Least-Squares Monte Carlo Simulation for High...

American option and VaR LLSM and Asymptotic Convergence Numerical studies Asymptotic Convergence Numerical Studies Conclusion

Least-Squares Monte Carlo Simulation forHigh-Dimensional Portfolio Problems

Hoi Ying WONG

Department of Statistics, The Chinese University of Hong KongPresentation based on collaborations

with J. Chen, T. Sit, and Z. LIThis research is supported by

Research Grants Council of Hong KongProject number: GRF 14303915

May, 2017

1/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG


Introduction

Background: Nonlinear Portfolios

A financial portfolio Π(t,X ) is a linear combination of traded securities.

Mathematically, one can define these portfolios as

Π(t,X ) = EQ[f {T ,X (T )} | Ft ],

where f is a deterministic payoff function and Q is a risk-neutral probability measure.

For portfolios with American feature or surrender rights, we can write

ΠA(t,X ) = supτ∈T

EQ[f {T ,X (τ)} | Ft ],

where T is a family of stopping times.



Introduction

Background

The two forms of portfolios cover most (if not all) financial portfolios in practice.

Examples:

• f {T ,X (T )} = e−r(T−t)∑pi=1 ωiXi (T )

Π(t,X ) reduces to a linear portfolio.

• f {T ,X (T )} = e−r(T−t) max[mini=1,...,p{Xi (T )/Xi (0)− 1} − K , 0]

Π(t,X ) represents a portfolio of holding a European rainbow option on pstocks, (or ΠA(t,X ) ... with an American rainbow option) which is nonlinear.



Introduction

Background

Two problems faced by financial institutions:

A. Pricing

B. Evaluation of the risk

of these (non-)linear portfolios.

In particular, they have to compute the regulatory capital/reserve subject to theBasel II/2.5/(going to be) III for banking industry (BIS, 2013a) or the Solvency IIfor the insurance industry (Bauer et al., 2012).

The regulatory margin charge has also been introduced to the OTC derviatves afterthe sub-prime mortgage crisis (BIS, 2013b).

The survey of Perignon and Smith (2010) shows that around 73% of VaR disclosedbanks use historical simulation and 21.5% Monte Carlo simulation.

All banks in the sample significantly overestimate VaR.



Introduction

Background · Problem A: Pricing

To compute the value of a portfolio which involves derivatives whose closed-formedprices are unknown, one can apply simulations to complete the task.

Figure 1 : Nested Simulation: Simulate N paths of underlying risk factors X from 0 to t1

under the P measure; Continue to simulate these paths from t1 to T under a Q measure



Introduction

Background · Problem A: Pricing

Nested simulations can be computationally intensive.

Longstaff and Schwartz (2001) proposed the use of least squares (LS)/regressionapproach to estimate the conditional expected payoff/values of (path-dependent)derivatives.

Figure 2 : Longstaff and Schwartz (2001) Approach: Regress discounted simulated values

on the underlying assets prices: e−r(T−t1)Z(T ) ∼ X (t1).



Introduction

Problem with OLS Regression

Consider the regression model:

Y =M∑

m=1

βm · φm(X ) + η. (1)

where η has zero mean and finite variance. {φm(X )}Mm=1 is the set of measurablereal-valued functions which serve as the basis functions for regression.

If ηi.i.d.∼ N(0, σ2), then standard regression analysis shows that the optimal rate of

convergence achieves

E [||Y − Y ||2] = E [||Y −M∑

m=1

βm · φm(X )||2] ∼ O(M

N),

where M is a known number of regressors for an unbiased estimator.A 100-asset (p) nonlinear portfolio approximated by a third-order polynomial needsnearly 1m basis functions (M = O(p3)).



Introduction

Background · Problem B: Risk Management

All existing regulatory capital calculations are related to Value-at-Risk (VaR), whichis a prerequisite for some coherent risk measures such as expected shortfall (ES) orconditional VaR (Artzner et al., 1999).

Recall that an h-day α-VaR of a portfolio Π evaluated at t0 is the quantile valuesuch that

Pr{Π(t0 + h)− Π(t0) < −VaR} = α, α ∈ (0, 1).

To calculate the VaR, the current practice in the industry tends to use Greekapproximations such as the Delta-normal and Delta-Gamma approximations (Jorion,2006).

