Derivatives-based portfolio decisions: an expected utility insight

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 theory 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.

Appendix A Proofs

1.1 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:

$$\begin{aligned} \sup _{{\varvec{\pi _t}}}\left\{ \frac{h'(T-t)}{1-\gamma }+r+{\varvec{\pi _t}}^T{\varvec{\Sigma _t\Lambda }}-\frac{\gamma }{2}({\varvec{\pi _t}}^T{\varvec{\Sigma _t\Phi \Phi }}^T{\varvec{\Sigma _t}}^T{\varvec{\pi _t}})\right\} =0, \end{aligned}$$

(A1)

Denote ${\varvec{\eta _t}}={\varvec{\Sigma _t}}^T{\varvec{\pi _t}}$, then problem (A1) can be rewritten as,

$$\begin{aligned} \sup _{{\varvec{\eta _t}}}\left\{ \frac{h'(T-t)}{1-\gamma }+r+{\varvec{\eta _t}}^T{\varvec{\Lambda }}-\frac{\gamma }{2}({\varvec{\eta _t}}^T{\varvec{\Phi \Phi }}^T{\varvec{\eta _t}})\right\} =0. \end{aligned}$$

(A2)

which implies the optimal strategy

$$\begin{aligned} \eta _t^*=\frac{({\varvec{\Phi \Phi }}^T)^{-1}{\varvec{\Lambda }}}{\gamma }. \end{aligned}$$

(A3)

With ${\varvec{\eta ^*_t}}={\varvec{\Sigma _t}}^T{\varvec{\pi ^*_t}}$, there are infinitely many choice of ${\varvec{\pi _t}}^*$ if $n>2$. Next, we substitute ${\varvec{\eta ^*_t}}$ into (A2) and derive the ordinary differential equation (ODE) for $h(T-t)$:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{h'(T-t)}{1-\gamma }+r+\frac{{\varvec{\Lambda }}^T({\varvec{\Phi \Phi }}^T)^{-1}{\varvec{\Lambda }}}{2\gamma }=0\\ h(T)=0. \end{array}\right. } \end{aligned}$$

(A4)

where the terminal condition results from $V(t,W)=U(W)$. The solution to (A4) is

$$\begin{aligned} h(T-t)=(1-\gamma )(r+\frac{{\varvec{\Lambda }}^T({\varvec{\Phi \Phi }}^T)^{-1}{\varvec{\Lambda }}}{2\gamma })(T-t). \end{aligned}$$

(A5)

1.2 A.2 Proof of Proposition 2

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

$$\begin{aligned}&\mathop {\mathrm {minimize}}\limits _{{\varvec{\pi _t}}}\quad {\Vert {\varvec{\pi _t}}\Vert _1}{}{}\nonumber \\&\mathop {\mathrm {subject\,to}}\limits \,\quad {{\varvec{\Sigma _t}}^T{\varvec{\pi _t}}={\varvec{\eta ^*_t}} }{}{} \end{aligned}$$

(A6)

According to the principle 4.5 in Rardin and Rardin (1998), problem (A6) is equivalent to

$$\begin{aligned} \mathop {\mathrm {minimize}}\limits _{\delta _t}\quad&{\mathbb {1}^T{\varvec{\delta _t}}}{}{}\nonumber \\ \mathop {\mathrm {subject\,to}}\limits \,\quad&{{\varvec{{\hat{\Sigma }}_t}}^T{\varvec{\delta _t}}={\varvec{\eta ^*_t }}}\nonumber \\&{{\varvec{\delta _t}}\ge 0} \end{aligned}$$

