Hedging goals

While modern portfolio theory (Markowitz 1952) posits that investors are risk averse and thus should seek to maximize their portfolios’ risk-adjusted returns, in reality, investors often find themselves in need of capital to finance future investment goals: a car, an apartment or their children’s college education. The importance of investment goals on a societal level can be appreciated in view of the exacerbating retirement problem in many Western countries, cf. Giron et al. (2018).

Goal-based investment strategies are not primarily concerned with risk preferences relating to portfolio volatility; instead, they are subject to the risk of falling short of reaching a goal by its maturity. Even exceeding an investment goal is not necessarily desirable; in this case, a strategy with less volatility could have led to an outcome matching the investment goal.

There are at least two ways to translate this practical problem into a mathematical optimization problem. Either, one attempts to maximize the probability of reaching an investment goal by a given maturity, or one tries to minimize the expected shortfall (or a function thereof).

This first approach was investigated in a series of papers by Browne (cf. Browne (1999b) and the references therein), who found the explicit portfolio allocation formula for the probability-maximizing strategy in the context of complete markets. In his articles, Browne used techniques from stochastic control theory as well as from Partial Differential Equations (PDEs). While highly appealing theoretically, the probability-maximizing paradigm suffers from the binary nature of its optimum: a goal missed by a hair’s breadth is still a goal missed, and any such strategy will be discarded. Rather, more and more leverage will be applied to attain the goal—even as the maturity draws closer—resulting in either success or bankruptcy. This indifference for the size of the shortfall constitutes a major drawback of probability-maximizing strategies for practical purposes.

Leukert (1999), Föllmer and Leukert (1999, 2000), and Föllmer and Schied (2016) treated the closely related problem of maximizing the probability of hedging contingent claims successfully when replication is attempted with less than the required initial capital (corresponding to the discounted value under the equivalent martingale measure). Their solution is based on a static optimization problem of Neyman–Pearson type. Another approach can be found in Spivak and Cvitanić (1999).

In practice, measuring and minimizing downward risk is arguably more significant than maximizing the probability of attaining a goal (in analogy with the dichotomy of Expected Shortfall versus Value-at-Risk, cf. Leukert 1999; Föllmer and Schied 2016). Downward risk can be quantified by the shortfall, i.e., the positive part of the distance between the profit a strategy has earned at maturity and the goal. Several authors have addressed this problem in the context of replicating contingent claims, cf. Leukert (1999), Föllmer and Leukert (1999, 2000), Pham (2002), Föllmer and Schied (2016), including Cvitanić and Karatzas (1999). The latter authors employ tools from convex duality to show that a solution exists and state explicit solutions for several special cases with a single risky asset. It is also interesting to note that quantile hedging (cf. Föllmer and Leukert 1999), i.e., the probability-maximizing paradigm, can be interpreted as the most risk-seeking limit of efficient hedging, cf. Föllmer and Leukert (2000). Nakano (2004) studied a similar problem, considering coherent risk measures instead of lower partial moments.

An intriguing and novel approach via optimal transport has recently been used to target prescribed terminal wealth distributions in Guo et al. (2021).

Bühler et al. (2019) introduced a flexible framework for hedging contingent claims by applying deep learning methods. This approach transcends the classical Black–Scholes model’s restrictions, e.g., the absence of transaction costs. Related reinforcement learning approaches can be found in Halperin (2020) and Szehr (2021). Ruf and Wang (2020) provide a comprehensive literature review regarding the application of neural networks for pricing and hedging purposes.

In our opinion, the potential that goal-based investing has for retirement saving and individual asset-liability management cannot be overestimated.

The theoretical foundations for the goal-based investment problem have been laid out in the—superficially unrelated—field of replicating contingent claims. Therefore, we regard adapting these results and making them accessible and palatable to practitioners as one of the main contributions of this paper. In particular, we show how risk preferences can be integrated into the original goal-based investment problem (cf., e.g., Proposition 7.1), drawing on results on efficient hedging derived by Föllmer and Leukert (2000).

Another important contribution is the adaptation of deep hedging techniques (cf. Bühler et al. 2019) to incorporate transaction costs into the optimization problems arising in goal-based investing.

