The authors declare no conflicts of interest regarding the publication of this paper.
The following paper will show an investigation regarding the hedging portfolios for the G2++ interest rate model which will be analyzed, in which a focus on the decomposition of its variations while using derived sensitivities from the instantaneous short-rate dynamics will also occur. Thus, this paper will provide the following information: a precise control of the associated approximation error, computed high-order sensitivities, an accurate polynomial approximation of the portfolio value variations with coefficients giving by the sensitives. The central aim of this paper is to provide a comprehensive, portfolio oriented presentation of the hedging methodology. Its structure will begin with defining the composition of the portfolios, bonds and swaps, to derive their decomposition in terms of sensitivities. Later, a clear presentation of the optimized problem that underlies the hedging strategy will be made, to end the work with a range of numerical experiments to demonstrate the efficiency of the proposed approach. Plenty of hedging methods exist in literature, however, our approach is more optimal in the sense, due to the way the hedger is determined based on an upper bound on the potential loss which is induced by the hedging strategy itself.
For the present work nowadays is solely devoted to enhancing and maintaining the problem of hedging a position that is sensitive to interest rate by means of a portfolio of bonds and swaps. Our methodology is based upon the solution approach rather than contributing to the exploration of the subject itself, which would be already considered by many.
To begin with, it is widely understood that there is no consensual theoretical approach for the hedging of a position that is sensitive to interest rate when compared with what is happening in the equality area. Due to the interest rate being derived which thus compels at least two difficulties: Firstly, which focuses on the need to simultaneously consider various types of interest rates; Secondly, which focuses on a given interest rate itself that does not appear to be present aged as a tradable asset. Furthermore, when it comes to financial literature, it often revolves on hedging a single position by one or few instruments, regardless the fact the most practitioners tend to be faced with covering a portfolio position by another one. Hence, a lack of clear and common theoretical reference on this aspect is seen, and due to the constant focus on of the development on studies cases, then consequently lack of evidence based on particular situations is seen. Thus, in depth analysis of the general mechanism of portfolio oriented hedging is often resulted as missing. So, we contribute in this work to fill the upcoming lack and gap.
Although, in literature, hedging in the inter temporal framework is well considered. However, continuously making infinite number of trades of the underlying asset used in the hedging is actually untenable in practice as required in the theory, which thusly grants practitioners an interest and confidence on rolling one periods of hedging position. Based on this observation, this paper will focus on studying one period hedging.
The background of hedging is just to compensate for the loss affecting the position to hedge, by the gain collected from the hedging instrument. And this can be achieved by the cash flows matching approach (e.g.: [
Additionally, relying on the considered position and underlying risk factors, multiple kinds of sensitivities might be defined which consequently will lead to different hedging results. For instance, the Coupon Bearing Bond (CBB), is the well established one and second order sensitivities known under the names of MacAulay or Fisher-Weil, with duration and convexity [
Although it is highly popular among academics and practitioners, the bond during and convexity, it is important to shed a light on the use of these sensitivities which underlines the strong assumption that at the hedging time horizon, the interest rate curve has made a parallel shift movement. To further clarify, in such context, explicit formulas for the sensitivity are made available as well as the hedging operation, and the performance of the matching sensitivities for the positions to hedge and the hedging instrument are easily done. So, the enhancement of this classical duration convexity approach and the corresponding hedging has been performed recently in [
Moreover, in general the parallel shift assumption appears to be unrealistic, since empirical observations show that at a given time horizon, the interest rate curve may have a very different shape. To elaborate further, it means that in order to perform a proper hedging strategy, a dynamic model for the interest rate must be introduced. This requirement has led to the introduction of stochastic duration [
The final notion is theoretically considered superior to the Macaulay’s or Fisher-Weil duration, which both require a yield curve to have made a parallel shift. Nevertheless, most empirical tests on bond immunization performance have not demonstrated any actual superiority of the stochastic duration when being compared to the simple classical duration [
Additionally, it is said to be unclear how the general portfolio and the extension of stochastic duration, as well as the convexity to interest rate which is based on instruments other than the CBBs, tend to be.
