Validation
In this chapter, we initiate a discussion on the strengths and weaknesses of the new theory and of its applicability.
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Validation
As a foreword to this chapter, we wish to recall that we have obviously no claim of overall superiority over Black–Scholes. But a careful analysis may reveal some interesting features, and some weaknesses of Black–Scholes that we avoid. Noticeably, in a time of great uncertainty and of crisis, one might be interested in a theory that does not make use of a probabilistic model of the market pricing process. It should also be noted that the Black–Scholes theory is too strongly entrenched to be challenged as a piece of “natural science” describing what will happen or why things happen the way they do. Rather, our analysis is always in the spirit of an “engineering” science of decision support, advising one about what to do.
10.1 Numerical Results and Comparisons 10.1.1 Numerical Computations We give here a few numerical results of our theory. Most computations were performed with the“standard” algorithm. Default values of the parameters are, unless otherwise specified, as in Table 10.1. Notice that it follows from (7.1)–(7.3) that if the relative rate of variation of u is τ , then that of S is τ + μ0 (Table 10.1). Figure 10.1 show plots of the functions vˇkh (·), the optimum hedging portfolio position in the underlying asset, and Wkh (·, 0), the worth of the portfolio, both as functions of the underlying asset’s (normalized) market price u, for tk = 0, 10, 20, 30, 40, and 44 = T . Concerning vˇkh , as tk increases, it gets closer to the separatrix of Fig. 8.2 and coincides with it for tk = T = 44. In a similar fashion, Wkh (u, 0) decreases as tk increases to become the graph of M for tk = T = 44. Recall that vˇ0h (u(0)) gives the initial portfolio of the hedging strategy and W0h (u(0), 0) the corresponding premium. Similarly, Fig. 10.2 shows the same graphs for a digital call. The peak in vˇkh is more pronounced the closer tk is to T . But because of transaction costs, it remains P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 10, © Springer Science+Business Media New York 2013
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10 Validation
Table 10.1 Default parameters for numerical computations
μ0 0.0123%
K 1
D 1
T 44
τ − + μ0 −5%
τ + + μ0 3%
C− −0.6986%
1.5 v
C+ 0.7014%
c− 0.5 ×C−
c+ 0.5 ×C+
w 0.3 0.25
1
0.2 0.15 0.5
0.1 0.05 0
0 0.5
u 0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
−0.05 0.5
u 0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
Fig. 10.1 Graphs of vˇ kh (u) and Wkh (u, 0) for tk = 0, 10, 20, 30, 40, 44 for a vanilla call 15 v
w D=1 0.8
10
0.6 0.4 5 0.2 0 0 0.5
0.6
0.7
0.8
0.9
1
1.1
u 1.2
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
u 1.2
Fig. 10.2 Graphs of vˇ kh (u) and Wkh (u, 0) for tk = 0, 10, 20, 30, 40, 44 for a digital call
bounded by D/C+ . Concerning Wkh (u, 0), the same remark applies that was made previously, but now with the step function M of a digital call.
10.1.2 Numerical Comparison with Black–Scholes The qualitative aspects of the graphs of W in Fig. 10.1 bear a notable similarity to
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