The remainder of this article is organized as follows.

After introducing the basic model in Sect. 4, we state the optimal policy for risk-neutral and risk-taking goal-based investors in Sect. 5. The optimal policy for risk-averse goal-based investors, whose utility is determined by a lower partial moment of the shortfall relative to the goal, can be found in Sect. 7. We discuss the shortcomings of the probability-maximizing paradigm in Sect. 6, where we also provide an illustrative example. To mitigate the risk inherent in the quantile and efficient hedging approaches, we propose a policy allowing for downward protection in Sect. 8.

Finally, in Sect. 9, we show that an artificial neural network can be trained to minimize the expected shortfall as well as lower partial moments, thereby approximating the optimal policies from Sects. 5 and 7.

The proofs of this paper can be found in Section A of the Appendix.

The policy minimizing the expected shortfall for hedging a zero-coupon bond paying out \(H\equiv 1\) at maturity was derived in Xu (2004). In what follows, we extend her approach to incorporate a constant risk-free rate \(r>0\) and an arbitrary constant payoff \(H\in \mathbb {R}_+\) subject to \(z < H_{0, T}\). Moreover, we show that the result of (Xu 2004, Sect. 2.2.1) is, in fact, equivalent to the one of Browne (1999b) for \(H \equiv 1\). In particular, the hedging strategy in the case of a single risky asset is indeed independent of its drift, which is not immediately obvious from the formulae stated in Xu (2004).

Let us assume that the investment universe consists of a single risky company share \(X=(X_t)_{t\in [0, T]}\) and a risk-less bank account \(B=(B_t)_{t\in [0, T]}\), cf. (1), (2). A digital (or binary) European call option on the underlying X with strike \(K>0\) is a financial derivative with payoff \(\mathbbm {1}_{\{X_T\ge K\}}\) at maturity T. Its Black–Scholes price is given byAccording to Corollary 5.4 (cf. Browne 1999b, Sect. 4), continuous rebalancing withreplicates this digital payoff starting from \(V_0=C(0;\, X_0,K)\) monetary units. By inspection, the initial price \(V_0=V_0(K)\) is monotonously decreasing withLet us assume that a financial investor owns \(c_0>0\) monetary units at time \(t=0\) and, by means of an admissible strategy in the investment universe, aims at owning \(c_T > c_0\) monetary units at time T. For simplicity, let us exclude intermediate income and consumption. To ensure that the mathematical problem is well posed, one needs to establish in what sense a certain strategy becomes optimal. In Browne (1999b, Theorem 3.1), the author proved the intriguing fact that replicating \(c_T\) digital call options with strikemaximizes the objective probability of reaching the goal. This result has an insightful economic interpretation; \(K^*\) coincides with the break-even point with respect to the strike where a single digital call option costs \(\frac{c_0}{c_T}\) at time 0. Notably, but also well known, the magnitude of the hardly ascertainable drift \(\mu \) does not affect \(K^*\). In fact, the above expression of \(K^*\) is only well-defined provided that the argument of \(\Phi ^{-1}\) is within (0, 1). In our setting, this prerequisite is only violated in the degenerate case \(c_0 \ge R_{0, T}\,c_T\), i.e., the goal can be super-replicated in terms of the bank account at no risk anyway. The maximized real-world probability of reaching the goal isFor real-world applications, the financial investor has two alternatives; either she buys it over-the-counter or she replicates the digital payoff herself. In the former case, she runs the risk of not getting the promised payoff due to the bankruptcy of the issuer. In the latter case, without further stop-loss measures in place, discrete rebalancing schedules imply the risk of arbitrarily large losses way beyond \(c_0\) due to the discontinuity of the payoff and, hence, the unbounded delta of the digital option. Notably, the strategy also requires an unlimited credit line at the bank which is collateralized only to an insufficient extent by the company share. Transaction costs exacerbate the situation. By approximating the digital payoff by a classical bull call spread and by diversifying the involved derivatives across several bona fide counterparties, the financial investor manages to deal with the mentioned impediments all the same. From a computational perspective, we lose analytical tractability with increasing degrees of complexity, e.g., additional constraints, more realistic price dynamics, transaction costs, etc. Despite all, and much more crucially, the all-or-nothing feature of the proposed optimal strategy is not feasible in many real-world applications such as traditional pension funds. For obvious reasons, retirement savings are not supposed to be a Bernoulli experiment. Therefore, we will consider further ways to control downward risk in Sect. 8.