Essentially, these approaches make use of a first/second order Taylor seriesexpansion to compute the corresponding VaR:

δΠ ≈ ∂Z

∂X

>(δX ) +

1

2(δX )>

∂2Z

∂X 2

>

(δX ) := ∆>(δX ) +1

2(δX )>Γ (δX ).



Introduction

Background · Problem B: Risk Management

Computations of ∆ (and even Γ) are mostly done numerically.

The approximation error is moderate for only short holding horizons. However, thesolvency capital ratio (SCR) for insurance companies is based on the 1-year VaR.

Bauer et al. (2012) propose the use of least-squares (LS) Monte Carlo (MC)method to compute the VaR.

The World Bank data shows that the total number of listed domestic companies inthe world was measured at 47,520 and in the USA at 4,102 in 2012.

The high-dimensional problem remains unsolved.



Introduction

How can we solve this problem?

The large number of basis functions included may result in highly volatile coefficientestimates, thus leads to bad VaR estimates.

Model selection methods such as shrinkage methods, contrained `1 minimizationcan be applied in order to reduce the number of parameters in the regression.

Figure 3 : Lasso: Regularized linear regression of e−r(T−t1)Z(T ) on L{X (t1)}.10/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG


Introduction

LASSO

The least absolute shrinkage and selection operator (LASSO) method proposed byTibshirani (1996) has become one of the standard approaches of estimatinghigh-dimensional regression parameters and for model selection.

An advantage of LASSO is its model selection consistency (Zhang and Huang,2008).

In a classical linear regression setting: Zi = X>i β + εi (i = 1, . . . , n). The estimatorβ is

β = arg minb∈Rp

n∑i=1

(Zi − X>i β)2 = (X>X )−1X>Z .

LASSO estimator considers another estimator which minimisesn∑

i=1

(Zi − X>i β)2 + λ

p∑j=1

|βj |,

where λ denotes the penalty.



Algoritm

Algorithm of LLSM · Pricing

For each path simulated, use a reduced model selected by LASSO to estimate thediscounted expected payoff at t1.

LLSM makes no model assumption on the underlying assets.



Algoritm

Algorithm · VaR

After fitting the regression model, we obtain n estimated values of Z(t1), sayZ(1)(t1) ≤ Z(2)(t1) ≤ . . . ≤ Z(n)(t1). Then an α VaR estimate can be obtained by

the empirical α quantile of the sample, i.e. Z(dnαe)(t1).

How about quantile regression?



Algoritm

American derivatives

With possible stopping times j = 1, . . . ,T − 1, the portfolio value is defined as,

U(j) := supτ∈Tj,k

E{Z(τ)|Fj}j=0,1,...,T .

Hence

U(j) =

{Z(T ), j = T

max[Zj ,E{U(j + 1) | Fj ], 0 ≤ j ≤ T − 1.

At t1, the value is define as,

U(t1) := E{U(j∗) | Ft1}, j∗ = min{j ∈ Tt1,T}

Therefore, the conditional expectations could be estimated using LASSO.



Convergence

Convergence Issues · Price Consistency

Ultimate goal of valuation convergence:

U [m,n](j)→ U(j) as m, n→∞.

The idea is to separate the goal into two steps:

limn→∞

U [m,n](j)→ U [m](j), limm→∞

U [m](j)→ U(j)



Convergence

Regularity conditions:

A1. For j = 1, . . . ,T − 1, the sequence {L`{X (j)}}`≥1 is total in L2[σ{X (j)}].A2. For j = 0, 1, . . . ,T − 1, if

∑m`=1 a`L`{X (j)} = 0 a.s., then a` = 0 for

` = 1, . . . ,m.

Theorem 1

Assume Condition A1 is satisfied, then for j = 1, 2, ...,T , we have

limm→∞

U [m](j) = U(j)

Remark: This is a direct result from Clement, Lumberton and Protter (2001).



Convergence

Convergence Issues · Price Consistency


A3. For j = 1, 2, . . . ,T − 1, the error terms in the linear regression model are i.i.d.with mean zero and finite second moment.

A4a. For j = 1, 2, . . . ,T − 1, there exists a non-singular matrix Cj such that thegram matrix in the regression model tends to Cj as N →∞.

A4b (Compatibility Condition). The compatibility condition is met for the set S0,if for some φ0 > 0 and for all α satisfying ‖αSc

0‖1 ≤ 3‖αS0‖1, it holds that

‖αS0‖21 ≤ (α>Σα)

s0

φ20

where s0 =card(S0) = |S0|.

