Robustness of shortfall risk minimising strategies in the binomial model

Gino Favero, Tiziano Vargiolu Università degli Studi di Padova Dipartimento di Matematica Pura ed Applicata via Belzoni 7, 35131 Padova – Italy

      favero@math.unipd.it, vargiolu@math.unipd.it

October 10, 2002

Abstract

In this paper we study the dependence on the loss function of the strategy which minimises the expected shortfall risk when dealing with a financial contingent claim in the particular situation of a binomial model. After having characterised the optimal strategies in the particular cases when the loss function is concave, linear or strictly convex, we analyse how optimal strategies change when we approximate a loss function with a sequence of suitable loss functions. Key Words: shortfall risk minimization, binomial model, Dynamic Programming algorithm, robustness

1 Introduction

Given a market with a vector S of asset prices (one of which is non-risky), let H_{N} be a liability to be hedged at some future time N. If V_{N}(π) is the value at time N of a portfolio corresponding to a self-financing investment strategy π, the shortfall risk minimization problem consists in determining J_{0}(S_{0}, V_{0}), where for every n = 0, 1, …, N,

J_{n}(s, v) := inf_{π} E {ℓ([H_{N} − V_{N}(π)]^{+}) | S_{n} = s, V_{n} = v}

for a suitable “loss function” ℓ, with S_{n} and V_{n} the (known) price at time n of the assets and the value at time n of the portfolio. The most applied side of the problem above consists in finding the optimal policy π^{∗} and, if possible, to determine it explicitly.

The loss function ℓ is classically considered to be increasing and such that ℓ(0) = 0. As it is customary to do in economic models, we shall furthermore suppose that ℓ is either concave or convex on ℝ_{+}, depending on the investor’s inclination or aversion (respectively) to taking risks.

In this paper, we want to analyse the dependence of the optimal policy π^{∗} on the loss function ℓ. In order to keep things simple, we restrict ourselves to the case in which the market follows the Cox-Ross-Rubinstein “binomial model”, where the non-risky asset is supposed (by considering forward prices) identically equal to 1, and the stock S follows the dynamics

S_{n+1} = S_{n}ω_{n+1}, n = 0, …, N − 1

where S_{0} > 0 is given and (ω_{n})_{n=1,...,N} are i.i.d. random variables taking values in the set {d, u} (with u and d known real numbers such that 0 < d < 1 < u) with probability law

p := P {ω_{n} = u} = 1 − P {ω_{n} = d}, n = 1, …, N.

To avoid technicalities, we shall take as our underlying probability space the minimal one for our model; that is, we let Ω = {u, d}^{N}, F = P(Ω).

An investment strategy is a sequence π = (π_{n})_{n=0,...,N−1}, where π_{n} = (α_{n}, β_{n}) is the decision to hold, at time n, β_{n} units of the non-risky asset and α_{n} units of the stock S_{n}. We shall always suppose our investment strategies to be self-financing, namely that

\begin{cases}V_{0} = β_{0} + α_{0}S_{0} \\ V_{n} := β_{n} + α_{n}S_{n} = β_{n+1} + α_{n+1}S_{n} \end{cases}

so that the dynamics of the portfolio value can be written as

V_{n+1}^{α} = V_{n+1}^{α}(V_{n}^{α}, S_{n}, α_{n}, ω_{n+1}) := V_{n}^{α} + α_{n}S_{n}(ω_{n+1} − 1).

By the last equality, then, the main concern in the hedging/replication problem is the determination of α_{n}, and this is why we use the notation V_{n}^{α} to stress the dependence of the portfolio dynamics upon the choice of α. Moreover, we shall suppose H to be a European contingent claim depending only on the final value S_{N} of the stock. Hence, the shortfall risk minimization problem can be written, for every n = 0, …, N, as

J_{n}(S_{n}, V_{n}) := inf_{α} E {ℓ([H(S_{N}) − V_{N}^{α}]^{+}) | S_{n}, V_{n}}.  (1.1)

The paper is organised as follows. Section 2 contains the definition of the Dynamic Programming Algorithm (DPA) and some interesting results that hold when the DPA is applied to the binomial model. Section 3 briefly recalls the main results of [1] and [8], which will be a useful reference in the sequel. Section 4 is dedicated to a self-contained treatment of the optimization in particular data structures that we call “recombining binomial trees”, and which is a key tool for the results in the next section. In Section 5, after the determination of the optimal control and optimal value for the case when ℓ is concave in ℝ_{+}, it will be surprisingly clear that the investor with “linear” loss function behaves in exactly the same way as a risk prone investor. Moreover, the same Section 5 explains also a way to approach the incomplete information case (i.e., the situation when p is unknown) for a risk prone investor. Section 6 treats the case when the loss function is convex, and the explicit determination of the optimal strategy is made using a Neyman-Pearson technique similar to [2]. In Section 7 we study the robustness of the optimal strategy with respect to the loss function ℓ; in particular, we shall study three cases, namely the case when we approximate a strictly convex loss function ℓ with a sequence (ℓ_{n})_{n} of strictly convex functions, the case when the functions in the approximating sequence (ℓ_{n})_{n} are strictly convex and ℓ is linear, and the case when ℓ and the ℓ_{n}s are concave.

2 The Dynamic Programming Algorithm

In [8], the key tool used for solving the shortfall risk minimization problem is the Dynamic Programming Algorithm (DPA for short). Since it will be also used in the present work, we summarize here its main features.

The DPA is computed according to the following backwards recursion:

\begin{aligned}J_{N}(s, v) & := ℓ([H(s) − v]^{+}) \\ J_{n}(s, v) & := inf_{α} E\{J_{n+1}(S_{n+1}, V_{n+1}^{α}) | S_{n} = s, V_{n}^{α} = v\}, n = 0, …, N − 1. \end{aligned}

Once solved, the DP algorithm allows us to compute the optimal value J_{0}(S_{0}, V_{0}) for our problem; moreover, if the inf operators in the various recursive steps are realized as min, the minimizing α is the optimal strategy.

For the DPA applied to the shortfall risk minimization in the binomial model, there are some remarkable results that do not depend on the choice of the loss function ℓ. First of all, there exists a unique equivalent martingale measure P^{∗} such that

p^{∗} = P^{∗}\{ω_{n} = u\} := \frac{1 − d}{u − d},  1 − p^{∗} = P^{∗}\{ω_{n} = d\} = \frac{u − 1}{u − d}.  (2.1)

In particular, for every n = 0, …, N, the arbitrage free price of H(S_{N}) at time n is V_{n}^{∗}(S_{n}) := E^{∗}{H(S_{N}) | S_{n}} (Cox-Ross-Rubinstein valuation formula). Note that, in particular, V_{N}^{∗}(S_{N}) = H(S_{N}). It is also well known that, if V_{0} ⩾ V_{0}^{∗}(S_{0}), there is a replicating strategy given by

α_{n}^{∗} := \frac{V_{n+1}^{∗}(S_{n}u) − V_{n+1}^{∗}(S_{n}d)}{S_{n}(u − d)}  (2.2)

As a consequence, the shortfall risk minimization problem for this binomial model is non trivial only in the case V_{0} < V_{0}^{∗}(S_{0}). Henceforth, we shall therefore suppose to be dealing with this case. The problem has already been solved in the case ℓ(x) = x in [8], even when there is incomplete information on the underlying model (see [1] and [8] again, and [4] for a different approach).

Other results on the DPA are gathered in the following

2.1 Proposition. In the notations of the present section and for every n = 0, …, N − 1,

1. J_{n}(s, v) is decreasing in v;

2. J_{n}(s, v) = 0 for v ⩾ V_{n}^{∗}(s);

3. there exist \overline{α} < α̃ such that

J_{n}(s, v) = inf_{α∈[\overline{α},α̃]} E\{J_{n+1}(S_{n+1}, V_{n+1}^{α}) | S_{n} = s, V_{n}^{α} = v\};

4. if v ⩽ V_{n}^{∗}(s), in (3) above one can choose

\overline{α} = α_{n}^{(1)}(s, v) := \frac{V_{n+1}^{∗}(sd) − v}{s(d − 1)} , α̃ = α_{n}^{(2)}(s, v) := \frac{V_{n+1}^{∗}(su) − v}{s(u − 1)} . (2.3)

Proof. We shall prove the result by backwards induction on n.

