_{1}

^{*}

In this article, we consider the exponential hedging and the mean-variance hedging in the basis-risk model. We construct hedging strategies for multiple units of claim and calculate hedging errors. We then observe how the hedge error risk increases when the investor raises trading volumes of the claim. Under our definition of the hedge error risk amount, the risk increases in a linear way, according to the claim volume for the mean-variance hedging. As to the exponential hedging, it does not,
* i.e., nonlinear *increment. The hedging error for the exponential hedging, however, tends to have the same properties to the mean-variance hedging when either risk-averse parameter or claim volume goes to zero. We numerically demonstrate this fact. Our numerical demonstration with the results of the previous researches verifies that the indifference price converges to the mean-variance hedging cost when the claim volume goes to zero under the basis-risk model.

In this article, we consider hedging problems for the European-type contingent claim taking into account the position of the claim on the basis-risk model. We consider both exponential hedging and mean-variance hedging for multiple units of claim. We only consider the seller’s problems for convenience, the position, thus implying the sold amount, which is also called claim volume in this study. In the previous studies, the exponential hedging problems and the mean-variance hedging problems have been solved for one unit of claim. However, in practice, many financial institutions trade great numbers of derivatives. If they want to maintain the solvency margin or the capital adequacy, they should manage or control their risks taking into account their positions. Hence, it is needed to consider the hedging problems for multiple units of the claim.

The basis-risk model is a typical example of the incomplete market model, which includes the model that the underlying asset of the contingent claim is not traded in the financial market. The pricing models for the weather derivative or the derivative written on the market index are recognized as one of the basis-risk models for instance. In the complete market (e.g., Black-Scholes model), any contingent claims are perfectly replicated with traded assets, and this simultaneously gives the price of the claim. On the other hand, the value of the claim is not surly attained with traded assets in the incomplete market setting. This means that the seller of the claim is exposed to have the hedge error risk, so she/he wants to control it with her/his preference. The exponential hedging and the mean-variance hedging have independently developed in the context of finding the optimal hedging strategy for the contingent claim in the incomplete market model. The significant difference of the both approaches is whether it includes the risk preference of the market participant or not. The exponential hedging reflects the investor’s attitude for the risk since it is based on the utility maximization with the exponential utility. The exponential hedging also has been developed in context of the utility indifference pricing with the exponential utility such as [

These methods do not only provide the optimal portfolio strategy, but also lead the pricing rule including the selection of the equivalent martingale measure. Davis [

It is recalled that we consider the hedging problems for multiple units of claim. We evaluate the hedge error risk with the squared root of the expectation of the quadratic hedge error (i.e.,

At this point, the asymptotic behaviors for both exponential hedging and utility indifference price have been considered in previous literatures. Ilhan et al. [

Reviewing previous researches makes us be aware that it has never verified the implication about the convergence goal of the indifference price and the exponential hedging when the claim volume goes to zero, for instance. This study considers and investigates that how the indifference price and the exponential hedging converge when the claim volume goes to zero. We implement the exponential hedging and observe its hedge error closing the claim volume zero. We then find that the hedge error of this hedging approach has linearity with respect to the claim volume for small claim volume. This property is also characterized for the mean-variance hedging as mentioned in the above. That is, the exponential hedging tends to have the same behavior to the one of the mean-variance hedging when the claim volume goes to zero. From our demonstration with the review of Ilhan et al. [

The rest of the paper is organized as follows: in Section 2, we set up the financial market model. We especially consider the basis-risk model. In Section 3, we solve the mean-variance hedging problem for the multiple units of claim. Also, we show the linear increment of the hedge error risk for the mean-variance hedging strategy. In Section 4, we construct the exponential hedging with the utility indifference price for multiple units of claim. In particular, we derive the exponential hedging strategy by asymptotic scheme. In Section 5, we implement the exponential hedging and numerically demonstrate behaviors of hedge error amounts for both hedging strategies. Finally, we conclude this study in Section 6.

We consider the basis-risk model (or non-traded asset model). That is, there are one risky asset S (typically the stock), one risk-free asset B (typically the bank account) with zero risk-free rate and one state level Y which is supposed to be not traded in the financial market. For instance, as to the weather derivative case, Y corresponds to a weather index such as the average temperature. Let us set the value process for above instruments. The uncertainty in this market is characterized by a probability space_{t} is the filtration generated by

The value process of the risk-free asset B is

with

for

We would price a European-type claim whose payoff function is denoted by

We use European put option in the numerical example.

