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We deal with the pricing of callable Russian options. A callable Russian option is a contract in which both of the seller and the buyer have the rights to cancel and to exercise at any time, respectively. The pricing of such an option can be formulated as an optimal stopping problem between the seller and the buyer, and is analyzed as Dynkin game. We derive the value function of callable Russian options and their optimal boundaries.

For the last two decades there have been numerous papers (see [

In this paper, we consider the pricing of Russian
options with call provision where the issuer (seller) has the right to call
back the option as well as the investor (buyer) has the right to exercise it.
The incorporation of call provision provides the issuer with option to retire
the obligation whenever the investor exercises his/her option. In their
pioneering theoretical studies on Russian options, Shepp and Shiryaev [

The paper is organized as follows. In Section

We consider the
Black-Scholes economy consisting of two securities, that is, the riskless bond
and the stock. Let

Russian option
was introduced by Shepp and Shiryaev [

Let

We define two
sets

Assume that

First, suppose that

By this lemma, we may apply Proposition 3.3 in Kifer
[

Should the
penalty cost

Set

We set

Suppose

Set

By regarding callable Russian options as a perpetual
double barrier option, the optimal stopping problem can be transformed into a
constant boundary problem with lower and upper boundaries. Let

Let

First, we prove (

We study the boundary point

For any

First, the derivative of the first
term is

We set

From
(

Since

Therefore, we obtain the following theorem.

The value function of callable Russian option

We can get (

From conditions (

In this
section, we present some numerical examples which show that theoretical results
are varied and some effects of the parameters on the price of the callable
Russian option. We use the values of the parameters as follows:

Figure

Penalty

0.01 | 0.1 | 0.09 | 0.3 | 0.5 | 1.04337 |

0.02 | 0.1 | 0.09 | 0.3 | 0.5 | 1.0616 |

0.03 | 0.1 | 0.09 | 0.3 | 0.5 | 1.07568 |

0.03 | 0.2 | 0.09 | 0.3 | 0.5 | 1.08246 |

0.03 | 0.3 | 0.09 | 0.3 | 0.5 | 1.09228 |

0.03 | 0.4 | 0.09 | 0.3 | 0.5 | 1.10842 |

0.03 | 0.5 | 0.09 | 0.3 | 0.5 | 1.14367 |

0.03 | 0.1 | 0.01 | 0.3 | 0.5 | 1.08092 |

0.03 | 0.1 | 0.05 | 0.3 | 0.5 | 1.07813 |

0.03 | 0.1 | 0.1 | 0.3 | 0.5 | 1.07511 |

0.03 | 0.1 | 0.3 | 0.3 | 0.5 | 1.06633 |

0.03 | 0.1 | 0.5 | 0.3 | 0.5 | 1.06061 |

0.03 | 0.1 | 0.09 | 0.1 | 0.5 | 1.02468 |

0.03 | 0.1 | 0.09 | 0.2 | 0.5 | 1.04997 |

0.03 | 0.1 | 0.09 | 0.4 | 0.5 | 1.1018 |

0.03 | 0.1 | 0.09 | 0.5 | 0.5 | 1.12833 |

0.03 | 0.1 | 0.09 | 0.3 | 0.1 | 1.18166 |

0.03 | 0.1 | 0.09 | 0.3 | 0.2 | 1.12312 |

0.03 | 0.1 | 0.09 | 0.3 | 0.3 | 1.09901 |

0.03 | 0.1 | 0.09 | 0.3 | 0.4 | 1.08505 |

Optimal boundary for the buyer.

The value function

The value function

Real line with dividend; dash line without dividend.

In this paper,
we considered the pricing model of callable Russian options, where the stock
pays continuously dividend. We derived the closed-form solution of such a
Russian option as well as the optimal boundaries for the seller and the buyer,
respectively. It is of interest to note that the price of the callable Russian
option with dividend is not equal to the one as dividend value

This work was supported by Grant-in-Aid for Scientific Research (A) 20241037, (B) 20330068, and a Grant-in-Aid for Young Scientists (Start-up) 20810038.