## 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.

### MSC:

 91B28 Finance etc. (MSC2000) 60H30 Applications of stochastic analysis (to PDEs, etc.) 91B24 Microeconomic theory (price theory and economic markets)

Zbl 0783.90011
Full Text: