Abstract
This paper develops variational inequality approach to establish optimal stopping problems arising from the pricing of stock loan and capped stock loan models. The valuations of the stock loans and ranges of reasonable values of key parameters such as loan sizes, interest rates, fees for providing such a service and quantity of this automatic termination clause and relationships among these parameters as well as the optimal terminable stopping times are explicitly derived. The paper also gives two numerical samples to portray the results and the difference between stock loan model and capped stock loan model.
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References
Borodin AN, Salminen P (2002) Handbook of Brownian motion-facts and formulae. Probability and its applications, 2nd edn. Birkhäuser, Basel
Broadie M, Detemple J (1995) American capped call options on dividend-paying asset. Rev Financ Stud 8(1):161–191
Broadie M, Detemple J (1997) The valuation of American options on multiple assets. Math Finance 7:241–286
Dai M, Xu ZQ (2010) Optimal redeeming strategy of stock loans with finite maturity. Mathematical Finance, doi:10.1111/j.1467-9965.2010.00449.x
Dayanik S, Karatzas I (2003) On the optimal stopping problems for one-dimensional diffusions. Stoch Process Appl 107(2):173–212
Gerber H, Shiu E (1996) Martingale approach to pricing perpetual American options on two stocks. Math Finance 3:87–106
Liang Z, Wu W, Jiang S (2010) Stock loan with automatic termination clause, cap and margin. Comput Math Appl 60:3160–3176
Karatzas I, Shreve SE (1991) Brownian motion and stochastic calculus. Springer, New York
Karatzas I, Shreve SE (1998) Methods of mathematical finance. Springer, New York
Øksendal B, Sulem A (2005) Applied stochastic control of jump diffusions. Springer, New York
Xia JM, Zhou XY (2007) Stock loans. Math Finance 17(2):307–317
Zhang Q, Zhou XY (2009) Valuation of stock loans with regime switching. SIAM J Control Optim 48(3):1229–1250
Liu G, Xu Y (2010) Capped stock loans. Comput Math Appl 59:3548–3558
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Liang, Z., Wu, W. Variational inequalities in stock loan models. Optim Eng 13, 459–470 (2012). https://doi.org/10.1007/s11081-011-9165-z
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DOI: https://doi.org/10.1007/s11081-011-9165-z