A general Multidimensional Monte Carlo Approach for Dynamic Hedging under stochastic volatility

Let $(S,\mathbf{F},\mathbb{P})$ be a financial market composed by a continuous $\mathbf{F}$-semimartingale $S$ which represents a discounted risky asset price process, $\mathbf{F}=\{\mathbf{F}_{t};0\leq t\leq T\}$ is a filtration which encodes the information flow in the market on a finite horizon $[0,T]$, $\mathbb{P}$ is a physical probability measure and $\mathcal{M}^{e}$ is the set of equivalent local martingale measures. Let $H$ be an $\mathbf{F}_{T}$-measurable contingent claim describing the net payoff whose the trader is faced at time $T$. In order to hedge this claim, the trader has to choose a dynamic portfolio strategy.

Under the assumption of an arbitrage-free market, the classical Galtchouk-Kunita-Watanabe (henceforth abbreviated by GKW) decomposition yields

where $L^{H,\mathbb{Q}}$ is a $\mathbb{Q}$-local martingale which is strongly orthogonal to $S$ and $\theta^{H,\mathbb{Q}}$ is an adapted process.

The GKW decomposition plays a crucial role in determining optimal hedging strategies in a general Brownian-based market model subject to stochastic volatility. For instance, if $S$ is a one-dimensional Itô risky asset price process which is adapted to the information generated by a two-dimensional Brownian motion $W=(W^{1},W^{2})$, then there exists a two-dimensional adapted process $\phi^{H,\mathbb{Q}}:=(\phi^{H,1},\phi^{H,2})$ such that

which also realizes

In the complete market case, there exists a unique $\mathbb{Q}\in\mathcal{M}^{e}$ and in this case, $L^{H,\mathbb{Q}}=0$, $\mathbb{E}_{\mathbb{Q}}[H]$ is the unique fair price and the hedging replicating strategy is fully described by the process $\theta^{H,\mathbb{Q}}$. In a general stochastic volatility framework, there are infinitely many GKW orthogonal decompositions parameterized by the set $\mathcal{M}^{e}$ and hence one can ask if it is possible to determine the notion of non-self-financing optimal hedging strategies solely based on the quantities (1.2). This type of question was firstly answered by Föllmer and Sonderman [9] and later on extended by Schweizer [23] and Föllmer and Schweizer [8] through the existence of the so-called Föllmer-Schweizer decomposition which turns out to be equivalent to the existence of locally-risk minimizing hedging strategies. The GKW decomposition under the so-called minimal martingale measure constitutes the starting point to get locally risk minimizing strategies provided one is able to check some square-integrability properties of the components in (1.1) under the physical measure. See e.g [12] and [26] for details and other references therein. Orthogonal decompositions without square-integrability properties can also be defined in terms of the the so-called generalized Föllmer-Schweizer decomposition (see e.g Schweizer [24]).

In contrast to the local-risk minimization approach, one can insist in working with self-financing hedging strategies which give rise to the so-called mean-variance hedging methodology. In this approach, the spirit is to minimize the expectation of the squared hedging error over all initial endowments $x$ and all suitable admissible strategies $\varphi\in\Theta$:

The nature of the optimization problem (1.3) suggests to work with the subset $\mathcal{M}^{e}_{2}:=\{\mathbb{Q}\in\mathcal{M}^{e};\frac{d\mathbb{Q}}{d\mathbb{P}}\in L^{2}(\mathbb{P})\}$. Rheinlander and Schweizer [22], Gourieroux, Laurent and Pham [10] and Schweizer [25] show that if $\mathcal{M}^{e}_{2}\neq\emptyset$ and $H\in L^{2}(\mathbb{P})$ then the optimal quadratic hedging strategy exists and it is given by $\big{(}\mathbb{E}_{\tilde{\mathbb{P}}}[H],\eta^{\tilde{\mathbb{P}}}\big{)}$, where

Here $\theta^{H,\tilde{\mathbb{P}}}$ is computed in terms of $\tilde{\mathbb{P}}$, the so-called variance optimal martingale measure, $\tilde{\zeta}$ realizes

and $V^{H,\tilde{\mathbb{P}}}:=\mathbb{E}_{\tilde{\mathbb{P}}}[H|\mathbf{F}_{\cdot}]$ is the value option price process under $\tilde{\mathbb{P}}$. See also Cerný and Kallsen [4] for the general semimartingale case and the works [16], [18] and [19] for other utility-based hedging strategies based on GKW decompositions.

