2021

These authors contributed equally to this work. These authors contributed
equally to this work.

[1]Yichen Zhu These authors contributed equally to this work.

[1]Department of Statistical and Actuarial Sciences, Western University, 1151
Richmond Street, London, N6A 5B7, Ontario, Canada

2]Department of Mathematics, Western University, 1151 Richmond Street, London,
N6A 5B7, Ontario, Canada

# Derivatives-based portfolio decisions. An expected utility insight

Marcos Escobar-Anel<EMAIL_ADDRESS>Matt Davison<EMAIL_ADDRESS><EMAIL_ADDRESS>*  [

###### Abstract

This paper challenges the use of stocks in portfolio construction, instead we
demonstrate that Asian derivatives, straddles, or baskets could be more
convenient substitutes. Our results are obtained under the assumptions of the
Black–Scholes–Merton setting, uncovering a hidden benefit of derivatives that
complements their well-known gains for hedging, risk management, and to
increase utility in market incompleteness. The new insights are also
transferable to more advanced stochastic settings.
The analysis relies on the infinite number of optimal choices of derivatives
for a maximized expected utility (EUT) agent; we propose risk exposure
minimization as an additional optimization criterion inspired by regulations.
Working with two assets, for simplicity, we demonstrate that only two
derivatives are needed to maximize utility while minimizing risky exposure. In
a comparison among one-asset options, e.g. American, European, Asian, Calls
and Puts, we demonstrate that the deepest out-of-the-money Asian products
available are the best choices to minimize exposure. We also explore optimal
selections among straddles, which are better practical choice than out-of-the-
money Calls and Puts due to liquidity and rebalancing needs. The optimality of
multi-asset derivatives is also considered, establishing that a basket option
could be a better choice than one-asset Asian call/put in many realistic
situations.

###### keywords:

Expected utility theory, Constant relative risk aversion (CRRA) utility,
Optimal derivative choice, Black-Scholes pricing

###### pacs:

[

JEL Classification]G11, G13, C61, C44, C63

##  1 Introduction

Financial derivatives such as futures, options and swaps play essential roles
in current financial markets. They are used for hedging and speculation as
well as for arbitrage opportunities. The history of derivatives is as old as
the history of commerce. Derivatives have grown into an indispensable asset
class since the 1970s, in part due to the increase in volatility and
complexity of global financial markets. The popularity of derivatives is such
that some market analysts, such as Maverick (2020), place the size of
derivatives at more than 10 times the total world gross domestic product
(GDP). Vanilla European and American options usually come to mind first when
discussing options. Beyond that, a wide variety of options are traded in
centralized exchanges or over-the-counter (OTC) markets, and some investors
are even able to define their own products and terms. The enduring appeal of
derivatives lies in their diversity and hence the capacity to fulfill their
needs of financial players.

This paper uncovers an additional benefit of derivatives. It addresses a
basic, yet poorly understood question for investors: what is the best
financial derivative to include in a portfolio? We challenge the common
practice of using the underlying stock; instead, we demonstrate that Asian
derivatives, straddles or baskets could be more convenient choices. Our
analysis and results are obtained under the safe assumptions of a
Black–Scholes–Merton model, which not only uncovers the hidden “in plain
sight” benefits of derivatives, but also highlights their potential for
applications in more advanced settings incorporating market incompleteness,
jumps and transaction costs, among others.

Derivative valuation in the context of continuously trading markets was
initiated by the seminal papers of Black and Scholes (1973) and Merton (1973).
The authors solved associated partial differential equations (PDEs) and
obtained the price function of a European option in closed form, when the
underlying asset follows a geometric Brownian motion (GBM). Their work (i.e.
Black–Scholes–Merton formula) laid the foundation for the development of
derivative pricing. Their results have been extended in many directions; most
relevant to this study are extensions to the pricing of many other types of
options, such as American options (see Bjerksund and Stensland (1993))
lookback options (see Goldman et al (1979)) and geometric average Asian
options (see Kemna and Vorst (1990)), to mention a few. The distinctive
exercise rights and structure of payoff reflect the complexity of financial
derivatives.

Our paper focuses on the benefits of derivatives from the portfolio investment
perspective for a maximized expected utility (EUT) agent. Investment
incorporating derivatives have been studied from multiple perspectives. Haugh
and Lo (2001) found a buy-and-hold strategy that minimizes the mean-squared
distance to the terminal wealth of Merton (1969) continuously rebalancing
portfolio. Moreover, an elasticity approach was introduced in Kraft (2003), by
which the author obtained the optimal strategy of a portfolio with path-
dependent options. Mi and Xu (2020) studied a portfolio of volatility
derivatives (options or swaps) for a constant absolute risk aversion (CARA)
investor. In additions, Liu and Pan (2003) investigated the optimal portfolio
in a stock-derivatives market with Heston’s SV model and jumps. Their results
demonstrated the improvement in performance when using derivatives to complete
the market, while showing that an infinite number of derivatives can be used
with the same optimal performance for the portfolio. In contrast to the
existing literature, our paper investigates how to best select derivatives.

For this purpose, in addition to a maximized EUT, we develop a second
optimization criterion for the portfolio manager as a way of selecting the
best derivative. We choose risky asset exposure minimization motivated by two
facts. First, the maximization of the utility can be achieved by an infinitely
number of equally optimal derivatives (as per Liu and Pan (2003)). Second,
minimizing risk exposure, as a second optimization criterion, aligns with
practical needs in the industry. In general, investment companies face many
constraints in the construction of their portfolios, many of which are imposed
by regulatory agencies. The key factor behind regulatory constraints is the
intention to control the exposure of a portfolio to risky assets, protecting
an investor’s capital in the case of a market crash. Some of these risks are
difficult to accurately model, which highlights the importance of minimizing
exposure.

Our findings demonstrate that derivatives can be used to reduce risk exposure
with no impact on the level of satisfaction of the investor (e.g. maximum
utility). We investigate the selection of derivatives in three specific option
classes: (i) American, European and Asian calls and puts; (ii) American,
European and Asian synthetic straddles; and (iii) basket options. We further
compare one-asset and multi-asset options in various realistic situations, and
we consider the relationship between risky asset exposure and portfolio
rebalancing frequency.

The contributions of the paper are summarized as follows:

  1. 1.

Given the infinite number of choices of equally optimal financial derivatives
for an EUT investor, we explore an additional optimization criterion, namely,
risk exposure minimization, to help investors make a practical derivative
selection.

  2. 2.

We demonstrate, in the context of two one-factor (e.g. GBM) assets, that the
minimum number of derivatives needed not only to maximize EUT performance, but
also to minimize risk exposure is exactly two.

  3. 3.

In a comparison of the most popular types of simple one-asset options (e.g.
American, European and Asian calls and puts), we illustrate that the deepest
out-of-the-money Asian products are the best choices for minimizing risk
exposure.

  4. 4.

To avoid illiquid out-of-the-money options, which also require plenty of
rebalancing, we explore optimal selections among straddles. We demonstrate the
existence of an optimal strike price for risk exposure minimization, which is
likely a better practical choice than out-of-the-money calls and puts.

  5. 5.

Given the setting of two assets in the portfolio, we study the optimality of
multi-asset derivatives. We determine that a basket option could be a better
selection than one-asset Asian calls and puts in many realistic situations.

  6. 6.

Several analyses are performed to solidify our findings; in particular, the
relationship between risky asset exposure and portfolio rebalancing frequency
is investigated, and the results are put to the test for a variety of
parametric choices.

The paper is organized as follows. Section 2 describes the EUT problem for an
investor allocating directly among financial derivatives. Given the vast
number of optimal derivatives available for this problem, Section 2 defines a
criterion based on an additional optimization problem that aids in selecting a
single optimal solution. Section 3 then explores two cases of one-asset
options available to a portfolio investor: (1) calls and puts and (2)
straddles. Thereafter, Section 4 focuses on the benefit of multi-asset
derivatives such as basket options, and Section 5 concludes this study.
Appendix A presents all the proofs and complementary analyses in support of
our findings.

##  2 Mathematical setting and results

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space with a
right-continuous filtration $\\{\mathcal{F}_{t}\\}_{t\in[0,T]}$. We consider a
frictionless market, in which trading occurs without transaction costs or
market impact, comprising a money market account $M_{t}$ and two risky assets
$\bm{S_{t}}=[S_{t}^{(1)},S_{t}^{(2)}]^{T}$, with the following dynamics:

$\begin{cases}\frac{\mathrm{d}M_{t}}{M_{t}}=r\mathrm{d}t\\\
\mathrm{d}\bm{S_{t}}=diag(\bm{S_{t}})\left[(r\cdot\bm{\mathbbm{1}}+diag(\bm{\sigma})\bm{\Lambda})\mathrm{d}t+diag(\bm{\sigma})\mathrm{d}\bm{B_{t}}\right]\end{cases}$
(1)

where $\bm{B_{t}}=[B_{t}^{(1)},B_{t}^{(2)}]^{T}$ are Brownian motions
modelling the risk of two underlying assets, whose correlation is denoted by
$\rho\in\left(-1,1\right)$. Here, $r$ is the risk-free rate and
$\bm{\mathbbm{1}}$ denote the vector of ones,
$\bm{\Lambda}=[\lambda^{(1)},\lambda^{(2)}]^{T}$ are constants capturing the
market price of risk of $\bm{B_{t}}$, and
$\bm{\sigma}=[\sigma^{(1)},\sigma^{(2)}]^{T}$ represents the volatility of the
two underlying assets.

We now introduce a set of admissible financial derivatives on the assets
$\bm{S_{t}}$, for a fixed $n\geq 1$:

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

where $\bm{\Sigma_{t}}$ represents the variance matrix of $O_{t}$. The element
$(i,j)$, $i=1,..,n$, $j=1,2$, of $\bm{\Sigma_{t}}$, denoted by $f_{t}^{ij}$,
represents the sensitivity of $O_{t}^{(i)}$ to the underlying asset
$S_{t}^{(j)}$, i.e. $f^{ij}=\frac{\partial O_{t}^{(i)}}{\partial
S_{t}^{(j)}}\frac{1}{O_{t}^{(i)}}S_{t}^{(j)}\sigma^{(j)}$. Note that
$\Omega_{O}$ is an infinite set, which could contain standardized exchange-
traded options and non-standardized OTC options available to a generic
investor. The reader should observe that $f_{t}^{ij}$ depends on derivative
type, style, underlying price, strike price, time to maturity and other
factors, and $\Sigma_{t}$ is a full-rank matrix, which allows us to continue
working in a complete market if $n\geq 2$. For simplicity, we also assume that
the derivatives in $\Omega_{O}^{(n)}$ will be rolled over, always keeping the
same time to maturity and a non-zero value.

