Valuation models
Exotic options
Cash and asset digitals, single and double knock-in and knock-out barriers, double touch and no-touch, arithmetic and geometric Asians with fixed or average strike, floating and fixed lookbacks, Geske compound options, simple choosers and diluted warrants, with Greeks.
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Methodology
Conventions
- Black-Scholes with the curve's rate and the dividend yield or foreign rate as carry b = r − q.
- Barriers are continuous, or shifted by 0.5826 σ √Δt for discrete monitoring.
- Asians average discrete fixings: geometric exactly, arithmetic by two-moment matching; an average strike is an exchange option on the two lognormals.
- Double barriers and double touches sum the exact series for paths that stay between the barriers.
- Lookbacks are continuously monitored; compound options follow Geske; warrants are diluted calls (Galai-Schneller).
References
- Haug, The Complete Guide to Option Pricing Formulas, ch. 4
- Geske (1979), The valuation of compound options
- Galai and Schneller (1978), Pricing of warrants
Formulas
- Cash digital
- V = C e^(−rT) N(d2)
- In-out parity
- V(in) + V(out) = V(vanilla)
- Chooser
- V = c(K, T) + p(K e^(−b(T−t)), t)
- Compound (call on call)
- V = S e^((b−r)T₂) M(z₁, y₁; ρ) − K₂ e^(−rT₂) M(z₂, y₂; ρ) − K₁ e^(−rT₁) N(y₂)
- Warrant
- W = k N / (N + k M) · c(S + M W / N, K, T)