(A7)

where ${\varvec{\delta _t}}=[\alpha _t^{(1)},\alpha _t^{(2)},\ldots ,\alpha _t^{(n)},\beta _t^{(1)}, \beta _t^{(2)},\ldots ,\beta _t^{(n)}]^T$ satisfies $\alpha _t^{(i)}=\frac{\vert \pi _t^{(i)}\vert +\pi _t^{(i)}}{2}$, and $\beta _t^{(i)}=\frac{\vert \pi _t^{(i)}\vert -\pi _t^{(i)}}{2}$, with

$$\begin{aligned} {\varvec{{\hat{\Sigma }}_t}}=\left[ \begin{array}{c} \Sigma _t \\ -\Sigma _t \end{array}\right] =\left[ \begin{array}{cc} f^{11}_t &{} f^{12}_t\\ \ldots &{}\ldots \\ f^{n1}_t &{} f^{n2}_t\\ -f^{11}_t &{} -f^{12}_t \\ \ldots &{}\ldots \\ -f^{n1}_t &{} -f^{n2}_t \end{array}\right] . \end{aligned}$$

(A8)

Theorems 2.3 and 2.4 in Bertsimas and Tsitsiklis (1997) list the necessary and sufficient conditions for the extreme point $\delta _t$, i.e.

• 1. ${\varvec{\delta _t}}=[\delta _t^{(1)}, \delta _t^{(2)},\ldots ,\delta _t^{(n)},\delta _t^{(n+1)}, \delta _t^{(n+2)},\ldots ,\delta _t^{(2n)}]^T$.

• 2. the ${\hat{q}}_{th}$ and ${\hat{p}}_{th}$ rows in $\hat{\Sigma _t}$ are linear independent, $\delta _t^{(i)}=0$ if $i\ne {\hat{q}}$ or ${\hat{p}}$.

• 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:

$$\begin{aligned} \begin{aligned} \delta _t^{[1]}&={\left\{ \begin{array}{ll} [\delta _t^{[1],(1)}, \delta _t^{[1],(2)},\ldots ,\delta _t^{[1],(n)},\delta _t^{[1],(n+1)}, \delta _t^{[1],(n+2)},\ldots ,\delta _t^{[1],(2n)}]^T\\ \delta _t^{[1],(i)}=0 \quad {if} i\ne q or p \end{array}\right. }\\ \delta _t^{[2]}&={\left\{ \begin{array}{ll} [\delta _t^{[2],(1)}, \delta _t^{[2],(2)},\ldots ,\delta _t^{[2],(n)},\delta _t^{[2],(n+1)}, \delta _t^{[2],(n+2)},\ldots ,\delta _t^{[2],(2n)}]^T\\ \delta _t^{[2],(i)}=0 \quad { if i\ne q+n or p} \end{array}\right. }\\ \delta _t^{[3]}&={\left\{ \begin{array}{ll} [\delta _t^{[3],(1)}, \delta _t^{[3],(2)},\ldots ,\delta _t^{[3],(n)},\delta _t^{[3],(n+1)}, \delta _t^{[3],(n+2)},\ldots ,\delta _t^{[3],(2n)}]^T\\ \delta _t^{[3],(i)}=0 \quad { if i\ne q or p+n} \end{array}\right. }\\\delta _t^{[4]}&={\left\{ \begin{array}{ll} [\delta _t^{[4],(1)}, \delta _t^{[4],(2)},\ldots ,\delta _t^{[4],(n)},\delta _t^{[4],(n+1)}, \delta _t^{[4],(n+2)},\ldots ,\delta _t^{[4],(2n)}]^T\\ \delta _t^{[4],(i)}=0 \quad { if i\ne q+n or p+n} \end{array}\right. } \end{aligned} \end{aligned}$$

(A9)

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 (A7).Now, theorem 2.7 in Bertsimas and Tsitsiklis (1997) guarantees that there is an optimal solution which is an extreme point for problem (A7).

With the second necessary and sufficient conditions of the extreme point, we know that an optimal solution $\delta _t^*$ for problem (A7) has at most two non-zero elements. This would imply an optimal solution, denoted by ${\varvec{\pi ^*_{t,n}}}=[\pi _{t,n}^{(1)}, \pi _{t,n}^{(2)},\ldots ,\pi _{t,n}^{(n)}]^T$, for problem (A6) with at most two non-zero elements, which would also be the optimal strategy for (9).