We are now ready to create a derivatives-based dynamic portfolio choice
problem for a risk-averse investor. The investor preference is measured by the
widely used and algebraically simple CRRA utility111The can easily be extended
to other utility functions. . We assume that an investor allocates in an
element of $\Omega_{O}$; that is, a specific
$\bm{O_{t}}=[O_{t}^{(1)},O_{t}^{(2)},...,O_{t}^{(n)}]^{T}$ $(n\geq 2)$. Note
that by arbitrage arguments, the dynamics of the derivatives-based assets and
the market account are as follows:

$\begin{cases}\frac{\mathrm{d}M_{t}}{M_{t}}=r\mathrm{d}t\\\
d\bm{O_{t}}=diag(\bm{O_{t}})\left[(r\cdot\bm{\mathbbm{1}}+\bm{\Sigma_{t}\Lambda})\mathrm{d}t+\bm{\Sigma}_{t}\mathrm{d}\bm{B_{t}}\right]\end{cases}$
(2)

The investor is not prohibited from trading on the underlying assets. This
would be equivalent to setting $n=2$ and taking $\bm{O_{t}}=\bm{S_{t}}$.

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

$\frac{\mathrm{d}W_{t}}{W_{t}}=(r+\bm{\pi_{t}}^{T}\bm{\Sigma_{t}\Lambda})\mathrm{d}t+\bm{\pi_{t}}^{T}\bm{\Sigma_{t}}\mathrm{d}\bm{B_{t}}.$
(3)

A CRRA utility function represents the investor’s preference on the terminal
wealth $W_{T}$, which is given by

$U(W_{T})=\frac{W_{T}^{1-\gamma}}{1-\gamma},$ (4)

Moreover, $\gamma>0$, $\gamma\neq 1$ measures the investor’s level of risk
aversion. The investor’s objective is to derive an investment strategy
$\bm{\pi_{t}^{\ast}}$ that maximizes the EUT of the terminal wealth $W_{T}$.
Then, the investor’s problem can be written as follows:

$V(t,W)=\max_{\bm{\pi_{s\geq
t}}\in\Omega_{\pi}^{(O)}}\mathbb{E}(U(W_{T})\mid\mathcal{F}_{t}),$ (5)

and $V(t,W)$ denotes the value function at time $t$. According to the
principles of stochastic control, we state the Hamilton-Jacobi-Bellman (HJB)
equation for the value function $V$:

$\sup_{\bm{\pi_{t}}}\left\\{V_{t}+W_{t}V_{W}(r+\bm{\pi_{t}^{T}\Sigma_{t}\Lambda})+\frac{1}{2}W_{t}^{2}V_{WW}(\bm{\pi_{t}^{T}\Sigma_{t}\Phi\Phi^{T}\Sigma_{t}^{T}\pi_{t}})\right\\}=0,$
(6)

where $\bm{\Phi}=\left[\begin{array}[]{cc}1&0\\\
\rho&\sqrt{1-\rho^{2}}\end{array}\right]$.

######  Proposition 1 (Solution for $V$ and $\bm{\pi_{t}^{*}}$).

The value function has the representation:

$V(t,W)=\frac{W^{1-\gamma}}{1-\gamma}\exp\left((1-\gamma)\left(r+\frac{1}{2}\frac{\bm{\Lambda}^{T}(\bm{\Phi\Phi}^{T})^{-1}\bm{\Lambda}}{\gamma}\right)(T-t)\right).$
(7)

Moreover, $\bm{\pi_{t}^{\ast}}$ is an optimal strategy if it satisfies
$\bm{\Sigma_{t}}^{T}\bm{\pi_{t}^{\ast}}=\bm{\eta_{t}^{\ast}}$, where
$\bm{\eta_{t}^{\ast}}$ is given by

$\bm{\eta_{t}^{\ast}}=\left[\begin{array}[]{c}\eta_{t}^{(1)}\\\
\eta_{t}^{(2)}\end{array}\right]=\frac{(\bm{\Phi\Phi}^{T})^{-1}\bm{\Lambda}}{\gamma}.$
(8)

Proof: See Appendix A.1.

Proposition 1 highlights three important implications. First, for any given
element in $\Omega_{O}^{(n)}$, if $n>2$ numerous, indeed infinitely many,
strategies, all produce the same maximum value function. This can be
interpreted as a redundant market case. Second, if $n=2$, then a unique
optimal strategy exists for the problem. Finally, if $n=1$, there is no
optimal solution: the value function cannot reach the global maximum; this is
actually an incomplete market situation.

In summary, as there are a host of tradeable derivatives in the financial
market, hence a myriad of elements exist in $\Omega_{O}^{(n)}$. This means
there are an infinite number of choices of the portfolio composition
$\bm{O_{t}}$ that can deliver the same optimal solution to the EUT problem.
The next section takes advantage of this pool of optimal solutions to design a
criterion that allows investors to select the best portfolio composition, with
the corresponding strategy. This extra criterion is motivated by investor
needs. Adding such a criterion will lead to an additional optimization
problem, the solution of which is explored below.

###  2.1 Derivative selection criterion: minimizing $\ell_{1}$ risk exposure

In this section, we propose a derivative selection criterion. Proposition 1
illustrates that, given $\bm{O_{t}}$ $\in\Omega_{O}^{(n)}$, with $n\geq 2$, an
optimal strategy $\bm{\pi_{t}^{\ast}}\in\Omega_{\pi}^{(O)}$ exists that
maximizes the EUT of terminal wealth. From the traditional dynamic portfolio
choice perspective, derivative selection does not benefit the investor,
because regardless of the derivative chosen, the optimal strategy always
achieves the same EUT.

In reality, investors are always concerned with the size of their risky
allocations. For example, market conditions may change over time, and a risky
investor could suffer large, unexpected losses especially during crisis
periods. Regulatory constraints also force the investor to keep increasingly
large percentages of wealth in cash. This means that strategies with smaller
exposure on the risky products are naturally preferable to the investor. In
this regard, we design a simple derivative selection criterion aimed at
capturing this practical dilemma:

$\underset{{\bm{O_{t}}\in\Omega_{O}^{(n)}}}{\min}\left\|\underset{\bm{\pi_{s\geq
t}}\in\Omega_{\pi}^{(O)}}{\arg\max}\mathbb{E}(U(W_{T})\mid\mathcal{F}_{t})\right\|_{1}$
(9)

where $\left\|\bm{\pi_{s\geq
t}}\right\|_{1}=\sum\limits_{i=1}^{n}\left|\pi_{t}^{(i)}\right|$ represents
the $\ell_{1}$ norm of allocations at time $t$. Note that this objective is
equivalent to maximizing the cash position while shorting less.

As we mentioned before, it is suboptimal for the investor to choose a
portfolio composition size $n<2$ based on the unhedgeable risk resources and
incompleteness of the market. However, the investor might be interested in a
redundant $(n>2)$ market situation, hoping to reduce their risky asset
exposure. In the next proposition, we demonstrate that the best choice of $n$,
for problem (9), is actually $n=2$.

######  Proposition 2.

Assume that an optimal solution for problem (9) exists for $n\geq 2$, then,
(9) leads to the same minimal $\ell_{1}$ norm for any $n\geq 2$. In addition,
an optimal strategy exists for problem (9) such that the number of non-zero
allocations is less than or equal to $2$.

Proof: See Appendix A.2.

Proposition 2 demonstrates that redundancy will not offer any additional help
with the investor’s risky asset exposure. In other words, working with $n=2$
is sufficient for problem (9). This allows us to work with the simplest case
given a complete market setting (i.e. $n=2$).

##  3 Applications to one-asset derivatives

In this section, we solve the derivative selection problem—that is, (9)—for
$n=2$, within subsets of the derivative set ${\Omega_{O}^{(2)}}$. The
derivative selection problem is rewritten as

$\underset{{\bm{O_{t}}\in\Omega_{O}^{(2,1)}}}{\min}\left\|\underset{\bm{\pi_{s\geq
t}}\in\Omega_{\pi}^{(O)}}{\arg\max}\mathbb{E}(U(W_{T})\mid\mathcal{F}_{t})\right\|_{1},$
(10)

where ${\Omega_{O}^{(2,1)}}$ captures one-asset (single-stock) derivatives,
which can be represented as follows:

$\Omega_{O}^{(2,1)}=\left\\{\bm{O_{t}}=[O_{t}^{(1)},O_{t}^{(2)}]^{T}\mid
O_{t}\in\Omega_{O}^{(2)}\text{, }O_{t}^{(i)}=g\left(S^{(i)}\right)\text{,
}i=1,2\text{, }t\in[0,T]\right\\}.$

In this situation, the option’s variance matrix for a portfolio composition
$\bm{O_{t}}=[O_{t}^{(1)},O_{t}^{(2)}]^{T}\in\Omega_{O}^{(2,1)}$ is defined by

$\bm{\Sigma_{t}}=\left[\begin{array}[]{cc}f^{(1)}_{t}&0\\\
0&f^{(2)}_{t}\end{array}\right],$ (11)

which, provided $f^{(1)}_{t}$ and $f^{(2)}_{t}$ are nonzero, is a rank $2$
(non-singular) diagonal matrix. By Itô’s lemma, the sensitivity of
$O_{t}^{(i)}$ is a function of the option Delta, spot price $S_{t}^{(i)}$,
stock volatility $\sigma^{(i)}$ and the option price $O_{t}^{(i)}$:

$f^{(i)}=\frac{\partial O_{t}^{(i)}}{\partial
S_{t}^{(i)}}\frac{S_{t}^{(i)}\sigma^{(i)}}{O_{t}^{(i)}}.$ (12)

The next proposition states the fundamental principle of one-asset option
selection.

######  Proposition 3 (Fundamental principle of one-asset option selection).

A portfolio composition:
$\bm{O^{*}_{t}}=[O_{t}^{(1),*},O_{t}^{(2),*}]^{T}\in\Omega_{O}^{(2,1)}$ is
optimal for problem (10) if and only if

$O_{t}^{(i),*}=\underset{O_{t}^{(i)}}{\arg\max}\left|\frac{\partial
O_{t}^{(i)}}{\partial S_{t}^{(i)}}\frac{1}{O_{t}^{(i)}}\right|.$ (13)

Proof: See Appendix A.3.

Note that both option Delta and price must be bounded away from $0$ and
$\infty$ to avoid the suboptimal incomplete market case, such that the option
sensitivity $f^{(i)}\in(0,\infty)$ and variance matrix $\Sigma_{t}$ is non-
singular. In other words, we could witness an infinitely large allocation as
$f^{(i)}\rightarrow 0$ (to be explained in Section 3.1), which would also lead
to a suboptimal solution (incomplete market) and hence a departure from the
investor’s target (i.e. maximizing utility and minimizing risk exposure).

The fundamental principle of one-asset option selection illustrates that the
selection of one-asset options is separable. Investors can first pick the
option with the largest relative sensitivity to $S_{t}^{(1)}$ among all one-
asset options on $S_{t}^{(1)}$ as $O_{t}^{(1),*}$, and they can then select
$O_{t}^{(2),*}$ in a similar way. Based on this principle, we consider the
case where $\Omega_{O}^{(2,1)}$ is a subset of put and call options in Section
3.1. Then, the best option style and strike price for minimizing
$\left\|\bm{\bm{\pi_{t}}}\right\|_{1}$ is quantified.

Selection of put and call options is studied first due to their popularity.
Nonetheless, calls and puts have a problem: their optimal is on the boundary
of the strike price range. This could lead to illiquid choices (high out-of-
the-money options) or, even worse, incomplete market suboptimality on the
limit as the strike price goes to zero (puts) or infinity (calls). In Section
3.2, we investigate derivative selection in a subset of straddles. Straddles
are also popular products, which avoid the boundary optimality of calls and
puts. Note that the selection of $O_{t}^{(1),*}$ and $O_{t}^{(2),*}$ is
independent, and the procedures are similar; hence, for simplicity, we only
present the result for $O_{t}^{(2),*}$.

The chosen parameters are presented in Table 1 222See Appendix A.9 for
analysis of other parameter choices.. These parameters are considered to be
plausible.

Table 1: Parameter Value Parameter | Value | Parameter | Value
---|---|---|---
$\sigma^{(1)}$ | 0.13 | $\sigma^{(2)}$ | 0.2
$r$ | 0.05 | $\rho$ | 0.4
$\lambda^{(1)}$ | 0.52 | $\lambda^{(2)}$ | 0.6
Investment horizon $T$ | 1 | Time to maturity of options $\hat{T}$ | 2
$S^{(1)}_{0}$ | 40 | $S^{(2)}_{0}$ | 30
$\eta_{t}^{(1)}$ | 0.083 | $\eta_{t}^{(2)}$ | 0.117
$\gamma$ | 4.0 |  |
|  |  |

The option variance matrix may not be solvable in closed form. Therefore, we
approximate the sensitivity of European, American and arithmetic average Asian
options via the finite difference method: the Delta of option $O_{t}$ is given
by

$Delta=\frac{O_{t}(S+\Delta S)-O_{t}(S-\Delta S)}{2\Delta S}.$ (14)

Here, $O_{t}(S)$ is the simulated option price given spot price $S$. In
addition, we estimate the sensitivity of American options with the generalized
infinitesimal perturbation analysis approach introduced by Chen and Liu
(2014).

###  3.1 Put and call options

We first consider the derivative selection problem (10) on the subset denoted
by $\Omega_{O}^{(2,put\ call)}$ that contains only European-, American- and
Asian-style put and call options:

$\begin{split}\Omega_{O}^{(2,put\
call)}=\left\\{\bm{O_{t}}=[O_{t}^{(1,j)},O_{t}^{(2,j)}]^{T}\mid
O_{t}^{(i,j)}=g^{(j)}\left(S^{(i)}\right)\text{,
}i=1,2,\,t\in[0,T]\right\\}.\end{split}$

For both practical and theoretical reasons, the strike price $K^{(i,j)}$ of a
given option $O_{t}^{(i,j)}$ is bounded within $[A^{(i,j)},B^{(i,j)}]$, where
$j\in\\{\textit{Euro Call, Asian Call, Amer Call, Euro Put, Asian Put, Amer
Put}\\}$. The put option strike price is bounded away from $0$; that is,
$A^{(i,j)}>0$, where $j\in\\{\textit{Euro Put, Asian Put, Amer Put}\\}$.
Similarly, the call option strike excludes $\infty$. Both conditions ensure
the non-zero option price assumption. For simplicity, we also assume that all
the options have the same time to maturity $\hat{T}=2$, and we search the
optimal portfolio composition in terms of option type, style and strike price.

European put and call prices and their sensitivities are solved by the well-
known Black–Scholes–Merton model (see Black and Scholes (1973)). Let
$O_{t}^{(i,Euro\ Call)}$ and $O_{t}^{(i,Euro\ Put)}$ be a call and put option
on $S_{t}^{(i)}$ given in (1). The Black–Scholes–Merton model indicates that

$\begin{split}O_{t}^{(i,Euro\ Call)}&=S_{t}^{(i)}N(d_{1})-K^{(i,Euro\
Call)}e^{-r(\hat{T}-t)}N(d_{2})\\\ O_{t}^{(i,Euro\ Put)}&=K^{(i,Euro\
Put)}e^{-r(\hat{T}-t)}N(-d_{2})-S_{t}^{(i)}N(-d_{1})\\\ \frac{\partial
O_{t}^{(i,Euro\ Call)}}{\partial
S_{t}^{(i)}}&=N(d_{1})\quad\quad\quad\frac{\partial O_{t}^{(i,Euro\
Put)}}{\partial S_{t}^{(i)}}=-N(-d_{1})\end{split}$ (15)

where $N$ is the cumulative distribution function of a standard normal random
variable and

$d_{1}=\frac{\ln{(S_{t}^{(i)}/K^{(i,Euro\
Call/Put)})}+(r+\frac{1}{2}(\sigma^{(i)})^{2})(\hat{T}-t)}{\sigma^{(i)}\sqrt{\hat{T}-t}},\quad
d_{2}=d_{1}-\sigma^{(i)}\sqrt{\hat{T}-t}.$ (16)

The proposition next demonstrates the existence of an optimal portfolio
composition $O_{t}^{\ast}\in\Omega_{O}^{(2,put\ call)}$, given the assumption
that $O_{t}^{(i,j)}\in\mathbb{C}^{1}$ and the Delta is non-zero.

######  Proposition 4 (Existence of optimal portfolio composition in put and
call subset).

Assume $O_{t}^{(i,j)}\left(S^{(i)},K^{(i,j)}\right)\in\mathbb{C}^{1}$ on
$(0,\infty)\times[A^{(i,j)},B^{(i,j)}]$333$\frac{\partial
O_{t}^{(i,j)}}{\partial K^{(i,j)}}$ on boundary is defined as the one-sided
derivative., the optimal portfolio composition for problem (10) within the
subset $\Omega_{O}^{(2,put\ call)}$ exists.

Proof: See Appendix A.4.

It is easy to show the expression for this optimal composition, in the case of
the European options, as clarified in the following corollary.

######  Corollary 5.

The risk exposure to an European call option decreases with $K^{(i,Euro\
Call)}$ and converges to $0$ as $K^{(i,Euro\ Call)}\rightarrow\infty$.
Similarly, the risk exposure to an European put option increases with
$K^{(i,Euro\ Call)}$ and converges to $0$ as $K^{(i,Euro\ Put)}\rightarrow 0$.
Therefore, the optimal European call is achieved when $K^{(i,Euro\
Call)}=B^{(i,Euro\ Call)}$ and the optimal European put is achieved when
$K^{(i,Euro\ Put)}=A^{(i,Euro\ Put)}$ .

Proof: See Appendix A.5.

Corollary 5 demonstrates that $\left\|\bm{\bm{\pi_{t}}}\right\|_{1}$ can
vanish if the European call option’s strike price approaches infinity. These
extreme options cannot be found in the market, but more importantly, they
would also lead to a violation of the non-singular matrix $\Sigma_{t}$
condition, creating an incomplete market situation. This is known as the
problem of “boundary optimality.” The corollary thus illustrates the
importance of our derivative selection.

Figure 1 (a) exhibits $\pi^{(2,j)}_{0}$—that is, the allocation on call
options $O_{t}^{(2,j)}$ at $t=0$—as a function of strike price and spot price
ratio $K^{(2,j)}/S_{t}^{(2)}$; for example, $K^{(2,j)}/S_{t}^{(2)}=1$
indicates at-the-money options. The analytical sensitivity of European calls
is obtained with (12) and (15), with which the optimal strategy is known
immediately. Note that the European call option reduces to $S_{t}^{(2)}$ when
$K^{(2,Euro\ Call)}=0$, and the allocation hence converges to the optimal
strategy on $S_{t}^{(2)}$. The allocation on the European call obtained by the
finite difference method is also plotted to verify the accuracy of numerical
approximation. Allocations obtained via analytical and numerical approaches
visually overlap except when $K^{(2,Euro\ Call)}$ is extremely large. The
allocation to Asian calls is similar to that of European calls when the strike
prices are small, but it decreases faster. Asian calls are consequently
preferable to European calls given the same upper bound on strike price; that
is, $B^{(2,Euro\ call)}=B^{(2,Asian\ call)}$.

The allocation on put options is illustrated in Figure 1 (b). The absolute
allocations for European, American444We assume stocks which pay no dividends,
so American call is identical to the European call in (a). In addition, the
allocation on American put is shown in (b) when $K^{(2,Amer\ Put)}<37.7$, i.e.
the region, in which the American put should be held and not yet exercised.
and Asian puts increase with strike price because of the decreasing
instantaneous sensitivity. The absolute allocation on Asian puts is the
smallest when $K^{(2,Asian\ Put)}$ is small, but the opposite occurs as the
strike price rises. Figure 1 not only confirms Corollary 5—that is
$\pi_{t}^{(2,Euro\ Call)}\rightarrow 0$ as $K^{(2,Euro\
Call)}\rightarrow\infty$ and $\pi_{t}^{(2,Euro\ Put)}\rightarrow 0$ as
$K^{(2,Euro\ Put)}\rightarrow 0$—but also demonstrates a similar conclusion
for Asian and American options.

In summary, the absolute allocation on a put or call option is monotone with
strike price; hence, the optimal choice of $O_{t}^{(2,j)}$ is on the boundary.
This means we only need to compare the allocation $\pi^{(2,j)}$ on the deepest
out-of-the-money option to obtain the option with the smallest absolute
allocation.

Next, we consider the case where options of different style555The actual range
of American put strike price is the intersection of $[A^{(i,Amer\
Put)},B^{(i,Amer\ Put)}]$ and the region of $K^{(i,Amer\ Put)}$ such that the
option is not exercised immediately, hence American option is not considered
when $[A^{(i,Amer\ Put)},B^{(i,Amer\ Put)}]$ is mutually exclusive with that
region.  share an identical boundary:

$\begin{split}&[S_{t}^{(2)},R^{B}S_{t}^{(2)}]=[A^{(2,j)},B^{(2,j)}],\quad
j\in\\{Euro\ Call,Asian\ Call\\}\\\
&[R^{A}S_{t}^{(2)},S_{t}^{(2)}]=[A^{(2,j)},B^{(2,j)}],\quad j\in\\{Euro\
Put,Asian\ Put,Amer\ Put\\}.\end{split}$

Here, $R^{A}\leq 1$, and $R^{B}\geq 1$. As we move the lower bound of put
option strike price $R^{A}S_{t}^{(2)}$ and the upper bound of call option
strike price $R^{B}S_{t}^{(2)}$, the optimal choice of $O_{t}^{(2,j)}$ is
shown in Figure 1 (c). The Asian option is preferable compared to American and
European options. Asian calls dominate when both $R^{A}$ and $R^{B}$ are
large, whereas Asian puts dominate when both $R^{A}$ and $R^{B}$ are small.

(a) allocation on call options

(b) allocation on put options

(c) Derivative selection region

Figure 1: Allocation on options versus strike price

Figure 2 displays the performance of the portfolio
$\bm{O_{t}}=[O_{t}^{(1,Euro\ Call)},O_{t}^{(2,Euro\ Call)}]^{T}$ versus strike
prices for different rebalancing frequencies. The portfolio performance is
measured by annualized certainty equivalent (CER), defined as

$U(W_{0}\exp{(CER*T)})=V(0,W_{0}).$ (17)

The theoretical optimal CER (orange wireframe) is plotted as the benchmark,
which can only be achieved by continuously rebalancing. We also present the
incomplete market CER (i.e. the green wireframe) for comparison purposes, for
example obtained through the lack of an asset to hedge the risk in
$B_{t}^{(2)}$. A portfolio with in-the-money call options is insensitive to
the rebalancing frequency, and the loss from occasional rebalancing is not
significant. On the other hand, the CER of a portfolio with two deep out-of-
the-money calls could be even smaller than that of an incomplete market CER
when the portfolio is only rebalanced $10$ times per year, whereas it
approaches complete market CER as the rebalancing frequency increases.
This is important for investors reducing their risk exposure with deep out-of-
the-money options. These products lack liquidity; hence, the trading strategy
might fail because investors cannot adjust their position quickly enough. In
summary, the best (out-of-the-money) options are those requiring more frequent
rebalancing, as Figure 2 demonstrates. This points at future research
addressing the trade-off between exposure and rebalancing.

