# Optimal market completion through financial derivatives with applications to
volatility risk.

Matt Davison Email address<EMAIL_ADDRESS>Department of Statistical and
Actuarial Sciences, Western University  Department of Mathematics, Western
University  Marcos Escobar-Anel Email address<EMAIL_ADDRESS>Department of Statistical and Actuarial Sciences, Western University  Yichen
Zhu Email address<EMAIL_ADDRESS>Department of Statistical and Actuarial
Sciences, Western University

(August 28, 2024)

###### Abstract

This paper investigates the optimal choices of financial derivatives to
complete a financial market in the framework of stochastic volatility (SV)
models. We introduce an efficient and accurate simulation-based method,
applicable to generalized diffusion models, to approximate the optimal
derivatives-based portfolio strategy. We build upon the double optimization
approach (i.e. expected utility maximization and risk exposure minimization)
proposed in Escobar-Anel et al. (2022); demonstrating that strangle options
are the best choices for market completion within equity options. Furthermore,
we explore the benefit of using volatility index derivatives and conclude that
they could be more convenient substitutes when only long-term maturity equity
options are available.
Keywords: Expected utility theory; Constant relative risk aversion (CRRA)
utility; Optimal derivative choice; Volatility risk; Volatility index(VIX)
options

##  1 Introduction

Financial markets are often modelled as a system of contingents on states
mirroring the real-world economy. This generates a concept widely used in
economic and finance literature, namely the complete market, which is simply
described as ‘a market for every good’. Earlier studies assumed that the
number of securities equals the number of states of nature and investigated
the optimal allocation, placing all of the capital at once (see Arrow (1964),
Arrow and Debreu (1954)). Recognizing that investors benefit from adjusting
allocation with a change of market status, more recent researchers have
focused on the idea of a dynamically complete market, which is defined as a
market wherein any contingent claim can be replicated by a self-financing
strategy.

The study of portfolio choice in a dynamically complete market under a
continuous-time framework can be traced back to the seminal work of Merton
(1969), who computed the optimal allocation and consumption policy with a
dynamic programming technique, assuming that the stock price follows a
geometric Brownian motion (GBM). In this framework, the uncertainty is
reflected in the Brownian motion, which captures the randomness of a stock’s
return; hence, investors can achieve the best portfolio performance with
investments only on the stock and a cash account.

However, the financial market is ever evolving, and becoming increasingly
complex; for instance, substantial evidence suggests that a single Brownian
motion or source of randomness is insufficient to explain the movements of a
single stock or index. Researchers have had to incorporate so-called stylized
facts such as stochastic volatility (SV) or stochastic interest rates in their
modelling to mimic this new reality. These stylized facts are captured via
adding new ‘state variables’ (e.g. a new random processes for SV). These state
variables have been recognized as important factors in the portfolio
allocation process.

The importance of adding financial derivatives into a portfolio for market
completion was demonstrated in Liu and Pan (2003), confirming that investors
can improve portfolio performance when adding as many linearly independent
equity options as new state variables in the portfolio composition. They do
this to hedge the risk of the new state variables, thereby achieving
significant improvement in portfolio performance compared to incomplete market
investment (e.g. investing solely on stock and cash account). This work was
extended in many directions. For example, Escobar et al. (2017) constructed an
optimal portfolios with the addition of options to hedge new state variables
accounting for stochastic correlation. Moreover, Li et al. (2018) solved
derivative-based strategies under an asset–liability management (ALM)
framework with the mean–variance criterion. In a similar setting, the optimal
complete and incomplete strategies for the 4/2 SV model were derived in Cheng
and Escobar-Anel (2021), which demonstrated the superiority of the complete
market portfolio.

Although the literature cited above strongly supports the addition of
derivatives to complete the market, investors may complete market in many ways
due to the variety of derivatives in the market. Therefore, investors
effectively have a non-unique solution to the problem (i.e. an infinite number
of strategies, each linked to a choice of derivative, producing the same
maximum expected utility). The problem of infinitely many solutions and the
optimal choice of derivatives was studied in the recent paper Escobar-Anel et
al. (2022) in the context of the Black—Scholes—Merton model. The paper
proposed an optimization criterion (i.e. additional to the maximization of the
utility, namely risk exposure minimization) to produce a unique, meaningful
solution, thus deriving a practical derivative selection methodology for
investors. The risk exposure minimization criterion can be motivated from many
angles, especially in terms of regulatory constraints intended to control
investors’ exposure to risky assets and hence to protect investors’ capital in
the event of a market crash. In this paper, we follow the same derivatives
selection framework and explore the optimal product for market completion in
the popular setting of SV models, with emphasis on the celebrated Heston model
(see Heston (1993)).

There are two major hurdles for our derivatives-based portfolio allocation
problem. First, given the complexity of advanced models with many state
variables jeopardizes the solvability of the utility maximization allocation
problem, closed-form solutions are often unavailable. This hurdle can be
overcome using approximation methods for dynamic portfolio choice problems.
Brandt et al. (2005), inspired by the least-squares Monte Carlo method (see
Longstaff and Schwartz (2001)), recursively estimated the value function and
optimal allocation following a dynamic programming principle. This method was
later named the BGSS and Cong and Oosterlee (2017) utilized the stochastic
grid bundling method for conditional expectation estimation, introduced in
Jain and Oosterlee (2015), further enhancing the accuracy of BGSS.
Additionally, Zhu and Escobar-Anel (2022) targeted unsolvable continuous-time
models, proposing an efficient and accurate simulation-based method, namely
the polynomial affine method for constant relative risk aversion utility
(PAMC). The second hurdle appears in the complexity of derivatives’ price
dynamics which, in contrast to traditional asset classes, could lead to highly
non-linear stochastic differential equations. In this paper, we overcome the
two hurdles simultaneously by unifying the PAMC and an options’ Greek
approximation technique. Notably, the broad applicability of this methodology
laid the foundation for the derivatives selection study within a generalized
model family.

As mentioned above, our focus is on investors who are particularly concerned
about volatility risk and seek the best derivatives to attain market
completion. The seminal paper by Heston (1993) recognized the mean-reverting
pattern of volatilities and introduced the well-known Heston (GBM 1/2) model.
Later, extensions, such as the GBM 3/2 (see Heston (1997)) and GBM 4/2 (see
Grasselli (2017)), were developed to better capture the volatility surface.
These lead to notable successes in the valuation of European equity options,
and semi-closed-form solutions for the option price and Greeks are generally
accessible using Fourier transformation. Popular equity options, such as call,
put, straddle and strangle options, are ideal products for investors to manage
the volatility risk. Furthermore, the volatility index (VIX), a measure of the
stock market’s volatility based on S&P 500 index options provided by the
Chicago Board Options Exchange (CBOE), affords investors an alternative way to
assess the volatility risk. The effectiveness of VIX products in the portfolio
performance enhancement has been confirmed in the literature: see Doran
(2020), Chen et al. (2011) and Warren (2012). Hence, in this paper we compare
two categories of derivatives, namely equity options and VIX options in terms
of optimal dynamic completion.

The contributions of the paper are as follows:

  1. 1.

The multitude of financial derivatives available in the market offers
investors non-unique optimal choice in terms of expected utility theory (EUT)
maximization. Hence we extend the additional optimization criterion proposed
in Escobar-Anel et al. (2022), namely risk exposure minimization, from the
family of GBM to SV models. This aids investors with practical derivative
selection in a popular stock markets modeling setting.

  2. 2.

The PAMC-indirect numerical method is proposed to approximate the optimal
allocation for a constant relative risk aversion (CRRA) investor investing in
the derivatives market. The superior accuracy and efficiency of the
methodology are verified on the Heston model.

  3. 3.

Targeting equity and volatility risk, we first consider the optimal choice
among equity options (e.g. calls, puts, straddles and strangles). We
demonstrate that strangles are the best options for minimizing risk exposure.

  4. 4.

We also investigate the usage of financial derivatives on the VIX as a means
of completing the market, and we conclude that investors would prefer VIX
options to equity strangles when only long-term maturity options are
available.

The remainder of this paper is organized as follows: Section 2 presents the
investor’s problem (i.e. the two criteria for optimal allocation [utility
maximization] and optimal market completion [risk exposure minimization]).
Section 3 details an efficient approximation method for derivatives-based
portfolio allocation. The optimal market completion targeting volatility risk
within an equity option and a VIX option is studied in Section 4, followed by
the conclusion in Section 5. Appendix A presents the mathematical proofs,
while Appendix B provides an alternative approximation method and a numerical
examination of accuracy and efficiency for the two methods.

