Hedging a commodity-linked portfolio Charles Ouanounou* Yves Rakotondratsimba** Mickael Sahnoun*** September 21, 2012
ECE Paris Graduate School of Engineering, 37 quai de Grenelle CS71520 75 725 Paris 15, France *
[email protected] ** w
[email protected] ***
[email protected] and corresponding author
Abstract With the recent developments in commodity markets, it appears to be crucial for people, coming both from the academic and financial industry, to grant some care to the hedging mechanism of a given commodity-linked portfolio. Various authors have considered and analyzed hedging strategies for a commodity long-term position with two shorter futures (see for instance [?, ?, ?, ?]). However it clearly seems there is no generic presentation and analysis for commodity-linked portfolio frameworks. Indeed even if authors consider portfolios their studies remain restricted to very particular data. Their approaches and results are mostly statistic-based. In this project we assume that a commodity-linked portfolio with its related instruments is given. Our purpose is to perform a general analysis for hedging against unfavorable price changes. We will illustrate it by a practical implementation of the initial portfolio. Indeed our main motivation is to provide a theoretical support of a practical hedging tool. The approach relies on a one-factor structure model for the future (or spot prices) as the one introduced by L. Clewlow and C. Strickland [?]. This model can fit the initial forward curve in contrast to the model introduced by Schwartz (1997). It reflects the mean reverting nature of commodity prices. However, it has the disadvantage of the constant volatility structure of forward prices. The use of this restrictive model is justified by the fact that the one-factor uncertainty model remains both in a theoretical and practical point of view as a benchmark reference. Next, it seems to us that this simple framework is suitable to present the main ideas underlying the hedging operation. The case related to another acceptable structure model (as the two-factor one) might also be performed but at the price of technical difficulties. We look forward to do so in a next project. Therefore we first derive the sensitivities of various basic products (futures, swaps, options on spot/future, cap/floor contracts) with respect to the shock responsible for the price changes. The latter is the one that underlies the single factor model under consideration. The point of this work, unlike some classical results, is about the sensitivities nature and the high orders considered. As a matter of fact it is common to make use of sensitivities with respect to the spot/future price or even the uncertainty factor but with limitations to the first order. The drawback of such standard methods is that hedging long maturity positions with short maturity contracts cannot be properly done as illustrated in our various experiments. We will show and illustrate that introducing higher order sensitivities leads to find more accurate hedging operations. Particularly, under a conservative viewpoint of the uncertainty factor, our approach enables to derive deterministic and point-wise estimates of the hedging error. This is economically meaningful in contrast to the standard hedging error in term of variance. The
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solution we provide for the portfolio-hedging problem has also the advantage of giving the suitable allocation of hedging instruments in term of security numbers, as it is usually required in practice. Contrarily to standard solutions where only proportions of the contracts to use are given, the hedger does not have to make any extra-decision concerning the security quantities. The method considers sensitivities with respect to the shocks that are related to the risk/opportunity factor. That might look a little bit unpleasant to the hedger that is mostly familiar with observable variables. Actually we show that once a high order for sensitivities is chosen the shock levels do not matter since all extreme potential losses/gains are under control and deterministically derived by our approach. In accordance with the real practice, our hedging approach relies on the combination of various instruments that the hedger has at disposal. Only the offsetting effects between the various sensitivities matter here. By contrast, we can observe that several papers dealing with hedging are especially focused on hedging a main given derivative by using associated underlying assets.
Keywords: commodity derivatives, sensitivities, hedging, one factor model JEL Classification: G11, G13.
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Contents 1
Introduction
2
Sensitivity results 2.1 The model . . . . . 2.2 Sensitivities related 2.3 Sensitivities related 2.4 Sensitivities related
4
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4 4 6 7 9
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Hedging mechanism 3.1 Profit& loss for a covered portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hedging and the sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 14
4
Numerical Illustrations 4.1 Experimental results on the sensibilities . . . . . . . . . . . . . . . . . . . 4.1.1 Influence of the order on the approximation of the change . . . . . 4.1.2 Influence of the absolute error on the approximation of the change 4.1.3 Influence of the maturity on the approximation of the change . . . 4.2 Portfolio hedging : scopes and applications . . . . . . . . . . . . . . . . . 4.2.1 Futures’ specifications . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Brief summary of the environment . . . . . . . . . . . . . . . . . . 4.2.3 Simulations and results . . . . . . . . . . . . . . . . . . . . . . . .