Concrete representations for the pure hedging strategies $\{\theta^{H,\mathbb{Q}};\mathbb{Q}=\hat{\mathbb{P}},\tilde{\mathbb{P}}\}$ can in principle be obtained by computing cross-quadratic variations $d[V^{H,\mathbb{Q}},S]_{t}/d[S,S]_{t}$ for $\mathbb{Q}\in\{\tilde{\mathbb{P}},\hat{\mathbb{P}}\}$. For instance, in the classical vanilla case, pure hedging strategies can be computed by means of the Feynman-Kac theorem (see e.g Heath, Platen and Schweizer [12]). In the path-dependent case, the obtention of concrete computationally efficient representations for $\theta^{H,\mathbb{Q}}$ is a rather difficult problem. Feynman-Kac-type arguments for fully path-dependent options mixed with stochastic volatility typically face not-well posed problems on the whole trading period as well as highly degenerate PDEs arise in this context. Generically speaking, one has to work with non-Markovian versions of the Feynman-Kac theorem in order to get robust dynamic hedging strategies for fully path dependent options written on stochastic volatility risky asset price processes.

In the mean variance case, the only quantity in (1.4) not related to GKW decomposition is $\tilde{Z}$ which can in principle be expressed in terms of the so-called fundamental representation equations given by Hobson [14] and Biagini, Guasoni and Pratelli [2] in the stochastic volatility case. For instance, Hobson derives closed form expressions for $\tilde{\zeta}$ and also for any type of $q$-optimal measure in the Heston model [13]. Recently, semi-explicit formulas for vanilla options based on general characterizations of the variance-optimal hedge in Cerný and Kallsen [4] have been also proposed in the literature which allow for a feasible numerical implementation in affine models. See Kallsen and Vierthauer [17] and Cerný and Kallsen [5] for some results in this direction. A different approach based on backward stochastic differential equations can also be used in order to get useful characterizations for the optimal mean variance hedging strategies. See e.g Jeanblanc, Mania, Santacrose and Schweizer [15] and other references therein.

In spite of deep characterizations of optimal quadratic hedging strategies and concrete numerical schemes available for vanilla-type options, to our best knowledge no feasible approach has been proposed to tackle the problem of obtaining dynamic optimal quadratic hedging strategies for fully path dependent options written on a generic multidimensional Itô risky asset price process. In this work, we attempt to solve this problem with a probabilistic approach. The main difficulty in dealing with fully path dependent and/or discontinuous payoffs is the non-Markovian nature of the option value and a priori lack of path smoothness of the pure hedging strategies. Usual numerical schemes based on PDE and martingale techniques do not trivially apply to this context.

The main contribution of this paper is the obtention of flexible and computationally efficient multidimensional non-Markovian representations for generic option price processes which allow for a concrete computation of the associated GKW decomposition $\big{(}\theta^{H,\mathbb{Q}},L^{H,\mathbb{Q}}\big{)}$ for $\mathbb{Q}$-square integrable payoffs $H$ with $\mathbb{Q}\in\mathcal{M}^{e}$. We provide a Monte Carlo methodology capable to compute optimal quadratic hedging strategies w.r.t general square-integrable claims in a multidimensional Brownian-based market model.

This article provides a feasible and constructive method to compute generalized Föllmer-Schweizer decompositions under full generality. As far as the mean variance hedging is concerned, we are able to compute pure optimal hedging strategies $\theta^{H,\tilde{\mathbb{P}}}$ for arbitrary square-integrable payoffs. Hence, our methodology also applies to this case provided one is able to compute the fundamental representation equations in Hobson [14] and Biagini, Guasoni and Pratelli [2] which is the case for the classical Heston model. In mathematical terms, we are able to compute $\mathbb{Q}$-GKW decompositions under full generality so that the results of this article can also be used to other non-quadratic hedging methodologies where orthogonal martingale representations play an important role in determining optimal hedging strategies.

