## The Russian option: Reduced regret (1993)

Venue: | Ann. Appl. Probab |

Citations: | 61 - 3 self |

### BibTeX

@ARTICLE{Shiryaev93therussian,

author = {A. N. Shiryaev and Shepp and A. N. Shiryaev},

title = {The Russian option: Reduced regret},

journal = {Ann. Appl. Probab},

year = {1993},

pages = {631--640}

}

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### Citations

3689 |
The Pricing of Options and Corporate Liabilities
- Black, Scholes
- 1973
(Show Context)
Citation Context ... and Wt is a standard Wiener process. The process X,swhich satisfies the stochastic differential equation dX = ouXdW + ,uXdt,sforms the basis for the famous option pricing theory of Black and Scholes =-=[4,s5]-=-. The parameters ,u, called the drift, and o-, the volatility, are assumedsknown.sWe solve the following mathematical problem, where r > 0 and s ? x aresgiven and we want to find a stopping time r E [... |

1121 |
Continuous Martingales and Brownian motion, Grundlehren der mathematischen Wissenschaften 293
- Revuz, Yor
- 1991
(Show Context)
Citation Context ...equality in (2.12) is thus an equality because of (2.15) and (2.4). Thessecond inequality will also be shown to be an equality if we can prove that Yt,s0 < t < r, is a uniformly integrable martingale =-=[15]-=-. We supply a (simpler)sdirect proof by showing thats(2.16) E sup Yt <oo,sO<t<cswhich will directly prove the equality we need.sTo prove (2.16) we note thatsYt= e-rt V(Xt,St)s(2.17) < e-rtV(St, St)s= ... |

128 | Rational theory of warrant pricing - Samuelson - 1965 |

85 | The Valuation of Option Contracts and a Test of Market Efficiency - Black, Scholes - 1972 |

38 |
Some solvable stochastic control problems
- BENEŠ, SHEPP, et al.
- 1980
(Show Context)
Citation Context ... insRussia. Second, our solution of the stopping problem (1.2) is derived by thesso-called principle of smooth fit, which was first enunciated by the greatsRussian mathematician A. N. Kolmogorov; cf. =-=[3]-=- and [10]. The Russiansoption is characterized by "reduced regret" because the owner is paid thesmaximum stock price up to the time of exercise and hence feels less remorsesat not having exercised at ... |

30 |
On Stefan’s problem and optimal stopping rules for Markov processes. Theory Probab
- Grigelionis, Shiryaev
- 1966
(Show Context)
Citation Context ...ia. Second, our solution of the stopping problem (1.2) is derived by thesso-called principle of smooth fit, which was first enunciated by the greatsRussian mathematician A. N. Kolmogorov; cf. [3] and =-=[10]-=-. The Russiansoption is characterized by "reduced regret" because the owner is paid thesmaximum stock price up to the time of exercise and hence feels less remorsesat not having exercised at the maxim... |

25 |
Sequential Tests for the Mean of a Normal Distribution IV
- Chernoff
- 1965
(Show Context)
Citation Context ...ed? The answer is thatswe used the "principle of smooth fit." This principle goes back to A. N.sKolmogorov, who discovered it in Russia in the 1950's, and it was latersindependently found by Chernoff =-=[6]-=- in the United States and also by Lindleysin Great Britain. It was used by Grigelionis and Shiryaev [10] and others [2,s18], though even now it is not appreciated widely. A new application tosBurkhold... |

20 |
Radon-Nikodym derivatives of Gaussian measures
- Shepp
- 1966
(Show Context)
Citation Context ...t K is given from (2.4) bys1s(2.18) K= (Y2ayl - y1aY2).sY2 - Y2sSo it is enough to show that supt exp( - rt)St is integrable; that is,s(2.19) fdyP supe-rtSt > y} < oo.sBy a well-known theorem of Doob =-=[15, 17]-=- for a > 0, 8 > 0,s(2.20) P{Wt < at + ,O? 0 < t < oo} = 1e-2a.sIf we chooses(2.21) a = (r-, + 2 )c, I 'log(?J)sthen from (1.3) for y > s, y > x,sP{supe-rtSt > ys(2.22) =P(su su |< t(uw + (i - 2 )u) -l... |

20 |
A New Look at Pricing of the “Russian Option”, Theory Probab
- Shepp, Shiryaev
- 1993
(Show Context)
Citation Context ...n the case ,u = o-2/2), although maybe it can be done,sand this might give an alternate derivation of (2.3) and (2.4) as has beenssuggested by several readers. (Note added in proof See our new papers =-=[19]-=-sand [20].)sSo assume r > max(0, ,u) as in (1.5) and let y = ym and Y = Y2, Yl < 0 <s1 < Y2, be the two roots of the quadratic equations22 J 2 +s_y _- - - =r, 22s(2.2) 22su2 /2 - ? (u2/2 - + 2ua2rsand... |