##  2 Investor’s problem

In this section, we introduce a market completion framework using financial
derivatives. We define a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$ with a right-continuous filtration
$\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$. The market is frictionless (i.e. no
transaction cost and market impact), and a risk-free cash account $M_{t}$, a
stock $S_{t}$ and an investor with constant relative risk aversion (CRRA)
utility, $U(W)=\frac{W^{1-\gamma}}{1-\gamma}$ exist. The market dynamics are
summarized as follows:

$\begin{cases}\frac{dM_{t}}{M_{t}}=rdt\\\
\frac{dS_{t}}{S_{t}}=(r+\lambda^{S}\sigma^{S})dt+\sigma^{S}dB_{t}^{S}\\\
dH_{t}=\mu^{H}dt+\sigma^{H}dB_{t}^{H}\\\
<dB_{t}^{S},dB^{H}_{t}>=\rho_{SH}dt.\end{cases}$ (1)

where $B_{t}^{H}$ and $B_{t}^{S}$ are Brownian motions with correlation
$\rho_{SH}\in(-1,1)$, and the interest rate $r$ is constant. State variable
$H_{t}$ follows a generalized diffusion process, where
$\mu^{H}=\mu^{H}(t,H_{t})$ denotes the drift and
$\sigma^{H}=\sigma^{H}(t,H_{t})$ denotes volatility. The market price of risk
and the volatility of stock could be functions of both the stock price and the
state variable, respectively; that is
$\lambda^{S}=\lambda^{S}(t,H_{t},\ln{S_{t}})$ and
$\sigma^{S}=\sigma^{S}(t,H_{t},\ln{S_{t}})$.

In this market, the number of investable risky assets is less than the number
of risk drivers, hence market incompleteness. To eliminate the welfare loss
resulting from the unhedgeable risk drivers, we introduce a set of financial
derivatives:

$\Omega_{O}^{(n)}=\left\\{\bar{O}_{t}=[O_{t}^{(1)},O_{t}^{(2)},...,O_{t}^{(n)}]^{T}\mid
O_{t}^{(i)}\neq 0\text{, }i=1,...,n\text{ and
rank}\left(\Sigma_{t}\right)=2,\,t\in[0,T]\right\\}.$

We assume that an investor allocates in an element of $\Omega_{O}$; that is, a
specific $\bar{O}_{t}=[O_{t}^{(1)},O_{t}^{(2)},...,O_{t}^{(n)}]^{T}$ $(n\geq
2)$. Note that by arbitrage arguments, the dynamics of the extended market are
as follows:

$\begin{cases}\frac{dM_{t}}{M_{t}}=rdt\\\
d\bar{O}_{t}=diag(\bar{O}_{t})\left[(r\cdot\mathbbm{1}+\Sigma_{t}\Lambda)dt+\Sigma_{t}dB_{t}\right]\\\
dH_{t}=\mu^{H}dt+\sigma^{H}dB_{t}^{H}\\\
<dB_{t}^{S},dB^{H}_{t}>=\rho_{SH}dt,\end{cases}$ (2)

where $B_{t}=[B_{t}^{S},B^{H}_{t}]^{T}$ and $\Sigma_{t}$ represents the
$n\times 2$ variance matrix of $\bar{O}_{t}$; the first column $(i,1)$
represents the sensitivity of $O_{t}^{(i)}$ to the underlying asset $S_{t}$
(i.e. $\frac{\partial O_{t}^{(i)}}{\partial
S_{t}}S_{t}\frac{1}{O_{t}^{(i)}}\sigma^{S}$); and the second column $(i,2)$
represents the sensitivity of $O_{t}^{(i)}$ to the state variable $H_{t}$
(i.e. $\frac{\partial O_{t}^{(i)}}{\partial
H_{t}}\frac{1}{O_{t}^{(i)}}\sigma^{H}$).
$\Lambda=[\lambda^{S},\lambda^{H}]^{T}$, where
$\lambda^{H}=\lambda^{H}(t,H_{t},\ln{S_{t}})$ denotes the market price of
volatility risk. Rank $2$ variance matrix $\Sigma_{t}$ guarantees the
completeness of the market. For simplicity, we also assume that the
derivatives in $\Omega_{O}^{(n)}$ will be rolled over, always maintaining the
same time to maturity and a non-zero value. Note that the investor is not
prohibited from trading on the stock, which is included in $\Omega_{O}^{(n)}$
as a special derivative.

Let $\Omega_{\pi}^{(O)}$ denote the space of admissible strategies satisfying
the standard conditions, where the element
$\pi_{t}=[\pi_{t}^{(1)},\pi_{t}^{(2)},...,\pi_{t}^{(n)}]^{T}$ represents the
proportions of the investor’s wealth in the derivatives
$\bar{O}_{t}=[O_{t}^{(1)},O_{t}^{(2)},...,O_{t}^{(n)}]^{T}$, with the
remaining $1-\mathbbm{1}^{T}\pi_{t}$ invested in the cash account $M_{t}$. The
investor’s wealth process $W_{t}$ satisfies

$\frac{dW_{t}}{W_{t}}=(r+\pi_{t}^{T}\Sigma_{t}\Lambda)dt+\pi_{t}^{T}\Sigma_{t}dB_{t}.$
(3)

The investor’s objective is to maximize the expected utility of their wealth
at terminal $T$; hence, their problem at time $t\in[0,T]$ can be written as

$V(t,W,H,\ln{S})=\max_{\pi_{s\geq
t}\in\Omega_{\pi}^{(O)}}\mathbb{E}(U(W_{T})\mid\mathcal{F}_{t}).$ (4)

The associated Hamilton-Jacobi-Bellman (HJB) equation for the value function
$V$ follows the principles of stochastic control and is given by

$\begin{split}\sup_{\pi_{t}}\bigg{\\{}V_{t}+W_{t}V_{W}(r+\pi_{t}^{T}\Sigma_{t}\Lambda)+\frac{1}{2}W_{t}^{2}V_{WW}(\pi_{t}^{T}\Sigma_{t}\Phi\Phi^{T}\Sigma_{t}^{T}\pi_{t})+W_{t}V_{WH}\sigma^{H}(\pi_{t}^{T}\Sigma_{t}A)+W_{t}V_{W\ln{S}}\sigma^{S}(\pi_{t}^{T}\Sigma_{t}B)\bigg{\\}}\\\
+V_{H}\mu^{H}+\frac{1}{2}V_{HH}(\sigma^{H})^{2}+V_{\ln{S}}(r+\lambda^{S}\sigma^{S})+\frac{1}{2}V_{\ln{S}\ln{S}}(\sigma^{S})^{2}+V_{H\ln{S}}\sigma^{H}\sigma^{S}\rho_{SH}=0,\end{split}$
(5)

where $\Phi=\left[\begin{array}[]{cc}1&0\\\
\rho_{SH}&\sqrt{1-\rho_{SH}^{2}}\end{array}\right]$, $A=[\rho_{SH},1]^{T}$ and
$B=[1,\rho_{SH}]^{T}$.

Next, we define a new artificial market, which consists of three assets: a
risk-free money account $M_{t}$ and two pure factor assets $S_{t}^{(S)}$ and
$S_{t}^{(H)}$:

$\begin{cases}\frac{dM_{t}}{M_{t}}=rdt\\\
\frac{dS_{t}^{(S)}}{S_{t}^{(S)}}=(r+\lambda^{S})dt+dB_{t}^{S}\\\
\frac{dS_{t}^{(H)}}{S_{t}^{(H)}}=(r+\lambda^{H})dt+dB_{t}^{H}\\\
dH_{t}=\mu^{H}dt+\sigma^{H}dB_{t}^{H}\\\
<dB_{t}^{S},dB^{H}_{t}>=\rho_{SH}dt.\end{cases}$ (6)

Compared to the original market, the market state variable is still $H_{t}$;
nonetheless, here the investor can put their money in the hypothetical pure
factor assets $S_{t}^{(S)}$ and $S_{t}^{(H)}$, which have a unit exposure on
$B_{t}^{S}$ and $B_{t}^{H}$, respectively. Let
$\eta_{t}=[\eta_{t}^{(1)},\eta_{t}^{(2)}]^{T}$ be the allocation on the pure
factors (also known as exposures in the literature: see Liu and Pan (2003));
$\hat{W}_{t}$ denotes the investor’s wealth process, and
$\hat{V}(t,\hat{W},H,\ln{S})$ represents the value function in the artificial
market. Similarly, the associated HJB equation is given by

$\begin{split}\sup_{\eta_{t}}\bigg{\\{}\hat{V}_{t}+\hat{W}_{t}\hat{V}_{\hat{W}}(r+\eta_{t}^{T}\Lambda)+\frac{1}{2}\hat{W}_{t}^{2}\hat{V}_{\hat{W}\hat{W}}(\eta_{t}^{T}\Phi\Phi^{T}\eta_{t})+\hat{W}_{t}\hat{V}_{\hat{W}H}\sigma^{H}(\eta_{t}^{T}A)+\hat{W}_{t}\hat{V}_{\hat{W}\ln{S}}\sigma^{S}(\eta_{t}^{T}B)\bigg{\\}}\\\
+\hat{V}_{H}\mu^{H}+\frac{1}{2}\hat{V}_{HH}(\sigma^{H})^{2}+V_{\ln{S}}(r+\lambda^{S}\sigma^{S})+\frac{1}{2}\hat{V}_{\ln{S}\ln{S}}(\sigma^{S})^{2}+\hat{V}_{H\ln{S}}\sigma^{H}\sigma^{S}\rho_{SH}=0.\end{split}$
(7)

If the solution of the associated HJB PDEs exists, then it is easy to verify
that,

$\displaystyle\hat{V}(t,\hat{W},H,\ln{S})$ $\displaystyle=V(t,W,H,\ln{S})$ (8)
$\displaystyle\hat{W}_{t}$ $\displaystyle=W_{t}$ (9)
$\displaystyle\Sigma_{t}^{T}\pi_{t}^{*}$ $\displaystyle=\eta_{t}^{*}.$ (10)

Furthermore, if the number of derivatives in $O_{t}$ is greater than $2$ (i.e.
$n\geq 2$), there are infinitely many optimal strategies, all producing the
same maximum value function.

