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On the impact of the penalty on the cancellable American options
Tsvetelin Zaevski ORCID icon link to view author Tsvetelin Zaevski details  

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https://doi.org/10.15559/25-VMSTA286
Pub. online: 21 October 2025      Type: Research Article      Open accessOpen Access

Received
13 January 2025
Revised
4 October 2025
Accepted
4 October 2025
Published
21 October 2025

Abstract

The cancellable American options, also known as game options, are financial instruments that give a canceling right to the option’s writer in addition to the existing such holder’s right. The writer owes some penalty above the usual option payoff for using this right. We assume that this penalty consists of three parts – a proportion of the usual payoff, some number of shares of the underlying asset, and a fixed amount. It turns out that a cancellable option can be of one of the following three types – a regular American option, an American-style derivative that expires either at the maturity or when the underlying asset reaches the strike, or a real cancellable option. In this paper, the impact of the penalty on the option’s type is investigated. The perpetual case is only explored having in mind that it determines the kind of the finite maturity instruments in some sense.

References

 
Baurdoux, E.J., Kyprianou, A.E.: Further calculations for Israeli options. Stoch. Int. J. Probab. Stoch. Process. 76(6), 549–569 (2004). MR2100021. https://doi.org/10.1080/10451120412331313438
 
Bensoussan, A., Friedman, A.: Nonlinear variational inequalities and differential games with stopping times. J. Funct. Anal. 16(3), 305–352 (1974). https://www.sciencedirect.com/science/article/pii/0022123674900767. ISSN 0022-1236. MR0354049. https://doi.org/10.1016/0022-1236(74)90076-7
 
Bensoussan, A., Friedman, A.: Nonzero-sum stochastic differential games with stopping times and free boundary problems. Trans. Am. Math. Soc. 231(2), 275–327 (1977). http://www.jstor.org/stable/1997905. ISSN 0002-9947. MR0453082. https://doi.org/10.2307/1997905
 
Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973). MR3363443. https://doi.org/10.1086/260062
 
Darling, D.A., Siegert, A.J.F.: The first passage problem for a continuous Markov process. Ann. Math. Stat. 624–639 (1953). MR0058908. https://doi.org/10.1214/aoms/1177728918
 
Dolinsky, Y.: On shortfall risk minimization for game options. Mod. Stoch. Theory Appl. 7(4), 379–394 (2020). ISSN 2351-6046. MR4195642. https://doi.org/10.15559/20-VMSTA164
 
Dumitrescu, R., Quenez, M.-C., Sulem, A.: Game options in an imperfect market with default. SIAM J. Financ. Math. 8(1), 532–559 (2017). MR3679314. https://doi.org/10.1137/16M1109102
 
Dynkin, E.B.: A game-theoretic version of an optimal stopping problem. Dokl. Akad. Nauk SSSR 185(1), 16–19 (1969). in Russian. MR0241121
 
Ekström, E.: Properties of game options. Math. Methods Oper. Res. 63(2), 221–238 (2006). ISSN 1432-5217. MR2264747. https://doi.org/10.1007/s00186-005-0027-3
 
Ekström, E., Peskir, G.: Optimal stopping games for Markov processes. SIAM J. Control Optim. 47(2), 684–702 (2008). MR2385859. https://doi.org/10.1137/060673916
 
Ekström, E., Villeneuve, S.: On the value of optimal stopping games. Ann. Appl. Probab. 16(3), 1576–1596 (2006). MR2260074. https://doi.org/10.1214/105051606000000204
 
Emmerling, T.J.: Perpetual cancellable American call option. Math. Finance 22(4), 645–666 (2012). MR2968279. https://doi.org/10.1111/j.1467-9965.2011.00479.x
 
Friedman, A.: Stochastic games and variational inequalities. Arch. Ration. Mech. Anal. 51(5), 321–346 (1973). MR0351571. https://doi.org/10.1007/BF00263039
 
Gapeev, P.V., Lerche, H.R.: On the structure of discounted optimal stopping problems for one-dimensional diffusions. Stoch. Int. J. Probab. Stoch. Process. 83(4–6), 537–554 (2011). MR2842594. https://doi.org/10.1080/17442508.2010.532874
 
