Monte Carlo Simulation

QuantNova provides a comprehensive Monte Carlo engine for pricing path-dependent derivatives.

Dynamics

Geometric Brownian Motion

dynamics = GBMDynamics(r, σ)

• r - Risk-free rate

• σ - Volatility

The asset follows: dS = rS dt + σS dW

Heston Stochastic Volatility

dynamics = HestonDynamics(r, v0, κ, θ, ξ, ρ)

• r - Risk-free rate

• v0 - Initial variance

• κ - Mean reversion speed

• θ - Long-term variance

• ξ - Vol of vol

• ρ - Correlation between spot and vol

Payoffs

Payoff	Constructor	Description
European Call	EuropeanCall(K)	max(S_T - K, 0)
European Put	EuropeanPut(K)	max(K - S_T, 0)
Asian Call	AsianCall(K)	max(avg(S) - K, 0)
Asian Put	AsianPut(K)	max(K - avg(S), 0)
Up-and-Out Call	UpAndOutCall(K, B)	Call that knocks out if S ≥ B
Down-and-Out Put	DownAndOutPut(K, B)	Put that knocks out if S ≤ B
American Put	AmericanPut(K)	Early exercise put
American Call	AmericanCall(K)	Early exercise call

Basic Pricing

S0 = 100.0  # Initial spot
T = 1.0     # Time to maturity
dynamics = GBMDynamics(0.05, 0.2)

result = mc_price(S0, T, EuropeanCall(100.0), dynamics;
    npaths=100000,    # Number of paths
    nsteps=252,       # Steps per path
    antithetic=true   # Variance reduction
)

result.price     # Monte Carlo estimate
result.stderr    # Standard error
result.ci_lower  # 95% CI lower bound
result.ci_upper  # 95% CI upper bound

Variance Reduction

Antithetic Variates

Enabled by default. Uses pairs of negatively correlated paths:

mc_price(S0, T, payoff, dynamics; antithetic=true)

Quasi-Monte Carlo

Uses Sobol sequences instead of pseudo-random numbers. Better convergence: O(1/N) vs O(1/√N).

# Direct QMC pricing
result = mc_price_qmc(S0, T, payoff, dynamics; npaths=10000)

# Generate Sobol-based normals directly
Z = sobol_normals(nsteps, npaths)  # (npaths × nsteps) matrix

American Options (LSM)

The Longstaff-Schwartz algorithm prices American options via regression:

result = lsm_price(S0, T, AmericanPut(100.0), dynamics;
    npaths=50000,
    nsteps=50  # Exercise dates
)

# Early exercise premium
eu_put = mc_price(S0, T, EuropeanPut(100.0), dynamics)
premium = result.price - eu_put.price

Monte Carlo Greeks

Compute sensitivities via pathwise differentiation:

using Enzyme
using QuantNova

# Delta only
delta = mc_delta(S0, T, payoff, dynamics;
    npaths=10000,
    nsteps=50,
    backend=EnzymeBackend()
)

# Delta and Vega
greeks = mc_greeks(S0, T, payoff, dynamics;
    npaths=10000,
    nsteps=50,
    backend=EnzymeBackend()
)
greeks.delta
greeks.vega

Backend Behavior

Backend	Method
ForwardDiffBackend()	Pseudo-random with fixed seed
PureJuliaBackend()	Finite differences
EnzymeBackend()	QMC (Sobol sequences)

Enzyme automatically uses QMC because it cannot differentiate through RNGs.

Heston Monte Carlo

heston = HestonDynamics(
    0.05,   # r
    0.04,   # v0
    2.0,    # κ
    0.04,   # θ
    0.3,    # ξ
    -0.7    # ρ
)

result = mc_price(100.0, 1.0, EuropeanCall(100.0), heston;
    npaths=50000,
    nsteps=252
)

# Compare to semi-analytical Heston
params = HestonParams(0.04, 0.04, 2.0, 0.3, -0.7)
analytic = heston_price(100.0, 100.0, 1.0, 0.05, params, :call)

Path Simulation

For custom payoffs or analysis, simulate paths directly:

# Single GBM path
path = QuantNova.MonteCarlo.simulate_gbm(S0, T, nsteps, dynamics)

# Antithetic pair
path1, path2 = QuantNova.MonteCarlo.simulate_gbm_antithetic(S0, T, nsteps, dynamics)

# Heston path
path = QuantNova.MonteCarlo.simulate_heston(S0, T, nsteps, heston_dynamics)

# QMC path (deterministic)
Z = sobol_normals(nsteps, 1)
path = QuantNova.MonteCarlo.simulate_gbm_qmc(S0, T, nsteps, dynamics, Z[1, :])