Without loss of generality, we assume $\pi _{t,n}^{(i)}=0$, $i\ne x,y$. ${\varvec{O_{t,2}}}=[O^{(x)}_t, O^{(y)}_t]$ and ${\varvec{\pi ^*_{t,2}}}=[\pi _{t,n}^{(x)}, \pi _{t,n}^{(y)}]^T$ 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 ${\varvec{{\hat{O}}_{t,n}}}=[{\hat{O}}^{(1)}_t, {\hat{O}}^{(2)}_t]$ and ${\varvec{{\hat{\pi }}^*_{t,2}}}=[{\hat{\pi }}_{t,2}^{(1)}, {\hat{\pi }}_{t,2}^{(2)}]^T$ such that $\Vert {\varvec{{\hat{\pi }}^*_{t,2}}}\Vert _1<\Vert {\varvec{\pi ^*_{t,2}}}\Vert _1$, then ${\varvec{{\hat{\pi }}^*_{t,n}}}=[{\hat{\pi }}_{t,2}^{(1)}, {\hat{\pi }}_{t,2}^{(2)},0,\ldots \ldots ,0]^T$ is a feasible strategy for (9) such that $\Vert {\varvec{{\hat{\pi }}^*_{t,n}}}\Vert _1<\Vert {\varvec{\pi ^*_{t,n}}}\Vert _1$, which is contradiction to our previous conclusion. Note that $\Vert {\varvec{\pi ^*_{t,2}}}\Vert _1=\Vert {\varvec{\pi ^*_{t,n}}}\Vert _1$, so problem (9) with $n=2$ and with $n\ge 2$ have the same minimum $\ell _1$ norm of allocation.

1.3 A.3 Proof of Proposition 3

Let $O_t\in \Omega _O^{(2,1)}$ with non-singular variance matrix ${\varvec{\Sigma _t}}$ (see (11)), the optimal strategy space $\Omega _{\pi }^{O}$ contains a unique strategy, i.e. ${\varvec{\pi _t}}=({\varvec{\Sigma _t}}^T)^{-1}{\varvec{\eta _t^{*}}}$, and $\ell _1$ norm of ${\varvec{\pi _t}}$ is given by

$$\begin{aligned} \left\| {\varvec{\pi _t}}\right\| _1=\left| \frac{\eta _t^{(1)}}{f_t^{(1)}}\right| +\left| \frac{\eta _t^{(2)}}{f_t^{(2)}}\right| . \end{aligned}$$

(A10)

A non-zero denominator is guaranteed by the non-singular variance matrix assumption. $O_t^{*}=[O_t^{{(1)},*},O_t^{{(2)},*}]^T\in \Omega _O^{(2,1)}$ with variance matrix

$$\begin{aligned} {\varvec{\Sigma _t^*}}=\left[ \begin{array}{cc} f^{(1),*}_t&{} 0 \\ 0 &{} f^{(2),*}_t \end{array}\right] . \end{aligned}$$

(A11)

For ${\varvec{O_t}}\in \Omega _O^{(2,1)}$, if $\left| f^{(1)}_t\right| \le \left| f_t^{(1),*}\right|$ and $\left| f^{(2)}_t\right| \le \left| f_t^{(2),*}\right|$ hold, then it is easy to see $\left\| {\varvec{\pi _t}}\right\| _1 \ge \left\| {\varvec{\pi _t}}^{*}\right\| _1$, so ${\varvec{O_t^{*}}}$ is a optimal portfolio composition.

If $\left| f^{(1)}_t\right| \le \left| f_t^{(1),*}\right|$ or $\left| f^{(2)}_t\right| \le \left| f_t^{(2),*}\right|$ does not hold, then there is a ${\varvec{O^{**}_t}}\in \Omega _O^{(2,1)}$, such that the corresponding strategy $\left\| {\varvec{\pi _t}}^{**}\right\| _1<\left\| {\varvec{\pi _t}}^{*}\right\| _1$, hence ${\varvec{O_t^{*}}}$ is not the optimal.