(n = N − 1). The expansion of the expected value gives

J_{N−1}(s, v) = inf_{α} E {ℓ([H(S_{N}) − V_{N}^{α}]^{+}) | S_{N−1} = s, V_{N−1}^{α} = v}

= inf_{α} {pℓ([H(su) − v − αs(u − 1)]^{+})+

+ (1 − p)ℓ([H(sd) − v − αs(d − 1)]^{+})},

so that (1) follows from the monotonicity of ℓ.

Write

\begin{aligned}j_{N}^{u}(s, v, α) & := pℓ([H(su) − v − αs(u − 1)]^{+}), \\ j_{N}^{d}(s, v, α) & := (1 − p)ℓ([H(sd) − v − αs(d − 1)]^{+}), \end{aligned}

so that J_{N−1}(s, v) = inf_{α}{j_{N}^{u}(s, v, α)+j_{N}^{d}(s, v, α)}. In this expression, note that for α > α_{N−1}^{(2)}(s, v) (as defined in (2.3)) one gets H(su) − v − αs(u − 1) < 0, so j_{N}^{u}(s, v, α) = 0. In the same way, j_{N}^{d}(s, v, α) = 0 for α < α_{N−1}^{(1)}(s, v). So, since j_{N}^{u} and j_{N}^{d} are, respectively, decreasing and increasing in α, we have that their sum is decreasing in α for α < α_{N−1}^{(1)}(s, v) and increasing in α for α > α_{N−1}^{(2)}(s, v). As a consequence, the inf of j_{N}^{u} + j_{N}^{d} must be attained in the interval with extremal points α_{N−1}^{(1)}(s, v) and α_{N−1}^{(2)}(s, v), and this proves (3).

To compare α_{N−1}^{(1)}(s, v) and α_{N−1}^{(2)}(s, v), note that

α_{N−1}^{(2)}(s, v) − α_{N−1}^{(1)}(s, v) = \frac{H(su) − v}{s(u − 1)} − \frac{H(sd) − v}{s(d − 1)}

= \frac{(1 − d)H(su) + (u − 1)H(sd) − (u − d)v}{s(u − 1)(1 − d)}

= \frac{u − d}{s(u − 1)(1 − d)}(V_{N−1}^{∗}(s) − v),

namely, that α_{N−1}^{(1)}(s, v) < α_{N−1}^{(2)}(s, v) if and only if v < V_{N−1}^{∗}(s), and this proves (4). Also, if v ⩾ V_{N−1}^{∗}(s) then any value of α ∈ [α_{N−1}^{(2)}(s, v), α_{N−1}^{(1)}(s, v)] realizes the absolute minimum of 0 for j_{u} + j_{d}, which implies that we are able to obtain perfect replication of the claim H at the final date N and that the shortfall risk minimization problem trivially becomes the CRR hedging problem. This also proves (2).

(n < N − 1). Expanding the expected value as before, one gets

J_{n}(s, v) = inf_{α} E{J_{n+1}(S_{n}, V_{n}^{α}) | S_{n} = s, V_{n}^{α} = v}

= inf_{α} {pJ_{n+1}(su, v + αs(u − 1))+

+ (1 − p)J_{n+1}(sd, v + αs(d − 1))},

and (1) follows from the (inductively assumed) monotonicity of J_{n+1}(s, ·).

As in the first part of the proof, define j_{n+1}^{u}(s, v, α) := pJ_{n+1}(su, v + αs(u − 1)), j_{n+1}^{d}(s, v, α) := (1 − p)J_{n+1}(sd, v + αs(d − 1)) and note that they are, respectively, decreasing and increasing in α. Since J_{n+1}(s, v) = 0 for v ⩾ V_{n+1}^{∗}(s) by the inductive hypothesis, we have that

α > α_{n}^{(2)}(s, v) ⇒ v + αs(u − 1) > V_{n+1}^{∗}(su) ⇒ j_{n+1}^{u}(s, v, α) = 0,

and in the same way that j_{n+1}^{d}(s, v, α) = 0 for α < α_{n}^{(1)}(s, v). Hence it follows that j_{n+1}^{u} + j_{n+1}^{d} is decreasing in α for α < α_{n}^{(1)}(s, v) and increasing in α for α > α_{n}^{(2)}(s, v). As before, this proves (3).

To compare α_{n}^{(1)}(s, v) and α_{n}^{(2)}(s, v), it suffices to compute

α_{n}^{(2)}(s, v) − α_{n}^{(1)}(s, v) = \frac{V_{n+1}^{∗}(su) − v}{s(u − 1)} − \frac{V_{n+1}^{∗}(sd) − v}{s(d − 1)}

= \frac{(1 − d)V_{n+1}^{∗}(su) + (u − 1)V_{n+1}^{∗}(sd) − (u − d)v}{s(u − 1)(1 − d)}

= \frac{u − d}{s(u − 1)(1 − d)}(V_{n}^{∗}(s) − v),

where the last equality comes from the fact that, for every n,

\begin{aligned}E^{∗}\{V_{n+1}^{∗}(S_{n+1}) | S_{n} = s\} & = E^{∗}\{E^{∗}\{H(S_{N}) | S_{n+1}\} | S_{n} = s\} \\ & = E^{∗}\{H(S_{N}) | S_{n} = s\} = V_{n}^{∗}(s). \end{aligned}  (2.4)

Following the same argument as in the first part of the proof, then, also (4) and (2) are proved.

The next two sections will make wide use of this result and of the structure of the functions to be minimized at each recursive step.

3 Preliminary results

This section briefly reviews the most important points of [1] and [8], which completely solve the problem in the case ℓ(x) = x. These results are reported here for the convenience of the reader, as they permit a better understanding of the meaning of the results and remarks contained in the next section.

Here, we shall often use the expressions defined in (2.3) in correspondence of the current values v = V_{n} and s = S_{n} for the portfolio and the risky asset respectively. For simplicity of notation, then, we shall write α_{n}^{(i)} instead of α_{n}^{(i)}(S_{n}, V_{n}). The reader should nevertheless keep in mind the dependence of the α_{n}^{(i)}s not only on the current time (via the Cox-Ross-Rubinstein valuation formula) but also on S_{n} and V_{n}.

In the case ℓ(x) = x, V_{0} < V_{0}^{∗}(S_{0}), the recursive step in the dynamic programming algorithm defined in previous section reduces, at each time n, to the minimization of a function which is piecewise linear in α on the three disjoint intervals (−∞, α_{n}^{(1)}), (α_{n}^{(1)}, α_{n}^{(2)}) and (α_{n}^{(2)}, +∞). As a consequence, the inf is attained as a min either for α = α_{n}^{(1)} or for α = α_{n}^{(2)}. A straightforward calculation (see [8, Theorem 4.1]) allows one to check that the optimal control α^{†} is calculated at each time n by choosing

α_{n}^{†} = \begin{cases}α_{n}^{(1)} & if p < p^{∗} \\ α_{n}^{(2)} & if p > p^{∗} \end{cases} .  (3.1)

(The case p = p^{∗} corresponds to S and V being martingales, so that any choice of the control would give the same results in the mean.) Note that the choice between the strategy α^{(1)} and α^{(2)} does not depend on n, so the optimal control is either α_{n}^{†} ≡ α_{n}^{(1)} for p < p^{∗} or α_{n}^{†} ≡ α_{n}^{(2)} for p > p^{∗}. In particular, the choice of the optimal control can be made “a priori”. The optimal value for the shortfall risk is then min\{\frac{p}{p^{∗}} , \frac{1−p}{1−p^{∗}} \}^{N}(V_{0}^{∗}(S_{0}) − V_{0})^{+}.

Note also that the strategies α^{(1)} and α^{(2)} are in some sense a modification of the optimal replicating strategy of Cox, Ross and Rubinstein (see the Introduction). Strategy α^{(1)} (respectively, α^{(2)}), indeed, is obtained from (2.2) by substituting u and V_{n+1}^{∗}(S_{n}u) (respectively, d and V_{n+1}^{∗}(S_{n}d)) by 1 and V_{n}. Moreover, note that p < p^{∗} (respectively, p > p^{∗}) if and only if E{S_{n} | S_{0}} is decreasing (respectively, increasing) in n. In some sense, then, the optimal strategy in the case V_{0} < V_{0}^{∗}(S_{0}) is built by replacing in the CRR formula the “less significant” event (i.e., the one against the mean tendency of S to decrease or increase) with the present value.

In [1], some additional properties of the strategies belonging to the class

Π := {(α_{n})_{n} | α_{n} ∈ {α_{n}^{(1)}, α_{n}^{(2)}} for every n}  (3.2)

are investigated. In particular, a slight modification of the proof of Theorem 4.1 in [8] shows that for every α ∈ Π,

E\{H(S_{N}) − V_{N}^{α} | S_{0}, V_{0}\} = (\frac{p}{p^{∗}} )^{λ_{N}} (\frac{1 − p}{1 − p^{∗}} )^{N−λ_{N}} [V_{0}^{∗}(S_{0}) − V_{0}]^{+}, (3.3)

where λ_{N} := #{n | α_{n} = α_{n}^{(1)}}. Moreover, it is possible to define a one to one correspondence between controls α ∈ Π and “events” ω = (ω_{n})_{n=0,...,N−1} by defining

(ω(α))_{n} := \begin{cases}u & if α_{n} = α_{n}^{(1)} \\ d & if α_{n} = α_{n}^{(2)} \end{cases} .  (3.4)

In other words, ω(α) is the only event such that the stock S goes up every time the investor “bets” on it going down on average, and the stock goes down every time the investor bets on it going up on average. Now, it is straightforward to prove that any strategy α ∈ Π gives perfect hedging in all events ω ≠ ω(α). This is quite a strong result, which has two main consequences. First of all, Π is a class of “quasi-replicating” controls, in the sense that every α ∈ Π gives perfect hedging with probability close to 1. More in detail, since E{(H(S_{N})−V_{N}^{α})^{+}} > 0 and H(S_{N})(ω) − V_{N}^{α}(ω) = 0 in all events ω ≠ ω(α), one has

P {H(S_{N}) − V_{N}^{α} > 0} = P {ω(α)} = p^{λ_{N}} (1 − p)^{1−λ_{N}}  (3.5)

with λ_{N} defined as above. Moreover, from (3.3), it follows that in the only “critical” event ω(α) we have

H(S_{N}(ω(α))) − V_{N}^{α}(ω(α)) = (\frac{1}{p^{∗}} )^{λ_{N}} (\frac{1}{1 − p^{∗}} )^{N−λ_{N}} [V_{0}^{∗}(S_{0}) − V_{0}]^{+},

which is generally much greater than the initial “lack” of capital.

4 Optimal paths in recombining binomial trees

This section contains some technical definitions and results that formalise a “branch-and-bound” type algorithm for the purpose of determining an optimal path in a particular directed graph, which we call a recombining binomial tree. These results will constitute a key tool for the proof of Theorem 5.2 concerning the solution of the shortfall risk minimization problem in the case when ℓ is concave.

4.1 Definition. For n ∈ ℕ, a recombining binomial tree (r.b.t. for short) of depth n is a directed graph T (n) with \frac{(n+1)(n+2)}{2} nodes τ_{k}^{i}, where k = 0, …, i, i = 0, …, n. This way, T (n) features i+1 nodes at each “depth” i. In particular, at the depth 0 there is only one node τ_{0}^{0}, which will be called the vertex of the tree T (n).

For i = 0, …, n − 1, each node τ_{k}^{i} is supposed to be connected with the two nodes τ_{k}^{i+1} and τ_{k+1}^{i+1}. The nodes at depth n are thus “terminal” nodes, and will be called the leaves of the tree. We shall refer to moving from τ_{k}^{i} to τ_{k}^{i+1} (respectively, to τ_{k+1}^{i+1}) with the expression branching left (respectively, branching right).

Since the graph is directed, the maximum length paths on the graph start from the vertex τ_{0}^{0} and reach a leaf τ_{k}^{n} for some k. Note that there is a one-to-one correspondence between maximum length paths and branching sequences (ϑ_{i})_{i=0}^{n−1} ∈ {l, r}^{n}, where ϑ_{i} = l (respectively, ϑ_{i} = r) means the decision to branch left (respectively, right) when passing from depth i to depth i + 1.

The problem we want to solve is the following. Suppose that each leaf τ_{k}^{n} of the r.b.t. T (n) is associated with some value r_{k} ∈ ℝ. Then, we want to find a branching sequence so as to reach a leaf associated with the minimum value, that we shall call, respectively, an optimal branching sequence (or strategy) and an optimal leaf. (The problem of reaching the maximum value can trivially be reformulated as a minimum problem by associating the values −r_{k} to the leaves.)

Note that, even if the correspondence between paths and branching sequences is one to one, in general the strategy leading to a chosen leaf τ_{k}^{n} is not unique. Actually, any strategy (starting from the vertex) which branches left n − k times and right k times in any order will end in leaf τ_{k}^{n}, and it is clear that there are \binom{n}{k} such strategies. Thus, the choice of the optimal branching strategy is not a consequence of the determination of the optimal leaf.

The algorithm we propose below solves the problem of choosing an optimal strategy by deciding the branching sequence while scanning the leaves to determine the minimum value.

4.2 Proposition. Let n ∈ ℕ, and T (n) be the recombining binomial tree of depth n, with values r_{k}, k = 0, …, n associated with the leaves.

Define

ϑ_{0} = \begin{cases}l & if r_{0} ⩽ r_{n} \\ r & if r_{0} > r_{n} \end{cases}

and then, recursively for every i = 1, …, n,

\begin{matrix}k_{ϑ,i} & := #\{j < i | ϑ_{j} = r\}, \\ ϑ_{i} = & \begin{cases}l & if r_{k_{ϑ,i}} ⩽ r_{n−i+k_{ϑ,i}} \\ r & if r_{k_{ϑ,i}} > r_{n−i+k_{ϑ,i}}. \end{cases} \end{matrix}

Then the leaf τ_{k_{ϑ,n}}^{n} is associated with the minimum value, and ϑ = (ϑ_{i})_{i=1}^{n} is an optimal branching sequence.

Proof. Induction on n.

(n = 1). In this case, there are only the two leaves τ_{0}^{1} and τ_{1}^{1} associated with the values r_{0} and r_{1} respectively. The thesis translates into the obvious decision to branch left and reach τ_{0}^{1} if r_{0} is the minimum value, and to branch right and reach τ_{1}^{1} otherwise.

(n > 1). To determine the branching at depth 0 the following considerations can be made. Note that the two outmost leaves τ_{0}^{n} and τ_{n}^{n} are both reached from the vertex by a unique branching strategy, namely, τ_{0}^{n} can only be reached by branching always left and τ_{n}^{n} can only be reached by branching always right. Any other leaf can be reached with a suitable branching sequence whatever ϑ_{0} is. Thus, the proposed strategy decides to branch so as to make unreachable the outmost leaf which is associated with the higher (and, thus, non-optimal) value and reach the node τ_{k_{ϑ,1}}^{1}.

One can now consider the“sub-tree” T_{1} of depth n − 1 with vertex τ_{k_{ϑ,1}}^{1} and leaves τ_{k}^{n}, k = k_{ϑ,1}, …, n − 1 + k_{ϑ,1}. Note that the minimum value has to be associated with one of the leaves of T_{1}, since the value associated with the “discarded” leaf is greater than a value associated with a leaf of T_{1}. The recursive step then corresponds to the first step in this sub-tree, and so the proposition follows by recursion.

5 Risk prone investor: the concave case

Throughout this section ℓ will be supposed to be a concave function on ℝ_{+}. As in Section 3, we use the convention of writing α_{n}^{(i)} instead of α_{n}^{(i)}(S_{n}, V_{n}).

Before starting the discussion, we need an extension of Proposition 2.1.

5.1 Proposition. If ℓ is concave then, in the notations of Section 2, the function J_{n}(s, ·)_{|(−∞,V_{n}^{∗}(s)]} is concave for every n = 0, …, N.

Proof. We proceed again by backwards induction on n. For n = N the statement follows immediately from the fact that V_{n}^{∗}(s) = H(s) and that ℓ is concave.

For n < N, choose v′ < \overline{v} < v′′ ⩽ V_{n}^{∗}(s), and let λ′, λ′′ be the convex combinators such that λ′ + λ′′ = 1, λ′v′ + λ′′v′′ = \overline{v}. Expanding the expected value as in the proof of Proposition 2.1 one gets (as explained below)

J_{n}(s, \overline{v}) = inf_{α} \{pJ_{n+1}(su, \overline{v} + αs(u − 1))+
+ (1 − p)J_{n+1}(sd, \overline{v} + αs(d − 1))\}

= inf_{α} {pJ_{n+1}(su, λ′(v′ + αs(u − 1)) + λ′′(v′′ + αs(u − 1)))+

+ (1 − p)J_{n+1}(sd, λ′(v′ + αs(d − 1)) + λ′′(v′′ + αs(d − 1)))}

⩾ inf_{α} {λ′[pJ_{n+1}(su, v′ + αs(u − 1)) + (1 − p)J_{n+1}(sd, v′ + αs(d − 1))]+

+ λ′′[pJ_{n+1}(su, v′′ + αs(u − 1)) + (1 − p)J_{n+1}(sd, v′′ + αs(d − 1))]} ⩾ λ′J_{n}(s, v′) + λ′′J_{n}(s, v′′).

The last inequality comes from the fact that whenever a, b are positive one has inf_{x}{af(x) + bg(x)} ⩾ a inf_{x}{f(x)} + b inf_{x}{g(x)}. The first inequality is not straightforward, because J_{n+1} is not concave on the entire real line, but can be proved as follows. By (3) of Proposition 2.1, in the computation of the inf we can restrict α to take values in the interval [α_{n}^{(1)}(s, \overline{v}), α_{n}^{(2)}(s, \overline{v})]. In this interval, both inequalities

\begin{matrix}\overline{v} + αs(d − 1) & ⩽ V_{n+1}^{∗}(sd) \\ \overline{v} + αs(u − 1) & ⩽ V_{n+1}^{∗}(su) \end{matrix}

hold, so that both j_{n+1}^{u} and j_{n+1}^{d} (in the same notations as for the proof to 2.1) are calculated in the (inductively assumed) concavity domain of J_{n+1}. This way j_{n+1}^{u} + j_{n+1}^{d}, being a sum of concave functions, is concave, and hence the first inequality follows.

The fact that the concavity “propagates” along the DP algorithm allows us to find a straightforward explicit solution for our problem.

5.2 Theorem. Let ℓ be a concave function and, for every n = 0, …, N, let J_{n} be the function defined in Equation (1.1). Suppose that V_{0} < V_{0}^{∗}(S_{0}). Then, for every n,

J_{n}(S_{n}, V_{n}) = min_{k=n,...,N} p^{k−n}(1 − p)^{N−k}ℓ (\frac{V_{n}^{∗}(S_{n}) − V_{n}}{(p^{∗})^{k−n}(1 − p^{∗})^{N−k}} ) .  (5.1)

Define, for every n = 0, …, N − 1,

α_{n}^{(1)} := \frac{V_{n+1}^{∗}(S_{n}d) − V_{n}}{S_{n}(d − 1)} , α_{n}^{(2)} := \frac{V_{n+1}^{∗}(S_{n}u) − V_{n}}{S_{n}(u − 1)} .

Then the optimal strategy is given by

α_{n}^{∗} = \begin{cases}α_{n}^{(1)} & if p^{N−n}ℓ (\frac{V_{n}^{∗}(S_{n})−V_{n}}{(p^{∗})^{N−n}} ) ⩽ (1 − p)^{N−n}ℓ (\frac{V_{n}^{∗}(S_{n})−V_{n}}{(1−p^{∗})^{N−n}} ) \\ α_{n}^{(2)} & if p^{N−n}ℓ (\frac{V_{n}^{∗}(S_{n})−V_{n}}{(p^{∗})^{N−n}} ) > (1 − p)^{N−n}ℓ (\frac{V_{n}^{∗}(S_{n})−V_{n}}{(1−p^{∗})^{N−n}} ) \end{cases} . (5.2)

5.3 Remark. As the proof of this theorem will make clear, we can always restrict the class of admissible controls to the class Π defined in (3.2). In particular, this allows a proof by induction via a straightforward calculation that from V_{0} ⩽ V_{0}^{∗}(S_{0}) follows V_{n} ⩽ V_{n}^{∗}(S_{n}) almost surely (i.e., for every event ω) for every n. This is the reason why in the expression for J_{n}(S_{n}, V_{n}) there is no need to consider the positive part of the argument of ℓ.

Proof of Theorem 5.2. For every n = 0, …, N − 1 define j_{n}^{u} and j_{n}^{d} as in the proof of Proposition 2.1, so that

J_{n}(S_{n}, V_{n}) = inf_{α}{j_{n+1}^{u}(s, v, α) + j_{n+1}^{u}(s, v, α)}.

By (3) and (4) of Proposition 2.1 we can restrict the computation of the inf to the interval [α_{n}^{(1)}, α_{n}^{(2)}] where, as already seen in the proof of Proposition 5.1, j_{n+1}^{u} + j_{n+1}^{d} is concave. So, since a concave function on an interval can attain its lowest value only at the extremal points, the problem reduces to comparing the values taken by the function at the extrema. Namely, on recalling that for α = α_{n}^{(1)} (respectively, α = α_{n}^{(2)}) one has j_{n+1}^{d} = 0 (respectively, j_{n+1}^{u} = 0), one finds that

J_{n}(S_{n}, V_{n}) = min{pJ_{n+1}(S_{n}u, V_{n} + α_{n}^{(1)}S_{n}(u − 1)),
(1 − p)J_{n+1}(S_{n}d, V_{n} + α_{n}^{(2)}S_{n}(d − 1))},
argmin[⋯] = α_{n}^{(1)} ⟺ pJ_{n+1}(S_{n}u, V_{n} + α_{n}^{(1)}S_{n}(u − 1)) ⩽
⩽ (1 − p)J_{n+1}(S_{n}d, V_{n} + α_{n}^{(2)}S_{n}(d − 1)),

and, as in [1, Section 2] (see also Section 3 here), we are allowed to restrict the class of admissible controls to the class Π defined in (3.2). Thanks to (2.4), it is straightforward to verify by backward induction on n that for every α ∈ Π, defining

λ_{n} := #{k ⩾ n | α_{k} = α_{k}^{(1)}}, μ_{n} := #{k ⩾ n | α_{k} = α_{k}^{(2)}} = N − n − λ_{n},

gives

E_{S_{n},V_{n}}\{ℓ([H(S_{N}) − V_{N}^{α}]^{+})\} = p^{λ_{n}}(1 − p)^{μ_{n}}ℓ (\frac{V_{n}^{∗}(S_{n}) − V_{n}}{(p^{∗})^{λ_{n}}(1 − p^{∗})^{μ_{n}}} ) . (5.3)

Hence it follows that the optimal value is as in (5.1).

To prove that the strategy given in (5.2) is optimal, we start by observing that from (5.3) it follows that, for the purpose of determining the shortfall risk, only the number of times that α_{n} = α_{n}^{(1)} matters, and not the particular sequence of choices. The evolution of the shortfall risk with respect to the chosen control from time n on can then be seen as a “recombining binomial tree” of depth N − n (see Definition 4.1 for the definition) as follows. For every m = n, …, N −1, associate the choice α_{m} = α_{m}^{(1)} (respectively, α_{m} = α_{m}^{(2)}) with the decision of branching left (respectively, right) at depth m − n, so as to set a one-to-one correspondence between strategies in the class Π and branching sequences for the tree. Formula (5.3) suggest associating with each leaf τ_{k}^{N−n} the value p^{k}(1 − p)^{N−n−k}ℓ(\frac{V_{n}^{∗}(S_{n})−V_{n}}{(p^{∗})^{k}(1−p^{∗})^{N−n−k}} ), i.e. the shortfall risk associated with any strategy leading to that leaf. The proof is now completed by observing that the proposed strategy is exactly the “optimal branching sequence” of Proposition 4.2.

5.4 Remark. At the end of Section 3 we noted that any strategy in the class Π is “quasi replicating”, i.e., it leads to perfect hedging in all events ω ≠ ω(α) as defined in (3.4). This result does not depend on the function ℓ taken into consideration, so we can conclude that the expected value (5.3) found in the proof to Theorem 5.2 corresponds to a shortfall of \frac{V_{n}^{∗}(S_{n})−V_{n}}{(p^{∗})^{λn} (1−p^{∗})^{μn}} in the only “critical” non-hedging event ω(α), whose probability is P (ω(α)) = p^{λ_{n}}(1 − p)^{μ_{n}} as seen in (3.5).

5.5 Remark. In the case of “complete information” (i.e., when the probability p is known by the investor), the optimal strategy can be chosen a priori instead of on a step-by-step basis. Actually, once the \overline{k} that minimizes p^{k}(1 − p)^{N−k}ℓ(\frac{V_{0}^{∗}(S_{0})−V_{0}}{(p^{∗})^{k}(1−p^{∗})^{N−k}} ) is determined, any strategy choosing \overline{k} times

α_{n} = α_{n}^{(1)} is optimal. Note also that all of these strategies have the same probability p\overline{k}(1−p)^{N−\overline{k}} of being non-hedging and the same shortfall ℓ(\frac{V_{0}^{∗}(S_{0})−V_{0}}{(p^{∗})\overline{k}(1−p^{∗})^{N−\overline{k}}} ) in the “critical” event, so that the choice of the “preferred” optimal strategy has to be made according to completely subjective criteria.

The expression (5.2) for the optimal control has been chosen because it can easily be adapted in the spirit of [1, Section 3] (summarized in Section 3) to elaborate optimal adaptive controls in the case of incomplete knowledge of the model.

5.6 Remark. In the case ℓ(x) = x examined in [1] and [8], a quite remarkable fact is that the (generally unique) optimal strategy consists of choosing either α ≡ α^{(1)} or α ≡ α^{(2)}. In other words, in this case the optimal leaf in the tree described in the proof of Theorem 5.2 is either at the extreme “left” or at the extreme “right”. This may not happen in the general case, as the following (perhaps quite artificial) example shows even in the very simple case of three possible final outcomes.

5.7 Example. Take ℓ(x) := \sqrt{x} + \sqrt[5]{x}, N = 2, p = .245, u = 3.17 and d = .8 (so that p^{∗} = .0845). Consider a contingent claim H such that V_{0}^{∗}(S_{0}) > 170 and choose V_{0} such that V_{0}^{∗}(S_{0}) − V_{0} = 170. The three values associated with the final leaves are then

\begin{aligned}p^{2}ℓ (\frac{V_{0}^{∗}(S_{0}) − V_{0}}{(p^{∗})^{2}} ) & ∼ 9.77, \\ p(1 − p)ℓ (\frac{V_{0}^{∗}(S_{0}) − V_{0}}{(p^{∗})(1 − p^{∗})} ) & ∼ 9.54, \\ (1 − p)^{2}ℓ (\frac{V_{0}^{∗}(S_{0}) − V_{0}}{(1 − p^{∗})^{2}} ) & ∼ 9.72. \end{aligned}

This way the optimal “leaf” is the central one, and two optimal controls can be built by choosing either α_{0} = α_{0}^{(1)}, α_{1} = α_{1}^{(2)} or α_{0} = α_{0}^{(2)}, α_{1} = α_{1}^{(1)}. Note also that the policy proposed in Theorem 5.2 will choose α_{1} = α_{1}^{(2)}.

5.8 Example. Consider the case when ℓ(x) = x^{κ} for some 0 ≤ κ ≤ 1. In particular, the case κ = 1 corresponds to minimising the mean shortfall risk, and, due to the assumption that ℓ(0) = 0, the case κ = 0 corresponds to minimising the probability of positive shortfall. These two particular cases have already been considered in [1] and [8]: when κ = 1, the optimal strategy is the α^{†} defined in (3.1), and when κ = 0 the optimal strategy α^{‡} is calculated by choosing

α_{n}^{‡} = \begin{cases}α_{n}^{(1)} & p < 0.5 \\ α_{n}^{(2)} & p > 0.5 \end{cases}

(when p = 0.5, all strategies α ∈ Π are optimal).

When 0 < κ < 1, the solution still takes an appearance close to the cited results, namely, the optimal policies always choose either α ≡ α^{(1)} or α ≡ α^{(2)}. Actually, in this case, for every n = 0, …, N − 1 the optimal control α^{∗} is computed by choosing

α_{n}^{∗} = α_{n}^{(1)} ⟺ \frac{p}{(p^{∗})^{l}} ⩽ \frac{1 − p}{(1 − p^{∗})^{κ}} i.e., \frac{p}{1 − p} ⩽ (\frac{p^{∗}}{1 − p^{∗}} )^{κ},

independent of n.

Moreover, let \overline{κ} = \frac{log(p)−log(1−p)}{log(p^{∗})−log(1−p^{∗})} (namely, the value for which \frac{p}{1−p} =

(\frac{p^{∗}}{1−p^{∗}} )\overline{κ}). Excluding the “undecidable” cases p = 0.5 (all events have the same probability), p^{∗} = 0.5 (division by zero in \overline{κ}) and p = p^{∗} (the stock and the portfolio are martingales under the real world probability measure), the following cases may then occur.

\begin{matrix}p^{∗} < p < 0.5 (0 < κ < 1) & α^{∗} = \{ \begin{matrix}α^{‡} = α^{(1)} & κ < \overline{κ} \\ α^{†} = α^{(2)} & κ > \overline{κ} \end{matrix} \\ p^{∗} < 0.5, p ≠ (p^{∗}, 0.5) (κ < 0 or κ > 1) & α^{∗} = α^{‡} = α^{†} = α^{(1)} \\ 0.5 < p < p^{∗} (0 < κ < 1) & α^{∗} = \{ \begin{matrix}α^{‡} = α^{(2)} & κ < \overline{κ} \\ α^{†} = α^{(1)} & κ > \overline{κ} \end{matrix} \\ p^{∗} > 0.5, p ≠ (0.5, p^{∗}) (κ < 0 or κ > 1) & α^{∗} = α^{‡} = α^{†} = α^{(2)} \end{matrix}

Note that if α^{‡} = α^{†}, then the optimal control α^{∗} for the shortfall risk minimization problem coincides with both α^{‡} and α^{†} for every κ. On the other hand, if α^{‡} ≠ α^{†}, the exponent 0 < \overline{κ} < 1 is “critical”, in the sense that α^{∗} = α^{‡} for κ ∈ (0, \overline{κ}) and α^{∗} = α^{†} for κ ∈ (\overline{κ}, 1). In other words, the study of the shortfall risk minimization problem with ℓ(x) = x^{κ} can be reduced to the two fundamental problems with κ = 0 and κ = 1.

6 Risk averse investor: the convex case

Throughout this section ℓ will be assumed to be a convex function on ℝ_{+}. As in the previous setting, we can give an extension of Proposition 2.1.

6.1 Proposition. If ℓ is convex then, in the notations of Section 2, the function J_{n}(s, ·) is convex for every n = 0, …, N.

Proof. We proceed by backwards induction on n.

(n = N). Since ℓ(0) = 0, the definition

ℓ̄(x) := \begin{cases}ℓ(x) & if x ⩾ 0 \\ 0 & if x < 0 \end{cases}

extends ℓ to a function which is convex on the entire real line, and ℓ(x^{+}) = ℓ̄(x). We can then write

J_{N}(s, v) = ℓ̄(H(s) − v)

and the conclusion follows from the convexity of ℓ̄.

(n < N). Write

J_{n}(s, v) = inf_{α} E{J_{n+1}(S_{n+1}, V_{n+1}^{α} | S_{n} = s, V_{n}^{α} = v}  (6.1)
= inf_{α}{pJ_{n+1}(su, v + αs(u − 1)) + (1 − p)J_{n+1}(sd, v + αs(d − 1))},

and define, as in the proof of Proposition 2.1,

\begin{aligned}j_{n+1}^{u}(s, v, α) & := pJ_{n+1}(su, v + αs(u − 1)), \\ j_{n+1}^{d}(s, v, α) & := (1 − p)J_{n+1}(sd, v + αs(d − 1)) \end{aligned}

so that J_{n}(s, v) = inf_{α}{j_{n+1}^{u}(s, v, α) + j_{n+1}^{d}(s, v, α)}. Since J_{n+1}(s, ·) is convex by the inductive hypothesis, it is straightforward to prove that j_{n+1}^{u}(s, ·, ·) and j_{n+1}^{d}(s, ·, ·) are convex, in the sense that for every v_{1} ⩽ \overline{v} ⩽ v_{2}, α_{1} ⩽ \overline{α} ⩽ α_{2} such that \overline{v} = λ_{1}v_{1} + λ_{2}v_{2}, \overline{α} = λ_{1}α_{1} + λ_{2}α_{2} (with λ_{1}, λ_{2} convex combinators) it is

\begin{matrix}j_{n+1}^{u}(s, \overline{v}, \overline{α}) & ⩽ λ_{1}j_{n+1}^{u}(s, v_{1}, α_{1}) + λ_{2}j_{n+1}^{u}(s, v_{2}, α_{2}) \\ j_{n+1}^{d}(s, \overline{v}, \overline{α}) & ⩽ λ_{1}j_{n+1}^{d}(s, v_{1}, α_{1}) + λ_{2}j_{n+1}^{d}(s, v_{2}, α_{2}). \end{matrix}

In particular, j_{n+1}(s, v, α) := j_{n+1}^{u}(s, v, α) + j_{n+1}^{d}(s, v, α) is convex in α, so the inf of (6.1) is realized as min.

Choose now v′ ⩽ ṽ ⩽ v′′, so that ṽ = λ′v′+λ′′v′′ for some convex combinators λ′,λ′′. By convexity of j_{n+1}(s, v, ·), there exist α̃, α′, α′′ such that

j_{n+1}(s, ṽ, α̃) = min_{α} j_{n+1}(s, ṽ, α) = J_{n}(s, ṽ)
j_{n+1}(s, v′, α′) = min_{α} j_{n+1}(s, v′, α) = J_{n}(s, v′)
j_{n+1}(s, v′′, α′′) = min_{α} j_{n+1}(s, v′′, α) = J_{n}(s, v′′),

and by convexity of j_{n+1}(s, ·, ·) and minimality of α̃ we have that

λ′J_{n}(s, v′) + λ′′J_{n}(s, v′′) = λ′j_{n+1}(s, v′, α′) + λ′′j_{n+1}(s, v′′, α′′)
⩾ j_{n+1}(s, ṽ, λ′α′ + λ′′α′′)
⩾ j_{n+1}(s, ṽ, α̃) = J_{n}(s, ṽ).

This proposition has as an immediate consequence the existence of an optimal strategy for the convex case. It also follows that, if ℓ is strictly convex, then the optimal strategy is unique. Nevertheless, when trying to determine explicitly the optimal strategy in general form by using dynamic programming arguments one is led to quite complex calculations. For the sake of simplicity, then, from now on we shall shift from a DPA-based approach to another one, with techniques similar to those in [2]. Notice that, for the case when ℓ is strictly convex, we could also use techniques based on convex duality as in [4] or [6]. However, due to the absence of lower bounds on the claim H(S_{N}) and on the portfolio X, it would not be possible to apply these techniques to the case when ℓ(x) = x.

6.2 Definition. We define the set of the modified contingent claims as

X := {X | X ⩽ H(S_{N}) (a.s.), E^{∗}{X} ⩽ V_{0}}.  (6.2)

Roughly speaking, X is the set of all the claims less than H(S_{N}) which can be replicated with initial capital (less than or equal to) V_{0} or, equivalently, the set of all the possible final states of adapted, self-financing strategies starting from initial capital (less than or equal to) V_{0}.

We can now consider the shortfall risk minimization problem from a “static” point of view:

min_{X∈X} E_{S_{0}} {ℓ(H(S_{N}) − X)}.  (6.3)

We now want to show that the modified contingent claim that solves (6.3) coincides with the payoff of the optimal portfolio for the shortfall risk minimization problem. We shall start from the linear case (ℓ(x) = x) and from this case we shall derive the solution for the strictly convex case.

The following lemma, a key tool for the proof of the main Theorem 6.4, shows how this approach can be applied to the study of the “mean shortfall risk minimization problem”, i.e., the shortfall risk minimization problem in the case ℓ(x) = x. Note that, in this case, (6.3) reduces to the problem max_{x∈X} E{X}.

6.3 Lemma. If X^{∗} ∈ X solves the problem max_{x∈X} E{X}, then the hedging strategy α^{∗} for the claim X^{∗} also solves the mean shortfall risk minimization problem min_{α} \{ E_{S_{0},V_{0}}\{(H(S_{N}) − V_{N}^{α})^{+}\}\}. Moreover, define c_{es} := min_{ω} \frac{dP}{dP^{∗}}, where P and P^{∗} are, respectively, the “real world” and the martingale probability measures. Set E := \{ω | \frac{dP}{dP^{∗}} (ω) = c_{es}\}, and

X^{∗} := H(S_{N})1_{E^{c}} + γ1_{E}

where γ is any random variable such that E^{∗}{X^{∗}} = V_{0}. Then X^{∗} solves the problem (6.3).

Proof. This proposition can be proved directly as in [9], but we give a shorter proof that uses the results of Section 3. Consider the strategy α^{†} defined in (3.1). Since the corresponding portfolio value V_{N}^{α^{†}} is the payoff of a self-financing strategy starting from the initial capital V_{0} and V_{N}^{α^{†}} ⩽ H(S_{N}), then V_{N}^{α^{†}} itself is a modified contingent claim in the sense of (6.2). Moreover, if X ∈ X, then one has E_{S_{0},V_{0}}{(H(S_{N})−X)^{+}} = E_{S_{0},V_{0}}{H(S_{N})}−E_{S_{0},V_{0}}{X}, so that the first part of the proposition follows.

As for the second part, if p = p^{∗} then all the modified contingent claims X ∈ X are solutions. If p ≠ p^{∗}, then from Section 3 we also deduce that the V_{N}^{α^{†}} is equal to H(S_{N}) on all events except for the “least probable” one, where the payoff is equal to

H(S_{N}) − \frac{1}{min \{(p^{∗})^{N}, (1 − p^{∗})^{N}\}}(V_{0}^{∗}(S_{0}) − V_{0}).

In particular, it is straightforward to check that E^{∗}{V_{N}^{α^{†}}} = V_{0}, and thus that V_{N}^{α^{†}} can be defined as in the statement above.

To apply this approach to the convex case, we make the further assumption that ℓ is strictly convex, continuously differentiable and such that ℓ′(0) = 0. Indeed, these properties appear to be the right ones to state the following results in a reasonable notation, but we believe that a generalization to non-C^{1} functions, in terms of the sub-differential, should be straightforward.

6.4 Theorem. Set I := (ℓ′)^{−1} and define the modified contingent claim

X^{∗} := H(S_{N}) − I (c^{∗} \frac{dP^{∗}}{dP} )

with c^{∗} > 0 chosen in such a way that E^{∗}{X^{∗}} = V_{0}. Then X^{∗} solves the “static” problem (6.3).

Proof. Fix an X^{∗} ∈ X and define, for every X ∈ X,

X_{ε} := (1 − ε)X^{∗} + εX, F_{X}(ε) := E {ℓ(H(S_{N}) − X_{ε})}.

Then X_{ε} ∈ X for every ε ∈ [0, 1] and it is clear that X^{∗} is optimal for the problem (6.3) if and only if

0 ⩽ F′_{X}(0^{+}) = E {(X^{∗} − X)ℓ′(H(S_{N}) − X^{∗})}

i.e., if and only if E {X^{∗}ℓ′(H(S_{N}) − X^{∗})} ⩾ E {Xℓ′(H(S_{N}) − X^{∗})}, for every X ∈ X.

Define now the measure Q by

\frac{dQ}{dP} := \frac{ℓ′(H(S_{N}) − X^{∗})}{E \{ℓ′(H(S_{N}) − X^{∗})\}},

so that X^{∗} minimizes the shortfall risk if and only if it is optimal for the problem max_{X∈X} E^{Q}{X}. According to Proposition 6.3, the latter maximization problem is solved by an X^{∗} of the form

X^{∗} := H(S_{N})1_{F^{c}} + γ1_{F} ,

where F := \{ω | \frac{dQ}{dP^{∗}} (ω) = c̄\}, c̄:= min_{ω} \frac{dQ}{dP^{∗}} (ω) and γ is such that E^{∗}\{X^{∗}\} = V_{0}.

Note that by definition of Q,

\frac{dQ}{dP^{∗}} = \frac{dQ}{dP} · \frac{dP}{dP^{∗}} = \frac{ℓ′(H(S_{N}) − X^{∗})}{E \{ℓ′(H(S_{N}) − X^{∗})\}} · \frac{dP}{dP^{∗}} .