This example shows that the probability-maximizing paradigm might be too rigid in the context of goal-based investing as it does not take into consideration the investor’s risk appetite. In the next section, we will discuss optimal policies for risk-averse investors.

We consider the case of \(p>1\), so that \((\ell _p)_{p>1}\) (cf. (7)) denotes a series of convex loss functions corresponding to increasing levels of risk aversion as p grows. According to (Leukert 1999, Lemma 11) the optimal strategy to minimize (6) consists in hedging the modified claimwhere the constant \(a_p\) is implicitly determined by the capital requirement \(\mathbb {E}^*[\varphi _p\, H] = z\).

The probability of reaching the target for the probability-maximizing policy, given by (Browne 1999b, Theorem 3.1)is the counter-probability of going bankrupt, which can be prohibitively high for practical purposes.

Clearly, this all-or-nothing strategy is too risky for most practical applications. Browne (Browne 1999b, Sect. 8.2) therefore proposed to control downside risk in the context of active portfolio management (cf. also Browne 1999a). We adapt his approach to goal-based investing as follows.

Note that, as \(\varepsilon \rightarrow 1\), the initial endowment \(x_\varepsilon \) tends to the discounted goal \(H_{0, T}\) minus the shortfall allowance \(\delta \,H_{0, T}\).

The investment strategies derived in the previous sections cannot be transferred to more realistic settings without further ado. The optimality fundamentally relies on the completeness of the financial market model as well as the simplistic distributional assumption on the price dynamics. More sophisticated price dynamics, for instance involving rough volatility, inevitably lead to incomplete market models. Furthermore, minimizing lower partial moments in such intricate environments may hardly be analytically tractable. It remains unclear whether the duality principle between the optimization problem and the hedging of a qualitatively similar payoff prevails. In contrast, simply applying the proposed delta hedging strategies for different price dynamics can be arbitrarily bad. Another impediment for applications in the real world are discrete hedging schedules and transaction cost. Therefore, we investigate whether we manage to circumvent these delicate issues by applying the striking approach of deep hedging as proposed in Bühler et al. (2019). Subsequently, we present our findings for the one-dimensional case.