Figure 2: Certainty equivalent rate (CER) versus strike price with different
rebalancing frequencies

###  3.2 Straddles

Next, we consider the derivative selection in a subset of options called
straddles:

$\Omega_{O}^{(2,straddle)}=\left\\{\bm{O_{t}}=[O_{t}^{(1,j)},O_{t}^{(2,j)}]^{T}\mid
O_{t}^{(i,j)}=g^{(j)}\left(S^{(i)}\right)\text{, }i=1,2\text{,
}\,t\in[0,T]\right\\}.$

where $j\in\\{Euro\ Strad,Asian\ Strad,Amer\ Strad\\}$. A straddle is an
option synthesized by simultaneously taking a long position in a call and a
put option; hence, the terminal payoff of a European straddle is
$\left|S_{T}^{(i)}-K^{(i,Euro\ Strad)}\right|$, where $K^{(i,Euro\ Strad)}$
denotes the strike price. The European straddle price and Delta are obtained
with the Black–Scholes– Merton model:

$\begin{split}O_{t}^{(i,Euro\ Strad)}&=S_{t}^{(i)}(2N(d_{1})-1)-K^{(i,Euro\
Strad)}e^{-r(\hat{T}-t)}(2N(d_{2})-1)\\\ &\frac{\partial O_{t}^{(i,Euro\
Strad)}}{\partial S_{t}^{(i)}}=2N(d_{1})-1.\end{split}$ (18)

$d_{1}$and $d_{2}$ are defined in (16). We substitute the straddle’s Delta
into (12), and the sensitivity of straddle $O_{t}^{(i,Euro\ Strad)}$ is given
by

$f^{(i,Euro\
Strad)}=\frac{(2N(d_{1})-1)S_{t}^{(i)}\sigma^{(i)}}{O_{t}^{(i,Euro\ Strad)}}.$
(19)

The non-singular variance matrix condition requires non-zero sensitivity
$f^{(i,Euro\ Strad)}$, such that $2N(d_{1})-1\neq 0$ and the feasible region
for the strike price is as follows:

$\begin{split}K^{(i,Euro\
Strad)}&\in\left[0,S_{t}^{(i)}\exp{(r+\frac{1}{2}(\sigma^{(i)})^{2})(\hat{T}-t)}\right)\\\
&\cup\left(S_{t}^{(i)}\exp{(r+\frac{1}{2}(\sigma^{(i)})^{2})(\hat{T}-t)},\infty\right).\end{split}$
(20)

It is easy to verify that $O_{t}^{(Euro\ Strad)}\in(0,\infty)$ and
$O_{t}^{(i,j)}\left(S^{(i)},K^{(i,j)}\right)\in\mathbb{C}^{1}$. We define the
feasible region for the straddle $O_{t}^{(i,j)}$ by analogy:
$K^{(i,j)}\in[0,A^{(i,j)})\cup(A^{(i,j)},\infty)$, where $j\in\\{Euro\
Strad,Asian\ Strad\\}$, and
$K^{(i,j)}\in[0,A^{(i,j)})\cup(A^{(i,j)},B^{(i,j)}]$, where $j\in\\{Amer\
Strad\\}$. Here, $A^{(i,Amer\ Strad)}$ is the point Delta of $O_{t}^{(i,Amer\
Strad)}$ equal to $0$, and $B^{(i,Amer\ Strad)}$ is the maximum strike price
such that the American put is not immediately exercised. The next theorem
shows the existence of an optimal straddle (i.e. an optimal strike price).

######  Proposition 6 (Existence of optimal portfolio composition in the
straddle subset).

There exists a portfolio composition
$O_{t}^{\ast}=[O_{t}^{(1),\ast},O_{t}^{(2),\ast}]^{T}$ with finite strike
prices $K^{(i,j,\ast)}$, such that $O_{t}^{\ast}$ is optimal for problem (10)
within $\Omega_{O}^{(2,straddle)}$. Here the optimal strike price,
$K^{(i,j,\ast)}$ is the solution of the equation:

$\frac{\partial^{2}O_{t}^{(i,j)}}{\partial S_{t}^{(i)}\partial
K^{(i,j)}}O_{t}^{(i,j)}-\frac{\partial O_{t}^{(i,j)}}{\partial
S_{t}^{(i)}}\frac{\partial O_{t}^{(i,j)}}{\partial K^{(i,j)}}=0.$ (21)

Proof: See Appendix A.6.

Figure 3: Allocation on straddle versus $\frac{K^{(i,j)}}{S_{0}^{(i,j)}}$

Next, we quantify the derivative selection within the subset of straddles.
Here, we only illustrate the optimal choice of $O_{t}^{(2,j)}$ because of the
separable selection within a one-asset option subset (see Proposition 3). We
first compute the optimal strike price of American straddle $B^{(i,Amer\
Strad)}=37.7$ and the unfeasible point $A^{(2,j)}$ for European, Asian and
American straddles; this is where the Delta of $O_{t}^{(2,j)}$ is equal to
$0$. From the formulas above, we have $A^{(2,Euro\
Strad)}=S_{t}^{(2)}\exp{(r+\frac{1}{2}(\sigma^{(2)})^{2})(\hat{T}-t)}=34.5$,
$A^{(2,Asian\ Strad)}=31.9$ and $A^{(2,Amer\ Strad)}=32.9$, which are all
obtained with Brent’s algorithm.

Figure 3 depicts the allocation $\pi_{0}^{(2,j)}$, where $j\in\\{Euro\
Strad,Asian\ Strad,Amer\ Strad\\}$ versus the ratio of spot price to strike
price. The allocation $\pi_{0}^{(2,j)}$ has the shape of a hyperbola, and it
approaches $\pm\infty$ as $K^{(2,j)}\uparrow\downarrow A^{(2,j)}$, which forms
a “cliff.” In contrast to the put and call option, the optimal straddle is
found in the interior regardless of the option style. The boundary optimality
issue is thus avoided, and it’s plausible that the straddle minimum risk
exposure will have acceptable liquidity. Moreover, the optimal straddle
minimizing the risk exposure lies on the left branch; that is,
$K^{(2,j)}\in[0,A^{(2,j)})$. The Asian straddle is superior to European and
American straddles because of its smaller absolute allocation.

The allocation on the European straddle $\pi_{t}^{(i,Euro\ Strad)}$ is solved
in closed form with (19):

$\begin{split}\pi_{t}^{(i,Euro\ Strad)}&=\frac{\eta_{t}^{(i)}}{f^{(i,Euro\
Strad)}}\\\ &=\frac{\eta_{t}^{(i)}(2S_{t}^{(i)}(2N(d_{1})-1)-K^{(i,Euro\
Strad)}e^{-r(\hat{T}-t)}(2N(d_{2})-1)}{(2N(d_{1})-1)S_{t}^{(i)}\sigma^{(i)}},\end{split}$
(22)

which is a function of the ratio of strike price to spot price $K^{(i,Euro\
Strad)}/S_{t}^{(i)}$.

In the subset of put and call $\Omega_{O}^{(2,put\ call)}$ (previous section),
the optimal option is always found at the boundary. Therefore, investors stick
to the option by rolling over with the same strike price. The optimal option
in $\Omega_{O}^{(2,straddle)}$ depends on the spot price and time-dependent
optimal ratio $K^{(i,Euro\ Strad)}/S_{t}^{(i)}$, and the investor should roll
from the current holding to new straddles at each rebalancing time to minimize
risk exposure. In this regard, Figure 4 (a) plots the optimal strike and spot
price ratio of European straddle $K^{(i,Euro\ Strad)}/S_{t}^{(i)}$ versus time
$t$. Both $K^{(1,Euro\ Strad)}/S_{t}^{(1)}$ and $K^{(2,Euro\
Strad)}/S_{t}^{(2)}$ increase with time $t$, while $K^{(2,Euro\
Strad)}/S_{t}^{(2)}$ grows faster.

The connection between portfolio CER and rebalancing frequency is demonstrated
in Figure 4 (b). As expected, the portfolio CER approaches the theoretical
result as rebalancing frequency rises. Note that the CER of the rolling
straddle portfolio is close to the theoretical CER even when the rebalancing
frequency is less than 10 times per year, suggesting that rebalancing even
relatively infrequently causes only a small loss. This is another benefit of
choosing straddles over out-of-the-money calls or puts.

In conclusion, straddles are an ideal option class for two reasons. First, the
optimal straddle for minimizing risk exposure happens to be an active and
liquid option. In addition, a rolling straddle portfolio is insensitive to the
rebalancing frequency, thus reducing an investor’s additional costs, such as
transaction costs (although transaction cost is not exactly modelled yet).

(a) Optimal ratio $\frac{K^{(i,Euro\ Strad)}}{S_{t}^{(i,Euro\ Strad)}}$ versus
time $t$

(b) CER versus rebalancing frequency

Figure 4: Straddle analysis

##  4 Multi-asset option selection

Multi-asset options are commonly traded in the OTC market. In this section, we
explore the benefits of including such options in an investor’s portfolio. A
subset of multi-asset options is defined as follows:

$\begin{split}\Omega_{O}^{(2,multi\
asset)}=\Big{\\{}\bm{O_{t}}&=[O_{t}^{(1,j_{1})},O_{t}^{(2,j_{2})}]^{T}\mid
O_{t}^{(1,j_{1})}=g^{(j_{1})}\left(S^{(1)}\right)\text{, }j_{1}=\textit{one-
asset option; }\\\
&O_{t}^{(2,j_{2})}=g^{(j_{2})}\left(S^{(1)},S^{(2)}\right)\text{,
}j_{2}=\textit{multi-asset option},\,t\in[0,T]\Big{\\}}.\end{split}$

Assume the portfolio composition $\bm{O_{t}}\in\Omega_{O}^{(2,multi\ asset)}$
consists of a one-asset option and a multi-asset option. The variance matrix
then has the representation

$\Sigma_{t}=\left[\begin{array}[]{cc}f^{(11)}_{t}&0\\\
f^{(21)}_{t}&f^{(22)}_{t}\end{array}\right].$ (23)

The next proposition states the fundamental principle of derivative selection
in the subset of multi-asset options.

######  Proposition 7 (Fundamental principle of multi-asset option selection).

If a portfolio composition $\bm{O^{*}_{t}}=[O_{t}^{(1),*},O_{t}^{(2),*}]^{T}$
is optimal for problem (10) within $\Omega_{O}^{(2,multi\ asset)}$, then

$O_{t}^{(1),*}=\underset{O_{t}^{(1,j_{1})}}{\arg\max}\left|\frac{\partial
O_{t}^{(1,j_{1})}}{\partial S_{t}^{(1)}}\frac{1}{O_{t}^{(1,j_{1})}}\right|.$
(24)

Proof: See Appendix A.7.

