Valuation of European Options Under an Uncertain Market Price of Volatility Risk

Figures & data

Figure 1. Truncated domain $\mathrm{\Omega }=[S_{min},S_{max}]\times [0,v_{max}]$.

Figure 2. Domain ${\mathrm{\Omega }}^{\mathrm{\prime }}$ after the transformation $S={\mathrm{e}}^x$.

Figure 3. Domain ${\mathrm{\Omega }}^{\mathrm{\prime }\mathrm{\prime }}$ with parameter values $\rho =0.5$, $\xi =0.7$

Figure 4. Value of a long butterfly position at t = 0 with T = 0.5, K = 50 and control set $L=[-2.4,-1.6]$.

Figure 5. Measurement of the effect of nonlinearity for a long butterfly position at $t\approx 0.39$ with T = 0.5, K = 50 and control set $L=[-2.4,-1.6]$. (a) Selected optimal control; which are throughout at the extreme points of the control set, (b) difference between solutions of a nonlinear problem and linear evolution problem with a fixed control $\lambda =-2.4$.

Figure 6. Measurement of effect of a diameter of a control set on the value function and its derivative. Control sets are symmetrical and centred at $-1.25$, measurements were made at t = 0. (a) Comparison of $V_{sup}$ and $V_{inf}$ at $(S,v)=(2.11,2.06)$, (b) comparison of $\mathrm{\partial }{V}_{sup}/\mathrm{\partial }S$ and $\mathrm{\partial }{V}_{inf}/\mathrm{\partial }S$ at $(S,v)=(53.12,0.75)$, (c) comparison of $\mathrm{\partial }{V}_{sup}/\mathrm{\partial }S$ and $\mathrm{\partial }{V}_{inf}/\mathrm{\partial }S$ at $(S,v)=(51.76,2.84)$, (d) comparison of $\mathrm{\partial }{V}_{sup}/\mathrm{\partial }S$ and $\mathrm{\partial }{V}_{inf}/\mathrm{\partial }S$ at $(S,v)=(51.43,0.23)$.

Figure 7. Comparison of plots of $\delta (V_{sup}-V_{inf})/\delta S$ at time t = 0 with control set $[-2.5,0.0]$. (a) Call option, (b) butterfly option.