On Minimization of Shortfall Risk in Tradable and Non-Tradable Assets ^†

Abstract

This paper studies shortfall risk minimization in an incomplete financial market driven by mixed Brownian–fractional Brownian motion. The contingent claim depends on a non-tradable asset, while trading is allowed only in a correlated tradable asset, making perfect hedging impossible. We consider a continuous-time mixed model with Hurst parameter $H>3/4$, for which the mixed process is a semimartingale and classical stochastic-calculus techniques can be applied. Two informational settings are investigated. First, the non-tradable asset is continuously observable, and the optimal trading strategy is derived using a suitable change of probability measure. Second, the non-tradable asset remains unobservable until the terminal time. In this case, the optimization problem is reformulated by conditioning on the hidden factor, yielding an optimal strategy expressed through conditional expectations. The results illustrate how the availability of information about the non-tradable asset affects optimal hedging.

1. Introduction

Financial markets are often incomplete because some sources of uncertainty cannot be traded directly. In particular, when a contingent claim depends on a non-tradable asset while trading is allowed only in a correlated tradable asset, perfect replication is generally impossible and residual hedging risk remains. In such settings, the problem becomes one of constructing an optimal trading strategy for managing the unhedgeable component of the claim.

Classical diffusion models driven by Brownian motion provide a natural framework for continuous-time financial modeling but do not capture long-range dependence and persistence observed in financial time series. Fractional Brownian motion provides a framework for incorporating such memory effects into asset-price dynamics [1,2,3]. However, for $H\ne 1/2$, fractional Brownian motion is not a semimartingale, which limits the direct application of classical stochastic calculus [2,4]. Moreover, pure fractional Brownian motion models may lead to arbitrage opportunities under continuous trading.

To overcome these difficulties, we consider the mixed Brownian–fractional Brownian motion introduced by Cheridito [5]. The mixed process combines a Brownian motion with a fractional Brownian motion and, when $H>3/4$, is a semimartingale with respect to its natural filtration [5]. This allows the use of classical stochastic-calculus and change-of-measure techniques while retaining the long-memory component of the model. Mixed fractional Brownian motion and related models have also been considered in recent financial applications; see, for example, [6,7].

Motivated by this framework, we study shortfall risk minimization in an incomplete market with two correlated assets driven by mixed Brownian–fractional Brownian motion. The contingent claim depends on the non-tradable asset, while trading is allowed only in the tradable asset. The investor seeks an optimal self-financing strategy maximizing the expected utility of terminal wealth after the claim is paid.

The problem of shortfall risk minimization was studied before in different aspects and for different models (see, for example, [8,9,10,11,12]).

We investigate two informational settings. First, the non-tradable asset is continuously observable throughout the trading horizon. Second, it remains unobservable until the terminal time, so that trading decisions are based only on the information generated by the tradable asset. The latter setting leads to an optimization problem under partial information, with the optimal strategy characterized through conditional expectations.

The remainder of the paper is organized as follows. Section 2 considers the case where the non-tradable asset is continuously observable and derives the corresponding optimal trading strategy. Section 3 treats the case where the non-tradable asset remains unobservable until the terminal time and characterizes the corresponding optimal strategy under partial information.

2. When the Untradable Asset Is Observable

We consider an incomplete financial market with two risky assets. The first asset is non-tradable and determines the payoff of a contingent claim, while trading is allowed only in the second asset. We assume throughout that the interest rate is zero. Let

$$(\mathsf{\Omega },\mathcal{F},{\left\{{\mathcal{F}}_t\right\}}_{0\le t\le T},\mathbb{P})$$

be a filtered probability space satisfying the usual conditions.

Let $W_1$ and $W_2$ be two correlated Brownian motions satisfying

$$d{W}_1\left(t\right)d{W}_2\left(t\right)=\rho dt,\left|\rho \right|<1,$$

and let $B_1^H$ and $B_2^H$ be independent fractional Brownian motions with Hurst parameter $H>3/4$, independent of the Brownian motions. Define the mixed Brownian–fractional Brownian processes

$$X_i\left(t\right)={\sigma }_i{W}_i\left(t\right)+{\nu }_i{B}_i^H\left(t\right),i=1,2.$$

Following the results of Cheridito [5], $X_i$ is a continuous semimartingale for $H>3/4$. Hence, it admits the decomposition

$$X_i\left(t\right)=M_i\left(t\right)+A_i\left(t\right),$$

where $M_i$ is a continuous local martingale and $A_i$ is a predictable finite-variation process.