Since ℓ′ is increasing and ℓ′(0) = 0, the above equality implies that \{\frac{dQ}{dP^{∗}} = 0} = {H(S_{N}) = X^{∗}}. As a consequence, if c̄ = 0, then P {H(S_{N}) = X^{∗}} > 0 and one would have X^{∗} = H(S_{N}) on the set \{\frac{dQ}{dP^{∗}} > 0\} = \{H(S_{N}) > X^{∗}\}, a contradiction. One then has that c̄ > 0, i.e., Q and P^{∗} are equivalent and, moreover, H(S_{N}) > X^{∗} almost surely. By definition of X^{∗}, it follows then that \frac{dQ}{dP^{∗}} = c̄ almost surely, so that X^{∗} = γ1_{F}, i.e., γ = X^{∗}. As a consequence,

\frac{dQ}{dP^{∗}} ≡ c̄ = 1 ⇒ \frac{dQ}{dP} \frac{dP}{dP^{∗}} = 1

⇒ \frac{dP^{∗}}{dP} = \frac{dQ}{dP} = \frac{ℓ′([H(S_{N}) − X^{∗}]^{+})}{c^{∗}}

⇒ ℓ′([H(S_{N}) − X^{∗}]^{+}) = c^{∗} \frac{dP^{∗}}{dP}

⇒ H(S_{N}) − X^{∗} = I (c^{∗} \frac{dP^{∗}}{dP} )

⇒ X^{∗} = H(S_{N}) − I (c^{∗} \frac{dP^{∗}}{dP} ) ,

with c^{∗} := E {ℓ′(H(S_{N}) − X^{∗})}. Since X^{∗} minimizes the shortfall risk, one must have that E^{∗}{X^{∗}} = V_{0}, i.e.,

E^{∗} \{I (c^{∗} \frac{dP^{∗}}{dP} )\} = E^{∗}\{H(S_{N})\} − V_{0}.  (6.4)

Since ℓ is convex and C^{1}, then ℓ′ is increasing and continuous and so is I. This means that the function c → E^{∗}\{I(c\frac{dP^{∗}}{dP})\} is continuous and increasing, thus there exists a unique c^{∗} such that (6.4) is satisfied.

6.5 Remark. Once the optimal modified contingent claim X^{∗} is determined, the optimal strategy is simply the Cox-Ross-Rubinstein replicating strategy for X^{∗} as in (2.2), that is,

α_{n}^{∗} = \frac{E^{∗}\{X^{∗}|S_{n+1} = S_{n}u\} − E^{∗}\{X^{∗}|S_{n+1} = S_{n}d\}}{S_{n}(u − d)}.

7 Robustness

In this section we assume that we are given a sequence (ℓ_{k})_{k} of loss functions that converge pointwise to a loss function ℓ. For each function ℓ_{k} (resp. for ℓ), we call X_{k}^{∗} (resp. X^{∗}) the optimal modified claim which solves the problem in (6.3), and α^{∗,k} (resp. α^{∗}) the corresponding optimal strategy. Throughout the section, we shall also write

Δ := E^{∗}{H(S_{N})} − V_{0} = V_{0}^{∗}(S_{0}) − V_{0}.

We distinguish three cases: the case when the limit loss function ℓ is strictly convex, the case ℓ(x) = x and the case when the limit loss function ℓ is concave. In the first case, we use a technique similar to [3].

7.1 Theorem. Let (ℓ_{k})_{k∈ℕ} be a sequence of strictly convex, C^{1} functions such that ℓ′_{k}(0) = 0 for every k. If lim_{k} ℓ_{k} = ℓ pointwise with ℓ strictly convex, C^{1} and such that ℓ′(0) = 0, then X_{k}^{∗} → X^{∗} almost surely.

Proof. We start by proving that c_{k}^{∗} → c^{∗}. To do this, define

φ_{k}(c) = E \{I_{k} (c\frac{dP^{∗}}{dP} )\}

and note that c^{∗} and the c_{k}^{∗}s have the property that Δ = φ_{k}(c_{k}^{∗}) = φ(c^{∗}) (where φ is defined in the obvious way). Since the probability space Ω is finite, it is straightforward to check that φ_{k}(c) < +∞ for all c ≥ 0 and that φ_{k} is continuous for all k.

Since I_{k} → I pointwise, we have that for every c ∈ ℝ+

I_{k} (c\frac{dP^{∗}}{dP} (ω)) ≤ sup_{k∈ℕ} I_{k} (c\frac{dP^{∗}}{dP} (ω)) ≤ sup_{ω∈Ω} sup_{k∈ℕ} I_{k} (c\frac{dP^{∗}}{dP} (ω)) =: K < ∞,

i.e., the sequence I_{k}(c\frac{dP^{∗}}{dP} (ω)) is dominated. We know from [3] that if ℓ_{k} → ℓ pointwise then I_{k} → I uniformly on compact sets. This means that φ_{k}(c) → φ(c) for all c ≥ 0. Since φ and the φ_{k} are continuous and strictly increasing, it follows that φ_{k}^{−1} → φ^{−1} pointwise. Moreover, these functions are continuous, so c_{k}^{∗} = φ_{k}^{−1}(Δ) → φ^{−1}(Δ) = c^{∗}.

Now we are able to prove the theorem. First of all notice that by Theorem 6.4

X_{k}^{∗} = H(S_{N}) − I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} ) .

Since I_{k} converges uniformly on compact sets to I, the right hand side converges to H(S_{N})−I(c^{∗} \frac{dP^{∗}}{dP}) almost surely, so that one gets X_{k}^{∗} → X^{∗} almost surely.

Now we consider the case ℓ(x) = x. For this case, since I cannot be defined, we can no longer use techniques similar to [3], and we have to develop an ad-hoc technique. Recall from Lemma 6.3 that an optimal claim in this case is given by

X^{∗} = H(S_{N}) − \frac{Δ}{P^{∗}(E)}1_{E}  (7.5)

with E = \{\frac{dP}{dP^{∗}} = c_{es}\}.

7.2 Theorem. Let (ℓ_{k})_{k∈ℕ} be a sequence of strictly convex, C^{1} functions such that ℓ′_{k}(0) = 0 for every k. If lim_{k} ℓ_{k}(x) = x pointwise, then X_{k}^{∗} → X^{∗} almost surely, where X^{∗} is defined as in (7.5).

Proof. Since ℓ_{k}(x) → x pointwise and the ℓ_{k} are convex, then ℓ′_{k}(x) → 1 uniformly on compact sets of ℝ_{+} (see [3] or [7]). Since the ℓ′_{k} are all increasing functions, it is easy to prove that

lim_{k→∞} I_{k}(x) = \begin{cases}0 & if x < 1 \\ +∞ & if x > 1 \end{cases}

The convergence is uniform on compact sets of (0, 1), and also the convergence to +∞ is “uniform” in the sense that for every ε > 0 and M > 0 there exists \overline{k} such that I_{k}(x) > M for all k > \overline{k} and x > 1 + ε.

The convergence above implies that c_{k}^{∗} → c_{es}. Indeed, for every ε > 0 there exists \overline{k} such that 0 < I_{k}(x) < ε for every k > \overline{k}, x ∈ (0, 1 − ε). Then

Δ = E^{∗} \{I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} )\} ≤ ε + E^{∗} \{I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} ) 1_{\{c_{k}^{∗} \frac{dP^{∗}}{dP} ⩾1−ε\}}\}

so E^{∗} \{I_{k}(c_{k}^{∗} \frac{dP^{∗}}{dP})1_{\{c_{k}^{∗} \frac{dP^{∗}}{dP} ≥1−ε\}}\} ≥ Δ − ε; in particular, \{c_{k}^{∗} \frac{dP^{∗}}{dP} ≥ 1 − ε\} is not

empty for all k > k̄, and this means that c_{k}^{∗} max_{ω} \frac{dP^{∗}}{dP} (ω) ≥ 1 − ε, so c_{k}^{∗} < (1 − ε) min_{ω} \frac{dP}{dP^{∗}} (ω) = (1 − ε)c_{es}. Conversely, for all ε > 0 and M > Δ/P^{∗}(E) there exists k̄ such that I_{k}(x) > M for all k > k̄, x > 1 + ε. This implies that

Δ = E^{∗} \{I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} )\} ≥ E^{∗} \{I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} ) 1_{E}\} = I_{k} (\frac{c_{k}^{∗}}{c_{es}} ) P^{∗}(E)

Thus we must have \frac{c_{k}^{∗}}{c_{es}} ≤ 1+ε. In fact, if \frac{c_{k}^{∗}}{c_{es}} > 1+ε, we obtain Δ ≥ M ·P^{∗}(E), but we took M > Δ/P^{∗}(E), so this is absurd. In conclusion, we have that for all ε, (1 − ε)c_{es} ≤ c_{k}^{∗} ≤ (1 + ε)c_{es} from a certain k̄ on, which yields c_{k}^{∗} → c_{es}.

