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jfuqua
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What is a free boundary problem and what is the optimal stopping time for American option?

January 29th, 2006, 8:24 pm

These might be of use:Van Moerbeke Pierre 'On Optimal Stopping & Free Boundary Problems' Archive for Rational Mechanics & Analysis 60(1975/76)Van Moerbeke Pierre 'Optimal Stopping & Free Boundary Problems' Rocky Mountain J. of Mathematics 1974 Wong D. 'Generalized Optimal Stopping Problems & Financial Markets' 97 Longman Pitman177 Springer 92 Xue X. 'Optimal Stopping of Continuous-Parameter Stochastic Processes' Proc. China-Japan Symp.Stats 84Boetius F., Michael Kohlmann 'Connections Between Optimal Stopping & Singular Stochastic Control' SP&A 77 (1998) <local time, options>Cutland Nigel et al 'Convergence of Snell Envelopes and Critical Prices in the American Put' (in Dempster M.,S. Pliska (ed) 'Math. of Derivative Securities'El Karoui Nicole, Ioanis Karatzas 'The Optimal Stopping Problem for a General American Put-Option' Mathematical Finance (ed) Davis SpringerFitt A., Paul Wilmott, Jeff Dewynne 'An Integral Equation for the Value of a Stop-Loss Option' Gatarek Dariusz, A. Swiech 'Optimal Stopping in Hilbert Spaces & Pricing of American Options' Math. Methods of O.R. 99Hesse C. 'Approximate Expected Hitting Times of Certain State Variables in Physics & Economics' App. Math. ModelingJacka Saul 'Optimal Stopping & the American Put' Mathematical Finance V.1, No 2 April 91 Jacka Saul, J. Lynn 'Finite Horizon Optimal Stopping,Obstacle Problems & Shape of Continutation Region' S&SR 1992Karatzas Ioannis, Hui Wang 'A Barrier Option of American Type' App. Math & Opt 2000 <optimal stopping/singular control,variational inequality>Mordecki Ernesto 'Optimal Stopping for a Compound Poisson Process with Exponential Jumps' <Amer. Options>Mordecki Ernesto 'Optimal Stopping for a Diffusion with Jumps' Finance & Stochastics 2/99, <Amer. Options>Mordecki Ernesto 'Perpetual Options for Levy Processes in the Bachelier Model' <option-perpetual, optimal stopping,American,prophet inequalities> 10/2000Mordecki Ernesto 'Ruin Probabilities & Optimal Stopping for a Diffusion with Jumps'Pham Huyen 'Optimal Stopping, Free Boundaries & American Option in a Jump-Diffusion Model' Appl. Math & Optim (97)Bank Peter 'Universal Exercise Signals for American Options: A New Approach to Optimal Stopping' Bachelier Conference 2004Bather John 'Bounds on Optimal Stopping Times for the American Put' U. Suxxex 1997 Bather John 'Optimal Stopping Problems for Brownian Motion' Advances in Appl. Prob. 1970 Battauz Anna, Maurizio Pratelli 'Optimal Stopping And American Options With Discrete Dividends And Exogenous Risk' Bachelier Conference 2004Ekstrom Erik 'Convexity Of The Optimal Stopping Boundary For The American Put Option' Bachelier Conference 2004Guo Xin, Larry Shepp 'Some Optimal Stopping Problems with Non-Trivial Boundaries for Pricing Exotic Options' J. Appl. Prob. 2002Marcozzi Michael, S. Choi, C.S. Chen 'RBF & Optimal Stopping Problems;An Application to the Pricing of Vanilla Options on One Risky Asset' Boundary Element Tech. XIV, ed. C. Chen 'Computational Mechanics Pub' 1999Mordecki Ernesto 'Optimal Stopping & Perpetual Options for Levy Proceses' F&S 2002
 
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ppauper
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What is a free boundary problem and what is the optimal stopping time for American option?

January 29th, 2006, 8:32 pm

QuoteOriginally posted by: ppauperQuoteOriginally posted by: CuchulainnIf we can define the problem in the usual way then a solution might be possible IMHO. Is there an article that does this (e.g. Detemple's book maybe)from your google link, the (2) papers by kwok seem to be the thingedit: google link deleted per request
 
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Cuchulainn
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What is a free boundary problem and what is the optimal stopping time for American option?