We have shown that, for any ${\varvec{O_t}}\in \Omega _O^{(2,1)}$, $\left| f^{(1)}_t\right| \le \left| f_t^{(1),*}\right|$ and $\left| f^{(2)}_t\right| \le \left| f_t^{(2),*}\right|$ is a sufficient and necessary condition for ${\varvec{O_t^{*}}}$ to be an optimal portfolio composition for problem (10). Therefore,

$$\begin{aligned} {\varvec{O_t^{(i),*}}}=\underset{O_t^{(i)}}{\arg \max }\left| \frac{\partial O_t^{(i)}}{\partial S_t^{(i)}}\frac{1}{O_t^{(i)}}\right| . \end{aligned}$$

(A12)

1.4 A.4 Proof of Proposition 4

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

$$\begin{aligned} O_t^{(i),*}={\arg \max }\left| \frac{\partial O_t^{(i,j)}}{\partial S_t^{(i)}}\frac{1}{O_t^{(i,j)}}\right| . \end{aligned}$$

(A13)

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

$$\begin{aligned} h(K^{(i,j)},j)= \left| \frac{\partial O_t^{(i,j)}}{\partial S_t^{(i)}}\frac{1}{O_t^{(i,j)}}\right| \end{aligned}$$

$h(K^{(i,j)},j)$ is continuous because $O_t^{(i,j)}\in {\mathbb {C}}^1$ and $O_t^{(i,j)}\ne 0$. 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 $\left\| {\varvec{\pi _t}}^*\right\| _1$. $O_t^{(i),*}$ 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 $\Omega _O^{(2,put\ call)}$.

1.5 A.5 Proof of Corollary 5

We first prove the following lemma,

Lemma 9

The inequality

$$\begin{aligned} \frac{\phi (x-c)}{N(x-c)} -c\le \frac{\phi (x)}{N(x)} \end{aligned}$$

(A14)

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:

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

We want to show

$$\begin{aligned} Y(x-c)-c\le Y(x). \end{aligned}$$

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

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

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

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

$$\begin{aligned} \frac{Y(x)-Y(x-c)}{c}\ge -1 \end{aligned}$$

for all x and c, which implies

$$\begin{aligned} Y(x)\ge Y(x-c)-c. \end{aligned}$$

$\square$

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,

$$\begin{aligned} \begin{aligned} \frac{\partial O^{(i,Euro\ Call)}_t}{\partial S_t^{(i)}}\frac{1}{O^{(i,Euro\ Call)}_t} \sigma ^{(i)}S_t^{(i)}&=\frac{N(d_1)\sigma ^{(i)}S_t^{(i)}}{S_t^{(i)}N(d_1)-Ke^{-r({\hat{T}}-t)}N(d_2)}\\ {}&=\frac{ \sigma ^{(i)} }{1-e^{-r({\hat{T}}-t)}\frac{K}{S^{(i)}_{t}}\frac{N(d_{2})}{N(d_{1})}}, \end{aligned} \end{aligned}$$

where

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

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:

$$\begin{aligned} \frac{\partial O^{(i,Euro\ Call)}_t}{\partial S_t^{(i)}}\frac{1}{O^{(i,Euro\ Call)}_t} \sigma ^{(i)}S_t^{(i)}=\frac{ \sigma ^{(i)}}{1-e^{\frac{b-x}{a}}G(x)} \end{aligned}$$

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:

