Utility Indifference Pricing with High Risk Aversion and Small Linear Price Impact

In financial markets, trading moves prices against the trader: buying faster increases execution prices, and selling faster decreases them. This aspect of liquidity, known as market depth [4] or price-impact has recently received increasing attention (see, for instance, [2, 5, 7, 9, 10, 12, 13, 14, 15, 16, 17, 18] and the references therein).

In [14] the authors showed that for a reasonable market model, in the presence of price impact, super–replication is prohibitively costly. Namely, in the presence of price impact, even in market models such as the Bachelier model or the Black–Scholes model (which are complete in the frictionless setup) there is no practical way to construct a hedging strategy which eliminates all risk from a financial position. This brings us to utility indifference pricing.

In this paper we study utility indifference pricing for vanilla European options in the multi–dimensional Bachelier model with linear price impact. Our main result is computing the asymptotic behavior of the exponential utility indifference prices where the risk aversion goes to infinity at a rate which is inversely proportional to the linear price impact which goes to zero. In addition we provide a family of asymptotically optimal hedging strategies.

This type of scaling limits goes back to the seminal work of Barles and Soner [8] which determines the scaling limit of utility indifference prices of vanilla options for small proportional transaction costs and high risk aversion. The present note provides an analogous analysis for the case of linear price impact which results in quadratic transaction costs, albeit using probabilistic techniques rather than taking a PDE approach as pursued in [8].

We divide the proof of our main result, namely Theorem 2.1 into two main steps: the proof of the lower bound and the proof of the upper bound.

The proof of the lower bound goes through a dual representation of the certainty equivalent. In the dual problem, there is only one player: a maximizing (adverse) player that controls the probability measure. A key ingredient in the proof of the lower bound is a construction of a family of probability measures which attain (in the asymptotic sense) the desired limit. This is done in Proposition 3.3.

The proof of the upper bound does not use duality and is based on a direct argument (Proposition 4.1). More precisely, we construct a family of trading strategies for which the expected utility converges to the scaling limit. Roughly speaking, these strategies are given by a reversion towards the $\Delta$–hedging strategy which corresponds to a modified European option and a modified stock price.

The rest of the paper is organized as follows. In the next section we introduce the setup and formulate the main results. In Section 3 we discuss the dual representation and prove the lower bound. In Section 4 we prove the upper bound.

Let $T<\infty$ be the time horizon and let $W=\left(W^{1}_{t},...,W^{d}_{t}\right)_{t\in[0,T]}$ be a standard $d$-dimensional Brownian motion defined on the filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\in[0,T]},\mathbb{P})$ where $(\mathcal{F}_{t})_{t\in[0,T]}$ is the (augmented) filtration generated by $W$. We consider a simple financial market with a riskless savings account bearing zero interest (for simplicity) and with $d$-risky asset $S=\left(S^{1}_{t},...,S^{d}_{t}\right)_{t\in[0,T]}$ with Bachelier price dynamics

where $s_{0}=(s^{1}_{0},...,s^{d}_{0})\in\mathbb{R}^{d}$ is the initial position of the risky assets, $\mu=(\mu^{1},...\mu^{d})\in\mathbb{R}^{d}$ is a constant vector (drift) and $\sigma=\{\sigma^{ij}\}_{1\leq i,j\leq d}\in M_{d}(\mathbb{R})$ is a constant nonsingular matrix (volatility). Without loss of generality we assume that the constant nonsingular matrix $\sigma$ is a positive definite matrix.

Following [1], we model the investor’s market impact, in a temporary linear form and, thus, when at time $t$ the investor turns over her position at the $i$–asset $\Phi^{i}_{t}$ at the rate $\phi^{i}_{t}:=\dot{\Phi}^{i}_{t}$ the execution price is $S_{t}+\frac{\Lambda}{2}\phi^{i}_{t}$ for some constant $\Lambda>0$. As a result, the profits and losses from a trading strategy $\phi=\left(\phi^{1}_{t},...,\phi^{d}_{t}\right)_{t\in[0,T]}$ with the initial position $\Phi_{0}=\left(\Phi^{1}_{0},...,\Phi^{d}_{0}\right)$ are given by

where, for convenience, we assume that the investor marks to market her position $\Phi_{t}=\Phi_{0}+\int_{0}^{t}\phi_{v}dv$ in the risky asset that she has acquired by time $t$. As usual, $\langle\cdot,\cdot\rangle$ and $||\cdot||$, denotes the standard scalar product and the Euclidean norm, respectively. In our setup, the natural class of admissible strategies is

