Valuation models
Basket, spread and rainbow options
Correlated lognormal Monte Carlo with antithetic paths and a geometric control variate, standard error, per-asset delta, gamma and vega, correlation sensitivity, and the moment-matched, Kirk or Stulz closed form alongside.
Inputs
| Asset | Spot | Weight | Vol % | Dividend % | |
|---|---|---|---|---|---|
Results
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Methodology
Conventions
- Correlated lognormal assets drawn exactly at expiry, in antithetic pairs, with a fixed seed.
- With all weights positive the geometric basket, which has a closed form, is the control variate.
- Greeks reuse the same random numbers, so bumps are not swamped by noise.
References
- Glasserman, Monte Carlo Methods in Financial Engineering
- Kirk (1995)
- Stulz (1982)
Formulas
- Assets
- S_i(T) = S_i exp((r − q_i − σ_i²/2)T + σ_i √T (LZ)_i), LLᵀ = ρ
- Payoff
- max(φ(Σ w_i S_i(T) − K), 0)
- Rainbow payoff
- max(φ(max_i or min_i w_i S_i(T) − K), 0)
- Kirk
- σ² = σ₁² − 2ρσ₁σ₂ F₂/(F₂+K) + (σ₂ F₂/(F₂+K))²