With respect to the shocks linked to the unobservable two uncertainty factors underlying the G2++ model, an analytical expression for the sensitivities for the coupon bearing bonds and interest rate swaps contracts are to be derived. As such, the hedging of a position sensitive to the interest rate, either by means of portfolio by bond or by swaps and in accordance with the market participants practice, it becomes easily transparent resulting from the equilibrium between the various involved sensitivities as seen in [
To conclude, this paper will be assembled as follows. Starting with our main results that are solely started in Section 2, a recalling of the primer notions on the interest rate model G2++ will be provided. Later, a presentation of the formulation of the short rate as stochastic formula is given. To continue with, the results of our article in Section 3 providing the basic identity portfolio change and its decomposition in Subsection 3.1 which is the main key for the hedging approach in Subsection 3.2 where a presentation of the overall expression (hedging portfolio and hedging instruments) changes is performed. To last but not least, state in Subsection 3.3, the optimization problem, the last nonlinear and integer optimization problem, which is solved using metaheuristics methods (such as genetic algorithms or using the commercial solver CPLEX). This last is used in our numerical illustrations that are displayed in Section 4. In that section, a given examples of hedging a CBB position by a portfolio made by CBBs and IRSs. To finally conclude the paper in Section 5.
The two-additive-factor Gaussian model G2++ ([
where
and
All of these dynamics are given under the risk-adjusted risk-neutral measure .Here
3.1.1. Portfolio Change of Coupon Bearing Bonds (CBB)
Let us denote by
Therefore there are
The bond
is assumed to have coupons with the coupon rate
paid at times
where
The future value
where
The zero order sensitivities of the portfolio of CBB, also referred as a residual term, is defined by
The portfolio of CBB sensitivity denoted by
The terms
The sensitivities
Therefore, the residual term
for all
It means that in the practical implementation, making use directly of the formula (8) and calling functions computing CBB sensitivities internally should not be the profitable way to compute the portfolio of CBB sensitivities. It may be convenient to pre-compute in advance the matrix of zero-coupon sensitivities and then to make appeal to the computed entries when they are really needed.
It should be emphasized that replacing the future change of the portfolio of CBB by a polynomial function (whose the coefficient is the initial portfolio sensitivities) is also interesting in the perspective of risk measurement and management. Indeed instead of re-evaluating the portfolio for each scenario of shocks considered (which is time and memory consuming) one has just to calculate the value of the polynomial function. Of course, this is meaningful whenever the remainder term is negligible. All the details for this decomposition and the remainder term are explained in details in [
3.1.2. Portfolio Change of Interest Rate Swaps (IRS)
Let us denote by
The swap
and have the ordered payment times
where
The future value
Therefore, the portfolio sensitivity of order zero or residual term, measuring the passage of time when the yield curve remains unchanged, is given by
The zero order sensitivities for the
Here the
The
The expression for
It should be emphasized that replacing the future change of the IRS portfolio by a polynomial function (whose the coefficient is the initial portfolio sensitivities) is also interesting in the perspective of risk measurement and management. Indeed, instead of re-evaluating the portfolio for each scenario of shocks considered (which is time and memory consuming) one has just to calculate the value of the polynomial function. Of course, this is meaningful whenever the remainder term is negligible.
Our main purpose in this Section is to show, when hedging a portfolio, how useful are the high order sensitivities, with respect to the shock factors underlying the structure of interest rate model taken into consideration.
First, we formulate the expression changes linked to both the portfolio to hedge and the hedging instruments. Next the involvement of the sensitivities and their offsetting effects are presented.