February 6th, 2006, 4:20 pm

QuoteOriginally posted by: ppauperQuoteOriginally posted by: CuchulainnProblems with several free boundaries (e.g. 2) are common in heat transfer problems. The the example of melting ice. In the one-phase case we have either ice or water phase and hence only 1 free boundary (like American options). But in some problems we have a 'mushy' region bewten ice and water. There are now two FB B1(t) and B2(t) and we can transform as before and use FDM, for example. We can define x1 = S/B1(t) and x2 = S/B2(t).also interesting would be an american straddle which I suspect would be like a block of ice melting both from above and belowYes I think this is a good analogy. The way I see it is a block of ice surrounded on both sides by water; a moving/free boundary on each side. Most examples in books only examine semi-infinite water/liquid interfaces, from what I have seen.For the American straddle we have one BS PDE and we expect movements DOWN or UP, so we need a put and a call. Each one will have a free boundary. On one side we get P = K - S (or C = S - K) when we can exercise while on the other side we have BS PDE. On the free boundaries we have 2 (usual) smooth pasting conditions, namely the value of the option there and its flux out of the PDE region. So we have 4 conditions which feels OK (2 for the option and 2 for B1 and B2). Means intuitively I can solve the nonlinear system.Now, a generalisation of the Landau transformation is now x = (S - B1(t))/(B2(t) - B1(t))and then we transform the linear PDE in a moving medium to a nonlinear one in the interval [0,1]. We can solve this numerically using standard numerical schemes.This would be thus a kind of front-fixing method.Is this the financial model you were thinking about?Edit: there might be a problems if the free boundaries 'touch' each other, like the ice cube forming a 'tip'. Then the variable x becomes large.
Last edited by Cuchulainn on February 5th, 2006, 11:00 pm, edited 1 time in total.
 
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ppauper
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What is a free boundary problem and what is the optimal stopping time for American option?

February 7th, 2006, 2:11 pm

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: ppauperalso interesting would be an american straddle which I suspect would be like a block of ice melting both from above and belowEdit: there might be a problems if the free boundaries 'touch' each other, like the ice cube forming a 'tip'. Then the variable x becomes large.that's exactly what happens when the dividend yield is equal to the risk free rate and both free boundaries start from the strike price
 
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Cuchulainn
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What is a free boundary problem and what is the optimal stopping time for American option?

February 21st, 2006, 7:22 pm

Can one cast a strip, for example as a free boundary problem?(would there be some 'perma' frost here)
 
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ppauper
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What is a free boundary problem and what is the optimal stopping time for American option?

February 23rd, 2006, 1:57 pm

QuoteOriginally posted by: CuchulainnCan one cast a strip, for example as a free boundary problem?(would there be some 'perma' frost here)why not ?if a strip solidifying on both sides perhaps ?
 
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Cuchulainn
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What is a free boundary problem and what is the optimal stopping time for American option?

March 4th, 2006, 2:38 pm

Another aspect with free boundaries is the presence of discrete dividends. Then the free boundary is not as smooth as a put with constant dividend. My question is: how is this manifested? Do we get different pasting conditions or is there a formula for the optimal stopping time?What is the analogy with ice? In this case the boundary between exercise/no exercise becomes blurred, a kind of 'mushy' region (bit ice, bit water).And can we get good approximations to the greeks (delta, gamma) in this case?
Last edited by Cuchulainn on March 3rd, 2006, 11:00 pm, edited 1 time in total.
 
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ppauper
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What is a free boundary problem and what is the optimal stopping time for American option?

March 13th, 2006, 1:25 pm

QuoteOriginally posted by: CuchulainnWhat is the analogy with ice? In this case the boundary between exercise/no exercise becomes blurred, a kind of 'mushy' region (bit ice, bit water).now I think about it, I vaguely remember having seen folks like herbert huppert and andrew woods talking about mushy layers.Anyone able to tell us in a few words (as opposed to linking one of huppert's papers which would take several hours to read) why there's a mushy interface instead of a crisp free boundary ?
 
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NamelessWonder
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What is a free boundary problem and what is the optimal stopping time for American option?

March 28th, 2006, 3:19 pm

I dont know if this link has been posted before but I thought this paper is useful in visualising the free boundary for american putsMeyer