Aside from the expected utility maximization, the investor is also concerned
with the size of their risky allocations. For instance, on the other hand, an
institutional investor may have to keep their gross allocation exposure under
a certain level due to regulatory constraints. On the other hand, a small
exposure is important for capital safety regarding unmodellable risk, such as
financial crisis. Hence, we consider an additional derivative selection
criterion, namely risk exposure minimization, introduced in Escobar-Anel et
al. (2022):

$\underset{{\bar{O}_{t}\in\Omega_{O}^{(n)}}}{\min}\left\|\underset{\pi_{s\geq
t}\in\Omega_{\pi}^{(O)}}{\arg\max}\mathbb{E}(U(W_{T})\mid\mathcal{F}_{t})\right\|_{1},$
(11)

where $\left\|\pi_{s\geq
t}\right\|_{1}=\sum\limits_{i=1}^{n}\left|\pi_{t}^{(i)}\right|$ represents the
$\ell_{1}$ norm of allocations at time $t$. Note that this objective is
equivalent to maximizing the cash position while shorting less. Escobar-Anel
et al. (2022) demonstrated that the redundancy offers no additional help with
either the investor’s expected utility or their risky asset exposure in the
case of two one-factor assets. In the next proposition, we demonstrate a
generalized conclusion, which applies to any diffusion model.

######  Proposition 1.

Assume that an optimal solution for Problem (11) exists for $n\geq 2$; then,
(11) leads to the same minimal $\ell_{1}$ norm for any $n\geq 2$. In addition,
an optimal strategy exists for Problem (11) such that the number of non-zero
allocations is less than or equal to $2$111The result can be easily extended
to higher dimension. When model contains $m\geq 2$ independent risk factors
(Brownian motions), an optimal strategy exists for Problem (11) such that the
number of non-zero allocations is less than or equal to $m$..

###### Proof.

See Appendix A.1.

∎

Proposition 1 demonstrates that investors do not need to consider the
portfolio composition $O_{t}$ with size $n>2$. Working with $n=2$ is
sufficient for both Problems (4) and (11). We hence only study the simplest
case given a complete market setting (i.e. $n=2$).

##  3 Polynomial affine method for CRRA utilities in financial derivatives
market

In this section, we introduce a methodology to compute derivatives-based
portfolio strategies. This method is required to find the optimal candidate
composition $\bar{O}_{t}\in\Omega_{O}^{(2)}$ for risk exposure minimization.

Complexity in assets’ dynamic models often jeopardizes the analytic
solvability of HJB PDE; this means that closed-form solutions are not always
available. Motivated by this fact, Zhu and Escobar-Anel (2022) proposed a
simulation-based method to approximate the optimal strategy for continuous-
time portfolios within EUT (i.e. the PAMC). The original PAMC method is only
applicable to asset classes, such as equity, fixed income and currency, where
assets’ dynamics are known explicitly. However, with proper modifications, the
PAMC is easily extended to financial derivatives markets. The new method,
namely the PAMC-indirect, is introduced in Section 3.1. Furthermore, an
alternative method is described in Appendix B. The performances of both
methodologies are demonstrated in the case of the Heston model, and the
comparison to the theoretical solution confirms the excellent accuracy and
efficiency of the PAMC-indirect method.

###  3.1 The PAMC-indirect

Inspired by the quadratic affine model family (see Liu (2006)), the PAMC
approach assumes that the value function has the following representation:

$V(t,W,H,\ln{S})=\frac{W^{1-\gamma}}{1-\gamma}f(t,H,\ln{S}),$ (12)

where $f(t,H,\ln{S})$ is approximated by an exponential polynomial function of
order $k$; that is, $\exp\\{P_{k}\\}$. The PAMC method utilizes the Bellman
equation and the fact that the value function at re-balancing time is the
conditional expectation of the value function at $t+\Delta t$; that is,

$V(t,W_{t},H_{t},\ln{S_{t}})=\max_{\pi_{t}}\mathbb{E}(V(t+\Delta t,W_{t+\Delta
t},H_{t+\Delta t},\ln{S_{t+\Delta t}})\mid\mathcal{F}_{t}).$

The PAMC expands the value function at $t+\Delta t$ with respect to wealth
$W$, state variable $H$ and log stock price $\ln{S}$, and it considers a
sufficiently small re-balancing interval $\Delta t$ such that the
infinitesimal $o(\Delta t)$ terms are omitted. Then, the value function
$V(t,W_{t},H_{t},\ln{S_{t}})$ is rewritten as a quadratic function of the
portfolio strategy, and the optimal strategy is immediately solved with the
first order condition given the information at $t+\Delta t$. Proposition 2
displays the estimation of optimal strategy $\eta_{t}^{*}$ in the artificial
pure factor market (6).

######  Proposition 2.

Given the approximation of the value function at the next re-balancing time
$t+\Delta t$ (i.e. $\frac{W^{1-\gamma}}{1-\gamma}\exp\\{P_{k}\\}(t+\Delta
t,H,\ln{S})$), the optimal strategy at time $t$ is given by

$\eta_{t}^{*}=\frac{1}{\gamma}(\Phi\Phi^{T})^{-1}(\Lambda+\frac{\partial
P_{k}}{\partial H}\sigma^{H}A+\frac{\partial
P_{k}}{\partial\ln{S}}\sigma^{S}B).$ (13)

###### Proof.

See Appendix A.2.

∎

The PAMC-indirect inherits the recursive approximation structure of the PAMC.
After the generation of paths of asset price and state variables, the optimal
pure factor strategies at last re-balancing time $T-\Delta t$ can be directly
computed with (13) because $P_{k}(T,H,\ln{S})=0$; the path-wise expected
utilities are obtained through simulation. Furthermore, the expected utilities
are regressed over stock price $S_{T-\Delta t}$ and state variable
$H_{T-\Delta t}$, and the regression function approximates the $V(T-\Delta
t,W,H,\ln{S})$. Then, the method moves backward, and similar procedures are
conducted at each re-balancing time until the optimal initial strategy of the
pure factor portfolio (i.e. $\eta_{0}^{*}$) is obtained.

Finally, the PAMC-indirect calculates the portfolio variance matrix
$\Sigma_{t}$, which depends on the option price $O_{t}$, Delta $\frac{\partial
O_{t}}{\partial S_{t}}$ and the sensitivity to the state variable
$\frac{\partial O_{t}}{\partial H_{t}}$. The optimal derivatives strategy
$\pi_{0}^{*}$ is solved with (10). Only in some special cases (e.g. the Black-
Scholes model) are option prices solved analytically. A variety of
approximation methods for option price and Greeks are available in the
existing literature. The choice of such methods should be determined by the
option style and underlying assets model. For example, an accurate Fourier
transform (FT) approximation is an ideal choice when the semi-closed-form
solution of an option is available (e.g. the Heston model, the
Ornstein–Uhlenbeck 4/2 model), while a simple Monte Carlo simulation is
universal for options with a deterministic exercise date; and a least-squares
Monte Carlo method is applicable when considering American style options.

We clarify the notation in Table 1 and detail the PAMC-indirect in Algorithm
1.

Notation | Meaning
---|---
$B_{t}^{m,S}$,$B_{t}^{m,H}$ | Brownian motion at time $t$ in $m^{th}$ simulated path
$S_{t}^{m}$ | Stock price at time $t$ in $m^{th}$ simulated path
$S_{t}^{m,S}$ | Pure factor asset $S_{t}^{S}$ at time $t$ in $m^{th}$ simulated path
$S_{t}^{m,H}$ | Pure factor asset $S_{t}^{H}$ at time $t$ in $m^{th}$ simulated path
$H_{t}^{m}$ | State variable Stock price at time $t$ in $m^{th}$ simulated path
$O_{t}^{m}$ | Derivatives price at time $t$ in $m^{th}$ simulated path
$n_{r}$ | Number of simulated paths
$\Sigma_{t}^{m}$ | Variance matrix of portfolio composition at time $t$ in $m^{th}$ simulated path
$N$ | Number of simulations to compute expected utility for a given set $(W_{0},S_{t}^{m},H_{t}^{m})$
$\hat{W}_{t+{\Delta t}}^{m,n}(\pi^{m})$ | The simulated wealth level at $t+{\Delta t}$ given the wealth, the allocation and other state variables
| at $t$ are $W_{0}$, $\pi^{m}$, $S_{t}^{m}$, and $H_{t}^{m}$
$\hat{S}_{t+{\Delta t}}^{m,n}$ | A simulated stock price at $t+{\Delta t}$ given $S_{t}^{m}$
$\hat{H}_{t+{\Delta t}}^{m,n}$ | A simulated state variable at $t+{\Delta t}$ given $H_{t}^{m}$
$\hat{O}_{t+{\Delta t}}^{m,n}$ | A simulated option price at $t+{\Delta t}$
$V(t,W,\ln{S},H)$ | Value function at time $t$ given wealth $W$, stock price $S$ and state variable $H$
$\hat{v}^{m}$ | Estimation of $P_{k}(t,\ln{S_{t}^{m}},H_{t}^{m})=\log(f(t,\ln{S_{t}^{m}},H_{t}^{m}))$ in (12). Regressand in regression;
| superscript ${m}$ indicates the corresponding regressor
$(\ln{S_{t}^{m}},H_{t}^{m})$
$L_{t}(H,\ln{S})$ | The regression function to be used to approximate $P_{k}(t,\ln{S},H)$
$\eta_{t}^{m}$ | Optimal strategy at time $t$ in $m^{th}$ simulated path
Table 1: Notation and definitions

