Class WhalleyWilmott

java.lang.Object
com.quantfinlib.hedging.WhalleyWilmott
public final class WhalleyWilmott extends Object
Whalley-Wilmott optimal hedge bands — the answer to the question every band hedger otherwise guesses: how wide should the no-trade band around delta be? Hedging continuously is bankruptcy by transaction costs; hedging never is a naked option. The asymptotic optimum (Whalley & Wilmott 1997, small proportional costs) puts the half-width at 
  band = ( (3/2) · e^{−r(T−t)} · k · S · Γ² / λ )^{1/3}
(Whalley and Wilmott 1997, Mathematical Finance 7(3) 307-324; the discount factor appears because costs paid today are compared with a terminal-wealth utility). Units: k is a fraction of traded notional, S and Γ are per share, and λ is the exponential-utility risk aversion in 1/currency (utility −e^{−λ·W}), so λ·S is dimensionless. A λ of 1 means a one currency unit change in wealth is already material; for a book measured in dollars a realistic λ is of order 1/(capital at risk), e.g. 1e-6. — wider when trading is expensive (k, the proportional cost) and when gamma churns the delta (Γ²), narrower when you are more risk averse (λ). The cube root is the interesting part: costs must move by 8x to move the band by 2x, which is why band width is remarkably stable across venues.

The policy is as important as the width: when the hedge drifts outside the band, trade back to the nearest edge, not to the center — hedging to delta itself throws away the band's whole point (you would pay the spread again on the next tick's drift). Zero gamma degenerates cleanly: zero band, always hedge exactly to delta.

Pairs with the band executors: trading.AutoHedger and crb.CrbAutoHedger take a band as configuration — this class is where that number should come from. Asymptotic result: for very large costs or tiny risk aversion the expansion degrades (stated, not hidden). Static, deterministic, research lane.

  • Method Details

    • bandHalfWidth

      public static double bandHalfWidth(double spot, double gamma, double costRate, double riskAversion)
      The optimal no-trade half-width around delta with zero discounting (r·(T−t) = 0); see bandHalfWidth(double, double, double, double, double, double).

    • bandHalfWidth

      public static double bandHalfWidth(double spot, double gamma, double costRate, double riskAversion, double rate, double timeToExpiry)
      The optimal no-trade half-width around delta, in shares.
      Parameters:
      spot - underlying price, > 0
      gamma - book gamma per share (signed; only |Γ| matters), finite
      costRate - proportional transaction cost k (e.g. 0.0005 = 5 bps), > 0
      riskAversion - λ > 0 in 1/currency (bigger = tighter hedging)
      rate - continuously compounded rate r, finite
      timeToExpiry - T − t in years, ≥ 0

    • rebalance

      public static WhalleyWilmott.Action rebalance(double currentHedge, double delta, double bandHalfWidth)
      The hedge-to-the-edge policy: hold inside the band, trade to the nearest edge outside it.
      Parameters:
      currentHedge - the position currently held against the book
      delta - the book's current delta (the band's center)
      bandHalfWidth - from bandHalfWidth(double, double, double, double), ≥ 0