section6 (2)

section6 (2) - PDE for Finance Notes Spring 2011 –...

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Unformatted text preview: PDE for Finance Notes, Spring 2011 – Section 6 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. For use only in connection with the NYU course PDE for Finance, G63.2706. Prepared in 2003, minor updates made in 2011. Revised and extended 4/3/2011. Optimal stopping and American options. Optimal stopping refers to a special class of stochastic control problems where the only decision to be made is “when to stop.” The decision when to sell an asset is one such problem. The decision when to exercise an Amer- ican option is another. Mathematically, such a problem involves optimizing the expected payoff over a suitable class of stopping times. The value function satisfies a “free boundary problem” for the backward Kolmogorov equation. We shall concentrate on some simple yet representative examples which display the main ideas, namely: (a) a specific optimal stopping problem for Brownian motion; (b) when to sell a stock which undergoes log-normal price dynamics; and (c) the pricing of a perpetual American option. At the end we discuss how the same ideas apply to the pricing of an American option with a specified maturity. My discussion of (a) borrows from notes Raghu Varadhan prepared when he taught PDE for Finance; my treatment of (b) is a dumbed- down version of one in Oksendal’s book “Stochastic Differential Equations;” the perpetual American put is discussed in many places. My main suggestion for further reading in this area is however F-R Chang’s excellent book “Stochastic Optimization in Continuous Time” (Cambridge Univ Press), which is on reserve in the CIMS library. It includes many examples of stochastic control, some close to the ones considered here and others quite different. *********************** Optimal stopping for 1D Brownian motion. Let y ( t ) be 1D Brownian motion starting from y (0) = x . For any function f , we can consider the simple optimal stopping problem u ( x ) = max τ E y (0)= x e- τ f ( y ( τ ) . Here τ varies over all stopping times . We have set the discount rate to 1 for simplicity. We first discuss some general principles then obtain an explicit solution when f ( x ) = x 2 . What do we expect? The x-axis should be divided into two sets, one where it is best to stop immediately , the other where it is best to stop later . For x in the stop-immediately region the value function is u ( x ) = f ( x ) and the optimal stopping time is τ = 0. For x in the stop-later region the value function solves a PDE. Indeed, for Δ t sufficiently small (and assuming the optimal stopping time is larger than Δ t ) u ( x ) ≈ e- Δ t E y (0)= x [ u ( y (Δ t ))] . By Ito’s formula E y (0)= x [ u ( y ( t )] = u ( x ) + Z t 1 2 u xx ( y ( s )) ds....
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section6 (2) - PDE for Finance Notes Spring 2011 –...

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