Valuation models
Equity exotics: forward starts to dispersion
Forward start options, Napoleons and reverse cliquets, Himalaya mountain ranges, outperformance options, corridor and conditional variance swaps, gamma swaps, correlation and dispersion swaps and accumulators, each written as a payoff script and valued by Monte Carlo under Black-Scholes, local vol, Merton jump-diffusion, Heston, Bates, Variance Gamma or NIG, with discrete cash dividends. Forward starts are checked against a Fourier-cosine price. The script that ran comes back with the result.
Inputs
| Asset | Spot | Vol % | Dividend % | |
|---|---|---|---|---|
Results
Every input and result then gets a + calc button.
Explain this result
Get a plain-English reading of the numbers above and what drives them.
Preview: rule-based reading. AI-assisted explanations run on the analytics core.
Methodology
Conventions
- Each product is written as a payoff script (shown with the result) and run on the scripted Monte Carlo.
- Levels are fractions of each asset's price today; variance uses daily log returns, annualized by the observations a year.
- Merton and Bates jumps, and Variance Gamma and NIG increments, are sampled exactly between dates; cash dividends drop the price on their ex-dates.
- Forward starts under Black-Scholes, Merton, Variance Gamma and NIG are also priced by Fourier-cosine expansion.
References
- Fang and Oosterlee (2008), A novel pricing method for European options based on Fourier-cosine series
- Bates (1996), Jumps and stochastic volatility
- Madan, Carr and Chang (1998), The Variance Gamma process
- Gatheral and Jacquier (2014), Arbitrage-free SVI volatility surfaces
Formulas
- Corridor variance
- σ² = (A/n) Σ rᵢ² 1{L ≤ Sᵢ₋₁/S₀ ≤ U}
- Gamma swap
- σ² = (A/n) Σ rᵢ² Sᵢ/S₀
- Dispersion
- Σ wᵢ σᵢ² − σ²_index
- Fourier-cosine
- V = e^{−rT} Σ′ Re[φ(uₖ) e^{iuₖ(x−a)}] Vₖ