PDE options: any payoff on a grid

Any payoff expression in the asset price (calls, puts, digitals, power options, straddles, butterflies), European, American or Bermudan exercise and continuous knock-out barriers with rebates, solved by Crank-Nicolson under Black-Scholes, CEV or local volatility and by Modified Craig-Sneyd ADI under Heston. Value, early exercise premium, exercise boundary, delta, gamma and vega.

Inputs

PayoffPayoff at expiry, in SAny expression in S and the parameters: + − * / **, comparisons, max, min, abs, exp, log, sqrt.Parameters

Currency

Spot

Dividend yield%

Expiryy

Units

Exercise

Knock-out below (0: none)

Knock-out above (0: none)

Model

Volatility%

Spot grid nodes

Time steps

Curve

Curve: flat rate% cc

Results

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Methodology

Conventions

Crank-Nicolson on ln S under Black-Scholes, CEV or local vol, with two implicit steps at the start and after each exercise date.
Heston on a (ln S, variance) grid by the Modified Craig-Sneyd ADI scheme.
Early exercise by projection on the payoff; barriers are grid edges held at the rebate.

References

in 't Hout and Foulon (2010), ADI schemes for the Heston model
Duffy, Finite Difference Methods in Financial Engineering

Formulas

PDE: Vₜ + ½σ²S²V_SS + (r − q)S V_S − rV = 0
Exercise: V = max(V, payoff(S)) on exercise dates