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\fancyhead[CO]{Pricing of callable put options with and without extra information: a special case}
\fancyhead[CE]{Neda Esmaeeli}
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{\Large \bf Pricing of callable put options with and without extra information: a special case}
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{\small Neda Esmaeeli}\index{Esmaeeli, Neda}\footnote{speaker}\\
{\small University of Isfahan, Isfahan, Iran}\\[2mm]
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\begin{abstract}
We consider a callable put on a financial market with an underlying following the Black-Scholes model. The callable put is simply a put that can be called by its issuer. We then will place ourselves in the framework of asymmetric information, a situation in which one party has more information compared to other one. We restrict ourselves in this paper to a simple case of information asymmetry and we will employ an algorithm for pricing callable put options with and without this extra information.
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\keywords{pricing, callable put option, asymmetric information.}
\subject{91A15, 91B24}
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\section{Introduction}
From a financial point of view, a callable put could be seen as a generalization of the notion of American put option, where not only has the buyer (or holder) the right to exercise at any time before the maturity, but also the seller (or issuer) has the right to cancel the contract at any time before the maturity time for a certain penalty. In \cite{Kuh07}, the callable put is characterised as a composite exotic option, and the value function is studied.
In the financial markets, the decisions of the participants are based on the information available to them. In this paper, we consider a callable put option in which the seller has additional information compared to the buyer.
While the buyer makes his decisions based on the public information flow $\mathbb{F}=(\mathcal{F}_{t})_{t\in[0,T]},$
the seller possesses the additional information modeled by some random variable $G$ known to him from the very beginning. We restrict ourselves to the simple case in which $G$ takes finitely many values.
Thus the information flow of the informed player is given by the initial enlargement of $\mathbb{F}$ with $G$.
When dealing with claims of American or callable type, one must solve an optimal stopping problem. Diffusion models for optimal stopping are difficult to solve using classical PDE methods such as finite difference methods. To remedy this problem, Monte-Carlo methods can be employed. The main difficulty encountered when applying these methods to the optimal stopping problem is the evaluation of conditional expectations.
For pricing American options, a Monte-Carlo algorithm is developed in \cite{Longstaff-Schwartz} which addresses the problem of evaluating conditional expectations by regressing them on a finite number of functions of the underlying.
This algorithm can be simplified by removing the least-squares approximation of the conditional expectation, and instead using the known continuation values for each path. This approach was taken by \cite{Chen-Shen}, who showed that it not only reduces the computational complexity, and therefore the running time of the algorithm, but also gives more accurate values than the least square estimator method.
In this paper, we employ the pricing algorithm for callable put options developed in \cite{Chen-Shen} to investigate the effect of this information asymmetry on the price of the callable puts.
\section{Main results}
Let $T>0$ represent a finite time. We consider a filtered probability space $\left(\Omega,\mathcal{F},\mathbb{F},\mathbb{P}\right)$, where $ \mathbb{F} = \left(\mathcal{F}_{t}\right)_{t \in [0,T]}$ is the reference filtration satisfying the usual conditions of right-continuity and completeness. Let $t \in \mathbb{R}^{+},$ we denote by $\mathcal{T}_{t,T}\left(\mathbb{F}\right)$ the set of $\mathbb{F}-$stopping times with values in $[t,T]$.
We write $\mathcal{T}\left(\mathbb{F}\right)$ for $\mathcal{T}_{0,T}\left(\mathbb{F}\right)$. \\
We consider the financial market with an underlying following
the Black-Scholes model. In other words, we suppose that the asset price process $S$ is given by the SDE
$dS_{t}=\mu S_{t}dt+\sigma S_{t}dB_{t},\,t\in[0,T],$ where $\mu$ is the drift and $\sigma>0$ the volatility.
Moreover, assume that $\delta=(\delta_{t})_{t\in[0,T]}$ is such that $\delta_{t}>0$ for $t\in[0,T)$ and $\delta_{T}=0$.
In addition, we assume that the seller terminates the contract at a stopping time $\gamma\in\mathcal{T}(\mathbb{F})$ and the buyer exercises his option at $\tau\in\mathcal{T}(\mathbb{F})$. Now, we are ready to define the callable put option formally.
\begin{definition}
A callable put which is a put option whose seller (issuer) can terminate it before its excersie time for a certain penalty $\delta=(\delta_{t})_{0\leq t\leq T}$ where $\delta$ is a positive process. Let $S=(S_{t})_{t\in[0,T]}$ denote the price process of the underlying asset and $K$ the strike price. Then a callable put option has the payoff function
\begin{equation*}
R(\gamma,\tau)=((K-S_{\gamma})^{+}+\delta_{\gamma})1_{\{ \gamma<\tau\}}+(K-S_{\tau})^{+}1_{\{ \tau \leq \gamma\}},\ \gamma, \tau \in \mathcal{T}_{t,T}(\mathbb{F})
\end{equation*}
which the seller pays to the buyer at time $\gamma\wedge\tau$.
\end{definition}
As a special case, we consider the situation in which $G:=1_{\{B_{T}\in[a,b]\}}$ for some $a