Proposition 7 demonstrates a necessary condition for multi-asset option
selection and reveals the sequential selection property for problem (10)
within $\Omega_{O}^{(2,multi\ asset)}$. Investors should pick the one-asset
option with the largest relative sensitivity to $S_{t}^{(1)}$ first, and they
should then search for the optimal multi-asset option (see Equation (43))
given a fixed $f^{(11)}_{t}$.

Now, we illustrate the multi-asset portfolio selection with an example of
basket options. The subset of basket option is given by

$\begin{split}\Omega_{O}^{(2,call\
basket)}&=\Big{\\{}\bm{O_{t}}=[O_{t}^{(1,j_{1})},O_{t}^{(2,j_{2})}]^{T}\mid
O_{t}^{(1,j)}=g^{(j_{1})}\left(S^{(1)}\right)\text{, }j_{1}=\textit{European
call; }\\\ &O_{t}^{(2,j_{2})}=g^{(j_{2})}\left(S^{(1)},S^{(2)}\right)\text{,
}j_{2}=\textit{Basket Call or Basket Put},\,t\in[0,T]\Big{\\}}.\end{split}$

Notably, $\Omega_{O}^{(2,call\ basket)}\subset\Omega_{O}^{(2,multi\ asset)}$.
The basket option, simultaneously hedging the risk on a combination of two
assets, has the following payoff666Investor can choose the weight on each
asset of basket option in OTC market, we only consider equal weighted case in
this paper. :

$\begin{cases}O_{T}^{(2,j_{2})}=\left(S_{T}^{(1)}+S_{T}^{(2)}-K^{(2,j_{2})}\right)^{+}&j_{2}=\textit{Basket
Call}\\\
O_{T}^{(2,j_{2})}=\left(K^{(2,j_{2})}-S_{T}^{(1)}-S_{T}^{(2)}\right)^{+}&j_{2}=\textit{Basket
Put}.\end{cases}$ (25)

Furthermore, $K^{(i,j_{i})}\in[A^{(i,j_{i})},B^{(i,j_{i})}]$, where
$j_{1}=Euro\ Call$ and $j_{2}\in\\{Basket\ Call,Basket\ Put\\}$, denotes the
strike price of call and basket options. The existence of optimal portfolio
composition within the subset of basket option is demonstrated in the next
proposition.

######  Proposition 8 (Existence of optimal portfolio composition in the
subset of basket option).

Let the basket option price $O_{t}^{(2,j_{2})}$ be a function of
$\left(S^{(1)}_{t},S^{(2)}_{t},K^{(2,j_{2})}\right)$ and
$O_{t}^{(2,j_{2})}\in\mathbb{C}^{1}$ on
$(0,\infty)\times(0,\infty)\times[A^{(2,j_{2})},B^{(2,j_{2})}]$, then the
optimal portfolio composition for problem (10) within the subset
$\Omega_{O}^{(2,call\ basket)}$ exists.

Proof: See Appendix A.8.

The risky asset exposure (see Equation (43)) is broken down into the
allocation on $O_{t}^{(1,j_{1})}$ and $O_{t}^{(2,j_{2})}$. Note that
allocation on $O_{t}^{(2,j_{2})}$ depends only on $f_{t}^{(22)}$ because
$B_{t}^{(2)}$ is solely hedged by the basket option $O_{t}^{(2,j_{2})}$.
Figure 5 (a) illustrates how risky asset exposure varies with $K^{(1,j_{1})}$
and $K^{(2,j_{2})}$. Allocation on $O_{t}^{(1,j_{1})}$ is scaled by the
relative sensitivity $f_{t}^{(11)}$ (see Equation (43)); hence, risky asset
exposure decreases with $K^{(1,j_{1})}$, and out-of-the-money
$O_{t}^{(1,j_{1})}$ is preferable regardless of the choice of
$O_{t}^{(2,j_{2})}$, for example one-asset or multi-asset options. In
addition, risky asset exposure is monotonic with $K^{(2,j_{2})}$ except when
$K^{(1,j_{1})}$ is extremely small. We plot the cross section of (a) in Figure
5 (b) for illustration purposes. Given an at-the-money or out-of-the-money
$O_{t}^{(1,j_{1})}$, $\left\|\bm{\pi_{t}}\right\|_{1}$ decreases when
$j_{2}=Basket\ Call$ and increases when $j_{2}=Basket\ Put$. Therefore, out-
of-the-money basket options minimize risky asset exposure. Moreover,
$\left\|\bm{\pi_{t}}\right\|_{1}$ is insensitive to $K^{(1,j_{1})}$ because
investing in basket options generally leads to a smaller allocation on
$O_{t}^{(1,j_{1})}$.

(a) $\ell_{1}$ norm of allocation versus strike price

(b) Cross section of (a), basket call

Figure 5: $\left\|\bm{\pi_{t}}\right\|_{1}$ versus strike price (Basket
option)

Similarly to put and call options, with the parameters listed in Table 1, the
optimal basket call is achieved when the strike price $K^{(2,j_{2})}$ is at
the upper bound, while the optimal basket put is achieved when strike price
$K^{(2,j_{2})}$ is at the lower bound.

Given the similarities between one-asset calls/puts and basket calls/puts,
investors could be interested in the best among those choices. To answer this
question, we fix the strike price of a European call $O_{t}^{(1,j_{1})}$,
letting the lower bound of a put option strike price and the upper bound of a
call option strike price be proportional to the spot price:

$\begin{split}&[S_{t}^{(2)},R^{B}S_{t}^{(2)}]=[A^{(2,j_{2})},B^{(2,j_{2})}],\quad
j_{2}\in\\{Euro\ Call,Asian\ Call\\}\mathchar 24635\relax\;\\\
&[S_{t}^{(1)}+S_{t}^{(2)},R^{B}(S_{t}^{(1)}+S_{t}^{(2)})]=[A^{(2,j_{2})},B^{(2,j_{2})}],\quad
j_{2}\in\\{Basket\ Call\\}\mathchar 24635\relax\;\\\
&[R^{A}S_{t}^{(2)},S_{t}^{(2)}]=[A^{(2,j_{2})},B^{(2,j_{2})}],\quad
j_{2}\in\\{Euro\ Put,Asian\ Put,Amer\ Put\\}\mathchar 24635\relax\;\\\
&[R^{A}(S_{t}^{(1)}+S_{t}^{(2)}),S_{t}^{(1)}+S_{t}^{(2)}]=[A^{(2,j_{2})},B^{(2,j_{2})}],\quad
j_{2}\in\\{Basket\ Put\\},\end{split}$

where $R^{A}\leq 1$ and $R^{B}\geq 1$.

By letting the ratio $R^{A}$ and $R^{B}$ vary, the optimal choice of
$O_{t}^{(2,j_{2})}$ is studied in Figure 6. One can observe that when
$O_{t}^{(1,j_{1})}$ is an at-the-money-option—that is, $K^{(1,j_{1})}=40$—a
one-asset Asian option dominates for a small $R^{B}$, while a basket call is
superior to other options when $R^{B}$ is large. However, basket calls become
less preferable as $K^{(1,j_{1})}$ increases. As mentioned above, compared
with one-asset options, investors have a smaller absolute allocation on
$O_{t}^{(1,j_{1})}$ and a larger absolute allocation on $O_{t}^{(2,j_{2})}$
with a basket option. Furthermore, the allocation on $O_{t}^{(1,j_{1})}$ is
scaled by the relative sensitivity $f_{t}^{(11)}$ (see Equation (35) and
(43)), which explains why a basket call has an advantage over one-asset
options when $K^{(1,j_{1})}$ is small but loses its dominant position as
$K^{(1,j_{1})}$ rises. The optimal choice of $O_{t}^{(2,j_{2})}$ with other
sets of parameters is demonstrated in Appendix A.9.

(a) $K^{(1,j_{1})}=40$

(b) $K^{(1,j_{1})}=50$

(c) $K^{(1,j_{1})}=60$

Figure 6: Derivatives selection $O_{t}^{(2,j_{2})}$

##  5 Conclusions

This paper reveals the benefit of using options to minimize the total risk
exposure of a portfolio, while maintaining an optimal level of utility. We
demonstrate that the farther out-of-the-money calls or puts are, the better
choices they are, particularly the Asian type. Given the lack of liquidity on
those type of options, we explored straddle options and found that optimal
choices are close to at-the-money options, which are hence likely liquid
products. We also explored multi-asset derivatives and can confirm that basket
options are preferable to one-asset options in terms of minimizing risk
exposure.

##  Appendix A Proofs

###  A.1 Proof of Proposition 1

We assume $V(t,W)=\frac{W_{t}^{1-\gamma}}{1-\gamma}\exp(h(T-t))$, which is
substituted into (6). Then, $h(T-t)$ satisfies:

$\sup_{\bm{\pi_{t}}}\left\\{\frac{h^{\prime}(T-t)}{1-\gamma}+r+\bm{\pi_{t}}^{T}\bm{\Sigma_{t}\Lambda}-\frac{\gamma}{2}(\bm{\pi_{t}}^{T}\bm{\Sigma_{t}\Phi\Phi}^{T}\bm{\Sigma_{t}}^{T}\bm{\pi_{t}})\right\\}=0,$
(26)

Denote $\bm{\eta_{t}}=\bm{\Sigma_{t}}^{T}\bm{\pi_{t}}$, then problem (26) can
be rewritten as,

$\sup_{\bm{\eta_{t}}}\left\\{\frac{h^{\prime}(T-t)}{1-\gamma}+r+\bm{\eta_{t}}^{T}\bm{\Lambda}-\frac{\gamma}{2}(\bm{\eta_{t}}^{T}\bm{\Phi\Phi}^{T}\bm{\eta_{t}})\right\\}=0.$
(27)

which implies the optimal strategy

$\eta_{t}^{*}=\frac{(\bm{\Phi\Phi}^{T})^{-1}\bm{\Lambda}}{\gamma}.$ (28)

With $\bm{\eta^{*}_{t}}=\bm{\Sigma_{t}}^{T}\bm{\pi^{*}_{t}}$, there are
infinitely many choice of $\bm{\pi_{t}}^{*}$ if $n>2$. Next, we substitute
$\bm{\eta^{*}_{t}}$ into (27) and derive the ordinary differential equation
(ODE) for $h(T-t)$:

$\begin{cases}\frac{h^{\prime}(T-t)}{1-\gamma}+r+\frac{\bm{\Lambda}^{T}(\bm{\Phi\Phi}^{T})^{-1}\bm{\Lambda}}{2\gamma}=0\\\
h(T)=0.\end{cases}$ (29)

where the terminal condition results from $V(t,W)=U(W)$. The solution to (29)
is

$h(T-t)=(1-\gamma)(r+\frac{\bm{\Lambda}^{T}(\bm{\Phi\Phi}^{T})^{-1}\bm{\Lambda}}{2\gamma})(T-t).$
(30)

###  A.2 Proof of Proposition 2

Let $\bm{O_{t,n}}=[O^{(1)}_{t},O^{(2)}_{t},...,O^{(n)}_{t}]^{T}$ with variance
matrix $\Sigma_{t}$ of rank 2 be an optimal subset of options for problem (9).
$\bm{\pi^{*}_{t,n}}$ is a strategy maximizing the expected utility if and only
if $\bm{\Sigma}^{T}_{t}\bm{\pi^{*}_{t,n}}=\bm{\eta^{*}_{t}}$. Therefore,
$\bm{O_{t,n}}$ and $\bm{\pi^{*}_{t,n}}$ is an optimal pair for (9) when
$\bm{\pi^{*}_{t,n}}$ is an optimal solution for