The hedging strategies are constructed by the self-financing rule. That is, the hedge portfolio value

with the hedging strategy

Definition 2.1. (Admissible) The portfolio strategy

Therefore, we denote by A the set of all admissible policies

Because of the incomplete market model both strategies are exposed to have hedging error. In this work, the hedging errors

For two-dimensional predictable process

We assume that

Then Z is a martingale under P. Z is a solution of

Defining an equivalent probability measure

then

then, from the Girsanov’s theorem,

In the present section, we consider the mean-variance hedging strategy for multiple units of claim. The result argued in this section is a basis for the main theorem. The purpose of the mean-variance hedging is to find a hedge portfolio strategy

then the value process of the hedge portfolio is represented by

since the initial hedging cost X(0) is

In the rest of the section, we construct the mean-variance hedging strategy for multiple units of claim (i.e., kH) and evaluate its hedging error (3.1).

The mean-variance hedging strategy is constructed with Galtchouk-Kunita-Watanabe decomposition of the claim H under so-called Variance Optimal Martingale Measure (VOMM). We denote VOMM by P^{*}. In particular, the initial hedging cost^{1} is given by the expected value of the discounted payoff of H under VOMM. We would like to recommend the reader to refer [^{*}. We define the VOMM according to [

Definition 3.1. (Variance-Optimal Martingale Measure: VOMM) The equivalent martingale measure

It is easy to find the VOMM for our basis-risk model.

Proposition 3.1. (Variance-Optimal Martingale Measure) The variance-optimal martingale measure P^{*} is given by

in our financial market model introduced in the previous section.

Proof. Under the real world measure P, the discount risky asset price

where

is then deterministic. Therefore, Lemma 4.7 in [

Q.E.D.

From Proposition 3.1, Z^{*} solves to

where

Remark 3.1. The variance optimal martingale measure P^{*} in our model coincides with the minimal martingale measure Q. ^{*}.

In this section we give the mean-variance hedging strategy. To this end, we first derive the perfect hedging strategy for the claim H under VOMM P^{*} by reference to [

The value processes S and Y are respectively driven by

under P^{*}, where^{*} is then given by

with

Both of ^{*}, ^{*}.

Remark 3.2. In the case of k units claim (k > 1), the Galtchouk-Kunita-Watanabe decomposition is directly given by

with

from (3.5) and (3.6), where

Now we solve

from Markov property. Feynman-Kac formula yields that

with

Substituting (3.7) into (3.8) we obtain

then it holds

By comparison between (3.6) and (3.10), we have

Theorem 3.1. (Schweizer [

for^{mvh} is the gain process for the mean-variance hedging strategy

In this section, we construct the mean-variance hedging strategy for the claim extending units of claim to multiple volumes.

Proposition 3.2. The mean-variance hedging strategy for k-claims is given by

for

Proof. For a constant k, kH remains in

Next, we verify that

for

Defining

From Ito’s formula and the orthogonal relation between

From Fubini’s theorem, we obtain

since

This yields

with

Q.E.D.

In this section, we solve the hedging error risk measured by

Theorem 3.2. (Heath et al., [

Let us consider the hedge error risk for multiple units of claim H. Define

again. Then

with

for^{*} and

From (3.15), Remark 3.2 and Theorem 3.2, it holds that

Therefore, we obtain the following result from (3.16).

Theorem 3.3. The risk amount measured by

In this section, we construct the exponential hedging strategy based on the utility indifference price for multiple units of claim. The former has already demonstrated by [

In this section, we derive the utility indifference price as the initial hedging cost in the exponential hedging. The indifference price is derived by solving two distinct utility maximization problems. The one is so-called Merton’s problem to maximize the expected utility from the terminal portfolio value, the other is one from terminal portfolio value equipped with claims. Delbaen et al. [

In order to derive the utility indifference price, we set utility maximization problems. The market participant has an exponential utility with the risk averse coefficient

for

We denote the set of all admissible policies

The problem to maximize the expected utility from terminal portfolio value is given by

where E_{t} denotes the expectation conditioned with the market information

This is the value function for the exponential hedging introduced in [

Definition 4.1. (Utility Indifference Price) The utility indifference price

Since the investor receives the premium p at the initial time, so p in Definition 4.1 implies the seller’s price. As argued in Section 5.3.2 in [

with the exponential utility for the general incomplete market, where

Theorem 4.1. The utility indifference price coincides with the mean-variance cost for small risk-aversion and claim volume in our basis-risk model.

