Valuation models

Local volatility

Dupire local vol read off an implied volatility surface, then vanilla, American, digital and barrier options valued by finite differences, against Black-Scholes at the surface's vol.

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Methodology

Conventions

  • Dupire's local vol is read off the implied surface's total variance on a grid of expiries and strikes.
  • The option is valued by Crank-Nicolson on ln S, with two implicit steps first to smooth the payoff.
  • Barriers are continuously monitored; knock-ins are the vanilla less the knock-out.

References

  • Dupire (1994), Pricing with a Smile
  • Gatheral, The Volatility Surface, ch. 1

Formulas

Local variance
σ²(loc) = ∂w/∂T / (1 − (y/w) ∂w/∂y + ¼(−¼ − 1/w + y²/w²)(∂w/∂y)² + ½ ∂²w/∂y²)
PDE
V_t + ½ σ²(loc) V_xx + (r − q − ½ σ²(loc)) V_x − r V = 0