Monte Carlo Methods for Derivatives Pricing

Random number generation, GBM simulation, path-dependent options, variance reduction, Longstaff-Schwartz for Americans, and convergence analysis.

When to Activate

• User pricing path-dependent or exotic derivatives (Asian, barrier, lookback)
• Pricing options on multiple underlyings (basket, rainbow, worst-of)
• American option pricing via Longstaff-Schwartz regression
• Evaluating variance reduction techniques for pricing efficiency
• Multi-factor model simulation (stochastic vol, rates, credit)
• Convergence analysis and error estimation for MC prices

Core Concepts

Monte Carlo Pricing Framework

The price of a derivative under risk-neutral pricing:

V_0 = exp(-rT) * E^Q[payoff(S_T)]

Monte Carlo estimates this expectation by:

1. Simulating N paths of the underlying under the risk-neutral measure
2. Computing the payoff for each path
3. Averaging the payoffs and discounting

V_MC = exp(-rT) * (1/N) * sum_{i=1}^{N} payoff_i

Standard error = sigma_payoff / sqrt(N), where sigma_payoff is the std dev of simulated payoffs.

GBM Path Simulation

Under risk-neutral GBM: dS = rSdt + sigmaSdW

Exact Simulation (log-normal): S(t+dt) = S(t) * exp((r - sigma^2/2)dt + sigmasqrt(dt)*Z) where Z ~ N(0,1)

• Exact: no discretization error regardless of dt
• Use for simple GBM dynamics (constant vol, no barriers)

Euler Discretization: S(t+dt) = S(t) + r*S(t)dt + sigmaS(t)*sqrt(dt)*Z

• Introduces discretization bias of order O(dt)
• Required for more complex SDEs (stochastic vol, local vol) where exact simulation is unavailable
• Can produce negative stock prices; use log-Euler: ln(S(t+dt)) = ln(S(t)) + (r - sigma^2/2)dt + sigmasqrt(dt)*Z

Milstein Scheme: S(t+dt) = S(t) + rS(t)dt + sigmaS(t)sqrt(dt)Z + 0.5sigma^2S(t)(Z^2 - 1)*dt

• Higher-order accuracy: O(dt) vs. O(sqrt(dt)) for Euler
• For GBM, Milstein = exact simulation (coincidence of the specific SDE)
• Useful for general SDEs where exact simulation is unavailable

Random Number Generation

• Use high-quality pseudo-random generators: Mersenne Twister (MT19937) is standard
• Box-Muller or inverse CDF to transform uniform to normal variates
• Quasi-random sequences (Sobol, Halton) for faster convergence:
  • QMC convergence: O(1/N) vs. O(1/sqrt(N)) for pseudo-random MC
  • Sobol sequences preferred for high dimensions (up to 1000+ dimensions)
  • Scrambled Sobol: adds randomization for error estimation while preserving QMC convergence
• Seed management: fix seeds for reproducibility; vary seeds for independence testing

Multi-Asset Simulation

For correlated assets with covariance matrix Sigma:

1. Cholesky decomposition: Sigma = L * L' where L is lower triangular
2. Generate independent Z = [Z_1, ..., Z_d] ~ N(0, I)
3. Correlated normals: W = L * Z
4. Simulate each asset using its correlated Brownian increment W_i
5. For d assets and N time steps: need d*N normal variates per path

Methodology

Variance Reduction Techniques

Antithetic Variates

• For each path using Z, also simulate the path using -Z
• Average the two payoffs: reduces variance when payoff is monotonic in Z
• Variance reduction factor: up to 2x for simple payoffs
• Free to implement (no additional random numbers needed)
• Works poorly for payoffs that are symmetric in Z (e.g., straddles)

Control Variates

• Use a correlated quantity with known expected value as a control
• Adjusted estimate: V_CV = V_MC + beta * (E[C] - C_MC)
• beta = -cov(payoff, C) / var(C) (optimal beta)
• Common controls: stock price itself (E[S_T] = S_0*exp(rT)), geometric Asian option (closed-form), BSM European option
• Can reduce variance by 10-100x for well-chosen controls
• Multiple controls: use regression to find optimal combination

Importance Sampling

• Change the probability measure to sample more from the region that matters
• V = E^P[payoff] = E^Q[payoff * dP/dQ] where Q is the importance sampling distribution
• Shift the mean of the GBM drift to make ITM paths more likely for OTM options
• Optimal importance distribution minimizes the variance of the weighted estimator
• Risk: poor choice of Q can increase variance; requires care

Stratified Sampling

• Divide the probability space into strata; sample uniformly within each stratum
• Ensures tails are sampled proportionally
• Latin hypercube sampling: generalization to multiple dimensions
• Typically combined with other techniques for maximum benefit