$\displaystyle\mathmakebox[\widthof{$\underset{\displaystyle\bm{\pi_{t}}}{\mathrm{subject~{}to}}$}][l]{\underset{\displaystyle\bm{\pi_{t}}}{\mathrm{minimize}}}\quad\|\bm{\pi_{t}}\|_{1}\hfil\hfil\hfil\hfil$
(31)
$\displaystyle\mathmakebox[\widthof{$\underset{\displaystyle\phantom{\bm{\pi_{t}}}}{\mathrm{subject~{}to}}$}][c]{{\mathrm{subject~{}to}}}\quad$
$\displaystyle\bm{\Sigma_{t}}^{T}\bm{\pi_{t}}=\bm{\eta^{*}_{t}}$

According to the principle 4.5 in Rardin and Rardin (1998), problem (31) is
equivalent to

$\displaystyle\mathmakebox[\widthof{$\underset{\displaystyle\delta_{t}}{\mathrm{subject~{}to}}$}][l]{\underset{\displaystyle\delta_{t}}{\mathrm{minimize}}}\quad\mathbbm{1}^{T}\bm{\delta_{t}}\hfil\hfil\hfil\hfil$
(32)
$\displaystyle\mathmakebox[\widthof{$\underset{\displaystyle\phantom{\delta_{t}}}{\mathrm{subject~{}to}}$}][c]{{\mathrm{subject~{}to}}}\quad$
$\displaystyle\bm{\hat{\Sigma}_{t}}^{T}\bm{\delta_{t}}=\bm{\eta^{*}_{t}},\hfil\hfil$
$\displaystyle\bm{\delta_{t}}\geq 0$

where
$\bm{\delta_{t}}=[\alpha_{t}^{(1)},\alpha_{t}^{(2)},...,\alpha_{t}^{(n)},\beta_{t}^{(1)},\beta_{t}^{(2)},...,\beta_{t}^{(n)}]^{T}$
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

$\bm{\hat{\Sigma}_{t}}=\left[\begin{array}[]{c}\Sigma_{t}\\\
-\Sigma_{t}\end{array}\right]=\left[\begin{array}[]{cc}f^{11}_{t}&f^{12}_{t}\\\
...&...\\\ f^{n1}_{t}&f^{n2}_{t}\\\ -f^{11}_{t}&-f^{12}_{t}\\\ ...&...\\\
-f^{n1}_{t}&-f^{n2}_{t}\end{array}\right].$ (33)

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.

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

  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}$ (34)

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 (32).Now, theorem 2.7 in Bertsimas
and Tsitsiklis (1997) guarantees that there is an optimal solution which is an
extreme point for problem (32).
With the second necessary and sufficient conditions of the extreme point, we
know that an optimal solution $\delta_{t}^{*}$ for problem (32) has at most
two non-zero elements. This would imply an optimal solution, denoted by
$\bm{\pi^{*}_{t,n}}=[\pi_{t,n}^{(1)},\pi_{t,n}^{(2)},...,\pi_{t,n}^{(n)}]^{T}$,
for problem (31) with at most two non-zero elements, which would also be the
optimal strategy for (9).
Without loss of generality, we assume $\pi_{t,n}^{(i)}=0$, $i\neq x,y$.
$\bm{O_{t,2}}=[O^{(x)}_{t},O^{(y)}_{t}]$ and
$\bm{\pi^{*}_{t,2}}=[\pi_{t,n}^{(x)},\pi_{t,n}^{(y)}]^{T}$ is a feasible
strategy for problem (9) with $n=2$. We show that it is an optimal pair by
contradiction.
If there is a feasible solution
$\bm{\hat{O}_{t,n}}=[\hat{O}^{(1)}_{t},\hat{O}^{(2)}_{t}]$ and
$\bm{\hat{\pi}^{*}_{t,2}}=[\hat{\pi}_{t,2}^{(1)},\hat{\pi}_{t,2}^{(2)}]^{T}$
such that $\|\bm{\hat{\pi}^{*}_{t,2}}\|_{1}<\|\bm{\pi^{*}_{t,2}}\|_{1}$, then
$\bm{\hat{\pi}^{*}_{t,n}}=[\hat{\pi}_{t,2}^{(1)},\hat{\pi}_{t,2}^{(2)},0,...,0]^{T}$
is a feasible strategy for (9) such that
$\|\bm{\hat{\pi}^{*}_{t,n}}\|_{1}<\|\bm{\pi^{*}_{t,n}}\|_{1}$, which is
contradiction to our previous conclusion. Note that
$\|\bm{\pi^{*}_{t,2}}\|_{1}=\|\bm{\pi^{*}_{t,n}}\|_{1}$, so problem (9) with
$n=2$ and with $n\geq 2$ have the same minimum $\ell_{1}$ norm of allocation.

###  A.3 Proof of Proposition 3

Let $O_{t}\in\Omega_{O}^{(2,1)}$ with non-singular variance matrix
$\bm{\Sigma_{t}}$ (see (11)), the optimal strategy space $\Omega_{\pi}^{O}$
contains a unique strategy, i.e.
$\bm{\pi_{t}}=(\bm{\Sigma_{t}}^{T})^{-1}\bm{\eta_{t}^{\ast}}$, and $\ell_{1}$
norm of $\bm{\pi_{t}}$ is given by

$\left\|\bm{\pi_{t}}\right\|_{1}=\left|\frac{\eta_{t}^{(1)}}{f_{t}^{(1)}}\right|+\left|\frac{\eta_{t}^{(2)}}{f_{t}^{(2)}}\right|.$
(35)

A non-zero denominator is guaranteed by the non-singular variance matrix
assumption.
$O_{t}^{\ast}=[O_{t}^{{(1)},\ast},O_{t}^{{(2)},\ast}]^{T}\in\Omega_{O}^{(2,1)}$
with variance matrix

$\bm{\Sigma_{t}^{*}}=\left[\begin{array}[]{cc}f^{(1),*}_{t}&0\\\
0&f^{(2),*}_{t}\end{array}\right].$ (36)

For $\bm{O_{t}}\in\Omega_{O}^{(2,1)}$, if
$\left|f^{(1)}_{t}\right|\leq\left|f_{t}^{(1),\ast}\right|$ and
$\left|f^{(2)}_{t}\right|\leq\left|f_{t}^{(2),\ast}\right|$ hold, then it is
easy to see
$\left\|\bm{\pi_{t}}\right\|_{1}\geq\left\|\bm{\pi_{t}}^{\ast}\right\|_{1}$,
so $\bm{O_{t}^{\ast}}$ is a optimal portfolio composition.
If $\left|f^{(1)}_{t}\right|\leq\left|f_{t}^{(1),\ast}\right|$ or
$\left|f^{(2)}_{t}\right|\leq\left|f_{t}^{(2),\ast}\right|$ does not hold,
then there is a $\bm{O^{\ast\ast}_{t}}\in\Omega_{O}^{(2,1)}$, such that the
corresponding strategy
$\left\|\bm{\pi_{t}}^{\ast\ast}\right\|_{1}<\left\|\bm{\pi_{t}}^{\ast}\right\|_{1}$,
hence $\bm{O_{t}^{\ast}}$ is not the optimal.

We have shown that, for any $\bm{O_{t}}\in\Omega_{O}^{(2,1)}$,
$\left|f^{(1)}_{t}\right|\leq\left|f_{t}^{(1),\ast}\right|$ and
$\left|f^{(2)}_{t}\right|\leq\left|f_{t}^{(2),\ast}\right|$ is a sufficient
and necessary condition for $\bm{O_{t}^{\ast}}$ to be an optimal portfolio
composition for problem (10). Therefore,

$\bm{O_{t}^{(i),*}}=\underset{O_{t}^{(i)}}{\arg\max}\left|\frac{\partial
O_{t}^{(i)}}{\partial S_{t}^{(i)}}\frac{1}{O_{t}^{(i)}}\right|.$ (37)

###  A.4 Proof of Proposition 4

$\bm{O_{t}^{\ast}}=[O_{t}^{(1),\ast},O_{t}^{(2),\ast}]^{T}$ is the optimal
portfolio composition for problem (10) if and only if it has the largest
absolute sensitivity (see Proposition 3), i.e.

$O_{t}^{(i),\ast}={\arg\max}\left|\frac{\partial O_{t}^{(i,j)}}{\partial
S_{t}^{(i)}}\frac{1}{O_{t}^{(i,j)}}\right|.$ (38)

For convenience, we write the absolute sensitivity as a function of strike
price $KK^{(i,j)}$.

$h(K^{(i,j)},j)=\left|\frac{\partial O_{t}^{(i,j)}}{\partial
S_{t}^{(i)}}\frac{1}{O_{t}^{(i,j)}}\right|$

$h(K^{(i,j)},j)$ is continuous because $O_{t}^{(i,j)}\in\mathbb{C}^{1}$ and
$O_{t}^{(i,j)}\neq 0$. According to the extreme value theorem, there is a
$\hat{K}^{(i,j)}\in[A^{(i,j)},B^{(i,j)}]$ that achieves the largest
$h(K^{(i,j)},j)$, hence minimizes $\ell_{1}$ norm of allocation
$\left\|\bm{\pi_{t}}^{*}\right\|_{1}$. $O_{t}^{(i),\ast}$ is the optimal for
problem (10) when the strike price $K^{(i,j)}$ is one of the $\hat{K}^{(i,j)}$
where:
$j\in\\{\textit{Euro Call, Asian Call, Amer Call, Euro Put, Asian Put, Amer
Put}\\}$
This guarantees the existence of optimal composition in $\Omega_{O}^{(2,put\
call)}$.

###  A.5 Proof of Corollary 5

We first prove the following lemma,

######  Lemma 1.

The inequality

$\frac{\phi(x-c)}{N(x-c)}-c\leq\frac{\phi(x)}{N(x)}$ (39)

holds for $\forall x\in(-\infty,\infty),c\in(0,\infty)$, where $\phi$ and $N$
are respectively the density function and distribution function of a standard
normal random variable.

Proof: Let us define the ”reversed hazard rate” function:

$Y(x)=\frac{\phi(x)}{N(x)}.$

We want to show

$Y(x-c)-c\leq Y(x).$

We first demonstrate $Y^{\prime}(x)\geq-1$ for all $x$. To see this, note:

$Y^{\prime}+1=\frac{\phi}{N^{2}}\left(\frac{\phi^{\prime}}{\phi}N-\phi+\frac{N^{2}}{\phi}\right)=\frac{\phi}{N^{2}}f.$

Using $\phi^{\prime}=-x\phi$, we have
$f^{\prime}=\left(1+\frac{xN}{\phi}\right)N$. It is not difficult to see that
$x+Y\geq 0$ (use the fact that $g=\left(x+Y\right)N\rightarrow 0$ as
$x\rightarrow-\infty$ and $g^{\prime}=N\geq 0$). Hence we know
$\left(1+\frac{xN}{\phi}\right)\geq 0$, which implies $f^{\prime}\geq 0$.
Moreover $f(x)\geq\underset{x\rightarrow-\infty}{\lim}f(x)=0$, therefore
$Y^{\prime}+1\geq 0$.