Theorem 2

Assume for each j = 1, 2, . . . ,T − 1, Pr[aj · L{X (j)} = Zj ] = 0 and Condition A1-A3 aresatisfied. Further assume that the compatibility condition holds the active set of variablesS0 = {`; a[m]

j` 6= 0, ` = 1, 2, . . . ,m}. Then, for the LASSO estimators a[m,n]j with suitable

penalty parameter λ : λn

= o(1) and λ = O( log mn

), U [m,n](j) converges to U(j) a.s.



Convergence

VaR Convergence · Optimal Stopping

Theorem 3

For j = 1, . . . ,T − 1, if conditions in the previous theorem are satisfied, then

VaR[m,n](j)→ VaR[m](j) as n→∞

where VaR[m,n](j) and VaR[m](j) is given by

VaR[m,n](j)4= inf

x∈IR

{Pr{U(0)− U [m,n](j) < −x} < α

}VaR[m](j)

4= inf

x∈IR

{Pr{U(0)− U [m](j) < −x} < α

}



Convergence


A5. We say Condition A5 holds for the average pricing error W if both of thefollowing are fulfilled.

(i) The joint pdf gN(·, ·) of U[M]t1

and W and its partial derivatives ∂∂u

gN(u,w),∂2

∂u2 gN(u,w) exist for each N and for all sets of (u,w).

(ii) For N ≥ 1, there exist non-negative functions p0,N(·), p1,N(·), p2,N(·) such thatfor all (u,w),

gN(u,w) ≤ p0,N(w),

| ∂∂u

gN(u,w)| ≤ p1,N(w),

| ∂2

∂u2gN(u,w)| ≤ p2,N(w).

In addition,

supN

∫ ∞−∞|w |rpi,N(w)dw <∞ for i = 0, 1, 2 , 0 ≤ r ≤ 4.



Convergence

VaR Convergence · Non-Optimal Stopping

Theorem 4

If Condition A1-A4b are satisfied, Condition A5 holds for the average pricing errorW , t1 6∈ T1,T , with LASSO estimators in the regression, then

|VaR[m,n](t1)− VaR[m](t1)| ≤ g(v)OP

(√s0 logm

nφ20

)+

∣∣∣∣dg(v)

dv

∣∣∣∣OP( s0 logm

nφ20

)+ oP

(1

n

).

With OLS estimators in the regression, if we substitute Condition A4a for A4b, theconvergence rate becomes,

|VaR[m,n](t1)− VaR[m](t1)| ≤ g(v)OP(√

m

n

)+

∣∣∣∣dg(v)

dv

∣∣∣∣OP(M

N

)+ oP

(1

n

),

where v = VaR[M](t1)− U(0), v ∈ [v − w/√n, v ], s0 is the cardinality of the active

set S0, φ0 is the compatibility constant defined in the Compatibility Condition.



Simulations

Simulation: Rainbow Options

A rainbow option is a typical example of derivatives with nonlinear payoff function.

Consider a European rainbow call option with 10 underlying stocks. The payoff atmaturity is

100 max

{min

i=1,...,10

(SiT

Si0

)− K , 0

},

where i = 1, ...10, K is strike price and T denotes the maturity.

To derive a benchmark, we assume the underlying stocks follow Black-ScholesModel (Black and Scholes, 1973).

Closed form solution for pricing options on the maximum or the minimum of severalassets under BS model is derived in Johnson (1987).



Simulations

Simulation · Rainbow Options

Estimation of drift and volatility for randomly chosen stocks are used in the B-Smodel to simulate 500 daily stock prices as historical observations.

Pretending we do not know the true value of parameters in the B-S model ofunderlying stocks, we use historical data to estimate parameters for simulation fromt = 0 and onward in each approach, including the Oracle approach.

All approaches are based on 10,000 sample paths. Each approach is repeated 500times to obtain sample statistics.



