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Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method

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  • In this work we develop partial differential equation (PDE) based computational models for pricing real options to contract the production or to transfer part/all of the ownership of a project when the underlying asset price of the project satisfies a geometric Brownian motion. The developed models are similar to the Black-Scholes equation for valuing conventional European put options or the partial differential linear complementarity problem (LCP) for pricing American put options. A finite volume method is used for the discretization of the PDE models and a penalty approach is applied to the discretized LCP. We show that the coefficient matrix of the discretized systems is a positive-definite $ M $-matrix which guarantees that the solution from the penalty equation converges to that of the discretized LCP. Numerical experiments, performed to demonstrate the usefulness of our methods, show that our models and numerical methods are able to produce financially meaningful numerical results for the two non-trivial test problems.

    Mathematics Subject Classification: Primary: 90-08, 65K15, 65M08; Secondary: 91G60, 90G20.


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  • Figure 5.1.  The values of European and American contracting options when $ \kappa = 0.5 $ for Test 1

    Figure 5.2.  The difference between the values of the European and American contracting options for Test 1

    Figure 5.3.  The value of the option to abandon for Test 2

    Figure 5.4.  Computed $ W - W^* $ and Greeks of $ W $ when $ \lambda = 0.5 $ for Test 2

    Figure 5.5.  The American option value and its optimal exercise curve for $ \lambda = 0.5 $ for Test 2

    Table 5.1.  Project, production and market data and functions used in Test 1

    $Q = 10^4$ million tons $B = 30\%$ per annum
    $c_0 = {\rm{US}} $ $25 $c(t) = c_0\times e^{0.005t} $
    $ R = 5\%$ per annum $r = 0.06 $ per annum
    $C = {\rm{US}}$ $ $5 \times 10^2$ million $T = 1$ year
    $\sigma = 30\% $ $\delta = 0.02$
    ${{q}_{0}}=\left\{ \begin{array}{*{35}{l}} \begin{align} &0.01Q\times {{e}^{0.007t}},\ \ t\le T_{1}^{*}, \\ &0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t\in (T_{1}^{*},{{T}^{*}}) \\ \end{align} \\ \end{array} \right.$ ${{q}_{1}}=\left\{ \begin{align} &0.01Q\times {{e}^{0.007t}},\ \ \ \ \ t <T\text{, } \\ &\kappa \times 0.01Q\times {{e}^{0.007t}},\ \ \ \ \ t\in [T,{{T}^{*}}) \\ \end{align} \right.$
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