Input: $S_{0}$,$W_{0}$,$H_{0}$

Output: Optimal trading strategy $\pi^{*}_{0}$

1 Initialization;

2 Generating $n_{r}$ paths of $B_{t}^{m,S}$, $B_{t}^{m,H}$,$S_{t}^{m}$,
$H_{t}^{m}$,$S_{t}^{m,S}$,$S_{t}^{m,H}$ $\quad for\quad m=1...n_{r}$;

3

4while  _ $t=T-\Delta t$ _ do

5 for  _$m=1...n_{r}$_ do

6 Directly compute optimal allocation $\eta^{m}_{T-\Delta t}$ with Equation
(13) where $P_{k}=1$ at time $T$;

7 for  _ $n=1...N$_ do

8 Generate $\hat{S}_{T}^{m,n,S}$ and $\hat{S}_{T}^{m,n,H}$ given
$S_{T-\Delta}^{m,S}$ and $S_{T-\Delta}^{m,H}$ ;

9 Compute wealth $\hat{W}_{T}^{m,n}(\eta^{m}_{T-\Delta t})$ at the terminal
time given the wealth at $W_{T-\Delta t}=W_{0}$, the transformed value
function is estimated by
$\hat{v}^{m}=\ln{[(1-\gamma)\frac{1}{N}\sum\limits_{n=1}^{N}U(\hat{W}_{T}^{m,n}(\pi^{m}_{T-\Delta
t}))]}-(1-\gamma)\ln{W_{0}}$ ;

10 Regress $\hat{v}^{m}$ over the polynomial of $H_{T-\Delta t}^{m}$ and
$\ln{S}_{T-\Delta t}^{m}$, and obtain the function $L_{T-\Delta t}(H,\ln{S})$;

11

12

13for  _$t=T-2\Delta t$ to $\Delta t$_ do

14 for  _$m=1...n_{r}$_ do

15 Directly compute optimal allocation $\eta^{m}_{t}$ with Equation (13) where
$P_{k}=L_{t+\Delta t}(H,\ln{S})$;

16 for  _ $n=1...N$_ do

17 Generate $\hat{S}_{t+\Delta t}^{m,n}$, $\hat{H}_{t+\Delta t}^{m,n}$,
$\hat{S}_{t+\Delta t}^{m,n,S}$ and $\hat{S}_{t+\Delta t}^{m,n,H}$ given
$S_{t}^{m}$, $H_{t}^{m}$, $S_{t}^{m,S}$ and $S_{t}^{m,H}$ ;

18 Compute wealth $\hat{W}_{T+\Delta t}^{m,n}(\eta^{m}_{t})$ at the terminal
given the wealth at $W_{t}=W_{0}$, the transformed value function is estimated
by $\hat{v}^{m}=\ln{[\frac{1}{N}\sum\limits_{n=1}^{N}(W^{m,n}_{t+\Delta
t}(\pi^{m}_{t}))^{1-\gamma}exp(L_{t+\Delta t}(\hat{H}_{t+\Delta
t}^{m,n},\ln{\hat{S}}_{t+\Delta t}^{m,n}))]}-(1-\gamma)\ln{W_{0}}$ ;

19 Regress $\hat{v}^{m}$ over the polynomial of $H_{t}^{m}$ and
$\ln{S}_{t}^{m}$, and obtain the function $L_{t}(H,\ln{S})$;

20

21

22while  _ $t=0$ _ do

23 $\eta_{0}^{*}$ is obtained with Equation (13) and where the
$P_{k}=L_{\Delta t}(H,\ln{S})$;

24 Apply approximation methods and obtain the price of
$O_{0}(H_{0},\ln{S_{0}})$ as well as its sensitivity $\frac{\partial
O_{0}}{\partial S_{0}}(H_{0},\ln{S_{0}})$ and $\frac{\partial O_{0}}{\partial
H_{0}}(H_{0},\ln{S_{0}})$ ;

25 Compute the variance matrix $\Sigma_{0}$, and the optimal allocation
$\pi_{0}^{*}=(\Sigma_{t}^{T})^{-1}\eta_{0}^{*}$;

26

27

return $\pi^{*}_{0}$

Algorithm 1 PAMC-indirect

##  4 Derivatives selection

In this section, we study derivative selection for market completion-that is,
(11)-for $n=2$ within subsets of the derivative set $\Omega_{O}^{(2)}$. The
derivative selection problem is rewritten as

$\underset{{\bar{O}_{t}\in\Omega_{O}^{(2,C)}}}{\min}\left\|\underset{\pi_{s\geq
t}\in\Omega_{\pi}^{(O)}}{\arg\max}\mathbb{E}(U(W_{T})\mid\mathcal{F}_{t})\right\|_{1},$
(14)

where $\Omega_{O}^{(2,C)}$ is a derivative set defined by

$\Omega_{O}^{(2,C)}=\left\\{\bar{O}_{t}=[S_{t},O_{t}^{(C)}]^{T}|O_{t}^{(C)}\in
C,t\in[0,T]\right\\}.$

The portfolio composition $\bar{O}_{t}\in\Omega_{O}^{(2,C)}$ consists of a
stock $S_{t}$ and a derivative security $O_{t}^{(C)}$; superscript $C$
represents the candidate set of derivative type; and $T_{op}$ denotes the time
to maturity of $O_{t}^{(C)}$. This setting coincides with a popular practical
strategic investment implementation (i.e. the elimination unhedgeable risk
factors of a pure-stock portfolio with financial derivatives). We use the
Heston SV model given in (15) as the proxy of the market dynamics.

$\begin{cases}\frac{dM_{t}}{M_{t}}=rdt\\\ \frac{dS_{t}}{S_{t}}=(r+\lambda
X_{t})dt+\sqrt{X_{t}}dB_{t}^{S}\\\
dX_{t}=\kappa^{X}(\theta^{X}-X_{t})dt+\sigma^{X}\sqrt{X_{t}}dB_{t}^{X}\\\
dO^{(C)}_{t}=(r+\lambda\frac{\partial O^{(C)}_{t}}{\partial
S_{t}}S_{t}X_{t}+\lambda^{X}\frac{\partial O^{(C)}_{t}}{\partial
X_{t}}\sigma^{X}X_{t})dt+\frac{\partial O^{(C)}_{t}}{\partial
S_{t}}S_{t}\sqrt{X_{t}}dB_{t}^{S}+\frac{\partial O^{(C)}_{t}}{\partial
X_{t}}\sigma^{X}\sqrt{X_{t}}dB_{t}^{X}\\\
<B^{S},B^{X}>_{t}=\rho_{S{X}}\end{cases}$ (15)

The Heston model is a specific case of the generalized diffusion model (1)
with $\lambda^{S}=\lambda\sqrt{X_{t}}$, $\lambda^{H}=\lambda^{X}\sqrt{X_{t}}$,
$\sigma^{S}=\sqrt{X_{t}}$, $\mu^{X}=\kappa^{X}(\theta^{X}-X_{t})$ and
$\sigma^{H}=\sigma^{X}\sqrt{X_{t}}$. We employed a representative market-
calibrated set of parameters (see Table 2), given in Liu and Pan (2003), to
investigate the best product to account for volatility risk. The optimal
allocation for the model (15) can be written explicitly with Equations (13)
and (10) as follows

$\displaystyle\pi^{S}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma(1-\rho_{SX}^{2})}(\lambda-\rho_{SX}\lambda^{X})-\pi^{O}_{t}\frac{S_{t}}{O^{(C)}_{t}}\frac{\partial
O^{(C)}_{t}}{\partial S_{t}}$ $\displaystyle\pi^{O}_{t}$ $\displaystyle=$
$\displaystyle\left(\frac{O^{(C)}_{t}}{\gamma\sigma^{X}(1-\rho_{SX}^{2})}(\lambda^{X}-\rho_{SX}\lambda)+\frac{O^{(C)}_{t}}{\gamma}\frac{\partial
P_{k}}{\partial X_{t}}\right)\frac{1}{\frac{\partial O^{(C)}_{t}}{\partial
X_{t}}}.$ (16)

The representation indicates that the optimal allocation on option
$\pi^{O}_{t}$ solely depends on the choice of option $O_{t}^{(C)}$ (i.e.
$\pi^{O}_{t}$ is a function of the option’s sensitivity to the instantaneous
variance and option price), while the optimal allocation on the stock
$\pi^{S}_{t}$ is determined by the ratio of the option’s sensitivity to the
instantaneous variance and the sensitivity to the stock.

