Theory and Modern Applications
Table 1 Values obtained from our approach and values from Monte-Carlo simulation. The baseline parameters are \(r=0.02\), \(\alpha =0.5\), \(D=100\), \(S^{1}_{0}=100\), \(S^{2}_{0}=100\), \(V_{0}=100\), \(k_{1}=k_{2}=k_{v}=1\), \(m_{1}=m_{2}=m_{v}=-2\), \(\rho _{12}=\rho _{1v}=\rho _{2v}=0.5\), and \(\eta _{1}=\eta _{2}=\eta _{v}=-0.7\). Pricing formula* in \(P_{0}\) denotes the value obtained from Theorem 1. ‘Av. run time’ denotes average CPU execution time for the computation when \(T=3\)
| Â | Â | Â | \(P_{0}\) | \(\tilde{P}^{\epsilon} (=P_{0}+P_{1}^{\epsilon} )\) | ||||
|---|---|---|---|---|---|---|---|---|
T | \(D^{*}\) | Â | Pricing formula* | Monte Carlo | R-err | Pricing formula | Monte Carlo | R-err |
Panel A. ϵ = 0.01 |  |  |  |  |  |  | ||
1 | 70 |  | 6.1275 | 6.1324 | 8.0 × 10−4 | 6.0996 | 6.1141 | 2.4 × 10−3 |
80 |  | 5.9549 | 5.9610 | 1.0 × 10−3 | 5.9069 | 5.9208 | 2.3 × 10−3 | |
90 |  | 5.4297 | 5.4353 | 1.0 × 10−3 | 5.3947 | 5.4077 | 2.4 × 10−3 | |
3 | 70 |  | 10.2437 | 10.2326 | 1.1 × 10−3 | 10.1801 | 10.2058 | 2.5 × 10−3 |
80 |  | 9.6883 | 9.6794 | 9.0 × 10−4 | 9.6293 | 9.6498 | 2.1 × 10−3 | |
90 |  | 8.9200 | 8.9118 | 9.0 × 10−4 | 8.8798 | 8.9014 | 2.4 × 10−3 | |
Panel B. ϵ = 0.02 |  |  |  |  |  |  | ||
1 | 70 |  | 6.1275 | 6.6.1324 | 8.0 × 10−4 | 6.0881 | 6.0596 | 4.7 × 10−3 |
80 |  | 5.9549 | 5.9610 | 1.0 × 10−3 | 5.8870 | 5.8639 | 3.9 × 10−3 | |
90 |  | 5.4297 | 5.4353 | 1.0 × 10−3 | 5.3802 | 5.3629 | 3.2 × 10−3 | |
3 | 70 |  | 10.2437 | 10.2326 | 1.1 × 10−3 | 10.1536 | 10.1374 | 1.6 × 10−3 |
80 |  | 9.6883 | 9.6794 | 9.0 × 10−4 | 9.6049 | 9.5900 | 1.6 × 10−3 | |
90 |  | 8.9200 | 8.9118 | 9.0 × 10−4 | 8.8631 | 8.8481 | 1.7 × 10−3 | |
Av. run time | (T = 3) |  |  |  | 0.0097 s | 26 min 48 s |  | |