American option and VaR LLSM and Asymptotic Convergence Numerical studies Asymptotic Convergence Numerical Studies Conclusion Least-Squares Monte Carlo Simulation for High-Dimensional Portfolio Problems Hoi Ying WONG Department of Statistics, The Chinese University of Hong Kong Presentation based on collaborations with J. Chen, T. Sit, and Z. LI This research is supported by Research Grants Council of Hong Kong Project number: GRF 14303915 May, 2017 1/ 61 LSMC Simulation for High-Dimensional Portfolios UCSB-CEMAR 2017 Hoi Ying WONG
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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
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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.
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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.
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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.
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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
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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.
on the underlying assets prices: e−r(T−t1)Z(T ) ∼ X (t1).
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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)).
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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 ).
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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.
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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
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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.
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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.
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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?
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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.
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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)
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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).
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Convergence
Convergence Issues · Price Consistency
Regularity conditions:
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.
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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
}
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Convergence
Regularity conditions:
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.
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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.
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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).
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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.
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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
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Simulations
Simulation - Rainbow Options
Table 2 : VaR of the Delta-normal Approach for Different Strikes
Remark Closed form solution is applied for valuation in this table.
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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.
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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,....
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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.
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Data Analysis
European Swaption · Correlation
Figure 4 : Instantaneous Correlation
This is the calibration result for instantaneous correlation matrix for forward rates.
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Data Analysis
European Swaption · Volatility
(a) Current Volatility (b) Volatility for different term structures
Figure 5 : Instantaneous Volatility
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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.
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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
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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
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Data Analysis
Bermudan Swaption · Increasing Stopping Times
Table 6 : VaR Trend for Increasing Number of Stopping Times
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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
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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
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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.
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• 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).
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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?
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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.
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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.
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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.
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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.
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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,
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
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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
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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)
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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.
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Asymptotic Convergence
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.
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Comparison for LS and LLS
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
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Comparison for LS and LLS
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:
Table 12 : Converged asset allocations for different initial guesses
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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.
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