Backward Induction for Bermudan Options in Black–Scholes

Summary

The document poses a pricing question about approximating an American option with daily Bermudan exercise dates in the Black–Scholes model. It sketches a backward-induction approach: start at maturity with the put payoff, then at each earlier exercise date compare immediate exercise value with the Black–Scholes value of continuing for one time step. The value at that date is the larger of the two.

The question proposes finding the exercise boundary numerically and then analytically integrating the resulting piecewise value function to step backward again. However, the document contains no answer or completed derivation, so it does not establish whether that integration approach works or provide a pricing formula. It is useful as a statement of the exercise-versus-continuation recursion and a potential computational idea, but leaves the method, boundary calculation, and accuracy relative to an American option unresolved.

Key ideas

• A Bermudan approximation allows exercise only at specified dates, such as daily intervals.
• At each exercise date, compare immediate exercise payoff with the continuation value.
• Backward induction can begin with the terminal option payoff and recurse to earlier dates.
• The document proposes a numerical exercise boundary but does not provide a solution or validation.

Tags

# Bermudan pricing in Black-Scholes


# Bermudan pricing in Black-Scholes












Is there an "analytical" method to price American options (approximated as daily Bermudans) in the Black-Scholes model using backward induction?

$$V_T(S) = \max(K-S, 0)$$ $$V_{T-\Delta t}(S) = \max(K-S, \operatorname{BS}(S, K, \Delta t, \sigma,r, q))$$

Then you could find the exercise boundary point (probably numerically) and knowing that, analyically integrate the piecewise function back another step to time $T-2\Delta t$...

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.