = P(t, T) \left[(L_t + \text{spread}) \Phi(d_1) - K \Phi(d_2)\right] \]

(Black-like formula with adjusted parameters)

Application: Calibrate \(\kappa, \sigma\) to cap/floor market.

Implementation¶

File: src/neutryx/models/hull_white.py

Key Functions: - calibrate_theta(forward_curve, kappa, sigma): Fit \(\theta(t)\) to market - bond_price(r, t, T, kappa, sigma, theta_func, forward_curve): Bond pricing - simulate(r0, T, n_steps, kappa, sigma, theta_func): Path simulation - caplet_price(...): Caplet pricing (closed-form)

Validation: - Perfect fit to initial term structure by construction - Caplet formula verified against [Brigo & Mercurio, 2006] - Calibration tested on yield curve data

References¶

For complete bibliography and detailed derivations, see:

Implementation files: src/neutryx/models/ and src/neutryx/calibration/