The AmBer Algorithm for Optimal Stopping

The AmBer Algorithm for Optimal Stopping Mike Curran

Duality-Based Methods M. Haugh and L. Kogan, approximating pricing and exercising of high-dimensional American options: A duality approach, Operations Research 52 (2004) 258-270 L. C. G. Rogers, Monte Carlo valuation of American options, Math. Finance 12 (2002) 271-286

But no prescription for obtaining hedging martingales.

AmBer (American/Bermudan) ● Combines martingale hedging with backward induction ● Martingales constructed by rolling over simple instruments ● Each time step involves two optimizations

Motivation via Trivial Example ● Time steps 0, t, t+1 ● Single early exercise opportunity (at time t) ● p: payoff process ● Hedge with european

Motivation via Trivial Example

Motivation via Trivial Example

…..just as expected.

Motivation via Trivial Example Note that a value of zero for lagrange multiplier gives perfect foresight solution. In this case, the payoffs at t+1 will be higher on average than the expectation at time t for paths that are continued. Paths that are exercised at t will have higher expectations on average than the subsequent payoffs. As the lagrange multiplier is increased, less foresight is allowed until unity where foresight is eliminated.

Motivation via Trivial Example Salient Points: ● Hedging strategy dominates payoff ● Lagrange multipliers drive excess profit of hedging strategy to zero conditional on continuation/exercise

Motivation via Trivial Example Salient Points: ● Primal: instead of making pathwise comparisons we separate objective and hedging PnL constraint ● Mean hedging cost must equal mean payoff for continuation paths

Two Optimizations per Time Step ● First optimization computes 2-step bermudan values for each path ● Second optimization updates exercise times to include current exercise

First Optimization ● Induction Hypothesis: exercise time for each path solved for (t+1,T). ● “Forward Looking” in that we solve for 2-step bermudan beginning at time t ● Decouple intervals (t,t+1) and (t+1,t+2)

First Optimization Hedges ● Hedge with one and two-step europeans leading to two constraints ● Additionally, impose that hedging with maximum european has mean cost equal to that of hedging with minimum european ● “Payoff” at t+1 taken as max of payoff and then one-step european

First Optimization Hedges Actual, “real life” hedge would more sensibly be with more valuable european since the bermudan must be a least as valuable as the more valuable european. For example, for vanilla payoff, in-the-money better hedged with one-step european while out-of-the-money better hedged with two-step european. But constrain so that mean hedging cost equal irrespective of hedging with max or min.

Second Optimization Hedges ● Roll over simpler instruments until next exercise for each path ● One-step europeans insure coverage of payoff at next step ● Two-step bermudans insure coverage of payoff OR establishment of subsequent one-step european rollover

Second Optimization Hedges So one-step europeans guarantee coverage of next step payoffs while two-step bermudans guarantee coverage of one-step europeans at the next time step as well. These work in concert in a staggered fashion to ensure that, at every step, the “pump” of the one-step european rollover process is “reprimed” at every step. Additionally we require the two-step bermudans to be conditionally consistent with both of their component europeans. These values are available because they have been estimated during previous iterations.

Computation ● Vast majority of computation is in estimating europeans at every time step and for every path ● Optimizations very fast and insensitive to dimension of problem ● Greatly mitigates curse of dimensionality