Table 2: Parameter value for the Heston model. Parameter | Value | Parameter | Value
---|---|---|---
$T$ | 1 year | $\rho_{SX}$ | -0.4
$\theta^{X}$ | 0.0169 | $\sigma^{X}$ | 0.25
$\kappa^{X}$ | 5.0 | $\lambda$ | 4.0
$\lambda^{X}$ | -7.1 | $T_{op}$ | 0.1 year
$\Delta t$ | $\frac{1}{60}$ | $period$ | 60
$r$ | 0.05 | $X_{0}$ | $\theta^{X}$
$S_{0}$ | 1.0 | $M_{0}$ | 1.0
$W_{0}$ | 1 | $\gamma$ | 4
$N$ | 2000 | $n_{r}$ | 100

###  4.1 Derivatives selection within options on stock

We start the selection among four popular equity options. Specifically, the
candidate set is given by

$C=\left\\{\textit{Call option, Put option, Straddle, Strangle}\right\\}.$

For simplicity, we only consider European-style derivatives. Call (i.e. payoff
$(S-K)^{+}$) and put (i.e. payoff $(K-S)^{+}$) options are the most common
products traded in the market. Additionally, a straddle (i.e. payoff
$(S-K)^{+}+(K-S)^{+}$) is a commonly used product when investors expect the
underlying asset to deviate from the spot price; hence, the long position of a
straddle is approximately a long position on volatility. Compared with a
straddle synthesized by purchasing a call and a put with the same strike price
and maturity, a strangle (i.e. payoff $(S-K_{1})^{+}+(K_{2}-S)^{+}$) has a
more flexible structure, as it takes long positions on out-of-the-money (OTM)
put and call, which is a cheaper way to acquire exposure to volatility
222Elements in variance matrix $\Sigma_{t}$, which are functions of option
prices and Greeks, can be obtained with numerical integration method (see
Rouah (2013) chapter 11). Specifically, we utilized the formula given in
Heston (1993) and compute numerical integration with the Newton-Cotes
formulas. .

Figure 1 displays the risk exposure $\|\pi_{t}\|_{1}$ of portfolios as a
function of derivative moneyness $K/S_{0}$, where $K$ is the strike price of
the options. Figure 1 (a) exhibits risk exposure given options with maturity
$T_{op}=0.1$, and Figure 1 (b) displays results when the option maturity is
$T_{op}=0.5$. In both cases, investors reduce their risk exposure with OTM put
and call options. Puts and calls could lead to illiquid choices, whereas a
straddle achieves minimum $\|\pi_{t}\|_{1}$ when it is near at-the-money
(ATM). The optimal moneyness of a straddle option shifts to the right as
maturity $T_{op}$ increases. The risk exposure with a strangle decreases as
its component put option moves deeper OTM. Furthermore, even the strangle
consisting of a near-ATM put and call outperforms other options. We
consequently conclude that the strangle minimizes the risk exposure.

(a) Maturity=0.1

(b) Maturity=0.5

Figure 1: The $\|\pi_{t}\|_{1}$ versus moneyness: The Y-axis is the risk
exposure of a portfolio containing different derivatives. The X-axis indicates
the moneyness $K/S_{0}$ of calls, puts and straddles.
The strangle is synthesized with an OTM put and an OTM call. Given moneyness
of the OTM put indicated by the X-axis, the strike price of the OTM call is
the one achieving minimum $\|\pi_{t}\|_{1}$ within the range
$[S_{0},110\%S_{0}]$.

The turning point on the left tail of the strangle’s risk exposure in Figure 1
is further studied in Figure 2, where we illustrate how the optimal moneyness
of an OTM call, an allocation on stock $\pi_{t}^{S}$ and an allocation on
strangle $\pi_{t}^{O}$ vary with the moneyness of an OTM put. Note the
practical range selected for the moneyness of an OTM call; that is,
$K^{Call}/S_{0}\in[S_{0},110\%S_{0}]$. It is shown that, if the strike price
of the put option starting at the spot price moves in the direction of OTM,
the corresponding optimal moneyness of the call option also becomes deeply
OTM. The OTM call reaches the boundary earlier than the put, which leads to
the turning point. Before the turning point, allocation on the stock
$\pi^{S}_{t}$ continues to be small, and $\pi^{O}_{t}$ gradually approaches
$0$; hence, the total risk exposure $\|\pi_{t}\|_{1}$ assumes a decreasing
trend. However, $\pi^{S}_{t}$ increases rapidly after the turning point, and
$\|\pi_{t}\|_{1}$ consequently rises as $\pi^{O}_{t}$ continues to drop.
Moreover, Figures 2 (a) and (b) compare strangles with maturity $T_{op}=0.1$
and $T_{op}=0.5$, respectively. The turning point for a longer maturity
strangle is more easily reached, which makes it less preferable.

(a) Maturity=0.1

(b) Maturity=0.5

Figure 2: Impact of the OTM put’s moneyness on the strangle. Left vertical
axis indicates the optimal moneyness of the OTM call within
$[100\%S_{0},110\%S_{0}]$. Right vertical axis indicates the allocation on the
stock and the strangle.

Equation (16) demonstrates that the allocation on the option is determined by
the ratio of the Vega to the option price. Therefore, in Figure 3, we
investigate the relationship between the Vega of the strangle and the time to
maturity to provide further insight for the comparison of maturity in Figures
1 and 2. Figure 3 (a) illustrates the Vega versus the maturity of an ATM
strangle (the moneyness of component put option $K/S_{0}=100\%$) and an OTM
strangle (the moneyness of component put option $K/S_{0}=95\%$). For an
especially short-term maturity strangle, the terminal payoff do not have
sufficient time to react to the change in volatility state, therefore, the
Vega is small. For the long-term maturity strangle, a change in the
instantaneous variance also has a small impact on the option price because of
its mean-reverting nature. Hence, the Vegas of both strangles are concave in
time to maturity, which peaks at around $0.3$ years. The impact from time to
maturity on the ratio of Vega to price is illustrated in Figure 3 (b), where
$\frac{\partial O^{(C)}_{t}}{\partial X_{t}}/O^{(C)}_{t}$ is always positive
and monotonically decreases with maturity, which leads to an increasing
$|\pi_{t}^{O}|$. In Figure 2, $\pi_{t}^{S}$ is close to $0$ before the
boundary, and $|\pi_{t}^{O}|$ increases with maturity; hence, we conclude that
a short-term maturity strangle is preferable.

(a) Vega $\frac{\partial O^{(C)}_{t}}{\partial X_{t}}$

(b) Vega to Price $\frac{\partial O^{(C)}_{t}}{\partial X_{t}}/O^{(C)}_{t}$

Figure 3: Sensitivity of strangle option price $O^{(C)}_{t}$ to instantaneous
variance $X_{t}$ versus time to maturity $T_{op}$. The legend indicates the
moneyness of component put, and the call option is the one achieving minimum
risk exposure. Note that there is no boundary for the strike price of the
component call.

###  4.2 Derivatives selection within VIX products

Next, we study an investor who has access to the VIX of the stock at hand,
such as the VIX for the S&P 500. In this case, the investor has direct access
to the volatility risk by investing in products based on the VIX. The VIX has
drawn investors’ attention since its origin in 1993; not only is it a real-
time indicator of the market sentiment, but also products such as VIX futures
and VIX options are popular for hedging volatility risk. In this section, we
explore products on the VIX. We consider a candidate set

$C=\left\\{\textit{VIX call, VIX put, VIX straddle, Strangle}\right\\}.$

Note that a strangle is the best option for minimizing risk exposure
considered in Section 4.1. VIX calls and VIX puts are call and put options,
respectively, based on the value of the VIX. A VIX straddle is an instrument
synthesized by the long position of a VIX call and a VIX put with the same
strike price.

Given the definition of VIX as specified in the CBOE white paper CBOE (2003),
Lin (2007) solved the VIX2 in closed-form as a function of instantaneous
variance $X_{t}$. Under the Heston model, we have

$\begin{split}\textit{VIX}^{2}_{t}&=\frac{1}{\tau}\left(a_{\tau}X_{t}+b_{\tau}\right)\\\
a_{\tau}=\frac{1-\exp{-\kappa_{v}^{*}\tau}}{\kappa_{v}^{*}},\quad
b_{\tau}=\theta_{v}^{*}(\tau-
a_{\tau})\quad&\kappa_{v}^{*}=\kappa_{v}+\lambda^{X}\sigma_{v},\quad\theta_{v}^{*}=\frac{\kappa_{v}\theta_{v}}{\kappa_{v}^{*}},\quad\tau=\frac{30}{365},\end{split}$
(17)

where VIX${}^{2}_{t}$ is linear with the instantaneous variance $X_{t}$.
Computing a VIX option’s price and Greeks is easy via Monte Carlo simulation;
this method enable us to find elements in the variance matrix $\Sigma_{t}$.

