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**Arbitrage pricing of Russian options and perpetual lookback options.**
*(English)*
Zbl 0783.90009

Summary: Let \(X=\{X_ t,\;t\geq 0\}\) be the price process of a stock, with \(X_ 0= x>0\). Given a constant \(s\geq x\), let \(S_ t= \max\bigl\{s,\sup_{0\leq u\leq t} X_ u\bigr\}\). Following the terminology of L. Shepp and A. N. Shiryaev [ibid. 3, No. 3, 631-640 (1993; Zbl 0783.90011)], we consider a “Russian option”, which pays \(S_ \tau\) dollars to its owner at whatever stopping time \(\tau\in [0,\infty)\) the owner may select. As in the option pricing theory of Black and Scholes, we assume a frictionless market model in which the stock price process \(X\) is a geometric Brownian motion and investors can either borrow or lend at a known riskless interest rate \(r>0\). The stock pays dividends continuously at the rate \(\delta X_ t\), where \(\delta\geq 0\).

Building on the optimal stopping analysis of Shepp an Shiryaev, we use arbitrage arguments to derive a rational economic value for the Russian option. That value is finite when the dividend payout rate \(\delta\) is strictly positive, but is infinite when \(\delta=0\). Finally, the analysis is extended to perpetual lookback options.

The problems discussed here are rather exotic, involving infinite horizons, discretionary times of exercise and path-dependent payouts. They are also perfectly concrete, which allows an explicit, constructive treatment. Thus, although no new theory is developed, the paper may serve as a useful tutorial on option pricing concepts.

Building on the optimal stopping analysis of Shepp an Shiryaev, we use arbitrage arguments to derive a rational economic value for the Russian option. That value is finite when the dividend payout rate \(\delta\) is strictly positive, but is infinite when \(\delta=0\). Finally, the analysis is extended to perpetual lookback options.

The problems discussed here are rather exotic, involving infinite horizons, discretionary times of exercise and path-dependent payouts. They are also perfectly concrete, which allows an explicit, constructive treatment. Thus, although no new theory is developed, the paper may serve as a useful tutorial on option pricing concepts.

### MSC:

91B28 | Finance etc. (MSC2000) |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

91B24 | Microeconomic theory (price theory and economic markets) |