Research Article | Open Access
Mean-Variance Hedging Based on an Incomplete Market with External Risk Factors of Non-Gaussian OU Processes
We prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semiexplicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly prove the no-arbitrage condition to be true for our financial market, which is used as an assumption in existing discussions. In doing so, we explicitly construct the square-integrable density process of the variance-optimal martingale measure (VOMM). Second, we derive a backward stochastic differential equation (BSDE) with jumps for the mean-value process of a given contingent claim. The unique existence of adapted strong solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution of the BSDE and the VOMM, we reach the justification of the global risk optimality for our hedging strategy.
In this paper, we justify the global risk optimality of the hedging strategy of contingent claim, which is explicitly constructed for an incomplete market defined on some filtered probability space . The financial market has primitive assets: one bond with constant interest rate and risky assets. The price processes of the assets are described by a generalized Black-Scholes model with coefficients driven by the market regime caused by leverage effect, and so forth. The financial market model includes the Barndorff-Nielsen and Shephard (BNS) volatility model proposed by Barndorff-Nielsen and Shephard  and further studied in Benth et al. , Benth and Meyer-Brandis , Lindberg , and so forth as a particular case. Our model is closely related to the one considered in Delong and Klüppelberg . As pointed out in Barndorff-Nielsen and Shephard , these models fit real market data quite well. Nevertheless, such models also induce incompleteness of the financial markets, which means that it is impossible to replicate perfectly contingent claims based on the bond and the primitive risky assets. A rule for designing a good hedging strategy is to minimize the mean squared hedging error over the set of all reasonable trading strategy processes: where is a random variable representing the discounted payoff of the claim, is the discounted price process of risky assets, is the initial endowment, and is the time horizon. Mathematically speaking, one seeks to compute the orthogonal projection of on the space of stochastic integrals.
To solve mean-variance hedging problem (1), we explicitly construct a trading strategy for the financial market and justify it to be the global risk-minimizing hedging strategy by using the following procedure.
First, we explicitly construct the square-integrable density process of a variance-optimal martingale measure (VOMM) . As a result, the set of equivalent (local) martingale measures with square-integrable densities, that is, is nonempty. Hence, our market is arbitrage-free (e.g, ). Second, we derive a BSDE with jumps and external random factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) type for the mean-value process of the option (i.e., ). The unique existence of adapted solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution to the BSDE and the VOMM, we get the optimal hedging strategy for our market.
The BSDE and VOMM based procedure is a mixed method of two typical approaches in solving mean-variance hedging problem: martingale approach stemmed from Harrison and Kreps  and stochastic control approach that views the problem as a linear-quadratic control problem and employs BSDEs to describe the solution (see, e.g., Yong and Zhou). This procedure is structured for a general semimartingale in Cĕrný and Kallsen  and explicitly (or semiexplicitly) presented for the current market in Dai . Some related and independent study can also be found in Jeanblanc et al. . More precisely, we have the following literature review and technical comparisons.
A closely related (local) risk-minimizing problem was initially introduced by Föllmer and Sondermann  under complete information, who also suggested an approach for the computation of a minimizing strategy in an incomplete market by extending the martingale approach of Harrison and Kreps . The basic idea of the approach was to introduce a measure of riskiness in terms of a conditional mean square error process where the discounted price process is a square-integrable martingale. Furthermore, the answer to the hedging problem is provided by the Galtchouk-Kunita-Watanabe decomposition of the claim. Then, this concept of local-risk minimization was further extended for the semimartingale case by Föllmer and Schweizer  and Schweizer [13, 14], where the minimal martingale measure and Föllmer-Schweizer (F-S) decomposition play a central role. Interested readers are referred to Fäollmer and Schweizer  and Schweizer  for more recent surveys about (local) risk minimization and mean-variance hedging.
Owing to the fact that one cares about the total hedging error and not the daily profit-loss ratios, the solution with respect to global risk minimization of the unconditional expected squared hedging error presented in (1) was considered (e.g., surveys in [16, 17]). Then, the study on global risk minimization was further developed by Cĕrný and Kallsen , who showed that hedging model (1) admits a solution in a very general class of arbitrage-free semimartingale markets where local-risk minimization may fail to be well defined. The key point of their approach is the introduction of the opportunity-neutral measure that turns the dynamic asset allocation problem into a myopic one. Furthermore, the minimal martingale measure relative to coincides with the variance-optimal martingale measure relative to the original probability measure . Recently, to overcome the difficulties appearing in Cĕrný and Kallsen  (i.e., a process appearing in Definition 3.12 is very hard to find and the VOMM in Proposition 3.13 is notoriously difficult to determine), the authors in Jeanblanc et al.  developed a method via stochastic control and backward stochastic differential equations (BSDEs) to handle the mean-variance hedging problem for general semimartingales. Furthermore, the authors in Kallsen and Vierthauer  derived semiexplicit formulas for the optimal hedging strategy and the minimal hedging error by applying general structural results and Laplace transform techniques. In addition to these works, some related studies in both general theory and concrete results in specific setups for the mean-variance hedging problem can be found in works, such as, Arai , Chan et al. , Duffie and Richardson , Gourieroux et al. , Heath et al. , Laurent and Pham , and references therein.
Comparing with the above studies, our contribution of the current research is threefold. First, we firmly prove the no-arbitrage condition to be true for our financial market; that is, the set defined in (2) is nonempty. This condition is used as an assumption for the existence of the VOMM in existing discussions (e.g., [8, 10, 18–20]). In doing so, we explicitly (or called semiexplicitly) construct a measure through identifying its explicit density by the general structure presented in Cĕrný and Kallsen . Then, we justify it to be the VOMM for our market model by proving the equivalent conditions given in Cĕrný and Kallsen . Second, in applying our VOMM to obtain the optimal hedging strategy, we derive a BSDE with jumps for the mean-value process of the option . Here, we lift the requirements that the contingent claims are bounded (e.g., [25, 26]) or satisfy Lipschitz condition (e.g., [20, 27]) to guarantee the corresponding integral-partial differential equation (IPDE) to have a classic or viscosity solution. Furthermore, the unique existence of an adapted solution to our derived BSDE is firmly proved under certain conditions while in the recent study of Jeanblanc et al.  such existence of an adapted solution to their constructed BSDE is only showed as an equivalent condition to guarantee the existence of an optimal strategy. More importantly, our BSDE can be solved by developing related numerical algorithms through the given terminal option (see, e.g., ). Third, from the purpose of easy applications, our discussion is based on a multivariate financial market model, which is in contrast to existing studies (e.g., [8, 10, 18, 20]). Therefore, unlike the studies in Hubalek et al.  and Kallsen and Vierthauer , our option is generally related to a multivariate terminal function and hence a BSDE involved approach is employed. Actually, whether one can extend the Laplace transform related method developed in Hubalek et al.  and Kallsen and Vierthauer  for single-variate terminal function to our general multivariate case is still an open problem.
Note that our study in this paper establishes the connection between our financial system and existing general semimartingale based study in Cĕrný and Kallsen  since we can overcome the difficulties in Cĕrný and Kallsen  by explicitly constructing the process and the VOMM as mentioned earlier. Furthermore, our objective and discussion in this paper are different from the recent study of Jeanblanc et al.  since the authors in Jeanblanc et al.  did not aim to derive any concrete expression. Nevertheless, interested readers may make an attempt to extend the study in Jeanblanc et al.  and apply it to our financial market model to construct the corresponding explicit results.
Finally, when the random variable in (1) is taken to be a constant (e.g., a prescribed daily expected return), the associated hedging problem reduces to a mean-variance portfolio selection problem as studied in Dai  by an alternative feedback control method. In this case, the optimal policies can be explicitly obtained by both the feedback control method in Dai  and the martingale method presented in the current paper. In the late method, the related BSDE is a degenerate one. From this constant option case, we can construct two insightful examples to provide the effective comparisons between the two methods. More precisely, our newly constructed hedging strategy can slightly outperform the feedback control based policy. However, the performance between the two methods is consistent in certain sense.
The remainder of the paper is organized as follows. We formulate our financial market model in Section 2 and present our main theorem in Section 3. Analytical and numerical examples are given in Section 4. Our main theorem is proven in Section 5. Finally, in Section 6, we conclude this paper with remarks.
2. The Financial Market
2.1. The Model
We use to denote a fixed complete probability space on which are defined a standard -dimensional Brownian motion with and -dimensional subordinator with and Càdlàg sample paths for some fixed (e.g., [31–33] for more details about subordinators and Lévy processes). The prime denotes the corresponding transpose of a matrix or a vector. Furthermore, , , and their components are assumed to be independent of each other. For each given , we let . Then, we suppose that there is a filtration related to the probability space, where for each .
The financial market under consideration is a multivariate Lévy-driven OU-type stochastic volatility model, which consists of assets. One of the assets is risk-free, whose price is subject to the ordinary differential equation (ODE) with constant interest rate : The other assets are stocks whose vector price process satisfies the following stochastic differential equation (SDE) for each : Here and in the sequel, the denotes the diagonal matrix whose entries in the main diagonal are with for a -dimensional vector and all the other entries are zero. is a Lévy-driven OU-type process described by the following SDE: where and . Now, define where with . Thus, we can impose the following conditions related to the coefficients in (4)-(5).
(C1) The functions and are continuous in and satisfy that, for each , where the norm takes the largest absolute value of all components of a vector or all entries of a matrix , and , , , , and are constants.
(C2) The derivatives and for all are continuous in and satisfy that, for each , where , , , and are some nonnegative constants.
We now introduce the conditions for each subordinator with , which can be represented by (e.g., Theorem 13.4 and Corollary 13.7 in ) Here and in the sequel, denotes a Poisson random measure with deterministic, time-homogeneous intensity measure . is the index function over the set . is the Lévy measure satisfying with taken to be a sufficiently large positive constant to guarantee all of the related integrals in this paper meaningful. Note that the condition in (12) is on the integrability of the tails of the Lévy measures (readers are referred to ([9, 30, 35–37]) for the justification of its reasonability).
2.2. Admissible Strategies
First, we use to denote the associated -dimensional discounted price process; that is, for each , Furthermore, we define to be the set of all -valued measurable stochastic processes adapted to such that . Thus, it follows from Lemma 10 that is a continuous -semimartingale. In addition, is locally in ; that is, there is a localizing sequence of stopping times with such that, for any ,
Second, let denote the set of -integrable and predictable processes in the sense of Definition 6.17 in page 207 of Jacod and Shiryaev . Furthermore, let denote the number of shares invested in stock at time and define . Then, we have the following definitions concerning admissible strategies.
Definition 1. An -valued trading strategy is called simple if it is a linear combination of strategies where are stopping times dominated by for some and is a bounded -measurable random variable. Furthermore, the set of all such simple trading strategies is denoted by .
Definition 2. A trading strategy is called admissible if there is a sequence of simple strategies such that in probability as for any and in as . Furthermore, the set of all such admissible strategies is denoted by .
3. Main Theorem
First, for each , define Note that the process presented in (19) is corresponding to the adjustment process defined in Lemma 3.7 of Cĕrný and Kallsen . Furthermore, the process presented in (20) is associated with the density process defined in Proposition 3.13 of Cĕrný and Kallsen . In addition, here and in the sequel, denotes the stochastic exponential for a univariant continuous semimartingale (e.g., pages 84-85 of ) with where denotes the quadratic variation process of .