18 |
Sequential decision in the control of a spaceship
- Bather, Chernoff
- 1967
(Show Context)
Citation Context ...red it in Russia in the 1950's, and it was latersindependently found by Chernoff [6] in the United States and also by Lindleysin Great Britain. It was used by Grigelionis and Shiryaev [10] and others =-=[2,s18]-=-, though even now it is not appreciated widely. A new application tosBurkholder-Gundy inequalities is in a paper in preparation [8]. It oftensenables one to obtain (see especially [2, 3, 6, and 18]) e... |

17 |
The Theory of Optimal Stopping
- Chow, Robbins, et al.
- 1971
(Show Context)
Citation Context ... problem of determining the price of option (3.3) is simpler than that ofs(1.4), but has apparently not been solved before despite its simplicity, al-sthough very similar problems have been discussed =-=[7]-=-. If we overlooked asprior solution, perhaps it has not been solved with the smooth fit principle,sbut the elementary solution could have been simply guessed in some othersway. Again the full power of... |

13 | Explicit Solutions to Some Problems of Optimal Stopping - Shepp - 1969 |

7 | A dual Russian option for selling short
- Shepp, Shiryaev
- 1996
(Show Context)
Citation Context ...e ,u = o-2/2), although maybe it can be done,sand this might give an alternate derivation of (2.3) and (2.4) as has beenssuggested by several readers. (Note added in proof See our new papers [19]sand =-=[20]-=-.)sSo assume r > max(0, ,u) as in (1.5) and let y = ym and Y = Y2, Yl < 0 <s1 < Y2, be the two roots of the quadratic equations22 J 2 +s_y _- - - =r, 22s(2.2) 22su2 /2 - ? (u2/2 - + 2ua2rsand sets/i- ... |

4 | A new look at the pricing of the Russian option - Shepp, Shiryaev - 1994 |

3 | Optimal control and replacement with state- dependent failure rate: an invariant measure approach - HEINRICHER, STOCKBRIDGE - 1993 |

2 |
On optimal stopping and maximal inequalities for Bessel processes. Theory of Probability and its Applications. Vol 38
- Dubins, Shepp, et al.
- 1993
(Show Context)
Citation Context ...Britain. It was used by Grigelionis and Shiryaev [10] and others [2,s18], though even now it is not appreciated widely. A new application tosBurkholder-Gundy inequalities is in a paper in preparation =-=[8]-=-. It oftensenables one to obtain (see especially [2, 3, 6, and 18]) explicit closed formssolutions to optimal stopping or optimal control problems in continuoussproblems where the discrete versions ca... |

2 | Total risk aversion, stochastic optimal control, and differential games - Jenson, Barron - 1989 |

2 |
Theorie de la speculation. Reprinted
- Bachelier
- 1900
(Show Context)
Citation Context ... purposes of comparison and to emphasize the mathematical nature ofsthe contribution here, we conclude the paper by analyzing an optimal stop-sping problem for the Russian option based on Bachelier's =-=[1]-=- original (1900)slinear model of stock price fluctuations:s(1.5) Xt = x + aWt + ,ut, t 2 O.sWe again introduce the running maximum as in (1.3):s(1.6) St = max{s, sup X>sO<u<tsTHE RUSSIAN OPTION: REDUC... |

1 | Th'eorie de la Speculation. Reprinted in Cootner - Bachelier - 1900 |

1 | Financial Options, From Theory to Practice, eds - Silker - 1990 |

1 | The Theory of Optimal Stopping, Dover Pubs - Chow, Robbins, et al. - 1991 |

1 | Arbitrage value of a Russian option, manuscript - Duffie, Harrison |

1 | Stockbridge (two papers in Ann - Heinricher |

1 | Total risk aversion and the pricing of options - Jenson, Barron - 1991 |

1 |
Arbitrage value of a Russian option. Unpublished manuscript
- DUFFIE, HARRISON
- 1993
(Show Context)
Citation Context ...that will maximize the expected (present) value of his future reward, where rsis the interest rate for discounting. Starting with our solution to the mathe-smatical problem (1.2), Duffie and Harrison =-=[9]-=- derive an "arbitrage price" forsthe Russian option. Their pricing analysis parallels the analysis of Europeanscall options by Black and Scholes [4]. Of necessity, this involves a morescomplete discus... |