For $H>3/4$, following the results of Cheridito [5], the mixed fractional Brownian motion is equivalent to a Brownian motion. In its semimartingale decomposition, the finite-variation component is absolutely continuous, while the martingale component is represented by ${\sigma }_i{W}_i$. Hence,

$$d{X}_i\left(t\right)={\sigma }_{i}d{W}_i\left(t\right)+{\dot{A}}_i\left(t\right)dt,i=1,2.$$

The prices of the two risky assets under $\mathbb{P}$ are given by

$${\displaystyle \frac{d{S}_1\left(t\right)}{S_1\left(t\right)}}={\mu }_{1}dt+d{X}_1\left(t\right),{\displaystyle \frac{d{S}_2\left(t\right)}{S_2\left(t\right)}}={\mu }_{2}dt+d{X}_2\left(t\right).$$

Using the semimartingale decomposition, these dynamics can be written in Itô form as

$${\displaystyle \frac{d{S}_1\left(t\right)}{S_1\left(t\right)}}=b_1\left(t\right)dt+{\sigma }_{1}d{W}_1\left(t\right),{\displaystyle \frac{d{S}_2\left(t\right)}{S_2\left(t\right)}}=b_2\left(t\right)dt+{\sigma }_{2}d{W}_2\left(t\right),$$

where

$$b_i\left(t\right)={\mu }_i+{\dot{A}}_i\left(t\right),i=1,2.$$

We represent the correlated Brownian motion $W_2$ as

$$d{W}_2\left(t\right)=\rho d{W}_1\left(t\right)+\sqrt{1-{\rho }^2}d{W}_3\left(t\right),$$

where $W_3$ is a Brownian motion independent of $W_1$.

Throughout this section, both asset prices are observable, although trading is permitted only in $S_2$. Let $\xi \left(t\right)$ denote the number of shares held in the tradable asset. The self-financing wealth process satisfies

$$d{X}_t=\xi \left(t\right)d{S}_2\left(t\right),$$

and the contingent claim is

$$f\left(S_1\left(T\right)\right),$$

where $f\in C^2$. We formulate shortfall risk minimization through utility maximization, since the objective is to maximize the utility of terminal wealth net of the contingent claim, consistent with the utility-based approach in Gundel and Weber [13] and Hegde et al. [14]. The investor chooses an admissible strategy to maximize

$$\underset{\xi }{\sup }{\mathbb{E}}^{\mathbb{P}}\left[U\left(X_T-f\left(S_1\left(T\right)\right)\right)\right],$$

where $U\in C^2$ is strictly increasing and strictly concave; see, e.g., ref. [15] for stochastic optimization in continuous-time financial models.

Define the market prices of risk by

$${\theta }_i\left(t\right)={\displaystyle \frac{b_i\left(t\right)}{{\sigma }_i}},i=1,2.$$

Suppose that the progressively measurable processes ${\lambda }_1$ and ${\lambda }_3$ satisfy Novikov’s condition [16], namely,

$${\mathbb{E}}^{\mathbb{P}}\left[\exp \left({\displaystyle \frac{1}{2}}{\int }_0^T\left({\lambda }_1^2\left(t\right)+{\lambda }_3^2\left(t\right)\right)dt\right)\right]<\infty .$$

Then the stochastic exponential

$$Z_t=\exp \left(-{\int }_0^t{\lambda }_1\left(s\right)d{W}_1\left(s\right)-{\int }_0^t{\lambda }_3\left(s\right)d{W}_3\left(s\right)-{\displaystyle \frac{1}{2}}{\int }_0^t\left({\lambda }_1^2\left(s\right)+{\lambda }_3^2\left(s\right)\right)ds\right)$$

is a martingale and defines an equivalent probability measure $\mathbb{Q}$ by

$${\left.{\displaystyle \frac{d\mathbb{Q}}{d\mathbb{P}}}\right|}_{{\mathcal{F}}_t}=Z_t.$$

By Girsanov’s theorem [2,16], under $\mathbb{Q}$,

$${\tilde{W}}_1\left(t\right)=W_1\left(t\right)+{\int }_0^t{\lambda }_1\left(s\right)ds,{\tilde{W}}_3\left(t\right)=W_3\left(t\right)+{\int }_0^t{\lambda }_3\left(s\right)ds$$

are independent Brownian motions. Choosing

$${\lambda }_1\left(t\right)={\theta }_1\left(t\right),{\lambda }_3\left(t\right)={\displaystyle \frac{{\theta }_2\left(t\right)-\rho {\theta }_1\left(t\right)}{\sqrt{1-{\rho }^2}}},$$

eliminates the Brownian drift of the tradable asset.