Unlike options on the stock, by investing in VIX products, the investor
acquires exposure only on the volatility risk; hence, the variance matrix
$\Sigma_{t}$ is diagonal. Moreover, the equity-neutral position of VIX
products leads to a specific case of (16):

$\displaystyle\pi^{S}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\gamma(1-\rho_{SX}^{2})}(\lambda-\rho_{SX}\lambda^{X})$
$\displaystyle\pi^{O}_{t}$ $\displaystyle=$
$\displaystyle\left(\frac{O^{(C)}_{t}}{\gamma\sigma^{X}(1-\rho_{SX}^{2})}(\lambda^{X}-\rho_{SX}\lambda)+\frac{O^{(C)}_{t}}{\gamma}\frac{\partial
P_{k}}{\partial X_{t}}\right)\frac{1}{\frac{\partial O^{(C)}_{t}}{\partial
X_{t}}}.$ (18)

In this case, the allocation on the stock is invariant to the choice of VIX
products, which thus becomes a natural lower bound for risk exposure (i.e.
$\|\pi_{t}\|_{1}\geq|\pi_{t}^{S}|$ ).

The risk exposure when investors hedge the volatility risk with VIX calls and
puts is displayed in Figure 4 (a). On the one hand, calls and puts on the VIX
have similar properties as those on the stock: OTM options tend to achieve
smaller risk exposure. On the other hand, a VIX straddle is less ineffective
in hedging the volatility risk because it is relatively insensitive to the
volatility, and a larger risk exposure $\|\pi_{t}\|_{1}$ is needed for
investors compared to the cases of VIX calls and puts. The risk exposure with
the equity strangle is displayed for comparison purpose; here, the turning
point resulting from the boundary of moneyness on the OTM call is still
evident. Moreover, the strangle achieves a much smaller risk exposure than the
VIX products. We therefore conclude that equity strangle is superior when the
time to maturity $T_{op}$ for candidate products is small ($T_{op}=0.1$).

Figure 4 (b) illustrates how the option maturity $T_{op}$ affects the risk
exposure $\|\pi_{t}\|_{1}$. It indicates that an OTM VIX call and an OTM VIX
put are preferable in (a), and a similar conclusion is verified numerically
for any $T_{op}\in(0,1]$. Therefore, risk exposure for the best VIX call
($K=105\%S_{0}$) and VIX put ($K=95\%S_{0}$) are plotted in Figure 4 (b). In
addition, the minimum risk exposure within a pre-specified region of moneyness
is also displayed. As the volatility time series exhibits a mean-reverting
property, the VIX options with long-term maturity are insensitive to the
instantaneous variance; hence, it has little effect in hedging the volatility
risk. The figure also suggests that a large allocation on the long-term
maturity VIX option is needed, such that the risk exposure increases rapidly
with maturity. A strangle achieves smaller risk exposure when short-term
maturity products are available in the market, aligning with the result in
Figure 4 (a).

According to Figures 1 and 2, the boundary of the OTM call is reached faster
as $T_{op}$ increases, and the boundary significantly restricts risk exposure,
thus reducing the effect of the strangle. This leads to a steep slope of risk
exposure for the strangle in Figure 4 (b). In summary, the investor should
make a choice between VIX products and an equity strangle, depending on the
situation. If the investor has access to short-term maturity options, then the
strangle is preferable. However, when only long-term maturity products are
available, the investor should choose call options on the VIX for market
completion.

(a) The $\|\pi_{t}\|_{1}$ versus moneyness: the time to maturity of both VIX
options and the strangle is $0.1$ year. The X-axis indicates moneyness of VIX
options and the OTM put in the strangle; the strike price of the OTM call is
the one achieving minimum $\|\pi_{t}\|_{1}$ within the range
$[S_{0},105\%S_{0}]$.

(b) The $\|\pi_{t}\|_{1}$ versus maturity: the strike price of VIX calls is
$105\%S_{0}$. The strike price of VIX puts is $95\%S_{0}$. The green line
shows the smallest $\|\pi_{t}\|_{1}$ is achieved by the strangle given the OTM
put strike price $K^{Put}\in[95\%S_{0},100\%S_{0}]$ and the OTM call strike
price $K^{Call}\in[100\%S_{0},105\%S_{0}]$.

Figure 4: The $\|\pi_{t}\|_{1}$ for VIX products

##  5 Conclusion

This paper explored optimal derivatives-based portfolios to complete a market
characterized by volatility risk as a state variable. An accurate and high-
speed approximation for optimal allocations is proposed, for the unsolvable
problem of optimal derivative exposure. In addition to the traditional
portfolio decision objective (i.e. EUT maximization), we work with an
additional criterion, namely risk exposure minimization, for derivative
selection. This aids in the selection of a meaningful product out of many that
maximize the utility. We found that strangle options are the best equity
option product for managing volatility risk. Moreover, we demonstrated that
options based on the VIX are superior to equity strangles in some realistic
situations.

There are many interesting potential extensions to this line of research. For
instance, we could incorporate multi-factor models considering the stochastic
interest rates, stochastic correlations, jumps and stochastic market prices of
risk, to mention a few. These are more realistic settings, solvable within our
numerical method, hence providing investors with valuable insight into optimal
high-dimensional portfolios and multi-asset derivatives for sensible practical
investment.

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##  Appendix A Proofs

###  A.1 Proof of Proposition 1

Let $O_{t,n}=[O^{(1)}_{t},O^{(2)}_{t},...,O^{(n)}_{t}]^{T}$ with variance
matrix $\Sigma_{t}$ of rank 2 be an optimal subset of options for problem
(11). $\pi^{*}_{t,n}$ is a strategy maximizing the expected utility if and
only if $\Sigma^{T}_{t}\pi^{*}_{t,n}=\eta^{*}_{t}$. Therefore, $O_{t,n}$ and
$\pi^{*}_{t,n}$ is an optimal pair for (11) when $\pi^{*}_{t,n}$ is an optimal
solution for

$\displaystyle\mathmakebox[c]{\underset{\displaystyle\pi_{t}}{\mathrm{minimize}}}\quad||\pi_{t}||_{1}\hfil\hfil\hfil\hfil$
(19) $\displaystyle\mathmakebox[c]{{\mathrm{subject\leavevmode\nobreak\
to}}}\quad$ $\displaystyle\Sigma^{T}_{t}\pi_{t}=\eta^{*}_{t}$

According to principle 4.5 in Rardin and Rardin (1998), problem (19) is
equivalent to

$\displaystyle\mathmakebox[c]{\underset{\displaystyle\delta_{t}}{\mathrm{minimize}}}\quad\mathbbm{1}^{T}\delta_{t}\hfil\hfil\hfil\hfil$
(20) $\displaystyle\mathmakebox[c]{{\mathrm{subject\leavevmode\nobreak\
to}}}\quad$
$\displaystyle\hat{\Sigma}^{T}_{t}\delta_{t}=\eta^{*}_{t},\hfil\hfil$
$\displaystyle\delta_{t}\geq 0$

where
$\delta_{t}=[\alpha_{t}^{(1)},\alpha_{t}^{(2)},...,\alpha_{t}^{(n)},\beta_{t}^{(1)},\beta_{t}^{(2)},...,\beta_{t}^{(n)}]^{T}$
satisfies $\alpha_{t}^{(i)}=\frac{|\pi_{t}^{(i)}|+\pi_{t}^{(i)}}{2}$, and
$\beta_{t}^{(i)}=\frac{|\pi_{t}^{(i)}|-\pi_{t}^{(i)}}{2}$, with

$\hat{\Sigma}_{t}=\left[\begin{array}[]{c}\Sigma_{t}\\\
-\Sigma_{t}\end{array}\right]=\left[\begin{array}[]{cc}f^{11}_{t}&f^{12}_{t}\\\
...&...\\\ f^{n1}_{t}&f^{n2}_{t}\\\ -f^{11}_{t}&-f^{12}_{t}\\\ ...&...\\\
-f^{n1}_{t}&-f^{n2}_{t}\end{array}\right].$ (21)

Theorems 2.3 and 2.4 in Bertsimas and Tsitsiklis (1997) lists the necessary
and sufficient conditions for the extreme point $\delta_{t}$, i.e.

  1. 1.

$\delta_{t}=[\delta_{t}^{(1)},\delta_{t}^{(2)},...,\delta_{t}^{(n)},\delta_{t}^{(n+1)},\delta_{t}^{(n+2)},...,\delta_{t}^{(2n)}]^{T}$.

  2. 2.

the $\hat{q}^{th}$ and $\hat{p}^{th}$ rows in $\hat{\Sigma_{t}}$ are linear
independent, $\delta_{t}^{(i)}=0$ if $i\neq\hat{q}$ or $\hat{p}$.

  3. 3.

$\delta_{t}$ is feasible solution.