Second, let denote the set of all -valued predictable processes (see, e.g., Definition 5.2 in page 21 of ) and let be the set of all -valued predictable processes satisfying Furthermore, let where is the -dimensional unit vector with the th component one. Then, we define
Definition 3. For a given random variable , a 3-tuple is called a -adapted strong solution of the BSDE: if is a Càdlàg process, , , and (27) holds a.s., where
To impose suitable condition on the option , we use for a positive integer to denote the set of all -valued, -measurable random variables satisfying .
Assumption 4. Consider and there exists a sequence of random variables satisfying in as and for all , where is a sequence of nondecreasing -stopping times satisfying a.s. as .
Theorem 5. Under conditions (C1), (C2), and (12) and Assumption 4, let be the unique -adapted strong solution of the BSDE in (27). Then, the optimal hedging strategy for (1) is given by where the pure hedge coefficient is given by In addition, is the unique solution of the SDE
Remark 6. The process appearing in Theorem 5 is actually the conditional mean-value process: Since it is not easy to be computed directly as the Markovian based conditional process , we turn to use the BSDE in (27) to evaluate the process , which is convenient for us to design the optimal hedging policy as explained in Section 1 of the paper.
4. Performance Comparisons
The material in this section is partially reported in the short conference version of the current paper (see, ). To be convenient and clear for readers, we refine it here. Note that the interest rate in (3) here is taken to be zero. Furthermore, the financial market is assumed to be self-financing, which implies that . In addition, the terminal option is taken to be a constant ; that is, . In this case, the optimal policies can be explicitly obtained by the feedback control method studied in Dai  and the martingale method presented in the current paper. In the late method, the related BSDE is a degenerate one, which can be easily observed from (34) in Remark 6. However, from this constant option , we can construct two insightful examples to provide the effective comparisons between the two methods.
More precisely, by (18) in Theorem 3.1 of Dai , we know that the terminal variance under the optimal policy stated in (15) of Theorem 3.1 of Dai  is given by In addition, by using Theorem 5 in the current paper and Theorem 4.12 in Cĕrný and Kallsen , we know that the hedging error under the optimal policy in (29) is given by For the purpose of performance comparisons, we calculate the differences between the optimal terminal variances in (35) and the optimal hedging errors in (36); that is, The result shown in the last inequality of (37) is intuitively right since the optimal strategy in (29) is taken over a general decision set given in Definition 2 and the one in (15) of Theorem 3.1 of Dai  is taken in an ad hoc approach. Nevertheless, the errors are very small as displayed in the following numerical examples.
Example 7. Here, we suppose that the financial market is given by the Black-Scholes model: where and are given constants. Owing to Definition 2.1.4(b) in pages 273-274 of Øksendal , the option (a positive constant) is not attainable and hence the associated hedging error can not be zero if the initial endowment . However, by the simulated results displayed in Figures 1 and 2, we see that the absolute error between the optimal variance based on the policy in (15) of Theorem 3.1 of Dai  and the optimal hedging error based on the strategy in (29) approaches zero as the terminal time increases. The rate of convergence is heavily dependent on the volatility . If is relatively large, the difference requires more time to reach zero. Nevertheless, if the millisecond is employed to represent the time unit in a supercomputer based trading system, the required time for the convergence makes sense in practice.
Example 8. Here, we assume that the financial market is presented by the BNS model: where and are given constants. Furthermore, owing to the remarks to the condition in (12) and owing to the discussions in Dai , we suppose that the driving subordinator with to the SDE in (5) is a compound Poisson process. The interarrival times of the process are exponentially distributed with mean and the jump sizes of the process are also exponentially distributed with mean . By the simulated results displayed in Figure 3, we see that similar illustration displayed in Example 7 also makes sense for the current example, where appearing in Figure 3 is the length of equally divided subintervals of . In addition, by the simulated results, we also see that, although perfect hedging is impossible in an incomplete market, the mean-variance hedging errors can be very small in many cases when terminal time increases.