Simulations

Simulation · Rainbow Options

Table 1 : 10-day 95% VaR of Rainbow option

Method Mean Median Std Dev. Back test Time (in sec)LSM 1.6646 1.6592 0.1207 0.0320 12.01LLSM 1.4778 1.4797 0.1069 0.0506 12.92Delta-normal 1.5509 1.5516 0.0203 0.0436 29.80Oracle 1.4882 1.4882 - 0.0500 8,858.67



Simulations

Simulation - Rainbow Options

Table 2 : VaR of the Delta-normal Approach for Different Strikes

Strike Delta-normal Back Testing Oracle0.8414 1.746 0.0330 1.56750.6414 2.251 0.0571 2.35080.4414 2.263 0.0606 2.42970.2414 2.263 0.0553 2.3472

Remark Closed form solution is applied for valuation in this table.



Data Analysis

Data analysis - European Swaptions

Consider a vanilla payer swaption. The maturity of the option is at T2. Theunderlying swap makes 18 annual coupon payments beginning from T2 to the finaltenor T20.

The payoff function at time T2 is

A∗(T2) = EQ2

[ 20∑j=3

D(T2,Tj){Lj(T2)− K}

]+ ∣∣∣∣F2

where D(Ti ,Tj) are discount factors, Lj(T2) is one year forward rates beginning atTj−1 measured at T2.



Data Analysis

Data analysis · European Swaptions

We adopt the Lognormal Forward LIBOR Model-LFM (Brigo et al., 2012) tomodeling forward interest rates.

The dynamics of forward rates Li (t) under forward-adjusted measure Qi are,respectively,

i < j , t ≤ Ti : dLj(t) = σj(t)Lj(t)

j∑k=i+1

ρkjδkσk(t)Lk(t)

1 + δkLk(t)dt + σj(t)Lj(t)dZj(t)

i = j , t ≤ Ti−1 : dLj(t) = σj(t)Lj(t)dZj(t)

i > j , t ≤ Tj−1 : dLj(t) = −σj(t)Lj(t)

j∑k=i+1

ρkjδkσk(t)Lk(t)

1 + δkLk(t)dt + σj(t)Lj(t)dZj(t)

where Z is a Brownian motion under measure Qi . Zi , Zj are Brownian motions ofdifferent forward rates Li (t), Lj(t) with instantaneous correlation ρ = (ρij)i,j=1,2,....



Data Analysis

Data analysis · European Swaptions

The instantaneous volatility is given by

σi (t) = ψi [((Ti−1 − t)γ1 + γ2)e−(Ti−1−t)γ3 + γ4].

where γ = (γ1, γ2, γ3, γ4) is a parameter set, ψi is a correction parameter to fit thevolatilities more closely to market data.

For instantaneous correlation ρ, its parameterized form is given by

ρij = e−β|i−j|.

where β is the only parameter in the correlation function.



Data Analysis

European Swaption · Correlation

Figure 4 : Instantaneous Correlation

This is the calibration result for instantaneous correlation matrix for forward rates.

For forward rates with same expiry, the correlation gets weaker as the gap in thematurity becomes larger.28/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG


Data Analysis

European Swaption · Volatility

(a) Current Volatility (b) Volatility for different term structures

Figure 5 : Instantaneous Volatility



Data Analysis

European Swaption · Specification

Market data which are the same as what described in Brigo et al. (2012) are usedto calibrate model parameters such as the instantaneous correlation, instantaneousvolatility, drift under P measure.

We choose n = 5, 000 in SLSM, GLSM, LLSM, Delta-normal approach.The numberof paths in the Oracle approach is 5, 000 ∗ 30, 000.

Summary statistics are calculated based on 500 iterations of each approach exceptthe Oracle approach.



Data Analysis

European Swaption · Results

Table 3 : 10-day 95% VaR of European Swaption

Method Mean Median Std Dev Back testing Time (in sec)SLSM 0.0089 0.0089 0.0014 0.0204 196.32GLSM 0.0198 0.0198 0.0011 0.0000 201.29LLSM 0.0080 0.0080 0.0014 0.0368 204.23Delta-normal 0.0089 0.0089 0.0000 0.0208 6,799.13Oracle 0.0072 0.0072 - 0.0500 33,965,121.77



Data Analysis

Data Analysis · Bermudan Swaption

Table 4 : Value of Bermudan Swaption

Time T2=year 2 t1=day 10Method Mean Median Std Dev Mean Median Std DevSLSM 0.0730 0.0730 8.521E-04 0.0695 0.0695 8.117E-04GLSM 0.0754 0.0729 8.674E-04 0.0718 0.0717 8.264E-04LLSM 0.0733 0.0732 8.749E-04 0.0698 0.0698 8.335E-04