3.2.1. Profit & Loss for a Covered Portfolio
Let us denote by
The idea underlying the hedging relies on using another portfolio, referred in the sequel as a hedging portfolio (or hedging instruments) such that, this last would lead to a nonnegative profit compensating the loss on the initial portfolio. Therefore instead of the absolute change
associated with the initial naked portfolio, at the horizon
The hedging portfolio
It means that
such that
where
with
In (20), we have used the fact that for instance the instrument value
and the corresponding security numbers
such that the value of
should be small as possible. The difficulty here is linked to the fact that the future values of the hedging instruments at time
3.2.2. Interaction between the Portfolio to Hedge and the Hedging Instruments
As presented in the next Section, under the G2++ model, the interest rate is governed by two uncertainty factors shocks, let us say
In this approximation, the shocks
is very small on the hedger’s point of view, as
where
where
Under the view of
and under the domain of definition of these two for some nonnegative real numbers
then the hedging problem previously presented above, is essentially driven by the minimization with respect to the objective function
where
Accurate statement and further development will be performed in the subsequent work. In this Section we can point the following facts:
1) An approximation as (21) plays an important role when performing hedging a given portfolio
2) No matter is the exact nature of instruments
that the challenge is just to get the corresponding approximations (21). This is performed in the sequel for the case of zero-coupon bonds, coupon-bearing bonds, interest rate swaps and their portfolios.
3) To introduce financial instrument sensitivities, with respect to the shock factors underlying the model used for the interest rate, as we do here seem to be not frequent in the financial literature. It has the advantage to not focus on a given asset to base the hedging. For instance, in the case of equity market, it is usual in stock option to take as a basis of reference for the hedging the underlying stock asset itself. A similar approach for the interest rate market is not so clear. So our approach about sensitivities with respect to the shock factors appear to be natural and interesting solution.
4) A main basic financial instrument in the interest rate market is the zero-coupon bond. Coupon-bearing bond and interest rate swap may be seen as linear combination of zero-coupon bonds. However, all of these instruments appear to be non-linear functions of the shock uncertainty factors underlying the interest rate model G2++ considered in this paper.
5) It is seen from this paper that considering just the first and second order sensitivities as the classical duration and convexity [
6) Interest rate option, as the one on zero-coupon bond, appears to be a non-linear function of zero-coupon bonds. It is tempting to apply the approach, as (21), we have used here for instrument linear on zero-coupon bonds. Investigation on this direction will be performed in one of the sequel works.
According to the above subsection, in a generic form, and by taking
with the constraint
and where
For convenience we can deal with vectorial notations such that the constraint (34) may be written as
where
and
For the function
where the components
Therefore, we obtain the following result.
where
for all
The objective function
In this Proposition, by the equivalence between
Observe that both the objective function and constraints associated with
The transaction cost parameters in the numerical illustrations are chosen to be small but strictly positive, reflecting typical orders of magnitude observed for bid–ask spreads and trading fees in liquid markets. These values are not calibrated to a specific market, but correspond to standard benchmark assumptions commonly used in the literature to illustrate the qualitative impact of transaction costs rather than to provide an exact empirical fit.
The portfolio to hedge is made by bonds in long and short positions, and there are both (payer and receiver) swaps and also bonds (in long and short positions) in the hedging portfolio. So our target here is to test how good or bad is our approach in such a situation.