The basis-risk model permits the explicit solutions for u_{0} and u respectively with the exponential utility, this leads explicit representation of p such as [

Proposition 4.1. The value function u is given by

where

Proof. Hamilton-Jacobi-Bellman (HJB) equation of the value function u_{0} is

The first order condition leads that the maximum of (4.3) is achieved at

Substituting

Now we set

where

This yields

with the terminal condition

Q.E.D.

On the other hand, the explicit solution of

Proposition 4.2.

where

Proof. HJB equation of the value function u is

The maximum of (4.6) is attained by

Substituting

By [

where

Feynman-Kac formula yields that

Plugging this into (4.9) concludes the proof.

Q.E.D.

Proposition 4.3. The utility indifference price

The exponential hedging has been considered by Delbaen et al. [

Proposition 4.4. The exponential hedging strategy

Proof. See [

Q.E.D.

Then, we define the hedge error for the exponential hedging as introduced in Section 3. The risk amount of the hedge error

Let us derive an asymptotic expansion of the exponential hedging strategy, i.e.,

Proposition 4.5. The utility indifference price

where

Proof. Taylor expansion for

where

The power series expansion for

with

if

Q.E.D.

From Proposition 4.5, we obtain a closed formula of the exponential hedging strategy (4.11) by calculating the first derivative of (4.12). From (4.12), we have

In this section, we demonstrate the exponential hedging discussed above by using Monte-Carlo simulation.

We also obtain main results of this work through the numerical simulations in this section.

Main ResultAs mentioned in Section 2, we consider the hedging problems for the put option written on Y. Its payoff function is

with the strike price K.

For the claim H presented by (5.1), we have

where

And also, from the fact that

it holds

We obtain the exponential hedging strategy in the closed form by substituting these into (4.16).

We first should check whether our model and parameters satisfy the condition (4.13). We specially select the upper of k to satisfy the condition (4.13). We use parameters described in

mulated results for

where N is the number of simulation times. And then, we simulate

The table shows that our parameters are valid up to k = 10. The fact that

Tables 3(a)-(d) show the risk amount

Needless to say, the hedge error risk increases according to the claim volume k for both hedging strategies. We are however interested in the increment of the risk amount rather than itself. That is, how the risk amount increases according to the claim volume k? To this end, we evaluate the proportion of the risk amount for multi- volume traded to the risk amount for a unit claim sold, i.e.,

As shown in Theorem 3.3, the risk amount of the hedge error varies in linear way for the mean-variance

Parameter | |||||||||
---|---|---|---|---|---|---|---|---|---|

Value | 100 | 0.01 | 0.25 | 100 | 0.12 | 0.30 | 100 | 1.0 | 0.75 |

Volume k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1.04 | 1.08 | 1.12 | 1.17 | 1.23 | 1.30 | 1.37 | 1.46 | 1.56 | 1.68 | |

Std. Error | 0.06 | 0.13 | 0.21 | 0.30 | 0.41 | 0.54 | 0.71 | 0.90 | 1.14 | 1.43 |

Upper Bound | 1.04 | 1.08 | 1.12 | 1.17 | 1.23 | 1.30 | 1.38 | 1.47 | 1.57 | 1.69 |

Volume k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

Risk Amount | 7.99 | 15.88 | 24.12 | 32.01 | 39.97 | 47.7 | 56.04 | 63.77 | 71.52 | 79.54 |

Bound | 7.94 - 8.04 | 15.79 - 15.98 | 23.98 - 24.27 | 31.81 - 32.21 | 39.72 - 40.21 | 47.41 - 48.00 | 55.70 - 56.38 | 63.38 - 64.16 | 71.09 - 71.96 | 79.05 - 80.03 |

a. The second line shows the risk amount of the hedge error R^{(k)}, and the third line lists the its confidence interval.

(b)

a. The second line shows the risk amount of the hedge error R^{(k)}, and the third line lists the its confidence interval.

(c)

a. The second line shows the risk amount of the hedge error R^{(k)}, and the third line lists the its confidence interval.

(d)

^{(k)}, and the third line lists the its confidence interval.