The optimal strategy is characterized by the following result.

Theorem 1. 

Assume that $U,f\in C^2$, that U is strictly increasing and strictly concave, and that the Novikov condition holds. Then the optimal self-financing trading strategy maximizing

$${\mathbb{E}}^{\mathbb{P}}\left[U\left(X_T-f\left(S_1\left(T\right)\right)\right)\right]$$

is

$${\xi }^{*}\left(t\right)=-{\displaystyle \frac{U^{\prime }\left(Y_t\right){\theta }_2\left(t\right)-\rho {\sigma }_1{S}_1\left(t\right)f^{\prime }\left(S_1\left(t\right)\right)U^{″}\left(Y_t\right)}{U^{″}\left(Y_t\right){\sigma }_2{S}_2\left(t\right)}},Y_t=X_t-f\left(S_1\left(t\right)\right).$$

(1)

Proof. 

Under the measure $\mathbb{Q}$ defined above, the Brownian drift of the tradable asset is removed. Let

$$Y_t=X_t-f\left(S_1\left(t\right)\right).$$

Applying Itô’s formula to $f\left(S_1\left(t\right)\right)$ and using the dynamics of $S_1$ and $S_2$ under $\mathbb{Q}$, we obtain

$$\begin{array}{cc}\hfill d{Y}_t=& \left(\xi \left(t\right){\sigma }_2{S}_2\left(t\right)\rho -f^{\prime }\left(S_1\left(t\right)\right){\sigma }_1{S}_1\left(t\right)\right)d{\tilde{W}}_1\left(t\right)\hfill \\ & +\xi \left(t\right){\sigma }_2{S}_2\left(t\right)\sqrt{1-{\rho }^2}d{\tilde{W}}_3\left(t\right)\hfill \\ & -\left(f^{\prime }\left(S_1\left(t\right)\right){\sigma }_1{S}_1\left(t\right)({\theta }_1\left(t\right)-{\lambda }_1\left(t\right))+{\displaystyle \frac{1}{2}}{f}^{″}\left(S_1\left(t\right)\right){\sigma }_1^2{S}_1^2\left(t\right)\right)dt.\hfill \end{array}$$

Consequently,

$$\begin{array}{cc}\hfill d{⟨Y⟩}_t=& [{\xi }^2\left(t\right){\sigma }_2^2{S}_2^2\left(t\right)-2\xi \left(t\right){\sigma }_2{S}_2\left(t\right)\rho f^{\prime }\left(S_1\left(t\right)\right){\sigma }_1{S}_1\left(t\right)\hfill \\ & +{\left(f^{\prime }\left(S_1\left(t\right)\right)\right)}^2{\sigma }_1^2{S}_1^2\left(t\right)]dt.\hfill \end{array}$$

Applying Itô’s formula to $U\left(Y_t\right)$ and then the product rule to $U\left(Y_t\right)/Z_t$, and using the drift-removal condition

$${\theta }_2\left(t\right)=\rho {\lambda }_1\left(t\right)+\sqrt{1-{\rho }^2}{\lambda }_3\left(t\right),$$

the drift term can be written as

$$\begin{array}{cc}\hfill \Gamma (t,\xi )=& U^{\prime }\left(Y_t\right)\left[\xi {\sigma }_2{S}_2\left(t\right){\theta }_2\left(t\right)-f^{\prime }\left(S_1\left(t\right)\right){\sigma }_1{S}_1\left(t\right){\theta }_1\left(t\right)-{\displaystyle \frac{1}{2}}{f}^{″}\left(S_1\left(t\right)\right){\sigma }_1^2{S}_1^2\left(t\right)\right]\hfill \\ & +{\displaystyle \frac{1}{2}}{U}^{″}\left(Y_t\right)[{\xi }^2{\sigma }_2^2{S}_2^2\left(t\right)-2\xi {\sigma }_2{S}_2\left(t\right)\rho f^{\prime }\left(S_1\left(t\right)\right){\sigma }_1{S}_1\left(t\right)\hfill \\ & +{\left(f^{\prime }\left(S_1\left(t\right)\right)\right)}^2{\sigma }_1^2{S}_1^2\left(t\right)].\hfill \end{array}$$

