Monte Carlo pricing

Use normal_sample to generate noise and euler_maruyama_fixed to simulate terminal prices, then average and discount the option payoffs.

The pattern

1. Generate N paths of the underlying asset price under the risk-neutral measure.
2. Compute the payoff for each path.
3. Average the payoffs and discount to present value.

Geometric Brownian motion paths

Under the risk-neutral measure, a stock price follows:

dS = r * S * dt + sigma * S * dW

This is a GBM SDE. Nautilus can simulate it using euler_maruyama_fixed from Nautilus.Sde:

import Nautilus.Sde (euler_maruyama_fixed)

def gbm_drift(y: f32, t: f32) -> f32 = mul(cast(0.05, f32), y)   -- r * S
def gbm_diff(y: f32, t: f32) -> f32 = mul(cast(0.2, f32), y)     -- sigma * S

def simulate_path[n](s0: f32, noise: tensor[n, f32]) -> f32 =
  euler_maruyama_fixed(gbm_drift, gbm_diff, s0,
    cast(0.0, f32), cast(1.0, f32), noise)

The noise tensor contains pre-drawn N(0,1) samples. Its length determines the number of timesteps.

Generating noise

The normal_sample function takes a key:

import Nautilus.Distributions (normal_sample)

def draw_noise[n](k: key, template: tensor[n, f32]) -> tensor[n, f32] =
  normal_sample(k, template, cast(0.0, f32), cast(1.0, f32))

The key fixes the underlying uniform stream: draw_noise(key_from_seed(42i64), template) returns the same draws on every run, and a different seed gives different draws. To draw several independent noise tensors, derive a child key per draw with split_key or split_keys rather than passing one key twice.

Computing the price

For a European call with strike K:

-- After simulating terminal price s_T:
payoff = if gt(s_T, k) then sub(s_T, k) else cast(0.0, f32)
discounted = mul(payoff, exp(neg(mul(r, t))))

The Monte Carlo estimate is the average of discounted over all paths. For a European call under this model, compare the estimate with the Black-Scholes price. A path-dependent payoff requires the intermediate path values, while euler_maruyama_fixed returns only the terminal value.

Practical considerations

• Key control. Reuse a seed to replay a calculation and derive child keys for distinct paths. For variance reduction (antithetic variates, control variates), structure the noise generation accordingly.
• Path count. Sampling error scales approximately as payoff standard deviation / sqrt(N). Measure the payoff variance to choose a path count; a fixed path count does not guarantee a fixed relative error.
• Euler vs Milstein. Milstein's correction, 0.5 * g * g' * (dW^2 - dt), vanishes only for additive noise (g' = 0). For GBM, g = sigma * S and g' = sigma, so milstein_fixed (with dg_dy returning sigma) raises the strong order from 0.5 to 1. Terminal-price payoffs depend only on weak accuracy, where the two schemes are both order 1.