First in
The hedging allocation is displayed in
The maximal amount
| maturity (years) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| yield (%) | 2.72 | 2.71 | 2.70 | 2.71 | 2.71 | 2.72 | 2.73 | 2.74 | 2.79 | 2.81 |
|
|
1.11% |
|
|
3.51% |
|
|
1.13% |
|
|
2.81% |
|
|
−7.21% |
|
|
90 days |
|
|
0.1% |
|
|
25% |
|
|
0 |
|
|
1 |
|
|
5 |
|
|
5 |
|
|
8 |
|
|
5% |
| type | number | cpn | maturity | unit value |
|
|
100,000 | 3% | 15 years | 600.18 |
|
|
150,000 | 5% | 7 years | 879.31 |
|
|
150,000 | 7% | 5 years | 997.07 |
|
|
175,000 | 4% | 10 years | 764.7 |
|
|
150,000 | 5% | 12 years | 809.38 |
|
|
1000 | 4% | 8 years | 803.72 |
|
|
9000 | 3.50% | 3 years | 913.71 |
|
|
1000 | 4.60% | 4 years | 920.21 |
| Type | number | face value | frequency | coupon | maturity | unit value |
|
|
|
100 | l year | 5.50% | 7 years | 90.62 |
|
|
|
100 | l year | 3.50% | 5 years | 85.29 |
| type | number | notional | frequency | maturity | rate swap |
|
|
|
1000,000 | 6 months | 5 years | 6.95% |
|
|
|
1000,000 | 6 months | 7 years | 7.11% |
|
|
|
|
|
Max Profit or Loss | ratio |
| 0 | 1267 | 1007 | 0 | 36,422,240 | 6.05% |
To conclude, it’s well established by practitioners and academics, throughout the years of methodology of hedging a portfolio, that it is a very severe issue. Multiple studies on hedging tend to essentially focus on its problems which relate to the usage of a single instrument, in which the management of the operations of a practitioners’ (whom are considered as dealers), should be thought in terms of portfolios. Therefore, based on that case, the work is often devoted to the issues of hedging a sensitive position to interest rate by means of portfolio of the bonds or swaps.
Here, a portfolio oriented hedging approach is proposed, in which it relies on the G2++ model. It can be considered as an alternative approach of the practitioner methodology which is based on bumping some pillars of the interest rate curves.
Following, it has been recognized by many standard practical approaches, that plenty of drawbacks those involving a great sum of sensitivities and a consumed considerable time computed are viewed. Regardless of the sensitivities limitations to the first and second orders, a lack of insurance of the reliability of the hedging is attained, as the hedger cannot evaluate the possible loss magnitude which results from the usage of the methodology.
As a comparison, the approach we tackle is solely based on two uncertain factors which lead to the consideration of the high order sensitivities with almost immediate time computation. Also, the hedger has the potential to fix or acknowledge in advance the magnitude of maximal loss, based on the results from the use of the hedging as presented here.
Therefore, our approach could be viewed as an optimal one, especially when compared with logically well-founded alternative methodologies, for which the hedger lacks prior control of the loss that could incur from the hedging approach.
Nevertheless, it is important to note that a continuous risk from the underlying G2++ model used will remain, despite the possibility to control such a loss.
Moreover, the G2++ model is still the most accurate and right one to be used, regardless of the presence of a bad estimation of the parameters which could affect the span of a bad hedging. Thus, it remains unknown to acquire a true data generating process, for the state variables.
Further, the suitability of the underlying model, which is assured to be fulfilled in this work, is considered as another aspect and issue that our scope tackles.
Here, the sensitivities we used in the hedging approach, involve the unobservable state variables that are considered by the underlying G2++ model. To clarify, having such views on these variables, is not necessarily so intuitive for the hedger. Rather, it could be recalled that these variables used in our paper, tend to follow Gaussian Laws, so a conservative values for the shocks as −5 and 5 can be used in our numerical illustrations. Based on that, it is ought to be noted that the statistical considerations and numerical tests are means to help in forming views on extreme values for the shocks.
Henceforth, we have attempted to provide explicit formulas for all the expressions involved in our hedging approach, regarding the sensitivities for the individual position or for portfolios. However, as for the final instrument of hedging allocation, it’s been deducted that the hedger needs a solver to solve the involved optimization problem.
In this paper, a usage of the CPLEX commercial solver is used, which may be a drawback if the hedger doesn’t have at her disposal such a tool. So, we came up with another alternative solution which mainly depends on the hedger software system.
Multiple extension and various questions can be raised after the presented work.
The G2++ model is solely the underlying model of our hedging approach, so it relies only on the two factor uncertainties, which makes it more appealing to explore the situation, where an alternative interest rate model has more uncertainty factors (such as, Arbitrage Free Dynamic Nelson-Siegel). Vaguely, it is considered as an open question of whether considering a general GATSM models as an underlying for the approach contributes to improve the hedging or if it simply implies only a complication from the practical point of view.
Henceforth, our work has been met with some limitation on the context, where just one interest rate curve is used. Nevertheless, most of the portfolios under consideration tend to generally contain instruments linked to multiple interest rate curves, as in the case of currency swaps, regardless if it’s the one to hedge or the one to use for the hedging. Thus, this raises questions about the performance of the hedging operation in the multi-curves setting.
Can one extend his hedging approach to handle such situation?
In our presented paper, a focus on the implicit assumption of the instruments being considered inside the portfolios are only exposed to the interest rate risks. Even-though, a CBB or IRS are also sensitive to credit as a counterparty risk, or an equity and foreign exchange risk and so on. So, the uncertainty risk factors approach, like the G2++ model dealt with, can be extended to take into account at least both the risks linked to interest rate and credit. Such a question is important to be explored.
Furthermore, we have focused on the sensitivities of the portfolio, those made by CBBs and IRSs, with respect to shocks associated to the uncertainty factors which are underlined in the used G2++ model. So, it is previewed that the model parameters solely depend on the hedging results. As the value of these parameters depend on the method used to their inference, it could also be intrigued to analyze the parameters sensitivities and to decide if it is possible to include the hedging the model tends to risk, based on the parameter uncertainties or instability for the time-period hedging considered.
Thusly, when considering the previously mentioned constraints, we have avoided to use generated model like Zero Coupon process without the economical sense. Yet, it still remains and issue when considering the weakness of the G2++ model, since its calibration is done on an improper manner as not discarding these ZC prices higher than 1.
Therefore, in order to maintain the hedging perspective, two questions can be raised in the manner: 1) What could happen if the calibration is carried in a consistent manner? 2) How could one modify the standard G2++ model to attain a new one that is consistent with the possibility to deal with interest rates near zero or below?
In that case, this paper has focused on the fundamental aspect of the hedging portfolio oriented one, and the rolling hedging positions. While, keeping close track to the market reality, the hedging costs are continually kept being taken into consideration. To end with the notion that it seen sought to be interesting to perform empirical illustrations to assess the efficiency of our approach, especially when a long period of time is examined under rolling the hedging operations.
Hedging a portfolio is an issue which has been considered for a long date by practitioners and academics, and this is always the case. Various available academical work on hedging are essentially focused on the hedging problems related to the use of one single instrument, though practitioners (as dealers for example) should manage their operations and think in term of portfolios. Considering this situation, the present work is devoted to the problem of hedging a position sensitive to interest rate by means of a portfolio of bonds and/or swaps.
Here we have proposed a hedging approach portfolio oriented which relies on the G2++ model. This can be seen as an alternative approach of the practitioner methodology based on bumping some pillars of the interest rate curves.
The standard practical approach has the drawback to involve a great number of sensitivities and to consume considerable time computation. More important, as these sensitivities are generally limited to the first and second orders, there is no insurance about the reliability of the hedging result as the hedger cannot a-priori evaluate the possible loss magnitude resulting from the use of the methodology.
For comparison, our approach is based only on two uncertainty factors and consequently high order sensitivities can be considered with almost immediate time computation. The hedger has the possibility to fix or know in advance the magnitude of maximal loss resulting from using the hedging results presented here.
Therefore, our approach may be seen as optimal when compared with alternative methodologies (possibly logically well-founded) but for which the hedger has no a-priori control of the loss which can incur from the hedging approach. Despite this possibility to control such a loss, it is important to note that there always remains the risk from the underlying G2++ model used. Indeed, if this last does not really well represent the market evolution then, the hedging approach based on it may behave badly. Moreover, even the G2++ is the right model to use, a bad estimates of the corresponding parameters can also span a bad hedging. Definitely, the true data generating process for the state variables remains unknown. The suitability of the underlying model (assumed to be fulfilled in this work) is another story and issue out of our scope here.
The sensitivities, used in our hedging approach, involve the unobservable state variables underlying the G2++ model considered. It means that having views on these variables as we largely used in this paper is not so intuitive for the hedger. But it would be recalled that these variables follow Gaussian laws, so a conservative values for the shocks as −5 and 5 can be used as we do over our numerical illustrations. We think that statistical considerations and numerical tests can help in forming views on extreme values for the shocks.
We have tried to provide explicit formulas of all of the expressions involved in our hedging approach, as the various sensitivities either for individual positions or for portfolios. But for the final instrument hedging allocation, the hedger needs a solver to solve the involved optimization problem. Here for the convenience, we have used the CPLEX commercial solver. This may be a drawback if the hedger does not have at her disposal such a tool. So another alternative solution has to be found depending on the hedger software system. Various extensions and questions can be raised after this present work.
The G2++ model is the underlying model of our hedging approach. As this model relies just on the two-factor uncertainties, it would be interesting to explore the situation where an alternative interest rate model with more uncertainty factors as the Arbitrage Free Dynamic Nelson-Siegel. More generally it is an open question whether considering a general GATSM model as an underlying for the approach contributes to improve the hedging or it implies just a complication from the practical point of view.
Our work has been limited on the context where just one interest rate curve is used. However usually the portfolios under consideration (either the one to hedge or the other used for the hedging) generally contains instruments linked to various interest rate curves, as in the case of currency swaps. This raises naturally the question of performing the hedging operation in the multi-curves setting. Can we extend our hedging approach to handle this situation?
Our present paper is focused on the implicit assumption that the instruments under consideration inside the portfolios are only exposed to interest rate risks. However, a CBB or IRS is also sensitive to credit (or counterparty) risk, equity and foreign exchange risk and so on. Probably the uncertainty risk factors approach, as the G2++ model we deal with, can be extended in order to take into account at least both the risks linked to interest rate and credit. This is an important question which deserves to be explored.
Our present investigation has been focused on the sensitivities of the portfolio (made by CBBs, IRSs) with respect to shocks associated to the uncertainty factors underlying the used G2++ model. It is seen that the model parameters are very determining on the hedging result. As the value of these parameters depend on the method used to their inference, it may be interesting to analyze the parameters sensitivities and to decide whether it makes sense to include in the hedging the model risk, coming from the parameter uncertainties or instability for the time-period hedging considered.
When considering the above-mentioned constraints, we have avoided to use generated model Zero Coupon prices without economical sense. But actually the weakness of the G2++ model remains an issue, for at least due to the fact that the model calibration is done on an improper manner as not discarding these ZC prices higher than 1. Always staying in the hedging perspective, two questions can be raised:
1) what does happen if the calibration is done in a consistent manner?
2) how to modify the standard G2++ model in order to get a new one which is consistent with the possibility to deal with interest rates near zero or below?
The present work has focused on the fundamental aspect of the hedging portfolio oriented and for a one-period step, as conform to the situation of rolling hedging positions as many practitioners do. To stick close on the market reality, we have taken into consideration the hedging costs. Then it may be interesting to perform empirical illustrations in order to assess the efficiency or not of our approach when a long time period is considered under rolling the hedging operations.
I would like to thank and acknowledge the American Community School of Beirut for providing an astounding educational environment that greatly nourishes talent. Additionally, I would like to thank Wolfram Alpha for providing the means to develop the figures utilized throughout the paper.
1The related notations for
2
3resp. for the swap
4Note that
5With