Continuous-Time Limits
This chapter is devoted to continuous-time limits. These limits are described without any probability, but only assuming that the magnitude of jumps per time period τ is of order \({\tau }^{\alpha }\) , \(\alpha\in[1/2,1]\) . Details related to models wit
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Continuous-Time Limits
15.1 Nonlinear Black–Scholes Equation Our models and results are most naturally adapted to a discrete-time context, which is not a disadvantage from a practical point of view, as all concrete calculations are anyway carried out on discrete data. However, for qualitative analysis, it is desirable to be able to see what is going on in the continuous-time limit. This limit can also be simpler and, hence, be used as an approximation to a less tractable discrete model. With this in mind, let us analyze possible limits as the time between jumps and their sizes tend to zero. Let us work with the general model of nonlinear jumps from Sect. 14.2, with the reduced Bellman operator of form (14.6). Suppose the maturity time is T . Let us decompose the planning time [0, T ] into n small intervals of length τ = T /n and assume gi (z) = z + τ α φi (z),
i = 1, . . . , k,
(15.1)
with some functions φi and a constant α ∈ [1/2, 1]. Thus the jumps during time τ are on an order of magnitude τ α . As usual, we assume that the risk-free interest rate per time τ equals
ρ = 1 + rτ , with r > 0. From (14.6) we deduce for the one-period Bellman operator the expression Bτ f (z) =
1 max pIi (z, τ ) f (z + τ α φi (z)), 1 + rτ I ∑ i∈I
(15.2)
where I are subsets of {1, . . . , n} of size |I| = J + 1 such that the family of vectors z+ τ α φi (z), i ∈ I, is in a general position and {pIi (z, τ )} is the risk-neutral probability law on such a family, with respect to ρ z, i.e.,
∑ pIi (z, τ )(z + τ α φi (z)) = (1 + rτ )z.
(15.3)
i∈I
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 15, © Springer Science+Business Media New York 2013
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15 Continuous-Time Limits
Let us deduce the Hamilton-Jacobi-Bellman (HJB) equation for the limit, as τ → 0, of the approximate cost function BτT −t , t ∈ [0, T ], with a given final cost fT , using the standard (heuristic) dynamic programming approach. That is, from (15.2) and assuming an appropriate smoothness of f we obtain the approximate equation ft−τ (z) =
∂ ft 1 max ∑ pIi (z, τ ) ft (z) + τ α φi (z) 1 + rτ I i∈I ∂z 1 2α ∂ 2 ft 3α + τ φi (z), φi (z) + O(τ ) . 2 ∂ z2
Since {pIi } are probabilities and using (15.3), this is rewritten as ∂ ft 1 ∂ ft 2 ft − τ + O(τ ) = ft (z) + rτ z, ∂t 1 + rτ ∂z 2 1 2α ∂ ft I + τ max ∑ pi (z) φi (z), φi (z) + O(τ 3α ), I i∈I 2 ∂ z2 where pIi (z) = lim pIi (z, τ ) τ →0
(clearly well-defined nonnegative numbers). This leads to the equation 2 1 ∂f ∂f ∂ f I + r(z, ) + max ∑ pi (z) φi (z), φi (z) rf = ∂t ∂z 2 I i∈I ∂ z2 in the case α = 1/2 and to the trivial first-order equation ∂f ∂f + r z, rf = , ∂t ∂z
(15.4)
(15.5)
with the obvious solution f (t, z) = e−r(T −t) fT (e−r(T −t) z),
(15.6)
in the case α > 1/2. Equation (15.4) is a nonlinear extension of the classic Black–Scholes equation. The well-posedness of the Cauchy problem for such a nonlinear parabolic equation in the class of viscosity solutions i
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