The starting point of this article is based on weak approximations developed by Leão and Ohashi [20] for one-dimensional Brownian functionals. They introduced a one-dimensional space-filtration discretization scheme constructed from suitable waiting times which measure the instants when the Brownian motion hits some a priori levels. In this work, we extend [20] to the multidimensional case as follows: More general and stronger convergence results are obtained in order to recover incomplete markets with stochastic volatility. Hitting times induced by multidimensional noises which drive the stochastic volatility are carefully analyzed in order to obtain $\mathbb{Q}$-GKW decompositions under rather weak integrability conditions for any $\mathbb{Q}\in\mathcal{M}^{e}$. Moreover, a complete analysis is performed w.r.t weak approximations for gain processes by means of suitable non-antecipative discrete-time hedging strategies for square-integrable payoffs, including path-dependent ones.

It is important to stress that the results of this article can be applied to both complete and incomplete markets written on a generic multidimensional Itô risky asset price process. One important restriction of our methodology is the assumption that the risky asset price process has continuous paths. This is a limitation that we hope to overcome in a future work.

Numerical results based on the standard Black-Scholes, local-volatility and Heston models are performed in order to illustrate the theoretical results and the methodology of this article. In particular, we briefly compare our results with other prominent methodologies based on Malliavin weights (complete market case) and PDE techniques (incomplete market case) employed by Bernis, Gobet and Kohatsu-Higa [1] and Heath, Platen and Schweizer [12], respectively. The numerical experiments suggest that pure hedging strategies based on generalized Föllmer-Schweizer decompositions mitigate very well the cost of hedging of a path-dependent option even if there is no guarantee of the existence of locally-risk minimizing strategies. We also compare hedging errors arising from optimal mean variance hedging strategies for one-touch options written on a Heston model with nonzero correlation.

The remainder of this paper is structured as follows. In Section 2, we fix the notation and we describe the basic underlying market model. In Section 3, we provide the basic elements of the Monte Carlo methodology proposed in this article. In Section 4, we formulate dynamic hedging strategies starting from a given GKW decomposition and we translate our results to well-known quadratic hedging strategies. The Monte Carlo algorithm and the numerical study are described in Sections 5 and 6, respectively. The Appendix presents more refined approximations when the martingale representations admit additional hypotheses.

In the sequel, we take $\mathbb{Q}\in\mathcal{M}^{e}$ and we set $W^{S}:=(W^{(1)},\ldots,W^{(d)})$ and $W^{I}:=(W^{(d+1)},\ldots,W^{(p)})$ where

is a standard $p$-dimensional Brownian motion under the measure $\mathbb{Q}$ and filtration $\mathbb{F}:=\{\mathcal{F}_{t};0\leq t\leq T\}$ generated by $W=(W^{(1)},\ldots,W^{(p)})$. In what follows, we fix a discounted contingent claim $H$. Recall that the filtration $\mathbb{F}$ is contained in $\mathbf{F}$, but it is not necessarily equal. In the remainder of this article, we assume the following hypothesis.

For a given $\mathbb{Q}$-square integrable claim $H$, the Brownian martingale representation (computed in terms of $(\mathbb{F},\mathbb{Q})$) yields

where $\phi^{H,\mathbb{Q}}:=(\phi^{H,\mathbb{Q},1},\ldots,\phi^{H,\mathbb{Q},p})$ is a $p$-dimensional $\mathbb{F}$-predictable process. In what follows, we set $\phi^{H,\mathbb{Q},S}:=(\phi^{H,\mathbb{Q},1},\ldots,\phi^{H,\mathbb{Q},d})$, $\phi^{H,\mathbb{Q},I}:=(\phi^{H,\mathbb{Q},d+1},\ldots,\phi^{H,\mathbb{Q},p})$ and

The discounted stock price process has the following $\mathbb{Q}$-dynamics

and therefore the $\mathbb{Q}$-GKW decomposition for the pair of locally square integrable local martingales $(\hat{V},S)$ is given by

At first, we fix once and for all $\mathbb{Q}\in\mathcal{M}^{e}$ and a $\mathbb{Q}$-square-integrable contingent claim $H$ satisfying (M). In the remainder of this section, we are going to fix a $\mathbb{Q}$-Brownian motion $W$ and with a slight abuse of notation all $\mathbb{Q}$-expectations will be denoted by $\mathbb{E}$. The choice of $\mathbb{Q}\in\mathcal{M}^{e}$ is dictated by the pricing and hedging method used by the trader.

In the sequel, $[\cdot,\cdot]$ denotes the usual quadratic variation between semimartingales and the usual jump of a process is denoted by $\Delta Y_{t}=Y_{t}-Y_{t-}$ where $Y_{t-}$ is the left-hand limit of a cadlag process $Y$. For a pair $(a,b)\in\mathbb{R}^{2}$, we denote $a\vee b:=\max\{a,b\}$ and $a\wedge b:=\min\{a,b\}$. Moreover, for any two stopping times $S$ and $J$, we denote the stochastic intervals $[[S,J[[:=\{(\omega,t);S(\omega)\leq t<J(\omega)\}$, $[[S]]:=\{(\omega,t);S(\omega)=t\}$ and so on. Throughout this article, $Leb$ denotes the Lebesgue measure on the interval $[0,T]$.

For a fixed positive integer $k$ and for each $j=1,2,\ldots,p$ we define $T^{k,j}_{0}:=0$ a.s. and

where $W:=(W^{(1)},\ldots,W^{(p)})$ is the $p$-dimensional $\mathbb{Q}$-Brownian motion as defined in (2.2).

For each $j\in\{1,\ldots,p\}$, the family $(T^{k,j}_{n})_{n\geq 0}$ is a sequence of $\mathbb{F}$-stopping times where the increments $\{T^{k,j}_{n}-T^{k,j}_{n-1};n\geq 1\}$ is an i.i.d sequence with the same distribution as $T^{k,j}_{1}$. In the sequel, we define $A^{k}:=(A^{k,1},\ldots,A^{k,p})$ as the $p$-dimensional step process given componentwise by

where

for $k,n\geq 1$ and $j=1,\ldots,p$. We split $A^{k}$ into $(A^{S,k},A^{I,k})$ where $A^{S,k}$ is the $d$-dimensional process constituted by the first $d$ components of $A^{k}$ and $A^{I,k}$ the remainder $p-d$-dimensional process. Let $\mathbb{F}^{k,j}:=\{\mathcal{F}^{k,j}_{t}:0\leq t\leq T\}$ be the natural filtration generated by $\{A^{k,j}_{t};0\leq t\leq T\}$. One should notice that $\mathbb{F}^{k,j}$ is a discrete-type filtration in the sense that

where $\mathcal{F}^{k,j}_{0}=\{\Omega,\emptyset\}$ and $\mathcal{F}^{k,j}_{T^{k,j}_{m}}=\sigma(T^{k,j}_{1},\ldots,T^{k,j}_{m},\eta^{k,j}_{1},\ldots,\eta^{k,j}_{m})$ for $m\geq 1$ and $j=1,\ldots,p$. Here, $\bigvee$ denotes the smallest sigma-algebra generated by the union. One can easily check that $\mathcal{F}^{k,j}_{T^{k,j}_{m}}=\sigma(A^{k,j}_{s\wedge T^{k,j}_{m}};s\geq 0)$ and hence

With a slight abuse of notation we write $\mathcal{F}^{k,j}_{t}$ to denote its $\mathbb{Q}$-augmentation satisfying the usual conditions.

Let us now introduce the multidimensional filtration generated by $A^{k}$. Let us consider $\mathbb{F}^{k}:=\{\mathcal{F}^{k}_{t};0\leq t\leq T\}$ where $\mathcal{F}^{k}_{t}:=\mathcal{F}^{k,1}_{t}\otimes\mathcal{F}^{k,2}_{t}\otimes\cdots\otimes\mathcal{F}^{k,p}_{t}$ for $0\leq t\leq T$. Let $\mathcal{T}^{k}:=\{T^{k}_{m};m\geq 0\}$ be the order statistics obtained from the family of random variables $\{T^{k,j}_{\ell};\ell\geq 0;j=1,\ldots,p\}$. That is, we set $T^{k}_{0}:=0$,

for $n\geq 1$. In this case, $\mathcal{T}^{k}$ is the partition generated by all stopping times defined in (3.1). The finite-dimensional distribution of $W^{(j)}$ is absolutely continuous for each $j=1,\ldots,p$ and therefore the elements of $\mathcal{T}^{k}$ are almost surely distinct for every $k\geq 1$. Moreover, the following result holds true.

With Lemma 3.1 at hand, we notice that the filtration $\mathbb{F}^{k}$ is a discrete-type filtration in the sense that

for $k\geq 1$ and $n\geq 0$. Itô representation theorem yields

where $\phi^{H}$ is a $p$-dimensional $\mathbb{F}$-predictable process such that

The payoff $H$ induces the $\mathbb{Q}$-square-integrable $\mathbb{F}$-martingale $X_{t}:=\mathbb{E}[H|\mathcal{F}_{t}];~0\leq t\leq T$. We now embed the process $X$ into the quasi left-continuous filtration $\mathbb{F}^{k}$ by means of the following operator

Since $X$ is a $\mathbb{F}$-martingale, then the usual optional stopping theorem yields the representation

Therefore, $\delta^{k}X$ is indeed a $\mathbb{Q}$-square-integrable $\mathbb{F}^{k}$-martingale and we shall write it as

where

and the integral in (3.1) is computed in the Lebesgue-Stieltjes sense.

Based on the Dirac process $\mathcal{D}^{j}\delta^{k}X$, we denote

In order to work with non-antecipative hedging strategies, let us now define a suitable $\mathbb{F}^{k}$-predictable version of $\mathbb{D}^{k,j}X$ as follows

One can check that $\mathbf{D}^{k,j}X$ is $\mathbb{F}^{k}$-predictable. See e.g [11], Ch.5 for details.

One should notice that (3.5) is reminiscent from the usual delta-hedging strategy but the price is shifted on the level of the sigma-algebras jointly with the increments of the driving Brownian motion instead of the pure spot price. For instance, in the one-dimensional case $(p=d=1)$, we have

and hence a natural procedure to approximate pure hedging strategies is to look at $\mathbf{D}^{k,1}X_{T^{k,1}_{1}}/S_{0}\sigma_{0}$ at time zero. In the incomplete market case, additional randomness from e.g stochastic volatilities are encoded by $\mathbb{E}[H|\mathcal{F}^{k}_{T^{k}_{1}}]$ where $T^{k}_{1}$ is determined not only by the hitting times coming from the risky asset prices but also by possibly Brownian motion hitting times coming from stochastic volatility.

In the next sections, we will construct feasible approximations for the gain and cost processes based on the ratios (3.5). We will see that hedging ratios of the form (3.5) will be the key ingredient to recover the gain process in full generality.

Based on (2.3), (2.1) and (2.5), let us denote

In order to shorten notation, we do not write $(\phi^{H,\mathbb{Q},S},\phi^{H,\mathbb{Q},I})$ in (3.6). The main goal of this section is the obtention of bounded variation martingale weak approximations for both the gain and cost processes, given respectively, by

We assume the trader has some knowledge of the underlying volatility so that the obtention of $\phi^{H,S}$ will be sufficient to recover $\theta^{H}$. The typical example we have in mind are generalized Föllmer-Schweizer decompositions, locally-risk minimizing and mean variance strategies as explained in the Introduction. The scheme will be very constructive in such way that all the elements of our approximation will be amenable to a feasible numerical analysis. Under very mild integrability conditions, the weak approximations for the gain process will be translated into the physical measure.

where $Y^{*}_{T}:=\sup_{0\leq t\leq T}|Y_{t}|$ and $1\leq p,q<\infty$ such that $\frac{1}{p}+\frac{1}{q}=1$. The subspace of the square-integrable $\mathbb{F}$-martingales will be denoted by $\text{H}^{2}(\mathbb{F})$. It will be also useful to work with $\sigma(\text{B}^{1},\Lambda^{\infty})$-topology given in [20]. For more details about these topologies, we refer to the works [6, 7, 20]. It turns out that $\sigma(\text{B}^{2},\text{M}^{2})$ and $\sigma(\text{B}^{1},\Lambda^{\infty})$ are very natural notions to deal with generic square-integrable random variables as described in [20].

In the sequel, we recall the following notion of covariation introduced in [20].