Deformation schemes help us solve portfolio problems over a larger interval∆T.
Suppose we wish to obtain a continuous time solution from an approximate solution ob- tained with∆T. Now we can choose∆T to be arbitrary and large enough then solve the problem withM1time steps. This will help us avoid interpolation errors and reduce computational time.
Then we compute the solution again with M2time steps where M2 > M1. We will continue to
do so obtaining a list of pairs involving the solution and time steps. Following a suitable basis set we can express the continuation behavior and find the limiting behavior when we approach a continuous time behavior. By continuation behavior we mean projecting our discrete solution sets to the required solution. This is discussed in detail in Chapter 5.
3.10. Towards robust and efficient lattice algorithms 47
Value function
risky fraction
Convergence
0.2 0.4 0.6 0.8 0.1065 0.1070 0.1075 0.1080 0.1085 0.1090 0.1095Figure 3.15: Deformation solution of value function with log-utility att = 0 by varyingγ` =
1 5, ..., 1 30 and parameters T = 1,N = 4, ω = 0.1, λ = µ = 0.01,s = e ω∆T, m = 0.14, σ = 0.3. WhereT is the time horizon for investment, N is the number of re-balancing nodes. ω is the continuous time risk-free rate, (λ, µ) are transaction cost factors, sis the risk free growth over the interval,mis the drift for continuous time GBM and σis the volatility for the continuous time GBM. The continuous time GBM implies a risky growth for the risky asset over the interval∆T. γ`is the deformation parameter.
the value function ( or optimal control law ) with respect to the problem parameters:
A-Discretization of time space
B-Discretization of state space
C-Discretization of probability model
The deformation schemes discussed earlier deal with discretization of probability model for risky growth. However, time space and state space introduce interesting continuation behavior the value of which should be exploited in an efficient algorithm to determine a continuous time solution. Probability deformation schemes help us solve over an arbitrary∆T, easing the pain of interpolation errors. This was the outermost loop in gaining numerical efficiency. In the inner loop we would seek continuation in time space. And in the innermost loop we would seek continuation in state space to determine the value function (or optimal control law ). As a result we now have three nested loops !
The above ideas are summarized in algorithmic form below:
Fori= 1,2, ...,N
For j= 1,2, ...,M
Fork =1,2, ...,L
P={(1i,1j,1k), ψ(γi,tj,Sk)}
48Chapter3. Introduction to discrete probability approximation and sketch of modeling approach
Value function
risky fraction
Convergence
0.2 0.4 0.6 0.8 1.058 1.059 1.060 1.061Figure 3.16: Deformation solution of value function with CRRA utility at t = 0 by varying
γ` = 15, ..., 301 and parametersT = 1,N = 4, ω = 0.1, λ = µ = 0.01,s = eω∆T,m = 0.14, σ =
0.3,V = 0.5. Where T is the time horizon for investment, N is the number of re-balancing nodes. ωis the continuous time risk-free rate, (λ, µ) are transaction cost factors, sis 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. γ` is the deformation parameter.
Next k Next j
Next i
Interpolate the listLand findL(0,0,0)
In the above algorithm ψ denotes the value function. Also we have i-th point discrete probability approximation, j-th point discrete time space approximation and k-th point state space approximation. So as when (1i, 1j, 1k) −→ (0,0,0) we have the required solution under an exact probability model and continuous-time domain. In simple words, we would make the discrete probability approximation, the time space division and state space division finer to see the corresponding continuous time solution.
In the thesis the state variables will typically be discretized using grids and the value func- tion numerically represented using polynomial basis via the grids. This definitely is the curse of dimensionality as far as state space is concerned because numerical representation of value function isO(Ns) wheresis the number of state variables21. Note the curse of dimensionality can never beremoved. It can only betamedby introducing an interesting set of new numerical issues to attain the desired accuracy which we must then deal with.
3.10. Towards robust and efficient lattice algorithms 49
Value function
risky fraction
Convergence
0.4 0.6 0.8 0.108 0.110 0.112 0.114Figure 3.17: Deformation solution of value function with log-utility and jump diffusion model att = 0 by varying γ` = 51, ...,201 and parameters: T = 1,N = 4, ω = 0.1, λ = µ = 0.01,s = eω∆T,m = 0.14, σ = 0.6, θ = 0.1, δ = 0.05. WhereT is the time horizon for investment, N is
the number of re-balancing nodes. ωis the continuous time risk-free rate, (λ, µ) are transaction cost factors,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. γ`is the deformation parameter.
It is also very important to frame a problem in the right state variables to allow decom- posability for the value function of the dynamic problems. For example, consider the same problem set-up as in section 3.7 but with CARA or exponential utility.