Now we complete the proof, using $Y^{\prime}\geq-1$ for all $x$, and the mean
value theorem, we conclude ”by contradiction” that

$\frac{Y(x)-Y(x-c)}{c}\geq-1$

for all $x$ and $c$, which implies

$Y(x)\geq Y(x-c)-c.$

Next, we show that absolute value of optimal allocation on an European call
decreases to $0$ as $K^{(i,Euro\ Call)}\rightarrow\infty$ and absolute value
of allocation on an European put increases with $K^{(i,Euro\ Put)}$ and
converges to $0$ as $K^{(i,Euro\ Put)}\rightarrow 0$. We first abbreviate
$K^{(i,Euro\ Call)}$ to $K$. Equation (12) and (15) shows the sensitivity of
an European call,

$\begin{split}\frac{\partial O^{(i,Euro\ Call)}_{t}}{\partial
S_{t}^{(i)}}\frac{1}{O^{(i,Euro\
Call)}_{t}}\sigma^{(i)}S_{t}^{(i)}&=\frac{N(d_{1})\sigma^{(i)}S_{t}^{(i)}}{S_{t}^{(i)}N(d_{1})-Ke^{-r(\hat{T}-t)}N(d_{2})}\\\
&=\frac{\sigma^{(i)}}{1-e^{-r(\hat{T}-t)}\frac{K}{S^{(i)}_{t}}\frac{N(d_{2})}{N(d_{1})}},\end{split}$

where

$\displaystyle d_{1}$ $\displaystyle=$
$\displaystyle\frac{\ln{(S^{(i)}_{t}/K)}+(r+\frac{1}{2}(\sigma^{(i)})^{2})(\hat{T}-t)}{\sigma^{(i)}\sqrt{\hat{T}-t}}=a\left(\ln{(S^{(i)}_{t}/K)}+r(\hat{T}-t)\right){+}b\quad$
$\displaystyle d_{2}$ $\displaystyle=$ $\displaystyle
d_{1}-\sigma^{(i)}\sqrt{\hat{T}-t}=a\left(\ln{(S^{(i)}_{t}/K)}+r(\hat{T}-t)\right){+}b-c$

with $c=\sigma^{(i)}\sqrt{\hat{T}-t}>0$, $b=\frac{1}{2}c$ and $a=\frac{1}{c}$.
Let us rewrite the sensitivity as follows:

$\frac{\partial O^{(i,Euro\ Call)}_{t}}{\partial
S_{t}^{(i)}}\frac{1}{O^{(i,Euro\
Call)}_{t}}\sigma^{(i)}S_{t}^{(i)}=\frac{\sigma^{(i)}}{1-e^{\frac{b-x}{a}}G(x)}$

where $x=a\left(\ln{(S_{t}/K)}+r(\hat{T}-t)\right){+}b\,$ and
$G(x)=\frac{N(d_{2})}{N(d_{1})}=\frac{N(x-c)}{N(x)}$.
We would like to show $H(x)=e^{\frac{b-x}{a}}G(x)$ is increasing in $K$, hence
decreasing in $x$. Its first derivative leads to:

$\displaystyle H^{\prime}(x)$ $\displaystyle=$
$\displaystyle-\frac{e^{\frac{b-x}{a}}}{a}G(x)+e^{\frac{b-x}{a}}G^{\prime}(x)$
$\displaystyle=$
$\displaystyle-\frac{e^{\frac{b-x}{a}}}{a}\frac{N(x-c)}{N(x)}+e^{\frac{b-x}{a}}\frac{n(x-c)N(x)-N(x-c)n(x)}{N^{2}(x)}$
$\displaystyle=$
$\displaystyle-\frac{e^{\frac{b-x}{a}}}{a}\frac{N(x-c)}{N(x)}+e^{\frac{b-x}{a}}\left(\frac{n(x-c)}{N(x-c)}-\frac{n(x)}{N(x)}\right)\frac{N(x-c)}{N(x)}$
$\displaystyle=$ $\displaystyle
e^{\frac{b-x}{a}}\left[-\frac{1}{a}+\left(\frac{n(x-c)}{N(x-c)}-\frac{n(x)}{N(x)}\right)\right]\frac{N(x-c)}{N(x)}$
$\displaystyle=$ $\displaystyle
e^{\frac{b-x}{a}}\left[-c+\left(\frac{n(x-c)}{N(x-c)}-\frac{n(x)}{N(x)}\right)\right]\frac{N(x-c)}{N(x)}$

It’s easy to see $H^{\prime}(x)<0$ with Lemma 1. The sensitivity of an
European call is positive and increases with $K$.

As $K\rightarrow\infty$, $O^{(i,Euro\ call)}_{t}\rightarrow 0$, so
$Ke^{-r(\hat{T}-t)}N(d_{2})\rightarrow 0$. Furthermore,

$\displaystyle H(x)$ $\displaystyle=$
$\displaystyle\frac{Ke^{-r(\hat{T}-t)}N(d_{2})}{S_{t}^{(i)}N(d_{1})}\xrightarrow{\text{L'Hôpital's
rule}}\frac{e^{-r(\hat{T}-t)}N(d_{2})-\phi(d_{1})\frac{S_{t}^{(i)}}{K\sigma^{(i)}\sqrt{\hat{T}-t}}}{-\frac{S_{t}^{(i)}\phi(d_{1})}{K\sigma^{(i)}\sqrt{\hat{T}-t}}}$
$\displaystyle\xrightarrow{\text{L'Hôpital's rule}}$
$\displaystyle\frac{-\phi(d_{1})\frac{S_{t}^{(i)}}{(K)^{2}\sigma^{(i)}\sqrt{\hat{T}-t}}-\phi(d_{1})\frac{d_{1}S_{t}^{(i)}}{(K\sigma^{(i)}\sqrt{\hat{T}-t})^{2}}+\frac{\phi(d_{1})S_{t}^{(i)}}{(K)^{2}\sigma^{(i)}\sqrt{\hat{T}-t}}}{-\phi(d_{1})\frac{S_{t}^{(i)}d_{1}}{(K\sigma^{(i)}\sqrt{\hat{T}-t})^{2}}+\frac{S_{t}^{(i)}\phi(d_{1})}{(K)^{2}\sigma^{(i)}\sqrt{\hat{T}-t}}}$
$\displaystyle=$
$\displaystyle\frac{-S_{t}^{(i)}\sigma^{(i)}\sqrt{\hat{T}-t}-S_{t}^{(i)}d_{1}+S_{t}^{(i)}\sigma^{(i)}\sqrt{\hat{T}-t}}{-S_{t}^{(i)}d_{1}+S_{t}^{(i)}\sigma^{(i)}\sqrt{\hat{T}-t}}\longrightarrow
1.$

Hence,

$\frac{\partial O^{(i,Euro\ Call)}_{t}}{\partial
S_{t}^{(i)}}\frac{1}{O^{(i,Euro\
Call)}_{t}}\sigma^{(i)}S_{t}^{(i)}=\frac{\sigma^{(i)}}{1-H(x)}\longrightarrow\infty.$

Moreover, the absolute value of allocation on $O_{t}^{(i,Euro\ Call)}$ is
decreasing with $K$ and

$\left|\pi^{(i,Euro\
Call)}_{t}\right|=\frac{\left|\eta_{t}^{(i)}\right|}{\left|\frac{\partial
O_{t}^{(i,Euro\ Call)}}{\partial S_{t}^{(i)}}\frac{1}{O_{t}^{(i,Euro\
Call)}}\sigma^{(i)}S_{t}^{(i)}\right|}\longrightarrow 0\quad\textit{ as
}K\longrightarrow\infty.$ (40)

The proof for European put follows similarly.

###  A.6 Proof of Proposition 6

Suppose for any $\bm{O_{t}}\in\Omega_{O}^{(2,straddle)}$, $O_{t}^{(i,j)}$
satisfies these four conditions:

  1. 1.

$\frac{\partial O_{t}^{(i,j)}}{\partial S_{t}^{(i)}}=0\iff
K^{(i,j)}=A^{(i,j)}$.

  2. 2.

$\left|\frac{\partial O_{t}^{(i)}}{\partial S_{t}^{(i)}}\right|$ has an upper
bound.

  3. 3.

$O_{t}^{(i,j)}\in(0,\infty)$ and $O_{t}^{(i,j)}\rightarrow\infty$ as
$K^{(i,j)}\rightarrow\infty$, where $j\in\\{Euro\ Strad,Asian\ Strad\\}$.

  4. 4.

$O_{t}^{(i,j)}\left(S^{(i)},K^{(i,j)}\right)\in\mathbb{C}^{1}$.

All four assumptions are not restrictive in a Black-Scholes setting.
Proposition 3 illustrates that
$\bm{O_{t}^{\ast}}=[O_{t}^{(1),\ast},O_{t}^{(2),\ast}]^{T}$ is the optimal
portfolio composition for problem (10) if it has the largest absolute
sensitivity, i.e.

$O_{t}^{(i),\ast}=\underset{O_{t}^{(i,j)}}{\arg\max}\left|\frac{\partial
O_{t}^{(i,j)}}{\partial S_{t}^{(i)}}\frac{1}{O_{t}^{(i,j)}}\right|.$ (41)

For convenience, we write the absolute sensitivity as a function of strike
price $K^{(i,j)}$.

$h(K^{(i,j)},j)=\left|\frac{\partial O_{t}^{(i,j)}}{\partial
S_{t}^{(i)}}\frac{1}{O_{t}^{(i,j)}}\right|$

With the four assumptions above, it’s easy to see,

$\begin{cases}h(0,j)\in(0,\infty)&j\in\\{Euro\ Strad,Asian\ Strad,Amer\
Strad\\}\\\ h(B^{(i,j)},j)\in(0,\infty)&j=Amer\ Strad\\\
h(K^{(i,j)},j)\rightarrow 0&\textit{as }K^{(i,j)}\uparrow\downarrow
A^{(i,j)},j\in\\{Euro\ Strad,Asian\ Strad,Amer\ Strad\\}\\\
h(K^{(i,j)},j)\rightarrow 0&\textit{as
}K^{(i,j)}\rightarrow\infty,j\in\\{Euro,Asian\\}.\end{cases}$ (42)