Table 5 : 10-day 95% VaR of Bermudan Swaption

Method Mean Median Std Dev Time (in sec)SLSM 0.0084 0.0085 0.0016 6,478.40GLSM 0.0210 0.0211 0.0015 6,415.70LLSM 0.0066 0.0066 0.0017 6,700.85Delta-normal 0.1896 0.0079 0.3720 226,835.75



Data Analysis

Bermudan Swaption · Increasing Stopping Times

Table 6 : VaR Trend for Increasing Number of Stopping Times

Stopping Times SLSM GLSM LLSM1 0.008797052 0.019784311 0.0079431414 0.008547246 0.021972795 0.0073746386 0.008314325 0.022834897 0.0070275398 0.008173179 0.022950451 0.00677894712 0.008041545 0.022971909 0.0065648118 0.007757233 0.022595665 0.006240398



Data Analysis

Bermudan Swaption · Results

Table 7 : 10-day 95% VaR of European Swaption

Method Mean Median Std Dev Back testing Time (in sec)SLSM 0.0089 0.0089 0.0014 0.0204 196.32GLSM 0.0198 0.0198 0.0011 0.0000 201.29LLSM 0.0080 0.0080 0.0014 0.0368 204.23Delta-normal 0.0089 0.0089 0.0000 0.0208 6,799.13Oracle 0.0072 0.0072 - 0.0500 33,965,121.77

Table 8 : 10-day 95% VaR of Bermudan Swaption

Method Mean Median Std Dev Time (in sec)SLSM 0.0084 0.0085 0.0016 6,478.40GLSM 0.0210 0.0211 0.0015 6,415.70LLSM 0.0066 0.0066 0.0017 6,700.85Delta-normal 0.1896 0.0079 0.3720 226,835.75



Data Analysis

Bermudan Swaption · A Boxplot of VaRs

(a) VaR boxplot for five approaches (b) VaR boxplot for four approaches

Figure 6 : VaR Boxplot for Bermudan Swaption



Data Analysis

Concluding Remarks

LLSM, as a modification of LSM, is a simulation-based efficient way to computeVaR, as well as the value of a high-dimensional nonlinear portfolio.

LLSM provides an objective way with oracle properties to reduce dimensionality.

Bias of VaR estimators in LSM can be adjusted with better-fitted regression inLLSM.

LLSM is robust to dimensionality and model assumption. Further variance reductionand bias correction method can be applied.



Data Analysis

Background: Portfolio optimization

• Single-period mean-variance portfolio (MVP) problem: Markowitz’s (1952, JF) →Nobel prize in Economics.

• Discrete-time multi-period MVP: Li and Ng (2000, MF).

• Continuous-time MVP: Zhou and Li (2000, AMO).

• Optioned portfolios using lattice approach: Liang, Zhang and Li (2008, MF).

• Time-consistency: Basak and Chabakoui (2010, RFS), Bjork and Murgoci (2010),Cui et al. (2012, MF).

• MVP-based pairs-trading: Chiu and Wong (2011, JEDC) and Chiu and Wong(2015, JCAM).



Data Analysis

Errors, Dimensionality and Constraints

• Estimation error issue: Jagannathan and Ma (2003, JF) discovered no-shortingMVP beats unconstrained MVP.

• Disappointing empirical performance of MVP: De Miguel et al. (2009, RFS) shownaive 1/p strategy beats MVP and a reason is the estimation error in mean.

• Controlling estimation error in mean by norm-constrained portfolio: De Miguel et al.(2009, MS) offer the first proposal; Fan et al. (2012, JASA) link to LASSO; Duboisand Veraart (2015, SIFIN) and Pun and Wong (2016, SIFIN) analyze the Mertonportfolio.

• Controlling estimation error in covariance matrix: Ledoit and Wolf (2003, JEF;2004, JPM) shrink the sample covariance matrix; Pun and Wong (2016, SIFIN)investigate the error effect for high-dimensional Merton portfolio and appliesconstrained `1 minimization to the product of precision and return vector; Chiu, Punand Wong (2017, RA) extend the `1 framework to continuous-time high-dimensionalMVP; Pun and Wong (2015, under review) consider high-dimensional multi-periodMVP.

Questions: How about constrained high-dimensional portfolios? Can the implementationbe extended to optioned portfolios?