Next, consider a vanilla European option with the payoff $f\left(S_{T}\right)$ where $f:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a Lipschitz continuous function. The investor will assess the quality of a hedge by the resulting expected utility. Assuming exponential utility with constant absolute risk aversion $\alpha>0$, the utility indifference price and the certainty equivalent price of one unit of the claim $f\left(S_{T}\right)$ (see, e.g., [11] for details on indifference prices) do not depend on the investor’s initial wealth and, respectively, take the well-known forms

and

If the risk aversion $\alpha>0$ is fixed, then by applying standard density arguments we obtain that for $\Lambda\downarrow 0$, the above indifference price converges to the unique price of the continuous time complete (frictionless) market given by (1). A more interesting limit emerges, however, if we re-scale the investor’s risk-aversion in the form $\alpha:=A/\Lambda$. Before we formulate the limit theorem we need some preparations.

For a given $A>0$ introduce the functions

and

where the vectors $x,y,W$ are considered as row vectors and for any row vector $z\in\mathbb{R}^{d}$, $z\sigma^{-1},z\sigma\in\mathbb{R}^{d}$ are the standard matrix products. The term $u^{A}(t,S_{t})$ represents the price at time $t$ of a European option with the payoff $g^{A}(S_{T})$ in the complete market given by (1). It is well known that $u\in C^{1,2}([0,T)\times\mathbb{R}^{d})$ solves the PDE

where $tr(\cdot)$ is the trace of the square matrix $\cdot$ and $D^{2}_{x}u^{A}$ is the Hessian matrix with respect to $x=(x^{1},...,x^{d})$ which is given by $[D^{2}_{x}u^{A}]_{ij}:=\frac{\partial^{2}u^{A}}{\partial x^{i}\partial x^{j}}$, $1\leq i,j\leq d$.

For a given $A,\Lambda>0$ consider the $d$–dimensional (random) ODE ($\Phi$ is a row vector)

where $D_{x}u^{A}:=\left(\frac{\partial u^{A}}{\partial x^{1}},...,\frac{\partial u^{A}}{\partial x^{d}}\right)\in\mathbb{R}^{d}$ is the gradient with respect to $x$. From the linear growth of $f$ it follows that for any $\epsilon>0$ the function $D_{x}u^{A},D^{2}_{x}u^{A}$ are uniformly bounded in the domain $[0,T-\epsilon]\times\mathbb{R}^{d}$. In particular $D_{x}u^{A}$ is Lipschitz continuous with respect to $x$ in the domain $[0,T-\epsilon]\times\mathbb{R}^{d}$. Hence, from the standard theory of ODE (see Chapter II, Section 6 in [19]) we obtain that for a given initial value $\Phi_{0}$ there exists a unique solution to (7) which we denote by $(\Phi^{A,\Lambda}_{t})_{0\leq t<T}$. Next, the Lipschitz continuity of $f$ implies that $g^{A}$ is a Lipschitz continuous function (with the same constant as $f$), and so $D_{x}u^{A}$ is uniformly bounded in $[0,T)\times\mathbb{R}^{d}$. This together with the mean reverting structure of the ODE (7) yields that $\lim_{t\rightarrow T-}\Phi^{A,\Lambda}_{t}$ exists and finite a.s. Thus, we can extend $\Phi^{A,\Lambda}$ to the interval $[0,T]$ by $\Phi^{A,\Lambda}_{t}:=\lim_{t\rightarrow T-}\Phi^{A,\Lambda}_{t}$ and we define $\phi^{A,\Lambda}\in\mathcal{A}$ by $\phi^{A,\Lambda}_{t}:=\dot{\Phi}^{A,\Lambda}_{t}$ for $t<T$ and $\phi^{A,\Lambda}_{T}=0$. Obviously,

We arrive at the main result of the paper which provides an explicit formula for the asymptotic behavior of the certainty equivalent and an optimal family (it should not be unique) of hedging strategies in the asymptotic sense.

From Theorem 2.1 we obtain immediately the following corollary which says that the asymptotic value of the utility indifference prices is equal to the price of the vanilla European option with the payoff $g^{A}(S_{T})$ and the shifted initial stock price $s_{0}-\sqrt{A}\Phi_{0}\sigma$.

The following remark provides a possible application of Corollary 2.2.

We end this section with the following example.

In this section we establish the inequality $\geq$ in (8).

We start with the following lemma.

Next, we fix $A>0$ and prove the following key result.