When $K^{(i,j)}\in[0,A^{(i,j)})$ and $j\in\\{Euro\ Strad,Asian\ Strad,Amer\
Strad\\}$, $h(K^{(i,j)},j)$ is continuous because
$O_{t}^{(i,j)}\left(S^{(i)},K^{(i,j)}\right)\in\mathbb{C}^{1}$. Besides, there
is a $Z$ such that $h(K^{(i,j)},j)<h(0,j)$ when $K^{(i,j)}\in(Z,A^{(i,j)})$.
According to the extreme value theorem, there is a $K^{(l,i,j)}$ such that $h$
attains the maximum in $[0,Z]$, hence $h(K^{(l,i,j)},j)\geq h(0,j)$.
$K^{(l,i,j)}$ is proven to be the maximum point for $h(K^{(i,j)},j)$ in
$[0,A^{(i,j)})$.
Let $M$ be a real number in $(A^{(i,j)},\infty)$ where $j\in\\{Euro\
Strad,Asian\ Strad\\}$, it’s obvious that $h(M,j)>0$. There is a $Z>0$ such
that $h(K^{(i,j)},j)<h(M,j)$ when
$K^{(i,j)}\in(A^{(i,j)},A^{(i,j)}+\frac{1}{Z})\cup(Z,\infty)$. According to
the extreme value theorem, there is a $K^{(r,i,j)}$ such that $h$ attains the
maximum in $[A^{(i,j)}+\frac{1}{Z},Z]$, i.e. $h(K^{(r,i,j)},j)>h(M,j)$. Then,
$K^{(r,i,j)}$ is the maximum point for $h(K^{(i,j)},j)$ in
$(A^{(i,j)},\infty)$.
As for American straddle ($j=Amer\ Strad$), $h(B^{(i,j)},j)>0$. There is a
$Z>0$ such that $h(K^{(i,j)},j)<h(B^{(i,j)},j)$ when
$K^{(i,j)}\in(A^{(i,j)},A^{(i,j)}+\frac{1}{Z})$. And there is a $K^{(r,i,j)}$
such that $h$ attains the maximum in $[A^{(i,j)}+\frac{1}{Z},B^{(i,j)}]$,
hence $h(K^{(r,i,j)},j)\geq h(B^{(i,j)},j)$. $h(K^{(r,i,j)},j)$ is the maximum
point on the right branch $[A^{(i,j)},B^{(i,j)}]$. $O_{t}^{(i),\ast}$ is the
optimal for the problem (10) when the strike price $K$ is either $K^{(l,i,j)}$
or $K^{(r,i,j)}$ where $j\in\\{Euro\ Strad,Asian\ Strad,Amer\ Strad\\}$. The
existence of the optimal composition in a straddle subset is proven.

###  A.7 Proof of Proposition 7

Let $\bm{O_{t}}\in\Omega_{O}^{(2,multi\ asset)}$ with non-singular variance
matrix $\bm{\Sigma_{t}}$ (see (23)), the optimal strategy space
$\Omega_{\pi}^{O}$ contains a unique strategy, i.e.
$\bm{\pi_{t}}=(\bm{\Sigma_{t}}^{T})^{-1}\bm{\eta_{t}^{\ast}}$. The allocation
and its $\ell_{1}$ norm can be written as

$\begin{split}\pi^{(1,j)}&=\frac{1}{f^{(11)}_{t}}(\eta_{t}^{(1)}-\frac{f^{(21)}_{t}}{f^{(22)}_{t}}\eta_{t}^{(2)})\quad\quad\pi^{(2,j)}=\frac{\eta_{t}^{(2)}}{f^{(22)}_{t}}\\\
\left\|\bm{\pi_{t}}\right\|_{1}&=\frac{1}{\left|f^{(11)}_{t}\right|}\left|\eta_{t}^{(1)}-\frac{f^{(21)}_{t}}{f^{(22)}_{t}}\eta_{t}^{(2)}\right|+\frac{\left|\eta_{t}^{(2)}\right|}{\left|f^{(22)}_{t}\right|}.\end{split}$
(43)

If
$\bm{O_{t}^{\ast}}=[O_{t}^{{(1),\ast}},O_{t}^{{(2),\ast}}]^{T}\in\Omega_{O}^{(2,multi\
asset)}$ achieves minimum $\ell_{1}$ norm of allocation and

$O_{t}^{(1),\ast}\neq\underset{O_{t}^{(1,j_{1})}}{\arg\max}\left|\frac{\partial
O_{t}^{(1,j_{1})}}{\partial S_{t}^{(1)}}\frac{1}{O_{t}^{(1,j_{1})}}\right|.$
(44)

Then there is a
$\bm{O_{t}^{\ast\ast}}=[O_{t}^{{(1),\ast\ast}},O_{t}^{{(2),\ast\ast}}]^{T}$,
such that

$\left|\frac{\partial O_{t}^{(1),\ast}}{\partial
S_{t}^{(1)}}\frac{1}{O_{t}^{(1),\ast}}\right|<\left|\frac{\partial
O_{t}^{(1),\ast\ast}}{\partial
S_{t}^{(1)}}\frac{1}{O_{t}^{(1),\ast\ast}}\right|.$ (45)

Therefore, let $O_{t}^{{(2),\ast\ast}}=O_{t}^{{(2)},\ast}$, this implies
$\left|f^{(11),\ast}_{t}\right|<\left|f^{(11),\ast\ast}_{t}\right|$,
$f^{(21),\ast}_{t}=f^{(21),\ast\ast}_{t}$ and
$f^{(22),\ast}_{t}=f^{(22),\ast\ast}_{t}$. Equation (43) indicates
$\left\|\bm{\pi_{t}}^{\ast}\right\|_{1}>\left\|\bm{\pi_{t}}^{\ast\ast}\right\|_{1}$,
which proves by contradiction that

$O_{t}^{(1),\ast}=\underset{O_{t}^{(1,j_{1})}}{\arg\max}\left|\frac{\partial
O_{t}^{(1,j_{1})}}{\partial S_{t}^{(1)}}\frac{1}{O_{t}^{(1,j_{1})}}\right|.$
(46)

is a necessary condition for $\bm{O_{t}^{\ast}}$ to be the optimal portfolio
composition for problem (10) within $\Omega_{O}^{(2,multi\ asset)}$.

###  A.8 Proof of Proposition 8

Recall $f_{t}^{(ik)}$ in the variance matrix $\Sigma_{t}$ can be written as

$f^{(ik)}=\frac{\partial O_{t}^{(i,j_{i})}}{\partial
S_{t}^{(k)}}\frac{S_{t}^{(k)}\sigma^{(k)}}{O_{t}^{(i,j_{i})}}.$ (47)

According to the non-singular variance matrix assumption,
$f_{t}^{(11)},f_{t}^{(22)}\neq 0$. In addition, $O_{t}^{(1,j_{1})}$ is an
European call option, hence $f_{t}^{(11)}$ is continuous with respect to
$K^{(1,Euro\ Call)}$ on $[A^{(1,Euro\ Call)},B^{(1,Euro\ Call)}]$.
$f_{t}^{(21)}$ and $f_{t}^{(22)}$ are continuous with respect to
$K^{(2,j_{2})}$ on $[A^{(2,j_{2})},B^{(2,j_{2})}]$ because
$O_{t}^{(2,j_{2})}\in\mathbb{C}^{1}$.
$\left\|\bm{\pi_{t}}\right\|_{1}$ (see (43)) is continuous on the closed set
$[A^{(1,Euro\ Call)},B^{(1,Euro\ Call)}]\times[A^{(2,j_{2})},B^{(2,j_{2})}]$,
so there is a portfolio with strike price $[\hat{K}^{(1,Euro\
Call)},\hat{K}^{(2,j_{2})}]^{T}$ that achieves minimum risky asset exposure
with any $j_{2}$. $\bm{O_{t}^{\ast}}$ is the optimal for problem (10) within
$\Omega_{O}^{(2,call\ basket)}$ when the strike price is in
$[\hat{K}^{(1,Euro\ Call)},\hat{K}^{(2,j_{2})}]^{T}$, where
$j_{2}\in\\{\textit{Basket Call, Basket Put}\\}$.

###  A.9 Comparison between the one-asset option and multi-asset option
subsets

In this section, we exhibit the optimal choice of $O_{t}^{(2,j_{2})}$ given
different set of parameters. In contrast to the two positive correlated
underlying assets considered in Section 4, i.e. $\rho=0.4$ (see Table 1), we
let correlation $\rho=-0.4$ while all other parameters remain unchanged, here
a similar derivatives selection is conducted.
Results are presented in Figure 7. Unlike the case of two positive correlated
underlying assets (see Figure 6), one-asset option is no longer preferable in
minimizing risky asset exposure while basket put becomes competitive.
Especially when $O_{t}^{(1,j_{1})}$ is an at-the-money European call, i.e.
$K^{(1,j_{1})}=40$, a basket put is superior to other options, i.e. larger
area.

(a) $K^{(1,j_{1})}=40$

(b) $K^{(1,j_{1})}=50$

(c) $K^{(1,j_{1})}=60$

Figure 7: Derivatives selection $O_{t}^{(2,j_{2})}$ ($\rho=-0.4$)

Next, we consider the case when the parameters of the two underlying assets
are exchanged, i.e. $\lambda^{(1)}=0.6$, $\lambda^{(2)}=0.52$,
$\sigma^{(1)}=0.2$, $\sigma^{(2)}=0.13$ while all other parameters are given
in Table 1, the optimal choice of $O_{t}^{(2,j_{2})}$ is shown in Figure 8.
Compared with Figure 6, basket put instead of basket call is selected in the
largest region. Furthermore, the one-asset Asian option is preferable when
$K^{(1,j_{1})}$ is lregardless of the underlying assets’ parameters.

(a) $K^{(1,j_{1})}=40$

(b) $K^{(1,j_{1})}=50$

(c) $K^{(1,j_{1})}=60$

Figure 8: Derivatives selection $O_{t}^{(2,j_{2})}$ (exchange assets’
parameter)

## References

  *   * Bertsimas and Tsitsiklis (1997) Bertsimas D, Tsitsiklis JN (1997) Introduction to linear optimization, vol 6. Athena Scientific Belmont, MA
  * Bjerksund and Stensland (1993) Bjerksund P, Stensland G (1993) Closed-form approximation of american options. Scandinavian Journal of Management 9:S87–S99
  * Black and Scholes (1973) Black F, Scholes M (1973) The pricing of options and corporate liabilities. Journal of political economy 81(3):637–654
  * Chen and Liu (2014) Chen N, Liu Y (2014) American option sensitivities estimation via a generalized infinitesimal perturbation analysis approach. Operations Research 62(3):616–632
  * Goldman et al (1979) Goldman MB, Sosin HB, Gatto MA (1979) Path dependent options:” buy at the low, sell at the high”. The Journal of Finance 34(5):1111–1127
  * Haugh and Lo (2001) Haugh MB, Lo AW (2001) Asset allocation and derivatives. Quantitative Finance 1:45–72
  * Kemna and Vorst (1990) Kemna AG, Vorst AC (1990) A pricing method for options based on average asset values. Journal of Banking & Finance 14(1):113–129
  * Kraft (2003) Kraft H (2003) Elasticity approach to portfolio optimization. Mathematical Methods of Operations Research 58(1):159–182
  * Liu and Pan (2003) Liu J, Pan J (2003) Dynamic derivative strategies. Journal of Financial Economics 69(3):401–430
  * Maverick (2020) Maverick J (2020) How big is the derivatives market? URL https://www.investopedia.com/ask/answers/052715/how-big-derivatives-market.asp
  * Merton (1969) Merton RC (1969) Lifetime portfolio selection under uncertainty: The continuous-time case. The review of Economics and Statistics pp 247–257
  * Merton (1973) Merton RC (1973) Theory of rational option pricing. The Bell Journal of economics and management science pp 141–183
  * Mi and Xu (2020) Mi H, Xu L (2020) Optimal investment with derivatives and pricing in an incomplete market. Journal of Computational and Applied Mathematics 368:112,522
  * Rardin and Rardin (1998) Rardin RL, Rardin RL (1998) Optimization in operations research, vol 166. Prentice Hall Upper Saddle River, NJ