5.8 Concluding remarks
6.1.3 Model output and validation
In this section logarithmic utility will be used for the purpose of calculations unless otherwise stated.
N = 1risky assets
In this section we will provide some results for one risky asset and validate them against ex- isting solution in the infinite horizon case by lettingT → ∞ 5. In all our subsequent results
the parameters and the shape of the state variable simplex is chosen to be the same as in [69].Figures 6.6 and 6.7 show output for parameters chosen to be λ = 0.001,T = 5,N =
500,∆T = T/N,s = e0.07∆T,m = 0.182, σ= 0.4. The paramter values are the same as used in
[69].
5Assuming a logarithmic utility and choosingu =em∆T+σ√∆T,d =em∆T−σ√∆T andp =1/2 we create an ap- proximation forMertonpoint using dynamic programming and find it to bea=− s(−∆T m+s−1)
6.1. Tree approximations for fixed transaction cost model 89
No-transaction region
sell-side boundary
buy-side boundary Sell region
Buy region
line tracing merton point
RISKY FRACTION AXIS
TIME AXIS 1 2 3 4 5 0.2 0.4 0.6 0.8 1.0
Figure 6.6: The sell-side and buy-side boundaries for the choice of parameters: λ= 0.001,T =
5,N = 500,∆T = NT,s = e0.07∆T,m = 0.182, σ = 0.4. Where T is the time horizon for investment, N is the number of re-balancing nodes. ω is the continuous time risk-free rate,
λ is the transaction cost factor, s is the risk free growth over the interval, m is the drift for continuous time GBM and σ is the volatility for the continuous time GBM. The continuous time GBM implies a risky growth for the risky asset over the interval∆T.
As seen by the graphs at infinite horizon discrete probability approximation solution im- plies risky fraction boundaries of (0.5425,0.834) which is reasonably close to a solution of (0.54,0.837) implied by solving a free-boundary problem as in [69] wherein the exact solution could be computed by solving a large system of nonlinear equations. The convergence to the infinite horizon solution took just few minutes with a simple discrete probability approximation approximation in contrast to more elaborate and mathematically complex free-boundary solu- tion procedure for high dimensions. Solution to corresponding continuous time PDEs could be fast too and the real advantage of discrete probability approximation lies in their simplicity and ease of implementation.
The exact distribution of risky growth as earlier stated is log-normal (and normal as∆T → 0) and this implies that we could exactly compute the risky fraction boundaries being estimated. However using the exact distribution is computationally very expensive. With just one period problem (which means∆T = 0.01 and we iterate the dynamic programming equations just once ) and the above choice of parameters as in Figure 6.6 and 6.7 the exact analytic distribution of risky growth implied risky fraction boundaries of (0.141,1) which is reasonably close to that implied by discrete probability approximation (0.137,1). As∆T → 0 the approximation solution converges to the exact solution.
Figure 6.8 shows the variation of risky fraction boundaries with the transaction cost pa- rameter. Plotting the width of no-transaction boundaries against λ in Figure 6.9 we observe
the boundary width to be of O(λ1/4) by performing a least squares fit of the width of bound-
90Chapter6. Moment based discrete probability approximation of transaction cost models
Iteration number Width of the no-transaction region
converging Width of boundary 100 200 300 400 500 0.290 0.295 0.300 0.305
Figure 6.7: Width of no-transaction region plotted against iteration depth: λ = 0.001,T =
5,N = 500,∆T = TN,s = e0.07∆T,m = 0.182, σ = 0.4. The iterations are the same as in the dynamic programming equation. We start from the last stage to go to the initial time. WhereT
is the time horizon for investment,N is the number of re-balancing nodes.ωis the continuous time risk-free rate,λis the transaction cost factor, sis the risk free growth over the interval,m
is the drift for continuous time GBM andσis the volatility for the continuous time GBM. The continuous time GBM implies a risky growth for the risky asset over the interval∆T.
This provides a basis for asymptotic expansion in multi-dimensional PDEs characterizing the transaction cost problem.
The error in the value function (or optimal control law) determination at each stage of dynamic recursion comes from three sources:
(1) Discretization of state space ( number of steps required for a given accuracy is obviously transaction cost and utility structure dependent )
(2) Approximation of value function at each stage of optimization. Especially adjusting the termination accuracy goal of optimization will impact the accuracy of controls at each stage and hence the accuracy in the value function.
(3) Approximation of real probability model of risky growth evolution by a by a discrete probability approximation.
In Figure 6.10 it is seen that for some choice of parameters binomial and trinomial approx- imations give a pretty close solution. Ifl(m)is the boundary based upon suitable discretization andmtime step divisions then we hypothesize||l(m)−l(0)|| ≤ k
mα for some suitable constants k
andα.
