On Russian options (Q2732674)

Let \(S=\{S_t:t\geq 0\}\) be a geometrical Brownian motion \(\text{GBM}(\mu,\sigma^2)\) started at \(x\) with parameters \(\mu\) and \(\sigma\): \( S_t=x \exp((\mu-\sigma^2/2)t+\sigma W_t)\), \(t\geq 0,\) where \(W\) is a standard Brownian motion. \(S\) is considered as the price process of a stock and it is assumed that there exists a riskless alternative to investments, e.g., bonds. The price process of a bond with the initial price \(y\) and the interest rate \(r>0\) is \(B=\{B_t:t\geq 0\},\) where \(B_t=y e^{r t}\). The Russian option as introduced by \textit{L. A. Shepp} and \textit{A. N. Shiryaev} [Ann. Appl. Probab. 3, No. 3, 631--640 (1993; Zbl 0783.90011)] is a perpetual American contingent claim with the payment process \(G_t=e^{-\lambda t}\max\{\sup_{s\leq t}S_s, \beta x\}\). In other words, the buyer of the Russian option can choose a time point \(\tau\) to exercise the option and receive the payment \(G_\tau\). The solution to the pricing problem of the Russian option given by \textit{L. A. Shepp} and \textit{A. N. Shiryaev} (loc. cit.) is considerably simplified by \textit{L. A. Shepp} and \textit{A. N. Shiryaev} [Theory Probab. Appl. 39, No. 1, 103--119 (1994); translation from Teor. Veroyatn. Primen. 39, No. 1, 130-149 (1994; Zbl 0829.60055)]. The proposed method is developed by \textit{A. N. Shiryaev, Yu. M. Kabanov, O. D. Kramkov} and \textit{A. V. Mel'nikov} [ibid. 39, No. 1, 14--60 (1994); resp. ibid. 39, No. 1, 23--79 (1994; Zbl 0833.60064) and ibid. 39, No. 1, 61--102 (1994); resp. ibid. 39, No. 1, 80--129 (1994; Zbl 0833.60065)] and the term ``dual martingale measure'' is introduced to describe it. NEWLINENEWLINENEWLINEIn this article a new approach to dual martingale measure based on excessive transforms or Doob's \(h\)-transforms is presented. Making use of a generalization of the Lévy theorem for a Brownian motion with drift the author proposes a shortcut to the result of \textit{L. A. Shepp} and \textit{A. N. Shiryaev} on the Russian option. The problem of determining the rational price of an American contingent claim is a problem of the optimal stopping [see \textit{I. Karatzas}, Appl. Math. Optim. 17, No. 1, 37--60 (1988; Zbl 0699.90010) and \textit{A. N. Shiryaev, Yu. M. Kabanov, O. D. Kramkov} and \textit{A. V. Mel'nikov} (loc. cit.)]. The corresponding optimal stopping problem is solved with the technique exploiting the representation theory of excessive functions as presented by the author [Math. Nachr. 124, 85--101 (1985; Zbl 0594.60080)]. The paper is concluded by indicating how to price American contingent claims with the payment process of the type \(t\mapsto e^{-\lambda t} S^a_t\sup_{s\leq t} S^b_s,\) where \(\lambda\geq 0\) and \(a\) and \(b\) are arbitrary real numbers.