Now we only have to prove that I_{k}(c_{k}^{∗} \frac{dP^{∗}}{dP}) → \frac{Δ}{P^{∗}(E)}1_{E}. Since c_{k}^{∗} → c_{es}, it follows that lim_{k→∞} c_{k}^{∗} \frac{dP^{∗}}{dP} = c_{es} \frac{dP^{∗}}{dP}, which is equal to 1 on E and less than one on E^{c}. Since I_{k} → 0 uniformly on compact sets of (0, 1), we have that I_{k}(c_{k}^{∗} \frac{dP^{∗}}{dP}) → 0 on E^{c}, and the limit is uniform. Thus for all ε > 0 there exists k̄ such that for all k > k̄ we have

Δ − E^{∗} \{I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} ) 1_{E}\} = E^{∗} \{I_{k} (c_{k}^{∗} \frac{dP^{∗}}{dP} ) 1_{E^{c}}\} ∈ (0, ε)

This means that I_{k}(\frac{c_{k}^{∗}}{c_{es}})P^{∗}(E) → Δ, and finally that I_{k}(c_{k}^{∗} \frac{dP^{∗}}{dP}) → \frac{Δ}{P^{∗}(E)}1_{E}. The proof is finished.

7.3 Corollary. Under the assumptions of Theorems 7.1 or 7.2, we have that α_{n}^{∗,k} → α_{n}^{∗} almost surely for all n = 0, …, N − 1.

Proof. Recall that the optimal strategy for the shortfall risk minimization problem with loss function ℓ_{k} is

α_{n}^{∗,k} = \frac{E^{∗}\{X_{k}^{∗}|S_{n−1}u\} − E^{∗}\{X_{k}^{∗}|S_{n−1}d\}}{S_{n−1}(u − d)}.

(see Remark 6.5). Since X_{k}^{∗} → X^{∗} almost surely and the probability space Ω is finite, the conditional expectations above converge almost surely to

\frac{E^{∗}\{X^{∗}|S_{n−1}u\} − E^{∗}\{X^{∗}|S_{n−1}d\}}{S_{n−1}(u − d)},

i.e., α_{n}^{∗,k} → α_{n}^{∗} almost surely for all n = 0, …, N − 1.

In the concave case similar results hold, but the formulation is not straightforward due to the fact that the optimal solution is in general not unique. Recall that in the concave case the strategy is given by (5.2) and thus the optimal modified contingent claim is given by

X^{∗} = H(S_{N}) − \frac{Δ}{P \{ω(α^{∗})\}}1_{\{ω(α^{∗})\}},

where ω(α^{∗}) is the “critical” event defined in (3.4).

7.4 Theorem. Let (ℓ_{k})_{k∈ℕ} be a sequence of concave functions such that ℓ_{k} → ℓ pointwise. Then there exists a (k_{h})_{h} such that X_{k_{h}}^{∗} → X^{∗} almost surely and α_{n}^{∗,k_{h}} → α_{n}^{∗} almost surely for all n = 0, …, N − 1, where X^{∗} is an optimal solution of problem (6.3) and α^{∗} is the corresponding optimal strategy.

Proof. We start by proving that, under the given hypotheses, there exist a limit strategy α^{∗} and a subsequence (α^{∗,k_{h}})_{h} such that α^{∗,k_{h}} ≡ α^{∗} from some \overline{h} on.

Define for every k ∈ ℕ,

φ_{k}^{0}(S_{0}, V_{0}) := p^{N}ℓ_{k} (\frac{V_{0}^{∗}(S_{0}) − V_{0}}{(p^{∗})^{N}} ) − (1 − p)^{N}ℓ_{k} (\frac{V_{0}^{∗}(S_{0}) − V_{0}}{(1 − p^{∗})^{N}} ) ,

so that according to (5.2) the optimal strategy at time 0 for the problem associated to ℓ_{k} is to choose α_{0}^{∗,k} = α_{0}^{(1)} (respectively, α_{0}^{∗,k} = α_{0}^{(2)}) if φ_{k}^{0} ⩽ 0 (respectively, φ_{k}^{0} > 0). Note also that, since α_{0}^{(1)} = \frac{V_{1}^{∗}(S_{0}d)−V_{0}}{S_{0}(d−1)} , α_{0}^{(2)} = \frac{V_{1}^{∗}(S_{0}u)−V_{0}}{S_{0}(u−1)} (see (2.3) for the definition) and V_{n}^{∗}(S_{n}) = E{H(S_{N}) | S_{n}} is independent of the loss function, α_{0}^{(1)} and α_{0}^{(2)} do not depend on the loss function either. We can now distinguish two cases:

• if either φ_{k}^{0} ⩽ 0 or φ_{k}^{0} > 0 from some \overline{k} on, then the sequence (α_{0}^{∗,k})_{k} is constant from \overline{k} on, and thus it converges to a limit α_{0}^{∗},

• if φ_{k}^{0} converges to 0 taking both positive and negative values infinitely many times, consider the subsequence (φ_{k_{h_{0}}}^{0} )_{h_{0}} formed either by the positive or with the non-positive values taken by (φ_{k}^{0})_{k} and the problem reduces to the previous case.

This way, we are dealing with a subsequence (ℓ_{k_{h_{0}}} )_{h_{0}} such that (α_{0}^{∗,k_{h_{0}}} )_{h_{0}} ≡ α_{0}^{∗} from some \overline{h}_{0} on. Note that, since V_{1}^{α} = V_{0} + α_{0}S_{0}(ω_{0} − 1) only depends on the chosen strategy and not on the loss function ℓ, the optimal portfolios of the problems associated to the ℓ_{k_{h_{0}}} s for h_{0} ⩾ \overline{h}_{0} all follow the same evolution in the first time interval.

The existence of the limit strategy α^{∗} now follows by induction on n in a similar way, i.e., defining at each step n = 1, …, N − 1

φ_{k}^{n}(S_{n}, V_{n}) := p^{N−n}ℓ_{k} (\frac{V_{n}^{∗}(S_{n}) − V_{n}}{(p^{∗})^{N−n}} ) − (1 − p)^{N−n}ℓ_{k} (\frac{V_{n}^{∗}(S_{n}) − V_{n}}{(1 − p^{∗})^{N−n}} )

for every k ∈ ℕ, and extracting from the sequence (ℓ_{k_{h_{n−1}}} )_{h_{n−1}} a subsequence

(ℓ_{k_{hn}})_{h_{n}} such that (α_{n}^{∗,k_{hn}})_{h_{n}} ≡ α_{n}^{∗} from some \overline{h}_{n} on.

Now remember that, by Remark 5.4 and by the results recalled in Section 3, in the “concave” case there is a one-to-one correspondence between optimal strategies belonging to the class Π defined in (3.2) and “critical” events defined as in (3.4). Note that this is equivalent to saying that there is a one to one correspondence between strategies α ∈ Π and modified contingent claims of the form

X(α) = H(S_{N}) − \frac{Δ}{P \{ω(α)\}}1_{\{ω(α)\}}.

In particular, since (α_{n}^{∗,k_{hn}})_{h_{n}} ≡ α_{n}^{∗} from some \overline{h}_{n} on, this implies that the optimal modified contingent claims X_{k_{h}}^{∗} corresponding to α^{∗,k_{h}} must converge to the contingent claim X^{∗} corresponding to α^{∗}.

It only remains to show that α^{∗} is an optimal strategy for the problem associated to ℓ. Note that, by Equation (5.1) and the independence between the strategies and the loss functions, the optimal values of the problems associated to the ℓ_{k}s converge to the shortfall associated to strategy α^{∗} under the loss function ℓ. On the other hand, since for every sequence of functions (f_{n})_{n} convergin pointwise to f one has lim inf f_{n} ⩽ inf f, this limit value necessarily must be the optimal value for the problem associated to ℓ.

7.5 Remark. Note that the optimal limit solution X^{∗} is not necessarily unique, i.e., there might be different subsequences of (X_{k}^{∗})_{k} converging to different optimal solutions to the problem (6.3). Note also that one does not need strict concavity of the ℓ_{k}s.

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