For fixed t, $\Gamma (t,\xi )$ is a quadratic function of $\xi$. Since U is strictly concave, $U^{″}\left(Y_t\right)<0$, and therefore $\Gamma (t,\xi )$ is strictly concave in $\xi$. Hence its unique maximizer is characterized by the first-order condition

$${\displaystyle \frac{\mathsf{\partial }\Gamma (t,\xi )}{\mathsf{\partial }\xi }}=0,$$

which gives

$$U^{\prime }\left(Y_t\right){\sigma }_2{S}_2\left(t\right){\theta }_2\left(t\right)+U^{″}\left(Y_t\right)\left[\xi {\sigma }_2^2{S}_2^2\left(t\right)-\rho {\sigma }_1{\sigma }_2{S}_1\left(t\right)S_2\left(t\right)f^{\prime }\left(S_1\left(t\right)\right)\right]=0.$$

Solving this equation yields

$${\xi }^{*}\left(t\right)=-{\displaystyle \frac{U^{\prime }\left(Y_t\right){\theta }_2\left(t\right)-\rho {\sigma }_1{S}_1\left(t\right)f^{\prime }\left(S_1\left(t\right)\right)U^{″}\left(Y_t\right)}{U^{″}\left(Y_t\right){\sigma }_2{S}_2\left(t\right)}},$$

which is the strategy stated in the theorem. Since both $S_1\left(t\right)$ and $S_2\left(t\right)$ are observable in this setting, ${\xi }^{*}\left(t\right)$ is progressively measurable with respect to the available information. The strict concavity of $\Gamma (t,\xi )$ ensures that this admissible candidate is the unique pointwise maximizer of the drift, and hence the unique optimal self-financing strategy. □

The optimal strategy in (1) can be decomposed into two components:

$${\xi }^{*}\left(t\right)=-{\displaystyle \frac{U^{\prime }\left(Y_t\right)}{U^{″}\left(Y_t\right)}}{\displaystyle \frac{{\theta }_2\left(t\right)}{{\sigma }_2{S}_2\left(t\right)}}+{\displaystyle \frac{\rho {\sigma }_1{S}_1\left(t\right)}{{\sigma }_2{S}_2\left(t\right)}}{f}^{\prime }\left(S_1\left(t\right)\right).$$

The first term represents the investment demand associated with the market price of risk, while the second term is the hedging demand generated by the dependence of the claim on the non-tradable asset. In particular, the hedging correction vanishes when $\rho =0$ or when the claim is independent of $S_1$.

3. When the Untradable Asset Is Unobservable

We now consider the case in which the non-tradable asset $S_1$ is not observable before the terminal time T. The investor observes only the tradable asset $S_2$, so that the available information is

$${\mathcal{G}}_t=\sigma \{S_2\left(s\right):0\le s\le t\},t<T.$$

The contingent claim, however, still depends on the unobservable terminal value $S_1\left(T\right)$. Thus, the investor must construct the hedging strategy using only the information contained in the tradable asset.

Using the correlation structure, the Brownian motion driving the non-tradable asset can be decomposed as

$$W_1\left(t\right)=\rho W_2\left(t\right)+\sqrt{1-{\rho }^2}{W}_3\left(t\right),$$

where $W_3$ is independent of $W_2$. Consequently, the terminal value of the non-tradable asset can be represented in the form

$$S_1\left(T\right)=S_2^{\kappa }\left(T\right)g(T,W_3\left(T\right)),\kappa ={\displaystyle \frac{\rho {\sigma }_1}{{\sigma }_2}},$$

for a suitable function g. Hence, the unobservable component of the claim is represented through the independent hidden factor $W_3\left(T\right)$.