Path-Dependent Option Pricing

Asian Options

• Payoff depends on the average price: max(avg(S) - K, 0) for fixed-strike Asian call
• Simulate full price path, compute the average at each monitoring date
• Arithmetic average: no closed-form, MC is the standard method
• Geometric average: closed-form exists, use as control variate for arithmetic
• Variance reduction: geometric Asian control variate reduces variance by 50-90%

Barrier Options

• Payoff depends on whether the path crosses a barrier
• Naive MC with discrete monitoring misses barrier crossings between time steps
• Brownian bridge correction: compute probability of crossing barrier between observed points
• Continuity correction (Broadie-Glasserman-Kou): adjust barrier by 0.5826sigmasqrt(dt) for discrete monitoring approximation of continuous barrier
• Importance sampling: shift drift toward the barrier to increase the frequency of barrier hits

Lookback Options

• Payoff depends on the maximum or minimum price over the path
• Simulate path and track running max/min
• Discretization bias: discrete monitoring underestimates continuous max/min
• Correction: apply Brownian bridge to estimate continuous extremum between discrete points

Longstaff-Schwartz for American Options

The LSM algorithm uses regression to estimate continuation value:

1. Simulate N paths of the underlying to maturity
2. At the final time step: exercise value = intrinsic value
3. Work backward from T-dt to dt: a. At time step t, identify in-the-money paths b. Regress discounted future cashflows on basis functions of current stock price (e.g., 1, S, S^2, or Laguerre polynomials) c. Fitted regression = estimated continuation value C(S) d. Exercise if intrinsic value > C(S); otherwise continue
4. Option price = average of discounted cashflows along optimal exercise paths

Key implementation details:

• Basis functions: polynomials of S (degree 3-5 usually sufficient), Laguerre or Hermite polynomials
• Only use ITM paths for regression (reduces noise)
• Forward pass after backward exercise decisions to get unbiased estimate
• Bias: LSM gives a lower bound (suboptimal exercise policy); use dual method for upper bound
• N = 50,000-200,000 paths typically needed for stable estimates

Examples

European Call via MC

S=100, K=100, r=5%, sigma=30%, T=1.0
Exact BSM price: $14.23

Standard MC (N=10,000): $14.35, SE=$0.28, 95% CI: [$13.80, $14.90]
Standard MC (N=100,000): $14.21, SE=$0.09, 95% CI: [$14.03, $14.39]
Antithetic (N=50,000 pairs): $14.24, SE=$0.06, 95% CI: [$14.12, $14.36]
Control variate + antithetic: $14.23, SE=$0.02, 95% CI: [$14.19, $14.27]

Variance reduction: 196x improvement from SE=$0.28 to SE=$0.02

Arithmetic Asian Option with Control Variate

S=100, K=100, r=5%, sigma=30%, T=1.0, monthly averaging (12 dates)

Geometric Asian (closed-form): $8.42
Arithmetic Asian (MC, N=100,000): $8.89, SE=$0.07
With geometric Asian control variate: $8.91, SE=$0.01

Control variate reduced SE by 7x (variance by 49x).
The correlation between arithmetic and geometric Asian payoffs is 0.998.

Longstaff-Schwartz American Put

S=100, K=100, r=5%, sigma=30%, T=1.0, monthly exercise (12 dates)
N=100,000 paths, basis functions: 1, S, S^2, S^3

European put (MC): $10.08
American put (LSM): $10.85
Early exercise premium: $0.77

Regression at t=0.5 (example):
  Continuation value = 2.31 + 0.15*S - 0.002*S^2 (fitted)
  At S=85: continuation = 2.31 + 12.75 - 14.45 = $0.61
  Intrinsic = 100 - 85 = $15.00
  Decision: exercise (intrinsic >> continuation)

Binomial tree (N=500) benchmark: $10.87 — LSM within $0.02.

Quality Gate

• Standard error must be reported with every MC price — a price without error bounds is meaningless
• SE target: less than 1% of option price, or less than $0.05, whichever is tighter
• Convergence check: verify that price stabilizes as N increases (no systematic drift)
• At least one variance reduction technique should be applied (antithetic is minimum)
• For barrier options: Brownian bridge correction must be applied; verify with analytical formula where available
• For American options (LSM): validate against binomial tree for simple cases; difference should be <$0.05
• Random number generator must pass standard tests (Mersenne Twister or better)
• For QMC (Sobol): use scrambled sequences and randomize for error estimation
• Discretization: use exact simulation for GBM; for other SDEs, verify convergence by halving dt
• Multi-asset: verify Cholesky decomposition reproduces target correlation matrix (simulate and measure)
• Seed must be documented for reproducibility