Data Analysis

Numerical approches

• Numerical PDE: Wang and Forsyth (2010, EJOR) and Dang and Forsyth (2016,EJOR) solved constrained MV problems via HJB Equations. Accurate results aredemonstrated for low-dimensional portfolios.

• Least-squared simulation: Cong and Oosterlee (2016, JEDC) numerically solvedconstrained MVP using bundling and the least squares Monte-Carlo (LS) method.



Motivation

Motivation

• If we consider multi-period mean-variance optimization problem with constraintsunder high-dimensional framework, numerical tests suggest that LS approach maynot work well.

• High-dimensional basis functions result in highly volatile estimates for regressioncoefficients and consequently lead to unstable estimation for the asset allocations.

• Instead of directly applying least squares regression, improvement could beincorporated to reduce the number of parameters in high-dimensional regression, toproduce consistent estimation results.



Motivation

Problem Formulation

Consider the discrete portfolio optimization problem:

Vt(Wt) = max{xs}T−1

s=t

{E [WT |Wt ]− λVar [WT |Wt ]} (2)

subject to the wealth process:

Ws+1 = Ws · (xsRs + r) + C , s = t, t + 1, ...,T − 1 (3)

where xt ∈ D ⊂ Rp denotes the asset allocation vector in period t, Rt ∈ Rp denotes theexcess return vector of risky assets in the portfolio, r denotes the risk-free return, and Cdenotes the consumption of the investor.



Motivation

Li and Ng (2000) proved that the optimal control for the problem defined in (1) is alsothe optimal control for the following problem:

Vt(Wt) = min{xs}T−1

s=t

{E [(WT −γ

2)2|Wt ]} (4)

where γ = 1λ

+ 2Ex∗ [WT |Wt ].Thus the problem converts to solve:

Vt(Wt) = minxt{E [Vt+1(Wt+1)|Wt ]} (5)

recursively with VT (WT ) = (WT − γ2

)2.



LLS Approach

Approach Description

1. Generate an initial guess of asset allocations {xt}T−1t=0 , and simulate N paths of the

wealth values {Wt(i)}Ni=1, t = 0, ...,T , thus obtain the corresponding utility values{VT (i)}Ni=1;For t = T − 1, ..., 0, backward in time:

2. Use quadratic regression model to estimate Vt+1(Wt+1) = ft+1(Wt+1);

3. Mathematically solve for the optimal asset allocations {xt(i)}Ni=1 from Vt+1(Wt+1);that is,

xt = arg minxt{E [Vt+1(Wt · (xtRt + r) + C)|Wt ]} (6)

4. Use reduced regression model selected by LASSO to estimate {Vt(i)}Ni=1;that is,

Vt(i) = E [Vt+1(i)|Wt = Wt(i), xt = xt(i)]} (7)

Let L(Wt , xt) be a vector of basis functions. LASSO means that

minβ∈Rp

{1

n‖Vt+1 − β · L(Wt , xt)‖2

2 + λ‖β‖1

}.

Iteratively, use the updated asset allocations {xt}T−1t=0 as the initial guess, and repeat

steps 1− 4;43/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG


LLS Approach

Why Regression in Step 4?Suppose Wt(i), xt(i) are obtained, we need to estimate:

Vt(i) = E [Vt+1(i)|Wt = Wt(i), xt = xt(i)]}Computational burden: N2 for each time step

0.0 0.5 1.0 1.5 2.0 2.5

02

46

810

time

Wea

lth

Figure 7 : Sub-paths Simulation



LLS Approach

Solution for Step 3

Suppose the function Vt+1(Wt+1) = ft+1(Wt+1) is such that

Vt+1(Wt+1) = a(Wt+1)2 + b(Wt+1) + c, a > 0

Recall thatxt = arg min

xt{E [Vt+1(Wt · (xtRt + r) + C)|Wt ]}

Proposition 5

Given the function ft+1(·), the optimal solution at time t is given by

xt = −2aWtr + 2aC + b

2aWtE [RtR

Tt ]−1E [Rt ] (8)



Asymptotic Convergence

Convergence Theorem

Theorem 6

Assume for j = 0, 1, ...,T − 1,, the conditions in Lemma (3)− (7) are satisfied. Denote

x[M,N]0 as the optimal asset allocation estimated by regression on N paths with M basis

functions chosen by LASSO with penalty λ : λ = O( logMN

), λN

= o(1).Then, there exists sufficiently large M, s.t.

x[M,N]0

P−→ x0, as N →∞

Idea of proof:

• The convergence of LASSO estimators;

• The convergence of value functions;

• The convergence of asset allocations.