5. Proof of Theorem 5
The proof consists of four parts presented in the subsequent four subsections: the justification of a proposition related to the discounted price process, the demonstration of a proposition related to the VOMM, the illustration of unique existence of solution to a type of BSDEs with jumps, and the remaining proof of Theorem 5.
5.1. The Proposition Related to the Discounted Price Process
Proposition 9. Under conditions (C1), (C2), and (12), one has that is a continuous -semimartingale; that is, where and are an -martingale and a predictable process of finite variation, respectively. Furthermore, is locally in in the sense as stated in (14).
We divide the proof of the proposition into two parts. First, we have the following lemma.
Lemma 10. Under (12), the unique adapted solution to the SDE in (5) for each , , and is given by Furthermore, under conditions (C1), (C2), and (12), there is a unique solution for (4)-(5), which is an -adapted and continuous semimartingale with In addition, for each ,
Proof. The claim concerning (41) directly follows from pages 316-317 in Applebaum . Furthermore, owing to conditions (C1) and (C2), we know that our market given by (4)-(5) satisfies the conditions as required by Lemma 4.1 in Dai . Thus, our market has a unique solution, which is -adapted, continuous, and mean square-integrable as stated in Lemma 10. In order to prove (43), let where, for any , Then, by condition (C1), there exists some nonnegative constant such that where we have used the facts that is nonnegative and nondecreasing in , the independence assumption among for , and Similarly, we can show that Note that and for are independent; is -martingale; and are -adapted. Then, it follows from Definition 4.1.1 in Øksendal  and the associated Itô’s formula (e.g., Theorem 4.1.2 in ) that given in (43) for each is the unique solution of (4).
Now, we show that for each is a square-integrable -semimartingale. To do so, we rewrite (4) in its integral formThen, the third term on the right-hand side of (51) is a square-integrable -martingale. In fact, it follows from (41) that, for each and , where the last equality in (52) holds in distribution. Thus, it follows from condition (C1) and (43) in Lemma 10 that where is some positive constant and we have used Theorem 39 in page 138 of Protter  and condition (12). Therefore, by Theorem 4.40(b) in page 48 of Jacod and Shiryaev , we know that the third term in (51) is a square-integrable -martingale.
Furthermore, by the same method, we can show that the second term on the right-hand side of (51) is of finite variation a.s. and is square-integrable over . Therefore, we conclude that for each is a square-integrable -semimartingale. Hence, we complete the proof of Lemma 10.
Proof of Proposition 9. It follows from Lemma 10 and Itô’s formula that, for each , Note that, by similar calculation as in (53), we have for all . Thus, it follows from Theorem 4.40(b) in page 48 of Jacod and Shiryaev  that is an -martingale. Furthermore, it follows from a similar explanation with the end of the proof for Lemma 10 that is a predictable process of finite variation and is square-integrable. Thus, we know that is a continuous -semimartingale. Moreover, it is locally in since we may take as the sequence of localizing times. Hence, we complete the proof of Proposition 9.
5.2. A Proposition Related to the VOMM
First of all, we use to denote the set of all signed -martingale measures in the sense that and with for a signed measure on and all . Then, we have the following proposition.
Proposition 11. Under conditions (C1), (C2), and (12), the following claims are true: (1) is a -martingale, where is given in (20);(2)the measure defined in (34) is an equivalent martingale measure (EMM) and that is defined in (2);(3)the measure is the VOMM in the sense that
We divide the proof of the proposition into demonstrating six lemmas as follows.
Proof. It follows from conditions (C1), (C2), and (41) that, for each , where for are some nonnegative constants and and are given by with . Then, based on an idea as used in Benth at al. , we can prove Lemma 12 by the following four steps.
First, by direct calculation, we know that is finite for any ; that is, where the nonnegative constant is given by Second, we prove that and the mapping for each is continuous. The continuity of for each can be shown as follows. Owing to condition (7) and fact (52), we know that By (12) and (49), we know that the function on the right-hand side of (68) is integrable for each fixed . Then, it follows from Lebesgue’s dominated convergence theorem that for each is continuous in terms of .