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

It’s easy to see $H^{\prime }(x)<0$ with Lemma 9. 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,

$$\begin{aligned} H(x)= & {} \frac{Ke^{-r({\hat{T}}-t)}N(d_2)}{S_t^{(i)}N(d_1)} \xrightarrow {\text {L'H}{\hat{\mathrm{o}}}\text {pital's rule}} \frac{e^{-r({\hat{T}}-t)}N(d_2)-\phi (d_1)\frac{S_t^{(i)}}{K\sigma ^{(i)}\sqrt{{\hat{T}}-t}}}{-\frac{S_t^{(i)}\phi (d_1)}{K\sigma ^{(i)}\sqrt{{\hat{T}}-t}}}\\&\xrightarrow {\text {L'H}{\hat{\mathrm{o}}}\text {pital's rule}}&\frac{-\phi (d_1)\frac{S_t^{(i)}}{(K)^2\sigma ^{(i)}\sqrt{{\hat{T}}-t}}-\phi (d_1)\frac{d_1S_t^{(i)}}{(K\sigma ^{(i)}\sqrt{{\hat{T}}-t} )^2} +\frac{\phi (d_1)S_t^{(i)}}{(K)^2\sigma ^{(i)}\sqrt{{\hat{T}}-t}}}{-\phi (d_1) \frac{S_t^{(i)}d_1}{(K\sigma ^{(i)}\sqrt{{\hat{T}}-t} )^2}+\frac{S_t^{(i)}\phi (d_1)}{(K)^2\sigma ^{(i)}\sqrt{{\hat{T}}-t}}} \\ {}= & {} \frac{-S_t^{(i)}\sigma ^{(i)}\sqrt{{\hat{T}}-t}-S_t^{(i)}d_1+S_t^{(i)}\sigma ^{(i)}\sqrt{{\hat{T}}-t}}{-S_t^{(i)}d_1+S_t^{(i)}\sigma ^{(i)}\sqrt{{\hat{T}}-t}}\longrightarrow 1. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{\partial O^{(i,Euro\ Call)}_t}{\partial S_t^{(i)}}\frac{1}{O^{(i,Euro\ Call)}_t} \sigma ^{(i)}S_t^{(i)}=\frac{ \sigma ^{(i)}}{1-H(x)}\longrightarrow \infty . \end{aligned}$$

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

$$\begin{aligned} \left| \pi ^{(i,Euro\ Call)}_t\right| =\frac{ \left| \eta _t^{(i)}\right| }{ \left| \frac{\partial O_t^{(i,Euro\ Call)}}{\partial S_t^{(i)}}\frac{1}{O_t^{(i,Euro\ Call)}} \sigma ^{(i)}S_t^{(i)}\right| }\longrightarrow 0 \quad \textit{ as } K\longrightarrow \infty . \end{aligned}$$

(A15)

The proof for European put follows similarly.

1.6 A.6 Proof of Proposition 6

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

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

$$\begin{aligned} O_t^{(i),*}=\underset{O_t^{(i,j)}}{\arg \max }\left| \frac{\partial O_t^{(i,j)}}{\partial S_t^{(i)}}\frac{1}{O_t^{(i,j)}}\right| . \end{aligned}$$

(A16)

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

$$\begin{aligned} h(K^{(i,j)},j)= \left| \frac{\partial O_t^{(i,j)}}{\partial S_t^{(i)}}\frac{1}{O_t^{(i,j)}}\right| \end{aligned}$$

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

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

(A17)

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 $h(K^{(l,i,j)},j)\ge h(0,j)$. $K^{(l,i,j)}$ 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 $h(K^{(r,i,j)},j)\ge h(B^{(i,j)},j)$. $h(K^{(r,i,j)},j)$ is the maximum point on the right branch $[A^{(i,j)},B^{(i,j)}]$. $O_t^{(i),*}$ 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.