Gapeev, P.V.: The spread option optimal stopping game. In: Exotic Option Pricing and Advanced Lévy Models. pp. 293–305. (2005). MR2343219
 
Gapeev, P.V., Li, L., Wu, Z.: Perpetual American cancellable standard options in models with last passage times. Algorithms 14(1), (2021). https://www.mdpi.com/1999-4893/14/1/3. ISSN 1999-4893. MR4213424. https://doi.org/10.3390/a14010003
 
Guo, I., Rutkowski, M.: Arbitrage-free pricing of multi-person game claims in discrete time. Finance Stoch. 21(1), 111–155 (2017). MR3590704. https://doi.org/10.1007/s00780-016-0315-1
 
Guo, P.: Pricing of the quanto game option with Asian feature. J. Finance Account. 8(3), 143 (2020). https://doi.org/10.11648/j.jfa.20200803.15
 
Guo, P., Chen, Q., Guo, X., Fang, Y.: Path-dependent game options: a lookback case. Rev. Deriv. Res. 17(1), 113–124 (2014). https://doi.org/10.1007/s11147-013-9092-6
 
Guo, P., Zhang, J., Wang, Q.: Path-dependent game options with Asian features. Chaos Solitons Fractals 141, 110412 (2020). MR4171636. https://doi.org/10.1016/j.chaos.2020.110412
 
Hamadène, S.: Mixed zero-sum stochastic differential game and American game options. SIAM J. Control Optim. 45(2), 496–518 (2006). MR2246087. https://doi.org/10.1137/S036301290444280X
 
Jacka, S.D.: Optimal stopping and the American put. Math. Finance 1(2), 1–14 (1991). http://dx.doi.org/10.1111/j.1467-9965.1991.tb00007.x.ISSN 1467-9965. https://doi.org/10.1111/j.1467-9965.1991.tb00007.x
 
Jacka, S.D.: Finite-horizon optimal stopping, obstacle problems and the shape of the continuation region. Stoch. Int. J. Probab. Stoch. Process. 39(1), 25–42 (1992). MR1293300. https://doi.org/10.1080/17442509208833761
 
Kallsen, J., Kühn, C.: Pricing derivatives of American and game type in incomplete markets. Finance Stoch. 8(2), 261–284 (2004). MR2048831. https://doi.org/10.1007/s00780-003-0110-7
 
Karatzas, I., Shreve, S.: Methods of Mathematical Finance. Springer, New York (1998). MR1640352. https://doi.org/10.1007/b98840
 
Karatzas, I., Sudderth, W.: Stochastic games of control and stopping for a linear diffusion. In: Random Walk, Sequential Analysis And Related Topics: A Festschrift in Honor of Yuan-Shih Chow, pp. 100–117. World Scientific, (2006). MR2367702. https://doi.org/10.1142/9789812772558_0007
 
Kifer, Y.: Game options. Finance Stoch. 4(4), 443–463 (Aug. 2000). ISSN 0949-2984. MR1779588. https://doi.org/10.1007/PL00013527
 
Kifer, Y.: Dynkin’s games and Israeli options. ISRN Probability and Statistics (2013, 2013)
 
Kim, I.J.: The analytic valuation of American options. Rev. Financ. Stud. 3(4), 547–572 (1990). ISSN 08939454, 14657368. http://www.jstor.org/stable/2962115. https://doi.org/10.1093/rfs/3.4.547
 
Kühn, C., Kyprianou, A.E.: Callable puts as composite exotic options. Math. Finance 17(4), 487–502 (2007). MR2352903. https://doi.org/10.1111/j.1467-9965.2007.00313.x
 
Kühn, C., Kyprianou, A.E., Van Schaik, K.: Pricing Israeli options: a pathwise approach. Stoch. Int. J. Probab. Stoch. Process. 79(1–2), 117–137 (2007). MR2290401. https://doi.org/10.1080/17442500600976442
 
Kunita, H., Seko, S.: Game call options and their exercise regions. Technical report, Nanzan Academic Society, Mathematical Sciences and Information Engineering, (2004).
 