For a fixed realization w of the hidden factor, define

$$Y_t\left(w\right)=X_t-f\left(S_2^{\kappa }\left(t\right)g(t,w)\right).$$

Since $W_3\left(T\right)$ is not available to the investor during the trading period, the optimization problem must be formulated using conditional expectations with respect to the available information ${\mathcal{G}}_t$. Since the hidden factor is independent of ${\mathcal{G}}_t$, these conditional expectations are obtained by averaging over the distribution of $W_3\left(T\right)$. Let

$${\phi }_T\left(w\right)={\displaystyle \frac{1}{\sqrt{2\pi T}}}\exp \left(-{\displaystyle \frac{w^2}{2T}}\right).$$

For any integrable function $h(t,w)$, define

$${\widehat{h}}_t:=\mathbb{E}\left[h(t,W_3\left(T\right))\mid {\mathcal{G}}_t\right]={\int }_{-\infty }^{\infty }h(t,w){\phi }_T\left(w\right)dw.$$

Thus, although the quantities $Y_t\left(w\right)$ and the corresponding first-order condition are defined for a fixed realization of the hidden factor, the actual trading strategy is obtained only after taking this conditional expectation. Consequently, the resulting strategy is ${\mathcal{G}}_t$-measurable and does not depend on the unobserved realization of $W_3\left(T\right)$.

Taking the conditional average of the first-order condition with respect to ${\mathcal{G}}_t$ yields

$${\xi }^{*}\left(t\right)={\displaystyle \frac{\widehat{U^{\prime }\left(Y_t\right)}{\theta }_2\left(t\right)-{\sigma }_1\rho S_2^{\kappa }\left(t\right)\widehat{U^{″}\left(Y_t\right)f^{\prime }\left(S_2^{\kappa }\left(t\right)g(t,W_3\left(T\right))\right)g(t,W_3\left(T\right))}}{-{\sigma }_2{S}_2\left(t\right)\widehat{U^{″}\left(Y_t\right)}}}.$$

(2)

Here, the notation $\widehat{U^{\prime }\left(Y_t\right)}$ and $\widehat{U^{″}\left(Y_t\right)}$ denotes conditional averaging over the hidden factor, namely

$$\widehat{U^{\prime }\left(Y_t\right)}={\int }_{-\infty }^{\infty }{U}^{\prime }\left(Y_t\left(w\right)\right){\phi }_T\left(w\right)dw,$$

and

$$\widehat{U^{″}\left(Y_t\right)}={\int }_{-\infty }^{\infty }{U}^{″}\left(Y_t\left(w\right)\right){\phi }_T\left(w\right)dw.$$

Similarly,

$$\begin{array}{cc}& \widehat{U^{″}\left(Y_t\right)f^{\prime }\left(S_2^{\kappa }\left(t\right)g(t,W_3\left(T\right))\right)g(t,W_3\left(T\right))}\hfill \\ & ={\int }_{-\infty }^{\infty }{U}^{″}\left(Y_t\left(w\right)\right)f^{\prime }\left(S_2^{\kappa }\left(t\right)g(t,w)\right)g(t,w){\phi }_T\left(w\right)dw.\hfill \end{array}$$

Thus, the main difference from the observable case is that the hedging component cannot depend directly on the unobservable asset $S_1\left(t\right)$ or on the unobserved realization of $W_3\left(T\right)$. Instead, its effect is incorporated through conditional expectations with respect to the investor’s available information. The resulting strategy is therefore ${\mathcal{G}}_t$-measurable and consists of the same market-price-of-risk component as in the observable case, together with a hedging correction averaged over the hidden source of uncertainty.

4. Numerical Illustration

To illustrate the theoretical results, we perform Monte Carlo simulations under the observable and unobservable informational settings. We use $T=1$, $H=0.80$, $\rho =0.50$, $S_1\left(0\right)=S_2\left(0\right)=100$, ${\mu }_1=0.05$, ${\mu }_2=0.06$, ${\sigma }_1=0.20$, ${\sigma }_2=0.25$, ${\nu }_1={\nu }_2=0.05$, $K=100$, $X\left(0\right)=30$, and $\gamma =0.1$. The claim is the European call $f\left(S_1\left(T\right)\right)={(S_1\left(T\right)-K)}^{+}$, and the utility function is $U\left(x\right)=-\exp (-\gamma x)$. The baseline experiment uses 3000 Monte Carlo paths with 252 time steps. The shortfall loss is defined by

$$L\left(T\right)={\left(f\left(S_1\left(T\right)\right)-X\left(T\right)\right)}^{+}.$$

Table 1 summarizes the terminal wealth, expected shortfall loss, probability of a positive shortfall, and expected utility for the considered strategies.

Table 1. Monte Carlo results under observable and unobservable information.