Volume k | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1.00 | 2.01 | 3.00 | 4.02 | 4.98 | 6.03 | 7.06 | 8.04 | 9.04 | 10.11 | |

1.00 | 2.00 | 3.06 | 4.13 | 5.23 | 6.37 | 7.62 | 8.86 | 10.19 | 11.66 | |

1.00 | 2.03 | 3.18 | 4.43 | 5.81 | 7.36 | 9.12 | 11.00 | 13.13 | 15.55 |

a. The risk amount tends to increase in nonlinear way according to the claim volume k for large risk-averse coefficient and claim volume.

hedging. Our numerical experiences thus show the convergence of the exponential hedging to the mean-variance hedging about the linear increment of the hedge error risk when the risk-averse coefficient and the claim volumego to zero. In fact, Tables 3(a)-(d) show that the difference in the risk amounts between the mean-variance hedging and the exponential hedging are very small for small risk-aversion γ and claim volume k. The risk amount of the hedge error for the exponential hedging with γ = 0.001 more closes with the one for the mean-va- riance hedging rather than the cases of γ = 0.005, 0.01. Such convergence has already been shown by [

We further add the characteristics about the increment of the risk amount for the both hedging strategies. In particular, we implement hedging strategies by varying ρ. The parameters used in this demonstration are described in

upper bound of

Tables 7(a)-(g) shows the hedge error risk amount for the mean-variance hedging, and the graph is described in

Furthermore, we consider the linearity of the hedge error risk, the linearity is one of the characteristics of the mean-variance hedging. Tables 9(a)-(g) shows the proportion of the risk amount

Parameter | |||||||||
---|---|---|---|---|---|---|---|---|---|

Value | 100 | 0.01 | 0.25 | 100 | 0.12 | 0.30 | 100 | 1.0 | 0.01 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.08 | 1.20 | 1.35 | 1.55 | 1.85 | |

Std. Error | 0.14 | 0.35 | 0.67 | 1.14 | 1.91 |

Upper Bound | 1.08 | 1.20 | 1.35 | 1.55 | 1.85 |

with k for

Summarizing results considered in the above, we have the following theorem.

Theorem 5.1. The exponential hedging with the utility indifference price as the initial hedging cost, converges to the mean-variance hedging when the claim volume k or the risk averse coefficient γ closes to zero.

In this work, we constructed both the mean-variance and exponential hedging strategies for multiple units of claim and calculated the hedge error risks for each risk-averse level and claim volume. The hedge error risk is measured by the squared root of the expectation of the quadratic hedge error. We then characterized the nonlinear increment of the hedge error risk with respect to the claim volume for the exponential hedging strategy; that is, the hedge error risk varies in a nonlinear way with respect to the claim volume. By contrast, the hedge error risk changes in the linear way for the mean-variance hedging. Our numerical examinations verified that the nonlinear increment is reduced to the linear increment when the risk-averse coefficient and the claim volume go to zero. That is, we showed that the exponential hedging converges to the mean-variance hedging from the point of

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Risk Amount | 7.84 | 15.68 | 23.56 | 31.49 | 39.29 |

Bound | 7.79 - 7.89 | 15.59 - 15.78 | 23.42 - 23.71 | 31.30 - 31.69 | 39.04 - 39.53 |

^{(k)}, and the third line lists the its confidence interval.

(b)

^{(k)}, and the third line lists the its confidence interval.

(c)

^{(k)}, and the third line lists the its confidence interval.

(d)

^{(k)}, and the third line lists the its confidence interval.

(e)

^{(k)}, and the third line lists the its confidence interval.

(f)

^{(k)}, and the third line lists the its confidence interval.

(g)

^{(k)}, and the third line lists the its confidence interval.

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

Risk Amount | 7.93 | 16.35 | 26.46 | 38.39 | 53.37 |

Bound | 7.88 - 7.98 | 16.25 - 16.44 | 26.08 - 26.83 | 38.00 - 38.77 | 52.51 - 54.21 |

^{(k)}, and the third line lists the its confidence interval.

(b)

^{(k)}, and the third line lists the its confidence interval.

(c)

^{(k)}, and the third line lists the its confidence interval.

(d)

^{(k)}, and the third line lists the its confidence interval.

(e)

^{(k)}, and the third line lists the its confidence interval.

(f)

^{(k)}, and the third line lists the its confidence interval.

(g)

^{(k)}, and the third line lists the its confidence interval.

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.06 | 3.34 | 4.84 | 6.73 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.09 | 3.32 | 4.79 | 6.67 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.03 | 3.14 | 4.35 | 5.67 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.00 | 3.02 | 4.09 | 5.20 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.03 | 3.15 | 4.35 | 5.67 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.08 | 3.37 | 4.81 | 6.48 |

Volume k | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1.00 | 2.06 | 3.28 | 4.86 | 6.64 |

the hedge error view. As mentioned in Section 1, it has been already shown that the utility indifference price with the exponential utility converges to the no arbitrage price when the claim volume goes to zero. Hence our results with the results of the previous researches lead a perspective that the utility indifference price with the exponential utility converges to the mean-variance hedging cost in the basis-risk model.