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Michael Brenndoerfer

Monte Carlo Simulation for Derivative Pricing › Multi-Asset Options

Michael BrenndoerferDecember 2, 20255 min read

Quantitative FinanceAll chapters

Part 6 of 15Numerical Methods for Derivative Pricing

Chapter 34Monte Carlo Simulation for Derivative Pricing › Multi-Asset Options

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Monte Carlo extends naturally to options on multiple correlated assets, such as basket options or spread options. The key challenge in multi-asset simulation is generating random variables with the correct correlation structure. For two assets with correlation $\rho$ , we generate correlated normal random variables using the Cholesky decomposition, a mathematical technique for expressing the correlation structure in a computationally convenient form.

If $Z_1, Z_2$ are independent standard normals, then the following transformation produces correlated standard normals:

\begin{aligned} \epsilon_1 &= Z_1 \\ \epsilon_2 &= \rho Z_1 + \sqrt{1 - \rho^2} Z_2 \end{aligned}

where:

$\epsilon_1, \epsilon_2$ : correlated standard normal variables
$Z_1, Z_2$ : independent standard normal variables
$\rho$ : correlation coefficient
$\rho Z_1$ : component of the second asset's return driven by the common factor
$\sqrt{1 - \rho^2} Z_2$ : idiosyncratic component unique to the second asset

This construction preserves unit variance for both $\epsilon_1$ and $\epsilon_2$ (since $\rho^2 + (1-\rho^2) = 1$ ) while ensuring the correct correlation. The intuition is that we decompose the second asset's randomness into two parts: a portion $\rho Z_1$ that comes from the same source as the first asset (the "common factor"), and a portion $\sqrt{1 - \rho^2} Z_2$ that is unique to the second asset (the "idiosyncratic factor"). When $\rho = 1$ , the second asset moves perfectly with the first. When $\rho = 0$ , they are independent. When $\rho = -1$ , they move in opposite directions.

For a basket call option on two assets with weights $w_1, w_2$ , the payoff is simply the call payoff applied to the weighted sum:

\text{Payoff} = \max(w_1 S_{1,T} + w_2 S_{2,T} - K, 0)

where:

$\text{Payoff}$ : cash flow received at expiration
$w_1, w_2$ : weights of the assets in the basket
$S_{1,T}, S_{2,T}$ : asset prices at time $T$
$K$ : strike price

In[28]:

Code

def monte_carlo_basket_call(
    S1_0, S2_0, K, T, r, sigma1, sigma2, rho, w1, w2, n_simulations
):
    """
    Price a call option on a weighted basket of two assets.

    Payoff = max(w1*S1_T + w2*S2_T - K, 0)
    """
    # Generate correlated normals
    Z1 = np.random.standard_normal(n_simulations)
    Z2 = np.random.standard_normal(n_simulations)

    eps1 = Z1
    eps2 = rho * Z1 + np.sqrt(1 - rho**2) * Z2

    # Simulate terminal prices
    S1_T = S1_0 * np.exp((r - 0.5 * sigma1**2) * T + sigma1 * np.sqrt(T) * eps1)
    S2_T = S2_0 * np.exp((r - 0.5 * sigma2**2) * T + sigma2 * np.sqrt(T) * eps2)

    # Basket value
    basket_T = w1 * S1_T + w2 * S2_T

    # Payoffs
    payoffs = np.maximum(basket_T - K, 0)

    discount_factor = np.exp(-r * T)
    price = discount_factor * np.mean(payoffs)
    std_error = discount_factor * np.std(payoffs) / np.sqrt(n_simulations)

    return price, std_error

In[29]:

Code

np.random.seed(42)

# Equal-weighted basket of two assets
basket_price, basket_se = monte_carlo_basket_call(
    S1_0=100,
    S2_0=100,
    K=100,
    T=1.0,
    r=0.05,
    sigma1=0.2,
    sigma2=0.3,
    rho=0.5,
    w1=0.5,
    w2=0.5,
    n_simulations=100000,
)
basket_ci_width = 1.96 * basket_se

Out[30]:

Console

Basket call price: $11.1603 ± $0.1007

The basket option benefits from diversification. Even though one asset has 30% volatility, the combined basket has lower volatility due to imperfect correlation. This makes basket calls cheaper than calls on the individual assets with equivalent total notional.

Out[31]:

Visualization

Scatter plot of two positively correlated terminal asset prices, with out-of-the-money points below the dashed basket-strike boundary and in-the-money points above it. — Joint distribution of correlated terminal asset prices for the basket option. The upward-sloping, right-skewed cloud reflects positive correlation between the lognormal asset prices. The dashed line shows where the weighted basket equals the strike price, separating in-the-money scenarios (upper right) from out-of-the-money scenarios.

For higher dimensions with more than two assets, the Cholesky decomposition of the correlation matrix generalizes this approach systematically:

\boldsymbol{\epsilon} = \mathbf{L} \mathbf{Z}

where:

$\boldsymbol{\epsilon}$ : vector of correlated random variables
$\mathbf{L}$ : lower triangular Cholesky matrix
$\mathbf{Z}$ : vector of independent standard normal variables
$\mathbf{R}$ : correlation matrix ( $\mathbf{R} = \mathbf{L}\mathbf{L}^T$ )

The Cholesky decomposition factors the correlation matrix $\mathbf{R}$ into the product of a lower triangular matrix $\mathbf{L}$ and its transpose. By multiplying a vector of independent standard normals by $\mathbf{L}$ , we obtain a vector of correlated normals with exactly the desired correlation structure. This procedure generalizes efficiently to any number of assets.

Key ParametersLink Copied

The key parameters for multi-asset basket options are:

S: Initial asset prices ( $S_{1,0}, S_{2,0}$ ).
σ: Volatilities of the individual assets.
ρ: Correlation coefficient between asset returns.
w: Weights of the assets in the basket.