Kwok, Y.-K.: Mathematical Models of Financial Derivatives. Springer-Verlag, (2008). MR2446710
 
Kyprianou, A.E.: Some calculations for Israeli options. Finance Stoch. 8(1), 73–86 (2004). MR2022979. https://doi.org/10.1007/s00780-003-0104-5
 
Lehoczky, J.P.: Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Probab. 601–607 (1977). MR0458570. https://doi.org/10.1214/aop/1176995770
 
McKean, H.P.: A free boundary problem for the heat equation arising from a problem in mathematical economics. Ind. Manag. Rev. 6(2), 32–39 (1965)
 
Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 141–183 (1973). MR0496534. https://doi.org/10.2307/3003143
 
Palmowski, Z., Stȩpniak, P.: Last-passage American cancelable option in Lévy models. J. Risk Financ. Manag. 16(2), 82 (2023). https://doi.org/10.3390/jrfm16020082
 
Peskir, G.: Optimal stopping games and Nash equilibrium. Theory Probab. Appl. 53(3), 558–571 (2009). MR2759714. https://doi.org/10.1137/S0040585X97983821
 
Shiryaev, A.N., Kabanov, Y.M., Kramkov, D.O., Mel’nikov, A.V.: Toward the theory of pricing of options of both European and American types. II. Continuous time. Theory Probab. Appl. 39(1), 61–102 (1995). MR1348191. https://doi.org/10.1137/1139003
 
Suzuki, A., Sawaki, K.: The pricing of perpetual game put options and optimal boundaries. In: Recent Advances in Stochastic Operations Research, pp. 175–188. World Scientific, River Edge, NJ, USA (2007). MR2313198. https://doi.org/10.1142/9789812706683_0012
 
van Moerbeke, P.: On optimal stopping and free boundary problems. Adv. Appl. Probab. 5(1), 33–35 (1973). MR2940361. https://doi.org/10.2307/1425961
 
Yam, S.C.P., Yung, S.P., Zhou, W.: Game call options revisited. Math. Finance 24(1), 173–206 (2014). MR3157693. https://doi.org/10.1111/mafi.12000
 
Zaevski, T.: Perpetual cancellable American options with convertible features. Mod. Stoch. Theory Appl. 10(4), 367–395 (2023). ISSN 2351-6046 (print), 2351-6054 (online). https://www.vmsta.org/journal/VMSTA/article/273/read. MR4655406. https://doi.org/10.15559/23-VMSTA230
 
Zaevski, T.: On the ϵ-optimality of American options. China Finance Review International, ahead–of–print, (2025). https://doi.org/10.1108/CFRI-06-2024-0361
 
Zaevski, T.S.: Perpetual game options with a multiplied penalty. Commun. Nonlinear Sci. Numer. Simul. 85, 105248 (2020). ISSN 1007-5704 (print), 1878-7274 (online). http://www.sciencedirect.com/science/article/pii/S1007570420300812. MR4074142. https://doi.org/10.1016/j.cnsns.2020.105248
 
Zaevski, T.S.: Discounted perpetual game call options. Chaos Solitons Fractals 131, 109503 (2020). ISSN 0960-0779 (print), 1873-2887 (online). http://www.sciencedirect.com/science/article/pii/S0960077919304552. MR4065328. https://doi.org/10.1016/j.chaos.2019.109503
 
Zaevski, T.S.: Discounted perpetual game put options. Chaos Solitons Fractals 137, 109858 (2020). ISSN 0960-0779 (print), 1873-2887 (online). http://www.sciencedirect.com/science/article/pii/S0960077920302587. MR4099368. https://doi.org/10.1016/j.chaos.2020.109858
 
Zaevski, T.S.: A new approach for pricing discounted American options. Commun. Nonlinear Sci. Numer. Simul. 97, 105752 (2021). ISSN 1007-5704 (print), 1878-7274 (online). https://www.sciencedirect.com/science/article/pii/S1007570421000630. MR4212896. https://doi.org/10.1016/j.cnsns.2021.105752
 
Zaevski, T.S.: Pricing finite maturity game call options with convertible features. Commun. Stat., Simul. Comput. 1–24 (2025). https://doi.org/10.1080/03610918.2025.2488972

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Keywords
Cancellable American options game options optimal boundaries optimal strategies impact of the penalty

MSC2020
42A38 60G40 60J65

Funding
This study is financed by the European Union-NextGenerationEU, through the National Recovery and Resilience Plan of the Republic of Bulgaria, project No BG-RRP-2.004-0008.

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