Next, we show that with for all exist and are continuous. In fact, consider an arbitrary but fixed point and take a compact set such that is in the interior of . Note that all points in can be assumed to be bounded by some positive constant . Thus, by (64), (41), (48), and (47), we have, for all , where denotes the process with the initial value at time . Owing to (12) and (49), the function on the right-hand side of (69) is integrable. Thus, it follows from Theorem 2.27(b) in Folland  that the partial derivative of in terms of for each exists. Hence, we have Again, by (12) and (49), we know that the function on the right-hand side of (70) is integrable. Therefore, by Theorem 2.27(b) in Folland , we can conclude that is differentiable with respect to . Furthermore, by (41), (70), and Lebesgue’s dominated convergence theorem, we obtain that the mapping for each is continuous. Hence, .
Third, we prove the square-integrable property (62) to be true. In fact, it follows from condition (12) that is a -finite measure since for any . In addition, it is easy to see that the nonnegative function is a measurable one on the product space . Hence, by the mean-value theorem, (69), (70), Jensen’s inequality, and the differentiability of in , we have where and are some positive constants. Furthermore, it follows from (66), (48), and (12) that (61) is true.
Fourth, we prove that satisfies the IPDE (59). In fact, for each , it follows from the time-homogeneity of that Since , it follows from Itô’s formula (see, e.g., Theorem 1.14 and Theorem 1.16 in pages 6–9 of ) that, for each fixed , Furthermore, let for each and . Then, is -predictable. Thus, owing to (62) (here we need to use an arbitrary but fixed to replace ), it follows from Theorem 4.2.3 in Applebaum  (or the explanation in pages 61-62 of ) that the last term in (73) is a semimartingale. Thus, taking expectations on both sides of (73), we get Then, by letting , we know that is in the domain of the infinitesimal generator of , which is denoted by ; that is, Now, by (66), we see that for each and all in a neighborhood of zero such that . Thus, we have where the second equality in (76) follows from the Markov property of (e.g., Proposition 7.9 in ). Then, we have Now, by the fundamental theorem of calculus, as , we a.s. have Furthermore, by the mean-value theorem, we have Since the function in the left-hand side of (78) is uniformly bounded by an integrable function, it follows from the dominated convergence theorem that the right derivative of at exists and satisfies Hence, by (72) and (80), we know that satisfies (59). In addition, we have where is some positive constant. Thus, by Lebesgue’s dominated convergence theorem, we can conclude that is continuous in . Therefore, it follows from (59) that is continuous in , which implies that . Hence, we complete the proof of Lemma 12.
Lemma 13. Let defined in (18). Then, under conditions (C1), (C2), and (12), is a -valued semimartingale with . Furthermore, define Then, is an -semimartingale and has the following canonical decomposition: where is defined in (25).
Proof. First, we show that is an -semimartingale. In fact, it follows from Itô’s formula (see, e.g., Theorem 1.14 and Theorem 1.16 in pages 6–9 of ) and Lemma 12 that Then, by Lemma 12 and the claim in pages 61-62 of Ikeda and Watanabe , we know that the third term in the right-hand side of (85) is an -martingale. Furthermore, by (63) and similar proof as that used for Lemma 10, we know that the second term on the right-hand side of (85) is of finite variation a.s. Hence, we get that is an -semimartingale. Thus, it follows from (85) and the definition of that (84) is true.
Second, defined as follows is an -martingale: In fact, by the mean-value theorem, (63), (41), (12), and the fact that is a -finite measure since for any , we have Thus, it follows from (87) and the claims in pages 61-62 of Ikeda and Watanabe  that is an -martingale. Therefore, we can conclude that is an -semimartingale. Hence, Lemma 13 is true.
Lemma 14. Let and