A Bayesian adaptive control approach to risk management in a binomial model

Wolfgang J. Runggaldier, Barbara Trivellato and Tiziano Vargiolu

Abstract. We consider the problem of shortfall risk minimization when there is uncertainty about the exact stochastic dynamics of the underlying. Starting from the general discrete time model and the approach described in Runggaldier and Zaccaria (1999), we derive explicit analytic solutions for the particular case of a binomial model when there is uncertainty about the probability of an “up-movement”. The solution turns out to be a rather intuitive extension of that for the classical Cox-Ross-Rubinstein model.

1. Introduction

In incomplete markets the superhedging criterion allows one to eliminate the risk completely, but it requires in general too much initial capital; it corresponds in fact to a min-max-type criterion. One may then ask by how much one can lower the initial cost if one is willing to accept some risk or, dually, what is the risk corresponding to an initial capital less than what is required for superhedging. The shortfall risk minimization approach allows one to deal with these issues.

Given a market with a non-risky and a certain number of risky assets, let H_{T} be a liability to be hedged at some fixed future time T. Denote by V_{T} (φ) the value at T of a portfolio corresponding to a self-financing investment strategy φ, possibly satisfying some additional constraints such as a shortselling prohibition. The problem is to find

J_{0}(S_{0}, V_{0}) := inf_{φ} E_{S_{0},V_{0}}^{P} {ℓ ([H_{T} − V_{T} (φ)]^{+})}  (1)

for a given initial value S_{0} of the asset(s) in the portfolio, for a given initial capital V_{0} and where ℓ(·) is a suitable increasing function such that ℓ(0) = 0 and ℓ(x) > 0 for all x > 0. For ℓ(z) = 1_{{z>0}}, the right hand side in (1) corresponds to the smallest shortfall probability. Problems of the type (1) have recently attracted considerable attention (see e.g. [2, 3, 4, 5, 7, 8, 9, 10, 16]). Let, for a given S_{0},

1991 Mathematics Subject Classification. 91B28, 93E35.

Key words and phrases. Risk management, shortfall risk minimization, restricted information, dynamic programming, Bayesian inference.

The authors wish to thank an anonymous referee for useful comments and suggestions.

V_{0}^{∗}(S_{0}) := inf{V_{0} | J_{0}(S_{0}, V_{0}) = 0}.  (2)

Since J_{0}(S_{0}, V_{0}) = 0 means that H_{T} ≤ V_{T} (φ^{∗}) P -almost surely (where φ^{∗} is the optimal strategy in (1)), V_{0}^{∗}(S_{0}) is the minimal initial capital needed to superhedge the claim. It follows that if V_{0} ≥ V_{0}^{∗}(S_{0}) then V_{T} (φ^{∗}) ≥ H_{T}, P -a.s. For superhedging, the choice of the underlying probabilistic model for the evolution of the risky assets is thus irrelevant as long as it induces probability measures that are equivalent. However for the more general problem of risk minimization in (1) the probabilistic structure of the underlying model matters, but the true model is almost never known exactly. A possibility is then to use a min-max-type criterion as in [4] trying to find

inf_{φ} sup_{P ∈P} E_{S_{0},V_{0}}^{P} {ℓ ([H_{T} − V_{T} (φ)]^{+})} ,

where P is a family of ”real world probability measures”. However, such a criterion does not allow one to incorporate additional information on the underlying model as it becomes successively available.

Thus, an adaptive approach, corresponding to a Bayesian-type criterion, appears more appropriate. Such adaptive approaches have already been dealt with in the literature (see e.g. [2, 3, 4, 5] and, in the context of portfolio optimization, in [1, 11, 12, 13]). In all these papers the uncertainty is only in the stock appreciation rate. The tools are mainly probabilistic in nature (involving also measure transformation) and are based on convex duality. An explicit solution is essentially possible only in simpler cases and transaction costs are not taken into account.

In the present paper we base ourselves on [17] following an approach along the lines of discrete time stochastic adaptive control. In that work the authors give a general description of this approach and they apply it, in particular, to a multinomial model for the risky assets where the probabilities are not known; for the specific case of a binomial model some numerical results are also presented. Here we focus our attention on the binomial model. By neglecting transaction costs (the portfolio is rebalanced only at discrete dates, which limits their impact) and imposing the self-financing requirement as the only constraint on the investment strategies, we succeed in deriving explicit solutions for the optimal investment strategy and for the corresponding minimal value of the shortfall risk in the case ℓ(x) = x. We do this both for the case when the probability p of an “up-movement” is known as well as when it is unknown and, according to the Bayesian approach, treated as a random variable with a Beta-type distribution. We obtain an analytic solution, that turns out to be an interesting variant of the Cox-Ross-Rubinstein (CRR) solution (see e.g. [15]), when the initial capital is insufficient for a perfect hedge.

The paper is organised as follows. In Section 2 we briefly recall some facts from the CRR binomial model, that are relevant for the sequel. In Section 3 we present a solution method to our problem which is based on backwards Dynamic Programming (DP). In Section 4, by assuming ℓ(x) = x, we compute the minimizing admissible strategy as well as an explicit evaluation formula for the minimal discounted shortfall risk in the case when p is known. In Section 5, again by assuming ℓ(x) = x, we compute the minimizing strategy as well as the minimal discounted shortfall risk in the case when p is unknown.

2. The Cox-Ross-Rubinstein binomial model

We consider a discrete-time market model with the set of dates 0, 1, …, N, and with two primary traded securities: a risky asset (a stock) S and a risk-free investment (a bond) B. We assume that the value of the bond is constantly equal to 1 through time, and that the stock price process S satisfies

S_{n+1} = S_{n}ω_{n}, n = 0, …, N,  (3)

where S_{0} > 0 is a given constant and {ω_{n}}_{n=0,...,N} is a sequence of i.i.d. random variables defined on a probability space (Ω, F, P), taking only two real values d and u satisfying 0 < d < 1 < u, with probability law

p := P {ω_{n} = u} = 1 − P {ω_{n} = d}, n = 0, …, N.

We can assume without loss of generality B_{n} ≡ 1 by letting S_{n} be the discounted prices of the asset. Let us then denote by φ_{n} = (η_{n}, ψ_{n}), n = 0, …, N, an investment strategy, where η_{n} stands for the amount of the nonrisky asset and ψ_{n} stands for the number of units of the risky asset that are held in the portfolio in period n. We assume that φ_{n} is adapted to the observation σ-algebra F_{n}^{S} := σ{S_{m}, m ≤ n}, for all n = 0, …, N, and that φ = {φ_{n}}_{n=0,...,N} satisfies the self-financing property

V_{0} = η_{0} + ψ_{0}S_{0}
V_{n+1} := η_{n+1} + ψ_{n+1}S_{n+1} = η_{n} + ψ_{n}S_{n+1}, n = 0, …, N − 1,

where V_{0} is a given constant, representing the initial value of the portfolio. We shall denote by A_{ad} the set of all self-financing strategies (the admissible investment strategies).

It is a rather classical result that V follows the dynamics

V_{n+1} = V_{n} + ψ_{n}S_{n}(ω_{n} − 1) =: V_{n+1}(V_{n}, S_{n}, ω_{n}, ψ_{n}),

so that one can restrict oneself to just the decision variable ψ_{n}.

Consider a European contingent claim H(S_{N}) and let P^{∗} be the martingale measure for our model. It is well known that P^{∗} corresponds to

P^{∗}\{ω_{n} = u\} = p^{∗} := \frac{1 − d}{u − d}, P^{∗}\{ω_{n} = d\} = 1 − p^{∗} = \frac{u − 1}{u − d}, n = 0, …, N,

(see e.g. [15, Ch. 2]). The arbitrage free price V_{n}^{∗} of H(S_{N}) at time n, where n = 0, …, N − 1, is given by the Cox-Ross-Rubinstein (CRR) evaluation formula

V_{n}^{∗}(S_{n}) = E^{∗}[H(S_{N})|S_{n}],  (4)

where by E^{∗} we denote the expectation with respect to P^{∗}. In particular, V_{0}^{∗}(S_{0}) is the minimal initial value of the portfolio needed to replicate the claim defined in (2). At time n, n = 0, …, N − 1, the replicating strategy ψ_{n} is given by

ψ_{n} = \frac{V_{n+1}^{∗}(S_{n}u) − V_{n+1}^{∗}(S_{n}d)}{S_{n}(u − d)}.  (5)

If V_{0} < V_{0}^{∗}(S_{0}), then the replication of the terminal payoff is not possible. In this situation, an investor may be interested in analyzing the shortfall risk defined as the expectation of the terminal deficit weighted by some loss function. Let us then introduce this problem in more detail.

Denote by V_{N}(φ) = η_{N} + ψ_{N}S_{N} the value of the portfolio at time N corresponding to an admissible investment strategy φ. The minimal shortfall risk is defined as

J_{0}(S_{0}, V_{0}) := min_{φ∈A_{ad}} E_{S_{0},V_{0}}^{P} {ℓ ([H(S_{N}) − V_{N}(φ)]^{+})} ,  (6)

for a given initial value S_{0} of the risky asset in the portfolio and a given initial capital V_{0} < V_{0}^{∗}(S_{0}), where ℓ(·) ≥ 0 is a suitable loss function, that is an increasing function such that ℓ(0) = 0 and ℓ(x) > 0 for all x > 0.

In this paper we consider the optimization problem (6) by assuming either that the probability p is known or that it is not. For the case when p is unknown, we adopt a Bayesian-type approach which allows us to incorporate additional information on the underlying model as it becomes successively available.

3. The dynamic programming algorithm

In this section we provide a DP algorithm to compute a solution to our problem (6) both for the case when p is known and when it is not (see [6] for an analogous algorithm for the case of superhedging with transaction costs). In the case where p is unknown, adopting the Bayesian point of view, we use the ordinary Bayes formula to successively update the initial (prior) density h(p) of p on the basis of {F_{n}^{S}}_{n=0,...,N}. This leads to what is called the Bayesian DP algorithm (see e.g. [14, 18]).

3.1. DP algorithm when p is known The DP algorithm proceeds backwards according to the following steps:

J_{N}(s, v) = ℓ((H(s) − v)^{+}),
J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{S_{n−1},V_{n−1}}{J_{n}(S_{n}, V_{n})} =
= inf_{ψ_{n−1}}{pJ_{n}(S_{n−1}u, V_{n−1} + ψ_{n−1}S_{n−1}(u − 1))
+(1 − p)J_{n}(S_{n−1}d, V_{n−1} + ψ_{n−1}S_{n−1}(d − 1))}, (7)

for n = N, …, 1.

3.2. DP algorithm when p is unknown The Bayesian DP algorithm proceeds similarly to the DP algorithm:

J_{N}(s, v) = ℓ((H(s) − v)^{+}),
J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{S_{n−1},V_{n−1}}^{P_{n−1}}{J_{n}(S_{n}, V_{n})}.  (8)

Now p too is a random variable, and its distribution depends on the information F_{n}^{S} up to time n. We incorporate this information in the probability measure P_{n}, that depends also on the distribution of p. Since P {ω_{n}|p} is the Binomial distribution, a conjugate family of distributions of p is that of the Beta distributions. With a prior density

h_{0}(p) ∝ p^{α_{0}}(1 − p)^{β_{0}},

with α_{0}, β_{0} ≥ 0, the posterior density in period n becomes

h_{n}(p) ∝ p^{α_{n}}(1 − p)^{β_{n}},

where, denoting by u_{n} the total number of “up-movements” (u_{0} := 0) cumulated up to time n,

α_{n} = α_{0} + u_{n} , β_{n} = β_{0} + n − u_{n}.

In particular, for n = 0 and α_{0} = β_{0} = 0 the prior density h_{0}(p) becomes the uniform density.

Since the values of p enter the DP recursions linearly, by the “smoothing property” of conditional expectations it is easily seen that, also in the present case, the DP recursions are given by the previous steps, except that p has to be replaced in J_{n−1}(S_{n−1}, V_{n−1}), n = N, …, 1, by E_{α_{n−1},β_{n−1}}[p] := E [p|F_{n−1}^{S}] and 1 − p by 1 − E_{α_{n−1},β_{n−1}}[p], where

E_{α,β}[p] = \frac{α + 1}{α + β + 2}.

4. Explicit solutions when p is known

In this section we are concerned with the evaluation of the minimal discounted shortfall risk (6) and the corresponding strategy in the case when ℓ(x) = x. Due to the possibility of making direct calculations on the DP algorithm steps, we can derive explicit evaluation formulas. These evaluation formulas are simple and meaningful, showing explicitly what was to be expected: that is, the shortfall risk is decreasing with respect to the initial capital (we will show that such dependence is linear), and it is always equal to zero when the level of the initial capital is greater than or equal to V_{0}^{∗}(S_{0}). We first consider the case when there is complete information on the underlying market and we compare our results with the wellknown results on perfect hedging of a European contingent claim.

In the following theorem we give some formulas for the optimal discounted shortfall risk and for the corresponding investment strategy.

Theorem 4.1. Consider a European contingent claim H on a stock whose price S is assumed to follow the CRR binomial model (3). Let V_{n}^{∗}(S_{n}), where n = 0, …, N − 1, be the arbitrage free price at time n defined by the CRR evaluation formula (4). Assume the loss function ℓ(·) in (6) is the identity function ℓ(x) = x. Then

i) if p > p^{∗}, then

J_{n}(S_{n}, V_{n}) = (\frac{1 − p}{1 − p^{∗}} )^{m} [V_{n}^{∗}(S_{n}) − V_{n}]^{+} ,  (9)

for n = 0, …, N − 1. In particular, for n = 0 the minimal discounted shortfall risk is

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

where V_{0}^{∗}(S_{0}) = C_{0}^{∗}(S_{0}). The minimizing investment strategy is given by

ψ_{n}^{1} = \frac{V_{n+1}^{∗}(S_{n}u) − V_{n}}{S_{n}(u − 1)} ,  (10)
for n = 0, …, N − 1;
ii) if p < p^{∗}, then

J_{n}(S_{n}, V_{n}) = (\frac{p}{p^{∗}} )^{m} [V_{n}^{∗}(S_{n}) − V_{n}]^{+} ,

for n = 0, …, N − 1. In particular, for n = 0 the minimal discounted shortfall risk is

J_{0}(S_{0}, V_{0}) = (\frac{p}{p^{∗}} )^{N} [V_{0}^{∗}(S_{0}) − V_{0}]^{+} ,

where V_{0}^{∗}(S_{0}) = C_{0}^{∗}(S_{0}). The minimizing investment strategy is given by

ψ_{n}^{2} = \frac{V_{n+1}^{∗}(S_{n}d) − V_{n}}{S_{n}(d − 1)} ,  (11)
for n = 0, …, N − 1.

Proof. We start from n = N − 1 by considering expression (7) of J_{N−1}(S_{N−1}, V_{N−1}). The function to be minimized in (7) is a linear combination of the two piecewise affine functions

[H(S_{N−1}u) − V_{N−1} − ψ_{N−1}S_{N−1}(u − 1)]^{+}  (12)

and

[H(S_{N−1}d) − V_{N−1} − ψ_{N−1}S_{N−1}(d − 1)]^{+}  (13)

The function in (12) is decreasing for ψ_{N−1} less than ψ_{N−1}^{1} in Equation (10), since u > 1 and S_{N−1} is positive, and thereafter it is equal to zero, while the function in (13) is equal to zero for ψ_{N−1} less than ψ_{N−1}^{2} in Equation (11), since d < 1 and S_{N−1} is positive, and from there on it is increasing. Therefore, if ψ_{N−1}^{1} ≤ ψ_{N−1}^{2} or, equivalently, if V_{N−1} ≥ E^{∗}[H(S_{N})], then both strategies ψ_{N−1}^{1} and ψ_{N−1}^{2} are optimal as well as any admissible strategy between them. If V_{N−1} < E^{∗}[H(S_{N})], then in order to establish the infimum in (7) it suffices to analyze the sign of the slope of

p[H(S_{N−1}u) − V_{N−1} − ψ_{N−1}S_{N−1}(u − 1)]+
+(1 − p)[H(S_{N−1}d) − V_{N−1} − ψ_{N−1}S_{N−1}(d − 1)],

which is given by the expression

S_{N−1}[p(d − u) + 1 − d].  (14)

If (14) is less than zero, or, equivalently, if p > p^{∗}, then the infimum in (7) is achieved at ψ_{N−1}^{1}. If this is the case, by putting ψ_{N−1}^{1} in (7) we obtain

J_{N−1}(S_{N−1}, V_{N−1}) = \frac{1 − p}{1 − p^{∗}} [C_{N−1}^{∗}(S_{N−1}) − V_{N−1}]^{+} .

Conversely, if p < p^{∗} then the infimum in (7) is achieved at ψ_{N−1}^{2}, and we obtain

J_{N−1}(S_{N−1}, V_{N−1}) = \frac{p}{p^{∗}} [C_{N−1}^{∗}(S_{N−1}) − V_{N−1}]^{+} .

This shows that formula (9) is true for n = N − 1.

We now proceed by backward induction with respect to n. Assume p > p^{∗} from now on. Assume that equality (9) holds for n, where n = N − 1, …, 1, with the minimizing strategy given by (10). We show that it also holds for n − 1, with the same strategy as (10) for n − 1. From (7) of the DP algorithm we have

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}}{p J_{n} (S_{n−1}u, V_{n}(V_{n−1}, S_{n−1}, u, ψ_{n−1})) +
+(1 − p) J_{n} (S_{n−1}d, V_{n}(V_{n−1}, S_{n−1}, d, ψ_{n−1}))},

which implies, by induction,

J_{n−1}(S_{n−1}, V_{n−1}) = (\frac{1 − p}{1 − p^{∗}} )^{m} inf_{ψ_{n−1}} \{p [V_{n}^{∗}(S_{n−1}u) − V_{n−1}−  (15)

−ψ_{n−1}S_{n−1}(u − 1)]^{+} + (1 − p) [V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1)]^{+}} .

Using the same arguments as in the first step, and taking into account that the slope of the expression

p [V_{n}^{∗}(S_{n−1}u) − V_{n−1} − ψ_{n−1}S_{n−1}(u − 1)] +
+(1 − p) [V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1)]

is again given by (14) with S_{N−1} replaced by S_{n−1}, we have that, under the assumption p > p^{∗}, the infimum in expression (15) is achieved at ψ_{n−1}^{1} given by (10). Putting ψ_{n−1}^{1} in (15) we easily obtain

J_{n−1}(S_{n−1}, V_{n−1}) = (\frac{1 − p}{1 − p^{∗}} )^{m+1} [V_{n−1}^{∗}(S_{n−1}) − V_{n−1}]^{+} .

This ends the proof of i). The proof of ii) can be obtained by using arguments similar to those of i). We only observe that, under the assumption p < p^{∗}, the infimum in the DP algorithm steps is achieved at ψ_{n−1}, for n = 0, …, N − 1, satisfying

V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1) = 0.
□

Remark 4.2. Notice that this approach is linked to the CRR model: in fact, by calculating the expected shortfall risk under the historical probability P, we arrive at an expression containing the expected price of the claim under the risk-neutral probability P^{∗}. Moreover, the hedging strategy in this case is similar to (5), which is the one of the CRR model: in fact in the CRR model the hedging strategy ψ is equal to the ratio between the difference of the expected prices of the claim in the two possible future outcomes and the different prices of the underlying; here ψ is equal to the ratio between the difference of the expected price of the claim in one of the possible future outcomes and the value the portfolio would have if it were invested in the bond B, and the difference between the price of the underlying in the same future outcome considered before and its present price as if it were invested in the bond B. In other words, it is as if we were hedging a claim having a payoff that in each state of nature could be the one of the original claim or the money corresponding to the present value of the portfolio; in the same way, the underlying could either assume the value corresponding to the value of the derivative, or the value corresponding to a riskless investment.

Remark 4.3. Differently from [8], here we have not imposed that V ≥ 0. This leads to different results: in fact, if we imposed V ≥ 0, we would have obtained (as they do) an optimal strategy equal to the replicating strategy of a modified claim that is between zero and the original claim; we have a different strategy, that in general gives an optimal expected shortfall lower than they have. In particular, our strategy succeeds in replicating perfectly the claim in all the states of nature but the least probable one (see [7] for an explicit proof), so that the expected shortfall comes entirely from this state of nature. However, if V_{0} is near V_{0}^{∗}, then V remains positive at all times prior to the maturity N, so the two strategies that we obtain by imposing or not the constraint V ≥ 0 coincide.

5. Explicit solutions when p is unknown

The formulas given in the following theorem for the optimal discounted shortfall risk are similar to those given in Theorem 4.1, and the minimizing investment strategies are the same. However, while in the previous case only two alternatives for the possible values of p^{∗} were considered, i.e. p^{∗} > p and p^{∗} < p, here at each step n we have to consider several alternatives according to the estimates of p, each of them leading to a different formula for J_{n}(S_{n}, V_{n}).

Theorem 5.1. Consider the assumptions of Theorem 4.1 for the case when p is unknown with a prior h_{0}(p) ∝ p^{α_{0}}(1 − p)^{β_{0}}, with α_{0}, β_{0} ≥ 0. Then

i) if p^{∗} < E_{α_{n},β_{n}+m−1} [p], then

J_{n}(S_{n}, V_{n}) = ( ∏_{j=0}^{m−1} \frac{1 − E_{α_{n},β_{n}+j}[p]}{1 − p^{∗}} ) [V_{n}^{∗}(S_{n}) − V_{n}]^{+} .  (16)

The minimizing investment strategy is given by (10). ii) if E_{α_{n}+i,β_{n}+m−1−i}[p] < p^{∗} < E_{α_{n}+i+1,β_{n}+m−2−i}[p], where i = 0, …, m −

2, then

J_{n}(S_{n}, V_{n}) = ( ∏_{j=0}^{m−2−i} \frac{1 − E_{α_{n},β_{n}+j}[p]}{1 − p^{∗}} ) ×

× (∏_{j=0}^{i} \frac{E_{α_{n}+j,β_{n}+m−1−i}[p]}{p^{∗}} ) [V_{n}^{∗}(S_{n}) − V_{n}]^{+} =  (17)

= (∏_{j=0}^{i} \frac{E_{α_{n}+j,β_{n}}[p]}{p^{∗}} ) ( ∏_{j=0}^{m−2−i} \frac{1 − E_{α_{n}+i+1,β_{n}+j}[p]}{1 − p^{∗}} ) [V_{n}^{∗}(S_{n}) − V_{n}]^{+}(18)

Both the strategies (10) and (11) are optimal as well as any admissible strategy between them. iii) if p^{∗} > E_{α_{n}+m−1,β_{n}} [p], then

J_{n}(S_{n}, V_{n}) = ( ∏_{j=0}^{m−1} \frac{E_{α_{n}+j,β_{n}}[p]}{p^{∗}} ) [V_{n}^{∗}(S_{n}) − V_{n}]^{+}.  (19)

The minimizing investment strategy is given by (11).

Proof. We start from n = N − 1 by considering the expression

J_{N−1}(S_{N−1}, V_{N−1}) =  (20)
= inf_{ψ_{N−1}} E_{α_{N−1},β_{N−1}} [p] [H(S_{N−1}u) − V_{N−1} − ψ_{N−1}S_{N−1}(u − 1)]^{+} +
+ (1 − E_{α_{N−1},β_{N−1}} [p]) [H(S_{N−1}d) − V_{N−1} − ψ_{N−1}S_{N−1}(d − 1)]^{+} ,

whose infimum is achieved, by using arguments similar to those in the proof of Theorem 4.1, at ψ_{n−1}^{1} in Equation (10) if E_{α_{N−1},β_{N−1}} [p] > p^{∗} and at ψ_{n−1}^{2} in Equation (11) if E_{α_{N−1},β_{N−1}} [p] < p^{∗}. Therefore, putting the minimizing strategy in (20) we have

J_{N−1}(S_{N−1}, V_{N−1}) = \frac{1 − E_{α_{N−1},β_{N−1}} [p]}{1 − p^{∗}} [C_{N−1}^{∗}(S_{N−1}) − V_{N−1}]^{+}

if E_{α_{N−1},β_{N−1}} [p] > p^{∗}, and

J_{N−1}(S_{N−1}, V_{N−1}) = \frac{E_{α_{N−1},β_{N−1}} [p]}{p^{∗}} [C_{N−1}^{∗}(S_{N−1}) − V_{N−1}]^{+}

if E_{α_{N−1},β_{N−1}} [p] < p^{∗}, showing that formulas (16)–(19) are true for n = N − 1 (let us observe that the validity of ii) is trivial for n = N − 1).

We now proceed by induction with respect to n. We assume that equalities (16)—(19) hold for n, where n = 1, …, N − 1, and we show that they also hold for n − 1. As regards alternative ii), we shall only prove formula (17). Indeed, it is not difficult to check (we omit calculations) the validity of the equality

( ∏_{j=0}^{m−2−i} \frac{1 − E_{α_{n},β_{n}+j} [p]}{1 − p^{∗}} ) (∏_{j=0}^{i} \frac{E_{α_{n}+j,β_{n}+m−1−i} [p]}{p^{∗}} ) =

= (∏_{j=0}^{i} \frac{E_{α_{n}+j,β_{n}} [p]}{p^{∗}} ) ( ∏_{j=0}^{m−2−i} \frac{1 − E_{α_{n}+i+1,β_{n}+j} [p]}{1 − p^{∗}} ) ,

corresponding to, respectively, (17) and (18) of ii). Let us remark that, as we shall see below, formula (17) (respectively, (18)) is obtained by always choosing strategy (10) (respectively, (11)) at each step n where, for some i ∈ {0, …, N − n − 2}, we have

E_{α_{n}+i,β_{n}+N−n−1−i} [p] < p^{∗} < E_{α_{n}+i+1,β_{n}+N−n−2−i} [p] .

This choice will be possible since, under the above condition for p^{∗}, both the strategies (10) and (11), as well as any admissible strategy between them, will be optimal. In fact, other representation formulas for J_{n}, different from (17) and (18), but equivalent to them, could be possible, each of them corresponding to a different procedure for selecting a minimizing strategy between (10) and (11).

From (8), we have

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{α_{n−1},β_{n−1}} [p] J_{n}(S_{n−1}u, V_{n}(V_{n−1}, S_{n−1}, u, ψ_{n−1}))+

+(1 − E_{α_{n−1},β_{n−1}}[p])J_{n} (S_{n−1}d, V_{n}(V_{n−1}, S_{n−1}, d, ψ_{n−1})) ,

which implies, by induction,

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{α_{n−1},β_{n−1}} [p] A (α_{n−1} + 1, β_{n−1}) ×

× [V_{n}^{∗}(S_{n−1}u) − V_{n} (V_{n−1}, S_{n−1}, u, ψ_{n−1})]^{+} + (1 − E_{α_{n−1},β_{n−1}} [p]) ×
×A (α_{n−1}, β_{n−1} + 1) [V_{n}^{∗}(S_{n−1}d) − V_{n}(V_{n−1}, S_{n−1}, d, ψ_{n−1})]^{+},

where (choosing e.g. (17) when taking into account alternative ii))

A(α, β) = ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α,β+j} [p]}{1 − p^{∗}} ) 1_{\{p^{∗}<E_{α,β+N−n−1}[p]\}}+

+ ∑_{k=0}^{N−n−2} ( ∏_{j=0}^{N−n−2−k} \frac{1 − E_{α,β+j} [p]}{1 − p^{∗}} ) (∏_{j=0}^{k} \frac{E_{α+j,β+N−n−1−k} [p]}{p^{∗}} ) ×

×1_{{E_{α+k,β+N−n−1−k}[p]<p^{∗}<E_{α+k+1,β+N−n−2−k}[p]}} +

+ ( ∏_{j=0}^{N−n−1} \frac{E_{α+j,β} [p]}{p^{∗}} ) 1_{\{p^{∗}>E_{α+N−n−1,β}[p]\}},

Proof of part i). If p^{∗} < E_{α_{n−1},β_{n−1}+N−n} [p] then it is not difficult to check that only the first indicator functions in both A(α_{n−1} +1, β_{n−1}) and A(α_{n−1}, β_{n−1} +1) are equal to one, while all the others are equal to zero. Therefore

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{α_{n−1},β_{n−1}}[p] ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α_{n−1}+1,β_{n−1}+j} [p]}{1 − p^{∗}} ) ×

× [V_{n}^{∗}(S_{n−1}u) − V_{n−1} − ψ_{n−1}S_{n−1}(u − 1)]^{+} +

+ (1 − E_{α_{n−1},β_{n−1}} [p]) ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α_{n−1},β_{n−1}+j+1} [p]}{1 − p^{∗}} ) ×

× [V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1)]^{+} .  (21)

The sign of the slope of this linear expression in ψ_{n−1} is given by

E_{α_{n−1},β_{n−1}} [p] ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α_{n−1}+1,β_{n−1}+j}[p]}{1 − p^{∗}} ) (1 − u)+  (22)

+ (1 − E_{α_{n−1},β_{n−1}} [p]) ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α_{n−1},β_{n−1}+j+1}[p]}{1 − p^{∗}} ) (1 − d).

After making elementary manipulations, one can rewrite (22) as the product of a suitable strictly positive term and (p^{∗} − E_{α_{n−1},β_{n−1}+N−n} [p]), which is less than zero by assumption. Therefore, using the same arguments as those in the proof of Theorem 4.1, we have that the infimum in (21) is achieved at ψ_{n−1}^{1} given by formula (10). Putting ψ_{n−1}^{1} in (21) we easily obtain

J_{n−1}(S_{n−1}, V_{n−1}) = ( ∏_{j=0}^{N−n} \frac{1 − E_{α_{n−1},β_{n−1}+j}[p]}{1 − p^{∗}} ) [V_{n−1}^{∗}(S_{n−1}) − V_{n−1}]^{+} ,

which gives formula (16) with n replaced by n − 1. This ends the proof of i).

Proof of part iii). It is not difficult to argue that the proof of iii) proceeds along the same lines as the proof of i). We only observe that if p^{∗} > E_{α_{n−1}+N−n,β_{n−1}}[p] (alternative iii) for n − 1) then only the last indicator functions in both A(α_{n−1} + 1, β_{n−1}) and A(α_{n−1}, β_{n−1} + 1) are equal to one, while all the others are equal to zero. Moreover, the sign of the slope of the linear expression which arises from the DP algorithm is positive, so that the infimum is achieved at ψ_{n−1} given by formula (11), i.e. satisfying

V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1) = 0.

Proof of part ii). Now we only have to prove ii). As we shall see, the linear expressions arising from the DP algorithm will have angular coefficients equal to zero, giving the possibility to choose, as minimizing strategy, any admissible strategy between (10) and (11). In particular, choosing strategy (10) we shall prove the validity of formula (17). If

E_{α_{n−1}+i,β_{n−1}+N−n−i} [p] < p^{∗} < E_{α_{n−1}+i+1,β_{n−1}+N−n−1−i} [p] ,

for some i ∈ {0, …, N −n−1}, then the last indicator function in A(α_{n−1}+1, β_{n−1}) and the first indicator function in A(α_{n−1}, β_{n−1} + 1) are equal to zero, i.e.

1_{{p^{∗}>E_{α_{n−1}+N−n,β_{n−1}} [p]}} ≡ 0

and

1_{{p^{∗}<E_{α_{n−1},β_{n−1}+N−n}[p]}} ≡ 0

Moreover, we have to distinguish between the following three alternatives: 1. If i = 0, then we have

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{α_{n−1},β_{n−1}} [p] ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α_{n−1}+1,β_{n−1}+j} [p]}{1 − p^{∗}} ) ×

× [V_{n}^{∗}(S_{n−1}u) − V_{n−1} − ψ_{n−1}S_{n−1}(u − 1)]^{+} +

+ (1 − E_{α_{n−1},β_{n−1}}[p]) ( ∏_{j=0}^{N−n−2} \frac{1 − E_{α_{n−1},β_{n−1}+1+j} [p]}{1 − p^{∗}} ) ×

×\frac{E_{α_{n−1},β_{n−1}+N−n} [p]}{p^{∗}} [V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1)]^{+} .

Since the slope of this linear expression in ψ_{n−1} is equal to zero (we omit calculations), the infimum is achieved at both (10) and (11) as well as at any admissible strategy between them. Choosing e.g. (10) we obtain

J_{n−1}(S_{n−1}, V_{n−1}) = ( ∏_{j=0}^{N−n−1} \frac{1 − E_{α_{n−1},β_{n−1}+j}[p]}{1 − p^{∗}} ) ×

×\frac{E_{α_{n−1},β_{n−1}+N−n}[p]}{p^{∗}} [V_{n−1}^{∗}(S_{n−1}) − V_{n−1}]^{+} ,

which gives formula (17) with n replaced by n − 1 and i = 0.

2. If i ∈ {1, …, N − n − 2}, then we have

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{α_{n−1},β_{n−1}} [p] ( ∏_{j=0}^{N−n−1−i} \frac{1 − E_{α_{n−1}+1,β_{n−1}+j} [p]}{1 − p^{∗}} )

× (∏_{j=0}^{i−1} \frac{E_{α_{n−1}+1+j,β_{n−1}+N−n−i} [p]}{p^{∗}} ) ×

× [V_{n}^{∗}(S_{n−1}u) − V_{n−1} − ψ_{n−1}S_{n−1}(u − 1)]^{+} +

+ (1 − E_{α_{n−1},β_{n−1}} [p]) ( ∏_{j=0}^{N−n−2−i} \frac{1 − E_{α_{n−1},β_{n−1}+1+j} [p]}{1 − p^{∗}} ) ×

× (∏_{j=0}^{i} \frac{E_{α_{n−1}+j,β_{n−1}+N−n−i} [p]}{p^{∗}} ) [V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1)]^{+} .

The slope of this linear expression in ψ_{n−1} is again equal to zero. Choosing e.g. (10) as ψ_{n−1} we obtain

J_{n−1}(S_{n−1}, V_{n−1}) = ( ∏_{j=0}^{N−n−1−i} \frac{1 − E_{α_{n−1},β_{n−1}+j} [p]}{1 − p^{∗}} ) ×

× (∏_{j=0}^{i} \frac{E_{α_{n−1}+j,β_{n−1}+N−n−i} [p]}{p^{∗}} ) [V_{n−1}^{∗}(S_{n−1}) − V_{n−1}]^{+} ,

which gives formula (17) with n replaced by n−1 and i ∈ {1, …, N−n−2}.

3. If i = N − n − 1, then we have

J_{n−1}(S_{n−1}, V_{n−1}) = inf_{ψ_{n−1}} E_{α_{n−1},β_{n−1}} [p] (\frac{1 − E_{α_{n−1}+1,β_{n−1}}[p]}{1 − p^{∗}}) ×

× ( ∏_{j=0}^{N−n−2} \frac{E_{α_{n−1}+1+j,β_{n−1}+1} [p]}{p^{∗}} ) ×

× [V_{n}^{∗}(S_{n−1}u) − V_{n−1} − ψ_{n−1}S_{n−1}(u − 1)]^{+} + (1 − E_{α_{n−1},β_{n−1}} [p]) ×

× ( ∏_{j=0}^{N−n−1} \frac{E_{α_{n−1}+j,β_{n−1}+1} [p]}{p^{∗}} ) [V_{n}^{∗}(S_{n−1}d) − V_{n−1} − ψ_{n−1}S_{n−1}(d − 1)]^{+} .

The slope of this linear expression in ψ_{n−1} is again equal to zero. Choosing e.g. (10) as ψ_{n−1} we obtain

J_{n−1}(S_{n−1}, V_{n−1}) = \frac{1 − E_{α_{n−1},β_{n−1}} [p]}{1 − p^{∗}}×

× ( ∏_{j=0}^{N−n−1} \frac{E_{α_{n−1}+j,β_{n−1}+1} [p]}{p^{∗}} ) [V_{n−1}^{∗}(S_{n−1}) − V_{n−1}]^{+} .

The inductive step is complete, and so is the proof. □

Corollary 5.2. Letting n = 0 in Theorem 5.1, the minimal discounted shortfall risk is

i) if p^{∗} < E_{α_{0},β_{0}+N−1} [p], then

J_{0}(S_{0}, V_{0}) = ( ∏_{j=0}^{N−1} \frac{1 − E_{α_{0},β_{0}+j} [p]}{1 − p^{∗}} ) [V_{0}^{∗}(S_{0}) − V_{0}]^{+} ;

ii) if E_{α_{0}+i,β_{0}+N−1−i}[p] < p^{∗} < E_{α_{0}+i+1,β_{0}+N−2−i}[p], where i = 0, …, N −2, then

J_{0}(S_{0}, V_{0}) = ( ∏_{j=0}^{N−2−i} \frac{1 − E_{α_{0},β_{0}+j} [p]}{1 − p^{∗}} ) ×

× (∏_{j=0}^{i} \frac{E_{α_{0}+j,β_{0}+N−1−i} [p]}{p^{∗}} ) [V_{0}^{∗}(S_{0}) − V_{0}]^{+} =

= (∏_{j=0}^{i} \frac{E_{α_{0}+j,β_{0}} [p]}{p^{∗}} ) ( ∏_{j=0}^{N−2−i} \frac{1 − E_{α_{0}+i+1,β_{0}+j} [p]}{1 − p^{∗}} ) [V_{0}^{∗}(S_{0}) − V_{0}]^{+} ;
iii) if p^{∗} > E_{α_{0}+N−1,β_{0}} [p], then

J_{0}(S_{0}, V_{0}) = ( ∏_{j=0}^{N−1} \frac{E_{α_{0}+j,β_{0}} [p]}{p^{∗}} ) [V_{0}^{∗}(S_{0}) − V_{0}]^{+} .

The minimizing investment strategies are those of Theorem 5.1.

Remark 5.3. In the case when p is unknown we obtain formulas that are similar to those of the case when p is known, with the following difference: while in the case when p is known we know immediately whether p > p^{∗} or not, and that relation either holds at all times n or does not hold, when p is unknown we cannot know whether p > p^{∗} or not, so we must use the Bayes estimators of p. Obviously these estimators change over time, so we obtain products of different factors depending on p^{∗} and on the Bayes estimators of p, while in the case when p is known these factors are all equal either to p/p^{∗} or to (1 − p)/(1 − p^{∗}). We also notice that the investment strategies are equal to those when p is known; this happens because

A Bayesian approach to risk management 15

the Bayes estimators of p enter linearly in the DP algorithm and do not modify the optimum.

Remark 5.4. Also when p is unknown, if we impose V ≥ 0, we obtain different results: in fact in this case too our strategy succeeds in replicating perfectly the claim in all the states of nature but one (see [7]); with the constraint V ≥ 0, it may happen that this is not possible, so that the optimal solution gives a shortfall higher than in our case. However, also in this case (as in the case when p is known), if V_{0} is near V_{0}^{∗}, then V remains positive at all times prior to the maturity N, so the two strategies that we obtain by imposing or not the constraint V ≥ 0 coincide.

References

[1] S. Browne and W. Whitt, Portfolio choice and the Bayesian Kelly criterion, Adv. Appl. Prob., 28 (1996), 1145–1176.

[2] J. Cvitanić, Methods of partial hedging, Asia-Pacific Financial Markets, 6 (1999), 7–35.

[3] J. Cvitanić, Minimizing expected loss of hedging in incomplete and constrained markets, SIAM J. Control Optimization, 38 (4) (2000), 1050–1066.

[4] J. Cvitanić and I. Karatzas, On dynamic measures of risk, Finance and Stochastics, 3 (4) (1999), 451–482.

[5] J. Cvitanić and G. Spivak, Maximizing the probability of a perfect hedge, The Annals of Applied Probability, 9 (4) (1999), 1303–1328.

[6] C. Edirisinghe, V. Naik and R. Uppal, Optimal replication of options with transactions costs and trading restrictions, J. of Financial and Quantitative Analysis, 28 (1993), 117–138.

[7] G. Favero, Shortfall risk minimization under model uncertainty in the binomial case: adaptive and robust approaches, Preprint, University of Padova, Department of Pure and Applied Mathematics (2000)

[8] H. Föllmer and P. Leukert, Quantile hedging, Finance and Stochastics, 3 (3) (1999), 251–273.

[9] H. Föllmer and P. Leukert, Efficient hedging: cost versus shortfall risk, Finance and Stochastics, 4 (2) (2000), 117–146

[10] P. Guasoni, Risk minimization under transaction costs, Preprint, Scuola Normale Superiore di Pisa, (1999).

[11] I. Karatzas and X. Zhao, Bayesian adaptive portfolio optimization, to appear in Advances in Mathematical Finance, Cambridge University Press, (1999).

[12] P. Lakner, Utility maximization with partial information, Stochastic Processes and their Applications, 56 (1995), 247–273.

[13] P. Lakner, Optimal trading strategy for an investor: the case of partial information, Stochastic Processes and their Applications, 76 (1998), 77–97.

[14] J. J. Martin, Bayesian decision problems and Markov chains, John Wiley and Sons, Inc., New York, 1967.

16 W. Runggaldier, B. Trivellato and T. Vargiolu

[15] M. Musiela and M. Rutkowski, Martingale methods in financial modelling, Springer, Berlin, 1997.

[16] H. Pham, Dynamic L^{p}-hedging in discrete time under cone constraints, SIAM J. Control Optimization, 38 (3) (2000), 665–682.

[17] W. J. Runggaldier and A. Zaccaria, A stochastic control approach to risk management under restricted information, Mathematical Finance, 10 (2000), 277–288.

[18] K. M. Van Hee, Bayesian control of Markov chains, Mathematical Centre Tracts 95, Amsterdam, 1978.

Department of Pure and Applied Mathematics University of Padova via Belzoni 7 I-35131 Padova, Italy

E-mail address: runggal@math.unipd.it
barbara@math.unipd.it
vargiolu@math.unipd.it
URL: http://www.math.unipd.it/~probab/home/probgroup.html