Without loss of generality, we assume the $p^{th}$ and $q^{th}$ rows in
$\Sigma$ are linear independent, and we consider 4 cases:

$\begin{split}\delta_{t}^{[1]}&=\begin{cases}[\delta_{t}^{[1],(1)},\delta_{t}^{[1],(2)},...,\delta_{t}^{[1],(n)},\delta_{t}^{[1],(n+1)},\delta_{t}^{[1],(n+2)},...,\delta_{t}^{[1],(2n)}]^{T}\\\
\delta_{t}^{[1],(i)}=0\quad\textit{ if $i\neq q$ or $p$}\end{cases}\\\
\delta_{t}^{[2]}&=\begin{cases}[\delta_{t}^{[2],(1)},\delta_{t}^{[2],(2)},...,\delta_{t}^{[2],(n)},\delta_{t}^{[2],(n+1)},\delta_{t}^{[2],(n+2)},...,\delta_{t}^{[2],(2n)}]^{T}\\\
\delta_{t}^{[2],(i)}=0\quad\textit{ if $i\neq q+n$ or $p$}\end{cases}\\\
\delta_{t}^{[3]}&=\begin{cases}[\delta_{t}^{[3],(1)},\delta_{t}^{[3],(2)},...,\delta_{t}^{[3],(n)},\delta_{t}^{[3],(n+1)},\delta_{t}^{[3],(n+2)},...,\delta_{t}^{[3],(2n)}]^{T}\\\
\delta_{t}^{[3],(i)}=0\quad\textit{ if $i\neq q$ or $p+n$}\end{cases}\\\
\delta_{t}^{[4]}&=\begin{cases}[\delta_{t}^{[4],(1)},\delta_{t}^{[4],(2)},...,\delta_{t}^{[4],(n)},\delta_{t}^{[4],(n+1)},\delta_{t}^{[4],(n+2)},...,\delta_{t}^{[4],(2n)}]^{T}\\\
\delta_{t}^{[4],(i)}=0\quad\textit{ if $i\neq q+n$ or
$p+n$}\end{cases}\end{split}$ (22)

It is clear that there is a non-negative strategy in $\delta_{t}^{[1]}$,
$\delta_{t}^{[2]}$, $\delta_{t}^{[3]}$ and $\delta_{t}^{[4]}$ because the
$i^{th}$ row in $\hat{\Sigma}$ is the opposite of the $(i+n)^{th}$ row, and
the non-negative strategy is feasible and an extreme point. This proves the
existence of an extreme point for problem (20). Now, theorem 2.7 in Bertsimas
and Tsitsiklis (1997) guarantees that there is an optimal solution which is an
extreme point for problem (20).
With the second necessary and sufficient conditions of the extreme point, we
know that an optimal solution $\delta_{t}^{*}$ for problem (20) has at most
two non-zero elements. This would imply an optimal solution, denoted by
$\pi^{*}_{t,n}=[\pi_{t,n}^{(1)},\pi_{t,n}^{(2)},...,\pi_{t,n}^{(n)}]^{T}$, for
problem (19) with at most two non-zero elements, which would also be the
optimal strategy for (11).
Without loss of generality, we assume $\pi_{t,n}^{(i)}=0$, $i\neq x,y$.
$O_{t,2}=[O^{(x)}_{t},O^{(y)}_{t}]$ and
$\pi^{*}_{t,2}=[\pi_{t,n}^{(x)},\pi_{t,n}^{(y)}]^{T}$ is a feasible strategy
for problem (11) with $n=2$. We show that it is an optimal pair by
contradiction.
If there is a feasible solution
$\hat{O}_{t,n}=[\hat{O}^{(1)}_{t},\hat{O}^{(2)}_{t}]$ and
$\hat{\pi}^{*}_{t,2}=[\hat{\pi}_{t,2}^{(1)},\hat{\pi}_{t,2}^{(2)}]^{T}$ such
that $||\hat{\pi}^{*}_{t,2}||_{1}<||\pi^{*}_{t,2}||_{1}$, then
$\hat{\pi}^{*}_{t,n}=[\hat{\pi}_{t,2}^{(1)},\hat{\pi}_{t,2}^{(2)},0,...,0]^{T}$
is a feasible strategy for (11) such that
$||\hat{\pi}^{*}_{t,n}||_{1}<||\pi^{*}_{t,n}||_{1}$, which is contradiction to
our previous conclusion. Note that
$||\pi^{*}_{t,2}||_{1}=||\pi^{*}_{t,n}||_{1}$, so problem (11) with $n=2$ and
with $n\geq 2$ have the same minimum $\ell_{1}$ norm of allocation.

###  A.2 Proof of Proposition 2

According to the Bellman equation, the value function can be rewritten as,

$\begin{split}V(t,W,\ln{S},H)&=\mathbb{E}_{t}(V(t+dt,W_{t+dt},H_{t+dt},\ln{S_{t+dt}})\mid
W,H,\ln{S})\\\
&=\max\limits_{\eta_{t}}\mathbb{E}_{t}(V(t+dt,W_{t+dt},H_{t+dt},\ln{S_{t+dt}})\mid
W,\eta,H,\ln{S}).\end{split}$ (23)

We expand $V(t+dt,W_{t+dt},H_{t+dt},\ln{S_{t+dt}})$ at $t+dt$ in terms of all
the variables.

$\begin{split}&V(t+dt,W_{t+dt},H_{t+dt},\ln{S_{t+dt}})=V(t+dt,W_{t},\ln{S_{t}},H_{t})+V_{W_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})d\hat{W}_{t}\\\
&+\frac{1}{2}V_{W_{t}W_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})(d\hat{W}_{t})^{2}+V_{\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})d\ln{S_{t}}+V_{H_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})dH_{t}\\\
&+\frac{1}{2}V_{\ln{S_{t}}\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})d\ln{S_{t}}d\ln{S_{t}}+\frac{1}{2}V_{H_{t}H_{t}}(t+dt,W_{t},\ln{S_{t}},H_{t})dH_{t}dH_{t}\\\
&+V_{W_{t}\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})d\hat{W}_{t}d\ln{S_{t}}+V_{W_{t}H_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})d\hat{W}_{t}dH_{t}\\\
&+V_{\ln{S_{t}}H_{t}}(t+dt,W_{t},\ln{S_{t}},H_{t})d\ln{S_{t}}dH_{t}+o(dt).\end{split}$
(24)

Substituting $d\hat{W}_{t}$, $d\ln{S_{t}}$, $dH_{t}$ which can be found in
equation (1), taking conditional expectation on both sides, and rewriting
$V(t,W_{t},H_{t},\ln{S_{t}})$ in a quadratic form with respect to $\eta$ leads
to

$\begin{split}V(t,W_{t},H_{t},\ln{S_{t}})&=\max_{\eta_{t}}\left(\sum_{i,j=1}^{2}f_{i,j}(t,W_{t},,H_{t}\ln{S_{t}})\eta^{(i)}_{t}\eta^{(j)}_{t}+\sum_{i=1}^{2}f_{i}(t,W_{t},H_{t},\ln{S_{t}})\eta^{(i)}_{t}+f_{0}(t,W_{t},H_{t},\ln{S_{t}})\right)\\\
f_{i,j}(t,W_{t},H_{t},\ln{S_{t}})&=\frac{1}{2}V_{W_{t}W_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})\hat{W}_{t}^{2}(\Phi\Phi^{T})_{i,j}dt\\\
f_{i}(t,W_{t},H_{t},\ln{S_{t}})&=V_{W_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})\hat{W}_{t}\Lambda_{i}dt+V_{W_{t}\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})\hat{W}_{t}\sigma_{S}B_{i}dt\\\
&+V_{W_{t}H_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})\hat{W}_{t}\sigma_{H}A_{i}dt\\\
f_{0}(t,W_{t},H_{t},\ln{S_{t}})&=V(t+dt,W_{t},H_{t},\ln{S_{t}})+V_{W_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})\hat{W}_{t}rdt\\\
&+V_{\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})(\lambda^{S}\sigma^{S}-\frac{1}{2}(\sigma^{S})^{2})dt+V_{H_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})\mu^{H}dt\\\
&+\frac{1}{2}V_{H_{t}H_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})(\sigma^{H})^{2}dt+\frac{1}{2}V_{\ln{S_{t}}\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})(\sigma^{S})^{2}dt\\\
&+V_{\ln{S_{t}}H_{t}}(t+dt,W_{t},H_{t},\ln{S_{t}})\sigma^{S}\sigma^{H}dt.\end{split}$
(25)

We assume a sufficiently small $dt$ so that $o(dt)$ terms are omitted when
taking conditional expectations. The optimal allocation is given by the
solution to the system of equations:

$\sum_{j=1}^{2}2f_{i,j}(t,W_{t},H_{t},\ln{S_{t}})\eta^{(*,j)}_{t}=-f_{i}(t,W_{t},H_{t},\ln{S_{t}}),\>i=1,2.$
(26)

With the representation of the value function in equation (12) and assuming
that $f(t,H,\ln{S})=exp(P_{k}(t,H,\ln{S}))$, the derivatives of value function
with respect to each stock and state variable can be rewritten as,

$\begin{split}V_{W}(t+dt,W_{t},H_{t},\ln{S_{t}})&=\hat{W}_{t}^{-\gamma}exp(P_{k}(t+dt,H_{t},\ln{S_{t}}))\\\
V_{WW}(t+dt,W_{t},H_{t},\ln{S_{t}})&=-\gamma\hat{W}_{t}^{-\gamma-1}exp(P_{k}(t+dt,H_{t},\ln{S_{t}}))\\\
V_{W\ln{S_{t}}}(t+dt,W_{t},H_{t},\ln{S_{t}})&=\hat{W}_{t}^{-\gamma}exp(P_{k}(t+dt,H_{t},\ln{S_{t}}))\frac{\partial
P_{k}(t+dt,H_{t},\ln{S_{t}})}{\partial\ln{S_{t}}}\\\
V_{WH_{t}}(t+dt,W,H_{t},\ln{S_{t}})&=\hat{W}_{t}^{-\gamma}exp(P_{k}(t+dt,H_{t},\ln{S_{t}}))\frac{\partial
P_{k}(t+dt,H_{t},\ln{S_{t}})}{\partial H_{t}}.\end{split}$ (27)