For any \(t\in \{0,1,2,\ldots ,N\}\) in some discrete time grid with horizon \(N\in \mathbb {N}\), we consider a feedforward neural networkwith some affine functionsand the sigmoid activation function \(\phi (x)=(1+e^{-x})^{-1}\). The input layer consists of the current holding \(\xi _{t\text {--}}\) before rehedging and the moneyness \(X_t/X_0\), where \(X_t\) is the marginal distribution of a geometric Brownian motion as considered above. The output layer reveals the outcome \(\xi _t\) of the rehedging at the time instance t, i.e.,Similarly as above, we aim at optimizing a function of the terminal wealth \(V_T\) that can be derived iteratively. Let \(b_{0\text {--}}\) denote the initial holdings in the bank account bearing the risk-free rate \(r\in \mathbb {R}\), \(\xi _{0\text {--}}\) denote the initial holdings in the underlying, \(\kappa \ge 0\) the coefficient for proportional transaction cost, and \(\tau >0\) the year fraction of a time step. Hence, the value of the portfolio before and after rehedging at time 0 is given bywhere \(b_0:=b_{0\text {--}}-\left( \xi _0-\xi _{0\text {--}}\right) X_0-\kappa \left| \xi _0-\xi _{0\text {--}}\right| X_0\) satisfies the self-financing principle. Then, we proceed consistently in terms of the iterationwhere \(b_{t\text {--}}=b_{t-1}e^{r\tau }\), \(\xi _{t\text {--}}=\xi _{t-1}\) and \(b_t=b_{t\text {--}}-\left( \xi _t-\xi _{t\text {--}}\right) X_t-\kappa \left| \xi _t-\xi _{t\text {--}}\right| X_t\) for \(t\in \{1,2,\ldots ,N-1\}\). At maturity, we have to bear the unwinding cost additionally. Hence,For experimental purposes, we chose similar parameters as in Fig. 2; a maturity \(T=10\), a discretization \(N=52T\) (i.e., weekly rehedging with \(\tau =1/52\)), a risk-free rate \(r=1\%\), a drift \(\mu =8\%\), and a volatility \(\sigma =30\%\). The initial state of the market and the wealth are standardized to \(X_0=100\), \(b_{0\text {--}}=70\) and \(\xi _{0\text {--}}=0\). The ultimate goal is to reach the deterministic payoff \(H=100\); this refers to as a continuously compounded return of \(h\approx 3.6\%\). Let \(J\in \mathbb {N}\) be a sufficiently large number³ of simulated paths \(X^{(j)}=(X_t^{(j)})_{t=0,1,2,\ldots ,N}\), e.g., \(J=10^4\). Given this parameter set, we seek to find optimal rehedging strategies. This can be achieved by applying a suitable backpropagation algorithm on the deep neural network architecture that consolidates the above feedforward neural network instances together with the intermediary accounting routines. A direct translation of the above concept is the minimization of the lossWe modify the loss function for two crucial reasons. Firstly, the function \(x\mapsto \max \{H-x,0\}\) is nondifferentiable at the point H and ignores any points beyond H. This raises concerns on the stability of the learning algorithm. Therefore, we replace the maximum with the softplus function \(\log {(1+e^{x})}\). Secondly, the natural extension of the loss function apparently has an undesirable local minimum for strategies with a deterministic equity portion \(\xi _t\equiv \xi \in [0,1]\); see Fig. 3 above.

Without further interventions, the learning algorithms often gets stuck in the suboptimal neighborhood of static strategies. Therefore, we also penalize deviations beyond H in terms offor a regularization parameter \(\lambda =0.1\). It needs to be noted that the introduction of the positive second summand does not alter the global optimum. The following charts exhibit the out-of-sample performance of a trained artificial financial agent for \(p\in \{1,1.5,5\}\) and \(\kappa \in \{0,0.005\}\). For the training, we relied on the default configuration of the Adam algorithm of TensorFlow Keras with a batch size of 64 over 500 epochs. All charts are generated with the same sample data. The training phase of the Jupyter notebook takes in each case approximately 2.5h on Google Colab. As a benchmark, we also show the performance of naively applying the continuous-time optimal hedging strategy on the same weekly time grid.

For \(p\in \{1,1.5\}\), deep hedging mitigates the risk of large losses. In the absence of transaction costs, our simulations suggest that deep hedging does not surpass the benchmark consistently, at least not for the selected parameters and without further measures. However, in the presence of transaction costs, the strength of deep hedging is particularly evident, cf. Table 1. Moreover, it could be extended to more realistic dynamics of the underlying for which analytical solutions are typically not available. The empirically derived expected terminal wealth, the value-at-risk to a significance of \(5\%\) as well as the success rates and the success ratios for the different investment strategies are lined up in Table 1. Remarkably, due to accounting for offsetting effects of an adjusted hedge and borne transaction cost, deep hedging leads to an improved value-at-risk in the presence of transaction cost (Table 1; cf. also Figs. 4, 5, 6, 7).

We have discussed two approaches to goal-based investing in this article. The first—analytical—approach yields several explicit continuous dynamic trading strategies that risk-taking, risk-neutral, and risk-averse investors need to implement to maximize their goal-based utilities.

In the real world, however, continuous-time trading is not feasible. We show that this drawback can be addressed with a more flexible deep hedging approach. Not only is this approach well-suited for discrete rebalancing, it also allows for the inclusion of transaction costs. Curiously, goal-based investing provides a use case for deep hedging with a probability-maximizing objective function, due to the problem’s equivalence with efficient hedging.

There are many ramifications of our work on hedging goals that we will investigate elsewhere. In particular, open research questions that we will address include: