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Derivatives Pricing and Computational Finance / Path-Dependent Exotics Beyond Barriers

Derivation — Reducing the Asian PDE to One Dimension

The Asian PDE in (S,A,t)(S,A,t) is honest but expensive: a three-dimensional grid (two spatial plus time) is the generic cost of path dependence. For one commercially important contract — the European arithmetic average-strike (floating-strike) call — TCF shows that a similarity transform collapses the problem to a single spatial variable R=A/SR = A/S. This note carries that reduction: the ansatz V=S H(R,t)V = S\,H(R,t), the one-dimensional PDE for HH, the terminal and boundary conditions, and how the result is read back to option prices. The algebra is the template examiners expect when they ask you to "reduce the dimension" of a path-dependent PDE.
Learning objectives. By the end of this note you should be able to:
  • define the normalised state Rt=At/StR_t = A_t/S_t for continuous arithmetic averaging;
  • verify the separation ansatz V(S,A,t)=S H(R,t)V(S,A,t) = S\,H(R,t) and derive the reduced PDE for HH;
  • state and justify the boundary conditions at R=0R = 0 and R→∞R \to \infty;
  • recover V(S,A,t)V(S,A,t) from the numerical solution H(R,t)H(R,t).

In this lesson

  1. Setup — floating-strike arithmetic call
  2. The similarity ansatz
  3. Boundary conditions
  4. The complete boundary-value problem
  5. What does not reduce

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