Substituting (27) into (26), the optimal strategy can be approximated as
follows:

$\begin{split}\sum_{j=1}^{2}g_{i,j}(t,W_{t},H_{t},\ln{S_{t}})\eta^{(*,j)}_{t}&=g_{i}(t,W_{t},H_{t},\ln{S_{t}}),\>i=1,2\\\
g_{i,j}(t,W_{t},H_{t},\ln{S_{t}})&=\gamma(\Phi\Phi^{T})_{i,j}\\\
g_{i}(t,W_{t},H_{t},\ln{S_{t}})&=\Lambda_{i}+\frac{\partial
P_{k}(t+dt,H_{t},\ln{S_{t}})}{\partial\ln{S_{t}}}\sigma^{S}B_{i}+\frac{\partial
P_{k}(t+dt,H_{t},\ln{S_{t}})}{\partial H_{t}}\sigma^{S}A_{i},\\\ \end{split}$
(28)

Then, the optimal strategy can be rewritten in matrix form:

$\eta_{t}^{*}=\frac{1}{\gamma}(\Phi\Phi^{T})^{-1}(\Lambda+\frac{\partial
P_{k}}{\partial H}\sigma^{H}A+\frac{\partial
P_{k}}{\partial\ln{S}}\sigma^{S}B).$ (29)

##  Appendix B Alternative approximation method and comparison

###  B.1 Direct method

We introduced an alternative method for derivatives-based portfolio strategy,
namely PAMC-direct method, which is straightforward application of the PAMC.
At each re-balancing time, the path-wise option price $O_{t}$, Delta
$\frac{\partial O_{t}}{\partial S_{t}}$ and the sensitivity to the state
variable $\frac{\partial O_{t}}{\partial H_{t}}$ are approximated, so the
instantaneous dynamics of derivatives are obtained. In this way, derivatives
can be taken as an asset with dynamics known explicitly, the PAMC method is
directly applied. Next proposition shows the estimation of optimal strategy
$\pi_{t}^{*}$ in PAMC-direct.

######  Proposition 3.

Given the approximation of the value function at the next re-balancing time
$t+\Delta t$ (i.e. $\frac{W^{1-\gamma}}{1-\gamma}\exp\\{P_{k}\\}(t+\Delta
t,H,\ln{S})$), the optimal strategy at time $t$ is given by

$\pi_{t}^{*}=\frac{1}{\gamma}(\Sigma_{t}\Phi\Phi^{T}\Sigma_{t}^{T})^{-1}(\Sigma_{t}\Lambda+\frac{\partial
P_{k}}{\partial H}\sigma^{H}\Sigma_{t}A+\frac{\partial
P_{k}}{\partial\ln{S}}\sigma^{S}\Sigma_{t}B).$ (30)

###### Proof.

Similar to Appendix A.2.

∎

We continue to use the notation in Table 1 and describe the step by step
algorithm of the PAMC-direct in Algorithm 2.

Input: $S_{0}$,$W_{0}$,$H_{0}$

Output: Optimal trading strategy $\pi^{*}_{0}$

1 initialization;

2 Generating $n_{r}$ paths of $B_{t}^{m}$, $B_{t}^{m,H}$,$S_{t}^{m}$,
$H_{t}^{m}$ $\quad for\quad m=1...n_{r}$;

3 Apply approximation methods and obtain the price of
$O_{t}(H_{t}^{m},\ln{S_{t}^{m}})$ as well as its sensitivity $\frac{\partial
O_{t}}{\partial S_{t}}(H_{t}^{m},\ln{S_{t}^{m}})$ and $\frac{\partial
O_{t}}{\partial H_{t}}(H_{t}^{m},\ln{S_{t}^{m}})$ $for$ $t=0,\Delta t,...T$ ;

4 while  _ $t=T-\Delta t$ _ do

5 for  _$m=1...n_{r}$_ do

6 Compute the variance matrix $\Sigma_{T-\Delta t}^{m}$ with derivatives price
and sensitivity obtained in step 3 ;

7 Directly compute optimal allocation $\pi^{m}_{T-\Delta t}$ with Equation
(30) where the $P_{k}=1$ at time $T$;

8 for  _ $n=1...N$_ do

9 Generate $\hat{S}_{T}^{m,n}$ and $\hat{H}_{T}^{m,n}$ given
$S_{T-\Delta}^{m}$ and $H_{T-\Delta}^{m}$ and obtain $\hat{O}_{T}^{m,n}$;

10 Compute wealth $\hat{W}_{T}^{m,n}(\pi^{m}_{T-\Delta t})$ at the terminal
given the wealth at $W_{T-\Delta t}=W_{0}$, the transformed value function is
estimated by
$\hat{v}^{m}=\ln{[(1-\gamma)\frac{1}{N}\sum\limits_{n=1}^{N}U(\hat{W}_{T}^{m,n}(\pi^{m}_{T-\Delta
t}))]}-(1-\gamma)\ln{W_{0}}$ ;

11 Regress $\hat{v}^{m}$ over the polynomial of $H_{T-\Delta t}^{m}$ and
$\ln{S}_{T-\Delta t}^{m}$, and obtain the function $L_{T-\Delta t}(H,\ln{S})$;

12

13

14for  _$t=T-2\Delta t$ to $\Delta t$_ do

15 for  _$m=1...n_{r}$_ do

16 Compute the variance matrix $\Sigma_{t}^{m}$ with derivatives price and
sensitivity obtained in step 3;

17 Directly compute optimal allocation $\pi^{m}_{t}$ with Equation (30) where
the $P_{k}=L_{t+\Delta t}(H,\ln{S})$;

18 for  _ $n=1...N$_ do

19 Generate $\hat{S}_{t+\Delta t}^{m,n}$ and $\hat{H}_{t+\Delta t}^{m,n}$
given $S_{t}^{m}$ and $H_{t}^{m}$ and obtain $\hat{O}_{t+\Delta t}^{m,n}$;

20 Compute wealth $\hat{W}_{t+\Delta t}^{m,n}(\pi^{m}_{t})$ at the terminal
given the wealth at $W_{t}=W_{0}$, the transformed value function is estimated
by $\hat{v}^{m}=\ln{[\frac{1}{N}\sum\limits_{n=1}^{N}(W^{m,n}_{t+\Delta
t}(\pi^{m}_{t}))^{1-\gamma}exp(L_{t+\Delta t}(\hat{H}_{t+\Delta
t}^{m,n},\ln{\hat{S}}_{t+\Delta t}^{m,n}))]}-(1-\gamma)\ln{W_{0}}$;

21 Regress $\hat{v}^{m}$ over the polynomial of $H_{t}^{m}$ and
$\ln{S}_{t}^{m}$, and obtain the function $L_{t}(H,\ln{S})$;

22

23

24while  _ $t=0$ _ do

25 Compute the variance matrix $\Sigma_{0}$ with derivatives price and
sensitivity obtained in step 3, and the optimal allocation $\pi_{0}^{*}$ is
obtained with Equation (30) and where the $P_{k}=L_{\Delta t}(H,\ln{S})$;

26

27

return $\pi^{*}_{0}$

Algorithm 2 PAMC-direct method

###  B.2 Comparison between the PAMC-direct method and the PAMC-indirect
method

In this section, we implement the PAMC-direct method and the PAMC-indirect
method on the Heston SV model given in (15) for comparison purpose. The
derivatives-based portfolio given the Heston model was first studied in Liu
and Pan (2003), where the author constructed a portfolio with derivative
securities and a stock so that volatility risk is able to be managed. The
optimal strategy stock-derivatives portfolio is solved in closed-form. The
accuracy and efficiency of the PAMC-direct and the PAMC-indirect are examined
in comparison with the analytical solution.

We continue to use the market-calibrated set of parameters in Table 2. For
simplicity, we let $O_{t}$ be a delta-neutral straddle because the delta-
neutral position keeps the straddle near-the-money, and the liquidity should
not be a concern.

Figure 5 (a) and (b) compares the optimal allocation on the stock and straddle
across different values of risk aversion level $\gamma$. We let the re-
balancing frequency of the PAMC-indirect method be $60$ times per year, i.e.
investors roughly adjust their positions weekly. Optimal allocation from the
PAMC-indirect method and theoretical solution (re-balancing continuously) are
visually overlapped, the PAMC-indirect method exhibits very excellent accuracy
in this case. The allocation from the PAMC-direct method with $60$ re-balances
per year is subject to a substantial error, on the other hand, the gap to the
theoretical solution shrinks if we let the re-balancing frequency be $300$
times per year (roughly daily re-balance). We expect the gap will vanish as
re-balancing frequency continues to increase. The computational times of the
PAMC-direct and PAMC-indirect methods are compared in figure 5 (c), the time
required for the PAMC-indirect method is significantly smaller than the time
for the PAMC-direct method. The PAMC-indirect is superior to the PAMC-direct
with regard to both accuracy and computational efficiency, we hence use only
the PAMC-indirect in section 4.

(a) allocation on stock

(b) allocation on straddle option

(c) Computational time

Figure 5: Allocation on straddle versus $\gamma$