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Convergence Analysis of BTM Dr. DAI Min Why convergence analysis is important? • To better understand the BTM and the continuoustime model – American options – Lookback options • To develop new schemes – FSGM – Finite difference scheme Consistency of BTM and PDE model for vanilla options (I) • PDE model: • BTM: • Consistency: Consistency of BTM and PDE model for vanilla options (II) • Consistency of assumptions – Continuous-time assumption: – BTM assumption: – Connection: Consistency of BTM and PDE model for vanilla options (III) Consistency of BTM and PDE model for American options • From BTM to continuous-time model Continuous-time model for American options • PDE model (variational inequality): • Optimal stopping problem: Derivation of PDE model for American options • Delta-hedging Consistency of BTM and PDE model for barrier options • V=V(S,t): up-out option price before the barrier is hit • PDE model: • BTM: PDE models for Asian options (I) PDE models for Asian options (II) • Derivation (arithmetic Asian) – Ito lemma: – Arithmetic average: – Delta hedging Consistency of BTM and PDE model for Asian options • Geometric Asian • Arithmetic Asian PDE model for lookback options • Pricing model (lookback maximum) – V=V(S,A,t) – For S<A, – Neumann boundary condition at S=A, Consistency of BTM and PDE model for lookback options • BTM: • Case i) consistent with the PDE • Case ii) consistent with the Neumann boundary condition Convergence analysis • Consistency does not mean convergence • Convergence proof (not required) Convergence analysis (continued) • The analysis works for all (linear) BTMs for Europeanstyle products. Convergence analysis of BTM for American options • The BTM for American options is nonlinear. • A criterion: • Usually we only check i) and ii), which hold for BTM for American options. FSGM • BTM – option pricing is a backward procedure; – the evolution of the underlying price is forward • FSGM: a modified BTM confined to a lattice – at any lattice point, we generate a single-period (forward) binomial tree through which a backward pricing applies – interpolation is likely needed Convergence analysis of FSGM (involving interpolation) • Does the interpolation affect the convergence? • Consistency: Convergence analysis of FSGM (continued) Convergence analysis of FSGM (continued) • Consistency: • Monotonicity: – holds for the nearest point or linear interpolation – no longer true for quadratic interpolation • Conclusion: – The FSGM with linear interpolation is recommended – Quadratic interpolation can improve the order of consistency, but damage the monotonicity. No convergence is guaranteed in this case! Convergence analysis of FSGM (continued)