Let X−
k+1 denote the wealth in the risky asset immediately before re-balancing and Y
−
k+1
the wealth in the risk-free asset. We assume the investor is allowed to both borrow and lend risk-free asset. Under singular controls the state evolution equations could be described by:
Xk−+1 = (Xk−+ ∆1k −∆2k)αk (3.67)
Yk−+1 =(Wk−−Xk−−(1+λk)∆1k+(1−µk)∆2k)sk (3.68)
Wk−+1 = s◦Wk−+F(Xk−) (3.69) where the stochastic functionalF(X−k) represents the investor behavior of buying, selling or choosing no to re-balance.
Define the value function as:
Jk(Wk−,X − k)=maxς E P (−e−zWN|F k) (3.70)
50Chapter3. Introduction to discrete probability approximation and sketch of modeling approach
Value function
Risky Fraction Convergence to the true value
function in continous time 0.2 0.4 0.6 0.8 1.057 1.058 1.059 1.060
Figure 3.18: Deformation solution of value function with CRRA utility and jump diffusion model at t = 0 by varying γ` = 15, ..., 201 and parameters: T = 1,N = 4, ω = 0.1, λ = µ =
0.01,s = eω∆T,m = 0.14, σ = 0.3, θ = 0.1, δ = 0.05,V = 0.5. Where T is the time horizon
for investment,Nis the number of re-balancing nodes. ωis the continuous time risk-free rate, (λ, µ) are transaction cost factors, sis the risk free growth over the interval, mis 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. γ`is the deformation
parameter. JN−1(WN−−1,X − N−1) = maxς E P (−e−z(sWN−−+F(X − N))|F k) (3.71) = e−zsWN−−1J(1)(X− N−1) (3.72)
We will show for CRRA utility in Chapter 6 that a similar decomposability with respect to the wealth occurs. In any case recursionktimes would lead to the following:
JN−k(WN−−k,X − N−k)=e −zskW− N−kJ(k)(X− N−k) (3.73)
Let us use the above example to summarize the numerical solution method in a few steps:
I- Find the relevant state variables. In the above they are (W−
k,X
−
k).
II- Sometimes we can simplify the problem in state space. In the above problem for in- stance the optimal control law depends only onXk−as amply illustrated by the structure of the problems.
III- Create a lattice /grid structure to carry out dynamic programming recursion using the state space selected in II. In the above problem, for instance, the relevant lattice contains only
3.11. Analysis of continuous time dynamic trading strategies 51
Xk−. We set the lower and upper limits so that the no-transaction boundaries always lie inside them. It turns out that theMertonline for the time varying CARA problem (see [23]) is m−ω
zσ2eω(T−t) so ifm≥ωfor continuous time risk free rateωthen we could chose the lower bound as being zero or less than zero. The upper bound would need to be set to a sufficiently high level with some experimentation to see for last stage whether no-transaction region lies inside the grid or not.
IV- Create an approximate discrete model for risky growth.
V- To avoid extrapolation errors the grid in the previous stage of dynamic program needs to be a factor lower than the next grid. Let (1+g) is the maximum factor by which the discrete probability approximation could grow for all the branches on the approximation. IfL(k) is the length of our grid at timek then L(0) = (1L+(gN))N. This is because the relevant state variable X
−
k
is directly tied to risky growths and at the upper boundary of the grid we are going to sell the risky asset rather than buy it.
VI- Starting from the last stage carry out recursion using the dynamic programming for- malism. We find optimal controls and value functions for each stage going all the way to the initial time.
We provide an approximate solution in Figure 3.19 for CARA utility using a fairly coarse grid. We set the lower boundary to 5, the upper boundary to 2000 and state space division length to 10. The purpose of this coarse grid is to illustrate the numerical solution. It is seen that the center of risky fraction boundaries is decreasing as we move closer to the initial time. Indeed this should be the case, since, under the no-transaction cost the optimal amount of wealth to hold in risky asset in continuous time is zσm2e−ωω(T−t) wheresis the continuous time risk free rate. In fact, for the parameters we chose in Figure 3.18 the no-transactionMertonamount of ' 632 lies in the center of no-transaction region at t = 0. This serves to show in many cases we can provide some intuitive validation for the numerical solution we have obtained via results in the academic literature.
The thesis will avoid the use of Monte-Carlo and PDE-based solution to dynamic portfolio problems. Monte-Carlo methods have been touted as being great in high dimensions. Stable numerical schemes to solve the non-linear PDEs associated with these set of problems are usually challenging to implement.