1.7 A.7 Proof of Proposition 7

Let ${\varvec{O_t}}\in \Omega _O^{(2,multi\ asset)}$ with non-singular variance matrix ${\varvec{\Sigma _t}}$ (see (23)), the optimal strategy space $\Omega _{\pi }^{O}$ contains a unique strategy, i.e. ${\varvec{\pi _t}}=({\varvec{\Sigma _t}}^T)^{-1}{\varvec{\eta _t^{*}}}$. The allocation and its $\ell _1$ norm can be written as

$$\begin{aligned} \begin{aligned} \pi ^{(1,j)}&=\frac{1}{ f^{(11)}_t}(\eta _t^{(1)}-\frac{f^{(21)}_t}{f^{(22)}_t}\eta _t^{(2)})\quad \quad \pi ^{(2,j)}=\frac{\eta _t^{(2)}}{f^{(22)}_t} \\ \left\| {\varvec{\pi _t}}\right\| _1&=\frac{1}{\left| f^{(11)}_t\right| }\left| \eta _t^{(1)}-\frac{f^{(21)}_t}{f^{(22)}_t}\eta _t^{(2)}\right| +\frac{\left| \eta _t^{(2)}\right| }{\left| f^{(22)}_t\right| }. \end{aligned} \end{aligned}$$

(A18)

If ${\varvec{O_t^{*}}}=[O_t^{{(1),*}},O_t^{{(2),*}}]^T\in \Omega _O^{(2,multi\ asset)}$ achieves minimum $\ell _1$ norm of allocation and

$$\begin{aligned} O_t^{(1),*}\ne \underset{O_t^{(1,j_1)}}{\arg \max }\left| \frac{\partial O_t^{(1,j_1)}}{\partial S_t^{(1)}}\frac{1}{O_t^{(1,j_1)}}\right| . \end{aligned}$$

(A19)

Then there is a ${\varvec{O_t^{**}}}=[O_t^{{(1),**}},O_t^{{(2),**}}]^T$, such that

$$\begin{aligned} \left| \frac{\partial O_t^{(1),*}}{\partial S_t^{(1)}}\frac{1}{O_t^{(1),*}}\right| <\left| \frac{\partial O_t^{(1),**}}{\partial S_t^{(1)}}\frac{1}{O_t^{(1),**}}\right| . \end{aligned}$$

(A20)

Therefore, let $O_t^{{(2),**}}=O_t^{{(2)},*}$, this implies $\left| f^{(11),*}_t\right| <\left| f^{(11),**}_t\right|$, $f^{(21),*}_t=f^{(21),**}_t$ and $f^{(22),*}_t=f^{(22),**}_t$. Equation (A18) indicates $\left\| {\varvec{\pi _t}}^{*}\right\| _1>\left\| {\varvec{\pi _t}}^{**}\right\| _1$, which proves by contradiction that

$$\begin{aligned} O_t^{(1),*}=\underset{O_t^{(1,j_1)}}{\arg \max }\left| \frac{\partial O_t^{(1,j_1)}}{\partial S_t^{(1)}}\frac{1}{O_t^{(1,j_1)}}\right| . \end{aligned}$$

(A21)

is a necessary condition for ${\varvec{O_t^{*}}}$ to be the optimal portfolio composition for problem (10) within $\Omega _O^{(2,multi\ asset)}$.

1.8 A.8 Proof of Proposition 8

Recall $f_t^{(ik)}$ in the variance matrix $\Sigma _t$ can be written as

$$\begin{aligned} f^{(ik)} = \frac{\partial O_t^{(i,j_i)}}{\partial S_t^{(k)}}\frac{S_t^{(k)}\sigma ^{(k)}}{O_t^{(i,j_i)}}. \end{aligned}$$

(A22)

According to the non-singular variance matrix assumption, $f_t^{(11)},f_t^{(22)}\ne 0$. 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 $O_t^{(2,j_2)}\in {\mathbb {C}}^1$.

$\left\| {\varvec{\pi _t}}\right\| _1$ (see (A18)) 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 $j_2$. ${\varvec{O_t^{*}}}$ 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 $j_2\in \{\textit{Basket Call, Basket Put}\}$.

1.9 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 Sect. 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 Fig. 7. Unlike the case of two positive correlated underlying assets (see Fig. 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.

Fig. 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 Fig. 8. Compared with Fig. 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.

Fig. 8

Derivatives selection $O_t^{(2,j_2)}$ (exchange assets’ parameter)