Information	Strategy	$\mathit{E}\mathbf{\left[}\mathit{X}\mathbf{\right(}\mathit{T}\mathbf{\left)}\mathbf{\right]}$	$\mathit{E}\mathbf{\left[}\mathit{L}\mathbf{\right(}\mathit{T}\mathbf{\left)}\mathbf{\right]}$	$\mathit{P}\mathbf{\left(}\mathit{L}\mathbf{\right(}\mathit{T}\mathbf{)}\mathbf{>}\mathbf{0}\mathbf{)}$	$\mathit{E}\mathbf{\left[}\mathit{U}\mathbf{\right(}\mathit{X}\mathbf{\left(}\mathit{T}\mathbf{\right)}\mathbf{\left)}\mathbf{\right]}$
Observable	Optimal	32.032	1.112	9.33%	−0.960
	Investment only	30.524	1.758	12.03%	−3.566
	No trading	30.000	2.058	13.10%	−5.224
Unobservable	Optimal	33.279	1.402	10.70%	−1.353
	Investment only	30.571	2.198	14.43%	−10.181
	No trading	30.000	2.546	16.00%	−18.506

In the observable case, the optimal strategy reduces the expected shortfall loss by approximately $45.9\%$ relative to no trading and by $36.7\%$ relative to the investment-only strategy. Under unobservable information, the corresponding reductions are approximately $44.9\%$ relative to no trading and $36.2\%$ relative to the investment-only strategy.

We further examine the sensitivity of the numerical results to the Hurst parameter H and the correlation parameter $\rho$, keeping the remaining parameters fixed. For each sensitivity experiment, 1000 Monte Carlo paths are used. The results are reported in Table 2 and Table 3.

Table 2. Sensitivity of shortfall loss to the Hurst parameter H.

H	$\mathit{E}\mathbf{\left[}\mathit{L}\mathbf{\right(}\mathit{T}\mathbf{\left)}\mathbf{\right]}$	$\mathit{P}\mathbf{\left(}\mathit{L}\mathbf{\right(}\mathit{T}\mathbf{)}\mathbf{>}\mathbf{0}\mathbf{)}$
0.76	1.019	9.70%
0.80	1.054	8.90%
0.85	0.817	8.10%
0.90	1.005	9.00%

Table 3. Sensitivity of shortfall loss to the correlation parameter $\rho$.

$\mathit{\rho }$	$\mathit{E}\mathbf{\left[}\mathit{L}\mathbf{\right(}\mathit{T}\mathbf{\left)}\mathbf{\right]}$	$\mathit{P}\mathbf{\left(}\mathit{L}\mathbf{\right(}\mathit{T}\mathbf{)}\mathbf{>}\mathbf{0}\mathbf{)}$	$\mathit{E}\mathbf{\left[}\mathit{U}\mathbf{\right(}\mathit{X}\mathbf{\left(}\mathit{T}\mathbf{\right)}\mathbf{\left)}\mathbf{\right]}$
0.00	1.735	12.10%	−2.069
0.25	1.630	11.90%	−1.828
0.50	0.807	7.60%	−0.616
0.75	0.352	4.00%	−0.364
0.90	0.137	3.20%	−0.195

The sensitivity results illustrate the dependence of the estimated shortfall measures on the Hurst parameter and the correlation between the tradable and non-tradable assets. As the reported values are obtained by Monte Carlo simulation, they should be interpreted as numerical estimates.

5. Conclusions

This paper studies utility-based shortfall risk minimization in an incomplete financial market driven by mixed Brownian–fractional Brownian motion. We considered two informational settings. When the non-tradable asset is observable, the optimal self-financing strategy is obtained explicitly through a change of probability measure and consists of an investment component and a hedging component associated with the non-tradable asset. When the non-tradable asset is unobservable, the hedging strategy cannot depend directly on the hidden asset or its unobserved realization; instead, the effect of the hidden factor enters through conditional expectations with respect to the investor’s available information. Thus, the main theoretical contribution is to characterize optimal hedging strategies under both full and partial information in a mixed fractional Brownian motion framework. From a practical perspective, the results illustrate how the availability of information about a non-tradable risk factor affects the form of the optimal hedging strategy. A limitation of the present analysis is the restriction to the mixed fractional Brownian motion setting with $H>3/4$ and the considered two-asset market structure. Future research could extend the results to more general Hurst parameters and richer market and information structures.