# Optimal Static Quadratic Hedging

###### Abstract

We propose a flexible framework for hedging a contingent claim by holding static positions in vanilla European calls, puts, bonds, and forwards. A model-free expression is derived for the optimal static hedging strategy that minimizes the expected squared hedging error subject to a cost constraint. The optimal hedge involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general Markov diffusion market. To illustrate the versatility of our approach, we present several numerical examples, including hedging path-dependent options and options written on a correlated asset.

Keywords: static hedging, leveraged ETF options, substitute hedging

JEL Classification: C52, D81, G11, G13

Mathematics Subject Classification (2010): 91G20, 91G80, 93E20

## 1 Introduction

Hedging derivatives using a static portfolio of standard financial instruments is a well-known alternative to dynamically hedging with the underlying asset. A static hedging portfolio is easy to construct and requires no continuous monitoring of the underlying or rebalancing over time. As such, static hedging strategies are more robust to significant underlying movements through market turbulence. Furthermore, static hedging portfolios are often useful for establishing no-arbitrage relationships or bounds for exotic derivatives. This idea dates back to Breeden and Litzenberger (1978) for standard options, and has been applied to exotic derivatives, such as basket options (see Hobson et al. (2005)).

A fundamental result on static hedging due to Carr and Madan (1998) shows that any European-style claim written on a single underlying asset can be perfectly replicated by holding a fixed number of bonds and forwards, along with a basket of European calls and puts with the same underlying. The importance of this result is that it provides a model-free, perfect, static replicating strategy. As such, it also gives a no-arbitrage price relationship between the contingent claims and the hedging instruments. Nevertheless, there are also a number of limitations. In particular, the static hedging strategy requires an unbounded continuous strip of European calls and puts and must include the bond and forward in the portfolio. In reality, calls and puts are available only at discrete strikes in a finite interval. This leads to a practical question: how can one optimally construct a static hedge with only a finite number of calls and puts, with or without forwards on the same underlying? More generally, when there are simply not enough traded standard instruments to achieve a perfect static hedge, or when the hedger faces a binding cost constraint, the result of Carr and Madan (1998) provides no direction on how one might proceed.

In this paper, we propose a flexible framework for hedging a contingent claim by choosing static positions in vanilla European calls, puts, bonds, and forwards. We are primarily interested in applications where the perfect static hedge is not available given a set of hedging instruments. To this end, we minimize the expected squared hedging error subject to a cost constraint. Our main result is a model-free expression for the optimal static hedging strategy, which involves computing a number of expectations that reflect the dependence among the contingent claim and the hedging assets. We provide a general method for approximating these expectations analytically in a general incomplete Markov diffusion setting that includes, but is not limited to, the well-known geometric Brownian motion (GBM), Heston CEV and SABR models.

Compared to Carr and Madan (1998), our framework includes a number of additional features. First, we allow for finite upper and lower bounds on the strikes of calls/puts used. Our static portfolio can involve any subset of the hedging assets among bonds, forwards, calls and puts, as opposed to include all of them. This gives the added flexibility to apply to underlying assets on which the forward contracts or some calls/puts are not written. Also, a cost constraint is incorporated into the hedging problem. When binding, this constraint may render a perfect static hedge impossible, and force the hedger to adjust the portfolio to minimize hedging error. While our methodology does not a priori assume the hedge is perfect, it can recover the perfect static hedge when it is available. This allows us to reconcile with the results in Carr and Madan (1998) as a special case of our framework.

In the recent literature, Carr and Wu (2013) work in a single-factor model and propose a finite approximation for the static hedging portfolio whose weights are computed based on the Gauss-Hermite quadrature rule. Also, there is a wealth of static hedging results specifically for barrier options under one-dimensional diffusion models; see Derman et al. (1995); Carr and Chou (1997); Carr et al. (1998); Carr and Lee (2009); Carr and Nadtochiy (2011); Bardos et al. (2010), among others. In contrast to these works, our framework applies to other exotic derivatives and multi-dimensional diffusion models. We illustrate the static hedging strategies in three examples: Asian options, leveraged exchange-traded fund (LETF) options, and options with an illiquid underlying.

The rest of this paper proceeds as follows: In Section 2, we formulate the optimal static hedging problem. In Section 3, we present our main results on hedging a contingent claim with a static portfolio of bonds, forwards and a strip of calls/puts. We also derive the optimal portfolio that consists of a finite set of assets. In both scenarios, we provide explicit, model-free optimal hedges. In Section 4, we discuss a practical method to numerically compute the hedging strategies in a general Markov diffusion setting. Lastly, in Section 5, we implement and illustrate our static hedging strategies in a number of applications.

## 2 Problem formulation

In the background, we fix a complete filtered probability space , where represents the physical probability measure and the filtration represents the price history of the assets in the market. The market is assumed to be arbitrage-free but may be incomplete. We take as given an equivalent martingale (pricing) measure , inferred from current market derivatives prices. For simplicity, we also assume a zero interest rate and no dividends.

Our static hedging problem involves a group of hedging assets , with being the index set. The number of hedging assets in may be finite, countably infinite or uncountably infinite. The hedging assets could be, for example, bonds, stocks, calls, puts, forwards or other derivative securities. The price of each asset at any time is denoted by .

We define a Static Portfolio as a signed measure such that the static portfolio value at any time is given by

(2.1) |

In other words, denotes the quantity of asset of type held in the static portfolio. Observe that may be negative, indicating a short position. Note that, while asset prices and the value of the static portfolio change with , the number of units remains constant for all .

###### Remark 2.1.

We will consider two main examples in this manuscript: (i) hedging with calls/puts with strikes in an interval , and (ii) hedging with a finite number of assets. In setting (i), the signed measure maps . In this case, we will assume that is absolutely continuous with respect to the Lebesgue measure and write where is a function that maps . In setting (ii), the signed measure maps . In this case, we will write where is a function that maps .

We now consider the contingent claim to be hedged at a future time . Its market price at any time is denoted by . If the claim expires at time , then is the terminal payoff. We are primarily interested in situations where perfect static replication is impossible with a given set of hedging assets. Our goal is to minimize the expected squared hedging error of the static portfolio at time subject to a possible cost constraint. We define the optimal static portfolio as the solution of the following optimization problem:

(2.2) |

That is, is the static portfolio that minimizes the expectation subject to the cost constraint . Note that the expectation in (2.2) is evaluated under the physical probability measure . Clearly, a perfect static hedge ( -a.s.) is possible if and only if . Note that the value of a portfolio at time can be expressed as

(2.3) |

since all assets are martingales under the pricing measure . Thus, the cost constraint involves computation under the pricing measure .

Naturally, the optimal hedging performance and the corresponding static portfolio depend on the hedging assets available in the market, as well as the underlying price dynamics. Our main objective is twofold: (i) we provide a model-free expression for the optimal static hedging strategy when the hedging assets include bonds, forwards, vanilla European calls and puts on the same underlying; (ii) we discuss the implementation of the hedging strategies for a number of claims under Markovian diffusion dynamics.

## 3 Methodology & Main Results

In this section, the set of hedging instruments contains a zero-coupon bond , which pays one unit of currency at time , a forward contract written on an underlying asset with payoff , and -maturity European puts and calls written on . We assume there is a put at every strike and call at every strike , with . Let us denote by the payoff the call/put with strike . That is

(3.1) |

While we observe and in practice and in our numerical examples, our model also allows for and so that we can reconcile with the results in Carr and Madan (1998) (see Sect. 3.1 below).

The terminal value of the static portfolio, composed of bonds, forwards and units of European calls/puts with strikes in the interval , is given by

(3.2) |

where , and may be either positive or negative (indicating a long or short position).

The cost constraint is given by

(3.3) |

Note that, since the cost to enter a forward contract at inception is zero, the number of forward contracts in the static portfolio plays no role in the cost constraint.

With given by (3.2), and cost constraint by (3.3), algebraic calculations show that the static hedging problem (2.2) is equivalent to solving for

(3.4) |

where

(3.5) | ||||

(3.6) | ||||

(3.7) | ||||

(3.8) |

and we have defined the expectations:

(3.9) |

In order to state and prove the optimal hedging strategy in this setting, we need the following Lemma. As preparation, it is convenient to introduce the probability density functions of under the physical (i.e., historical) and risk-neutral probability measures

(3.11) |

###### Lemma 3.1.

Assume the random variable has a strictly positive density . Recall the function as defined in (LABEL:eq:psi), and let be . Then the solution of the integral equation

(3.12) |

is given by

(3.13) |

###### Proof.

In what follows, let be the anti-derivative of and be the anti-derivative of so that and . Observe from (3.1) that

(3.14) |

Let us further observe that . Then, equation (3.12) implies

(3.15) | |||||

(by (LABEL:eq:psi)) | (3.16) | ||||

(3.17) | |||||

(3.18) | |||||

(3.19) | |||||

(3.20) | |||||

(3.21) | |||||

(integrate by parts) | (3.22) | ||||

(by (3.14)). | (3.23) |

To obtain (3.13), simply divide (3.23) by , differentiate both sides twice, and use . ∎

Using Lemma 3.1, we can now state and prove the optimal hedging strategy. To this end, we define the function

(3.24) |

where , , the densities and are defined in (3.11) and the function is given in (LABEL:eq:psi).

###### Theorem 3.2.

Assume the random variable has a strictly positive density under and a density under . Assume further that . Finally, assume the matrix inverses defined in (3.26) and (LABEL:eq:soln.3) are well defined. Then the optimal strategy that solves the optimal static hedging problem (3.4) is given by

(3.25) |

where

(3.26) |

and

(3.27) |

###### Proof.

First, we define the Lagrangian associated with (3.4):

(3.29) | ||||

(3.30) | ||||

(3.31) | ||||

(3.32) |

where acts on functions in . The Karush-Kuhn-Tucker (KKT) conditions, necessary for optimality, are (below, is an arbitrary function satisfying )

stationarity: | (3.33) | ||||

(3.34) | |||||

(3.35) | |||||

stationarity: | (3.36) | ||||

(3.37) | |||||

stationarity: | (3.38) | ||||

(3.39) | |||||

comp. slackness: | (3.40) | ||||

(3.41) |

Note that (3.35) is of the form (3.12) with . Thus, using Lemma 3.1 we obtain

(3.42) |

Next, noticing that

(3.43) | ||||

(3.44) | ||||

(3.45) | ||||

(3.46) | ||||

(3.47) | ||||

(3.48) | ||||

(3.49) |

and substituting these expressions into (3.42), we see that in (3.42) coincided with the expression given in (3.24). Next, inserting expression (3.24) into the KKT conditions, (3.37), (3.39), and (3.41), gives the following system of three equations

(3.50) | ||||

(3.51) | ||||

(3.52) |

The above system has two possible solutions corresponding respectively to the cases and . For the case , the triplet is given by (3.26). On the other hand, when , the triplet is given by (LABEL:eq:soln.3). Finally, the KKT conditions are necessary conditions. Since both the objective function and constraint are convex, and the primal problem is feasible (Slater’s condition), the KKT conditions are also sufficient for optimality (see, e.g. Zalinescu (2002, Theorem 2.9.3)). ∎

In Theorem 3.2, the solution corresponds to the unconstrained optimization problem. But if the associated cost is less than , that is,

(3.53) |

then must coincide with the solution of the constrained optimization problem with initial cost (upper bound) . On the other hand, if the unconstrained optimization problem admits a cost greater than , then the corresponding constrained optimization problem has the solution where, by construction the constraint is binding: .

In fact, from (3.24), the first term in parenthesis in the optimal strategy can be interpreted as a conditional expectation. Heuristically, we have

(3.54) |

In other words, the number of units of call/put held at strike involves computing the expected terminal claim conditioned on the terminal price of the underlying taking value .

###### Remark 3.3.

Although we considered hedging with European call/puts on a single asset , the results of this section can be extended to the case where one hedges with European calls/puts on assets . The only difficulty that may arise is in solving the equations that result by imposing the KKT conditions.

### 3.1 Connection to Carr and Madan (1998)

Let us recall the main result in Carr and Madan (1998): if satisfies , then

(3.55) |

As such, a contingent claim with payoff can be perfectly hedged by holding bonds, forward contracts and a basket of puts and calls, where the weight of the put/call with strike is . The following corollary proves that equation (3.55) is indeed a special case of our Theorem 3.2.

###### Corollary 3.4.

###### Proof.

We must show that , given by (3.56), satisfies (3.50) and (3.51). First, we observe that

(3.57) | ||||

(3.58) |

Thus, using , we see from (3.24) that

(3.59) |

Next, dividing equation (3.50) by two and rearranging terms we find

(3.60) |

This equation will clearly be satisfied if and . To see this, simply take the expectation of (3.55). Next, dividing equation (3.51) by two, and rearranging terms we have

(3.61) |

This equation will also be satisfied if and . To see this, simply multiply (3.55) by and take an expectation. ∎

### 3.2 Connection to static hedging with finite assets

Our framework can be related to static hedging with a finite number of assets. In this case, the set has a finite number units of hedging assets, and the static portfolio value can be expressed as a finite sum:

(3.62) |

This is indeed the discrete version of the static portfolio in (2.1).

###### Assumption 3.5.

We assume that the random variables are elements of and are linearly independent. Stated in financial terms, this assumption simply requires that none of the hedging instruments is redundant, as defined in (Duffie, 2001, Chaper 2).

With given by (3.62), a direct computation shows that the static hedging problem amounts to determining the optimal strategy

(3.63) |

where and are given by

(3.64) |

and

(3.65) |

Compared to the “continuous” case in (3.2)–(LABEL:eq:psi), the objective function again involves the expectations of products of payoffs, namely, and . Note that in this discrete case we can consider general claims, not limited to forwards, puts, and calls, and continue to derive explicitly the optimal static portfolio.

###### Proposition 3.6.

We provide a proof in Appendix A. The vector corresponds to the optimal strategy without a cost constraint. If the unconstrained optimization problem has a cost , then is the optimal strategy for the constrained optimization problem. On the other hand, if the unconstrained optimization problem has a cost , then the solution of the constrained optimization problem is given by which, by construction, has a cost equal to , that is, . Lastly, we emphasize that the optimal static hedging strategy with discrete strikes can be quite different than the optimal strategy when continuous strikes available but implemented at discretized strikes. We will visualize the difference in Section 5.1.

###### Remark 3.7 (Relation to Markowitz mean-variance portfolio optimization).

In his seminal work, Markowitz (1952) solves the problem of minimizing portfolio variance for a given level of expected return. Mathematically, the minimization problem is given by

(3.67) |

where are the portfolio weights to be found, and are, respectively, the covariance matrix and expected returns of a group of assets, and is the minimum level of expected return. Interestingly, the portfolio optimization problem (3.67) has the same structure as the static hedging problem (3.63), which, in matrix notation, is given by

(3.68) |

Though, clearly, the economic interpretations of (3.67) and (3.68) are distinct.

## 4 Implementation Under a Markov Diffusion Framework

Thus far, we have made no assumption about the dynamics of the underlying . To illustrate the performance of our static hedging strategies, we now present the calculations and numerical implementation under a general incomplete Markov diffusion market. The analytic approximations we present below are useful when the claim to be hedged is European-style. Specifically, the payoff may be some function of the final value of a -dimensional Markov diffusion . Note, by allowing components of to be the quadratic variation or running average of other components, our definition of European-style claims allows for path dependence and includes both Asian options and options on variance/volatility (e.g., a variance swap). Extending the approximations to cases where is a barrier-style claim or look-back option is not trivial and is well beyond the scope of this paper.

Let be a Markov diffusion satisfying the following stochastic differential equations (SDEs) under and , respectively

(under ) | (4.1) | ||||

(under ) | (4.2) |

Here, (resp. ) is an -dimensional Brownian motion under (resp. ), and the functions , and map

(4.3) |

Let us suppose that the terminal values of the hedging assets from Section 3.2, the stock from Section 3 and the claim to be hedged are given by

(4.4) |

where the function maps and for each the function maps . Since is traded, in order to preclude arbitrage, we must have

(4.5) |

In order to implement the optimal hedging strategies (Theorems 3.2 and 3.6), we must compute the expectations defined in equations (LABEL:eq:psi) and (3.65). For general dynamics of the form (4.1)-(4.2), closed-form expressions for these expectations are not available. Moreover, computing these expectations via Monte Carlo simulation is not practical, since, in the case of Theorem 3.2, the expectations appear in the integrands of various integrals. As such, we provide here a method for obtaining analytic approximations the expectations in (LABEL:eq:psi) and (3.65). The methods that we describe below were developed first formally in a scalar setting in Pagliarani and Pascucci (2012) and later extended to multiple dimensions with rigorous error bounds in Lorig et al. (2015b) and Lorig et al. (2015a). Here, we give a concise review of these methods and also provide some extensions, which are needed to implement Theorems 3.2 and 3.6.

We fix a time and consider an expectation of the general form

(4.6) |

Under mild conditions on the drift , diffusion coefficient and terminal data the function satisfies the Kolmogorov backward equation. Omitting -dependence below to ease notation, we have

(4.7) |

where is the generator of under probability measure . Explicitly, the operator is given by