Results forN ≥ 2risky assets
Results are provided for both the cases of independent and correlated stock returns. As re- marked in [69], [6] and [3] the continuation regionis similar to an ellipse in two dimensions
6.1. Tree approximations for fixed transaction cost model 91 Λ
risky fraction
boundaries
0.0002 0.0004 0.0006 0.0008 0.0010 0.55 0.60 0.65 0.70 0.75 0.80Figure 6.8: Risky fraction boundaries at initial time with respect to transaction cost parameter
λ: λ = 0.001,T = 3,N = 500,∆T = TN,s = e0.07∆T,m = 0.182, σ = 0.4. Where T is the
time horizon for investment, N is the number of re-balancing nodes. ωis the continuous time risk-free rate, λis the transaction cost factor, s is the risk free growth over the interval,m is the drift for continuous time GBM andσ is the volatility for the continuous time GBM. The continuous time GBM implies a risky growth for the risky asset over the interval∆T.
and ellipsoid in three dimensions. The correlation factor serves to stretch the ellipsoid. It seems that above a certain correlation factor the region hits risky fraction axis on either side. This might pose problems for existing methods to solve it as in [71], [6] and [83] for a con- strained state variable simplex where the region of inaction is assumed to lie inside and not on the boundary of the simplex. Comparison of our result in the infinite horizon case is done using the method in [69] and gives a reasonably close answer. It is slightly offfrom the region implied by the asymptotic analysis of [6] and [3]. As depicted in Figure 6.16, our result for infinite horizon is sufficiently close to the exact result in the literature and reasonably close to approximate asymptotic results. This provides an argument in favor of discrete probability ap- proximation methods as far as accuracy is concerned. Figure 6.14 and 6.15 show no-transaction boundaries at terminal time for independent and correlated stocks respectively. Figure 6.17 and 6.18 shows the geometry of inactionat infinite horizon for high positivecorrelation between two stocks. It is seen that the region drifts towards and eventually hits it the risky fraction axis as time progresses. This might pose problems for free boundary PDE solving procedures as the region has sharp boundaries and also for boundary update procedures since the the region lies on the boundary of risky fraction axis [51]. It must be mentioned in context of PDE solving procedures that the boundary of the solution is not known and must be computed as part of the solution itself.
This section provided results exclusively for the fixed cost transaction structure. However, discrete probability approximation based methods seem very promising for arbitrary transac- tion cost structures. For structures of the Davis Norman type as in [28] and the parameters as in
92Chapter6. Moment based discrete probability approximation of transaction cost models
Width of boundary
Λ
Curve fitted to data points
0.0002 0.0004 0.0006 0.0008 0.0010
0.15 0.20 0.25
Figure 6.9: Variation of the width of risky fraction boundaries in the last stage with respect to transaction cost parameter and parameters: T = 3,N = 100,∆T = NT,s = e0.07∆T,m =
0.182, σ=0.4. Mathematica’sFindFit[]command used to directly do non-linear least squares optimization. Where T is the time horizon for investment, N is the number of re-balancing nodes. ωis the continuous time risk-free rate,λis the transaction cost factor,sis the risk free growth over the interval,mis the drift for continuous time GBM andσis the volatility for the continuous time GBM. The continuous time GBM implies a risky growth for the risky asset over the interval∆T.
[72] the discrete probability approximation solution implies a solution with a fairly coarse time stepping which agrees with the infinite horizon solution through boundary update procedure as in [51]. An infinite horizon risky fraction boundary of (0.276,0.606 ) implied by the probability approximation methodology was based upon a∆T =3/40 it just took couple of seconds for 80 iterations ! This is reasonably close to the value quoted as in [73]. We note that getting exact solution using analytic techniques could be very fast tough but discrete probability approxima- tion based method possesses the advantage of being intuitive and easy to implement. We have also had promising results in higher dimensions using numerical probability approximation for problems of the Davis Norman type. We believe from the structure of problem that the ODEs and PDEs might become stifffor some parameters. This is supported by the fact that in Figure 6.18 shows a very small no transaction region for some choice of parameters which means the boundaries conditions are very rapidly changing! We note that if implicit PDE techniques are used stiffness is not an issue. However, implicit techniques for free-boundary problems might be very elaborate and complex. In contrast a discrete probability approximation method is simple, intuitive and easily implementable.
6.1. Tree approximations for fixed transaction cost model 93
risky fraction boundaries (Trinomial solution is dashed and binomial solid ) time 1 2 3 4 5 0.2 0.4 0.6 0.8 1.0
Figure 6.10: Risky boundary approximation: Comparing trinomial approximation for approx- imate normal with binomial approximation for exact log-normal with λ = 0.001,T = 5,N =
500,∆T = NT,s= e0.07∆T,m= 0.182, σ=0.4. WhereT is the time horizon for investment,Nis
the number of re-balancing nodes. ω is the continuous time risk-free rate,λis the transaction cost factor, s is the risk free growth over the interval,mis the drift for continuous time GBM andσis the volatility for the continuous time GBM. The continuous time GBM implies a risky growth for the risky asset over the interval∆T.