Convergence of LASSO estimators

(1) For j = 1, ...,T − 1, the sequence {Lm(Xj)}m≥1 is total in L2(σ(Xj)).

(2) For j = 0, ...,T − 1, if∑M

m=1 amLm(Xj) = 0 a.s., then am = 0 for m = 1, ...,M.

(3) For j = 1, 2, ...,T − 1, the error terms in the linear regression model are i.i.d. withmean zero and finite second moment.

(4) For j = 1, 2, ...,T − 1, there exists a non-singular matrix Cj , such that the grammatrix in the regression model tends to Cj as N tends to infinity.

Lemma 7

Assume for j = 1, 2, ...,T − 1, Pr {a[M]j · L[M](Xj) = Vj} = 0. Furthermore, conditions

(1)− (4) are satisfied. Then, for the LASSO estimators a[M,N]j with penalty parameter

λ : λN

= o(1), we have

a[M,N]j

a.s.−→ a[M]j , as N →∞




Convergence of Value Functions

Lemma 8

Assume for j = 1, 2, ...,T − 1, we have x[M,N]j −→ x

[M]j , as N →∞. Further assume that

the basis functions are smooth. Then we have

V[M,N]j

a.s.−→ V[M]j , as N →∞

Numerical tests suggest that polynomial basis functions up to third order work well.

Lemma 9

Assume for j = 1, 2, ...,T − 1, we have x[M]j −→ xj , as M →∞. Further assume that

V[M]j+1

a.s.−→ Vj+1, as M →∞. Then we have

V[M]j

a.s.−→ Vj , as M →∞

Notice that for j = T − 1, we have V[M,N]T = V

[M]T = VT .




Convergence of Asset AllocationsDefine the functions V

[M]t+1(Wt+1) = f [M](Wt+1) and Vt+1(Wt+1) = f (Wt+1);

Lemma 10

Assume for j = 0, 1, ...,T −1, we have V[M,N]j+1

a.s.−→ V[M]j+1 , as N →∞. Further assume that

(1) the function f [M](·) is a quadratic function;(2) the quadratic regression errors are i.i.d.;Then we have

x[M,N]j

P−→ x[M]j , as N →∞

Lemma 11

Assume for j = 0, 1, ...,T − 1, we have V[M]j+1

a.s.−→ Vj+1, as M →∞. Further assume that

(1) the function f [M](·) is a quadratic function;(2) ∃ x∗ and d0, s.t. f

[M](x) −→ f (x), ∀x ∈ B(x∗, d0);(3) ∃ x0 ∈ B(x∗, d0), s.t. f ′′(x0) > 0;Then we have

x[M]j

P−→ xj , as M →∞



Comparison for LS and LLS

One-Dimensional CaseConsider a portfolio consists of one risky asset and one risk-free asset.The dynamic of the risky asset reads:

logSt+1

St= (r − σ2/2 + εσ)dt + σ

√dt · ηt , ηt ∼ N(0, 1)

The asset allocation for the risky asset is within [0, 1.5];The parameter setting is as follows:

Parameters Valuesr 0.03ε 0.33σ 0.15dt 1/4W0 1C 0.05T 20N 10000

Table 9 : Parameters setting for comparison 1




By varying γ, the efficient frontier is as follows:

Figure 8 : Efficient Frontier for comparison 151/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG



Multi-Dimensional CaseConsider a portfolio consists of five risky assets and one risk-free asset.The dynamic of each risky asset reads:

logS it+1

S it

= (r − σ2i /2 + εiσi )dt + σi

√dt · ηt , ηt ∼ N(0, 1), i = 1, ..., 5

For simplicity, assume the Brownian Motions are independent;The asset allocation for each risky asset is within [−1.5, 1.5];The parameter setting is as follows:

Parameters Values Parameters Valuesε1 0.06 σ1 0.15ε2 0.045 σ2 0.13ε3 0.08 σ3 0.25ε4 0.068 σ4 0.17ε5 0.049 σ5 0.14r 0.03 dt 1/4W0 1 C -0.1T 20 N 10000

Table 10 : Parameters setting for comparison 252/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG



By varying γ, the efficient frontier is as follows:

Figure 9 : Efficient Frontier for comparison 253/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG


High-Dimensional Constrained Case

High-Dimensional with Box Constraint

Consider a portfolio consists of ten risky assets and one risk-free asset.The dynamic of each risky asset reads:

logS it+1

S it

= (r − σ2i /2 + εiσi )dt + σi

√dt · ηt , ηt ∼ N(0, 1), i = 1, ..., 10

For simplicity, assume the Brownian Motions are independent, except for ρ12 = 0.5;The asset allocation for each risky asset is within [−1, 1];




The parameter setting is as follows:

Parameters Values Parameters Valuesε1 0.15 σ1 0.055ε2 0.35 σ2 0.095ε3 0.23 σ3 0.07ε4 0.2 σ4 0.065ε5 0.38 σ5 0.1ε6 0.25 σ6 0.075ε7 0.33 σ7 0.09ε8 0.3 σ8 0.085ε9 0.28 σ9 0.08ε10 0.18 σ10 0.06r 0.03 dt 1W0 1 C 0.01T 20 N 10000

Table 11 : Parameters setting for box constraint




By fixing γ and setting various initial guesses:

0 10 20 30 40 50 60

010

2030

4050

60

Standard Deviation

Exp

ecte

d R

etur

n

Figure 10 : Updating iterations for different initial guesses




Converged Asset Allocations

Compare converged asset allocations for each risky asset at t = 0:

Initial Guess 1 2 3 4 5 6 7 8 9 10-0.9 -1 0.618 1 -1 1 1 1 1 1 -1-0.7 -1 0.617 1 -1 1 1 1 1 1 -1-0.5 -1 0.618 1 -1 1 1 1 1 1 -1-0.3 -1 0.618 1 -1 1 1 1 1 1 -1-0.1 -1 0.619 1 -1 1 1 1 1 1 -10.1 -1 0.617 1 -1 1 1 1 1 1 -10.3 -1 0.617 1 -1 1 1 1 1 1 -10.5 -1 0.618 1 -1 1 1 1 1 1 -10.7 -1 0.618 1 -1 1 1 1 1 1 -10.9 -1 0.619 1 -1 1 1 1 1 1 -1

Table 12 : Converged asset allocations for different initial guesses



American Option Included Case

American Option Included

Consider a portfolio consists of five risky assets and one risk-free asset.The dynamic of each risky asset reads:

logS it+1

S it

= (r − σ2i /2 + εiσi )dt + σi

√dt · ηt , ηt ∼ N(0, 1), i = 1, ..., 5

For simplicity, assume the Brownian Motions are independent. The allocation for eachrisky asset is within [0, 1]. In addition, we consider that

• the portfolio consists of an American Option on S1 with the strike price set atK = 3 and maturity same as the investment horizon.

• The allocation for American Option is within [0, 0.1].

• The American option is valued using LS simulation where the basis functions are{1, S1, L2(S1), · · · }. In this way, the correlation between S1 and the American optioncan easily be obtained from the coefficient of S1 in the regression.




The parameter setting is as follows:

Parameters Values Parameters Valuesε1 0.25 σ1 0.25ε2 0.30 σ2 0.27ε3 0.35 σ3 0.30ε4 0.40 σ4 0.32ε5 0.45 σ5 0.35r 0.03 dt 1W0 1 C 0T 20 N 10000

Table 13 : Parameters setting for option included




Converged Allocations

Compare converged allocations for each risky asset and the option at t = 0:

Initial Guess 1 2 3 4 5 Option0.1 0.1586 0.1636 0.1598 0.1597 0.1521 0.10.3 0.1586 0.1636 0.1598 0.1597 0.1521 0.10.5 0.1586 0.1636 0.1598 0.1597 0.1521 0.10.7 0.1586 0.1636 0.1598 0.1597 0.1521 0.10.9 0.1586 0.1636 0.1598 0.1597 0.1521 0.1

Table 14 : Converged allocations with different initial guesses



Conclusion

• Extend the multi-period mean-variance portfolio optimization problem intohigh-dimensional framework.

• Apply model selection criterion (LASSO) to improve LS approach. LLS approach isrobust to dimensionality and model assumption.

• Asymptotic convergence is discussed under particular assumptions. Numericalstudies show the consistency of the estimation result by MC simulation.

• American Option can be included to the portfolio.

• Further research could be considered by imposing complicated constraints to thecontrol.


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