18 18 19 20 21 21 22 22 23
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Conclusion
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A Appendix A : Proofs of Results A.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26 26 26 27
B Appendix B : Tables and Plots B.1 Influence of the order on the approximation of the change . . . . . B.1.1 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.2 Spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1.3 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Influence of the maturity on the approximation of the change . . . B.2.1 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2.2 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Influence of the absolute error on the approximation of the change B.3.1 Futures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.2 Spots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3.3 Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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C Appendix C : Implementations and codes
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Introduction 1. Various work in literature essentially deal with modeling energy spot and other key variables, such as the convenience yield on the asset and interest rate. Here our focus is rather on the hedging aspect starting with a well-established model. Our concern is not on the appropriateness or not of the underlying model itself, but rather on the accurateness when this last is used in the hedging view point. 2. This is in line with the willing to improve the use of existing models, a concern taken into consideration by various academics, regulators and practitioners following the financial crisis of 2007/08. 3. In this paper we consider the pricing framework model, introduced by Clewlow and Strickland [Cl-St; 1999], which enables the valuation of general energy contingent claims. This Clewlow and Strickland model can be seen as an extension of the first model by Schwartz [Sc; 1997], for which the curve is endogenously determined. For the model considered here, the evolution of the entire forward curve conditional on the initially observed forward curve is simultaneous modeled, allowing a unified approach to the pricing and risk management of portfolio of energy derivative positions. 4. First we derive the sensitivities of various instruments that we considered (futures, spots and swaps) with respect to the shock responsible for the price changes. It is common to use the sensitivities but with limitations to the first order, the point of this work is to introduce high order sensitivities when decomposing an instrument value. We will show that introducing higher order sensitivities leads to find lower estimation errors on valorization independently of the shock, and better performances in terms of portfolio assessment. 5. In accordance with the real practice, our hedging approach relies on the combination of various instruments that the hedger has at disposal. Furthermore it also has the advantage of giving suitable allocation of hedging instruments in term of security numbers, so that the hedger does not have to make extra-decision concerning the security quantities.
2 2.1
Sensitivity results The model
In this paper t and T represent times such that 0 < t < T. As introduced by L. Clewlow and C. Strickland [Cl-St; 1999], the energy forward curve F (t, T ) is assumed to be driven by the stochastic differential equation dF (t, T )(·) = σ(t, T )F (t, T )dWt (·)
(1)
where Wt (·) is standard Brownian motion. After performing integral calculus, the forward value is given by Z Z t 1 t 2 F (t, T )(·) = F (0, T ) exp − σ (u, T )du + σ(u, T )dWu (·) . (2) 2 0 0 To obtain a Markov spot price process, the volatility process σ(u, T ) must have a negative exponential form as h i σ(u, T ) ≡ σ exp −α(T − u) for 0 < u < T (3) where the model two parameters σ and α are assumed to be non-negative constants and independent of the times t and T . Here σ determines the level of spot and forward prices’ volatility, and α corresponds to the rate at which the volatility of increasing maturity forward prices declines. It is called the speed 4
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of mean reversion of the spot price. As detailed in [Cl-St; 1999], these parameters can be estimated directly from the prices of options on the spot of energy or forward contracts. Alternatively it may be done by best fitting to historical volatilities of forward prices. Under the specification (3), it appears that Z
t
σ 2 (u, T )du = σ 2
0
o 1 n exp[−2α(T − t)] − exp[−2αT ] 2α
and t
Z 0
Z t σ(u, T )dWu (·) = σ exp[−αT ] exp[αu]dWu (·) 0 r n o 1 = σ exp[−αT ] exp[2αt] − 1 εt (·) 2α
where εt (·) ≡ ε(·; t, α) is a random variable following a standard Gaussian normal law, which depends only on the time t and the model parameter α 1 . It may be emphasized that εt does not depend on the maturity T of the forward price F (t, T ). As a consequence, under (3), we have h i F (t, T )(·) = F (0, T ) exp −g1 (T − t, T ; α, σ) + g2 (T − t, T ; α, σ)εt (·) (4) where g1 (u, T ; α, σ) ≡ σ 2 and
o 1 n exp[−2αu] − exp[−2αT ] 4α
r g2 (u, T ; α, σ) ≡ σ
o 1 n exp[−2αu] − exp[−2αT ] . 2α
(5)
(6)
It may be noted that g1 (u, T ; α, σ) and g2 (u, T ; α, σ) are positive terms satisfying g1 (T, T ; α, σ) = 0 and g2 (T, T ; α, σ) = 0. The dynamic, described by (2) and (3), should be seen as a general version of the one-factor model considered by E. Schwartz [Sc; 1997], where the spot energy price and the forward price curves have particular forms. Indeed by considering the spot price process St ≡ F (t, t) we retrieve the E. Schwartz [Sc; 1997] single factor model n o dSt (·) = α µ(t) − ln[St ] dt + σdWt (·) St where µ(t) =
o 1 ∂ σ2 n ln F (0, t) + ln F (0, t) + 1 − exp[−2αt] α ∂t 4
(7)
(8)
in order the model to attain consistency with the initial forward curve F (0, T ). It may be noted that similarly to (4), under the model (1), the dynamic of the spot price is given by h i S(t)(·) = F (0, t) exp −g1 (0, t; α, σ) + g2 (0, t; α, σ)εt (·) .
(9)
In this paper, as in [Cl-St; 1999], we assume the interest rates are deterministic such that the futures and forward prices remain to be the same. 1
Indeed it is well-known that the random variable n o Rt 1 and variance 0 exp[2αu]du = 2α exp[2αt] − 1 .
Rt 0
exp[αu]dWu (·) is a Gaussian random variable with zero-mean
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Sensitivities related to the forward/future price
First we assume that at the present time-0 the forward price F (0, T ), with the maturity T , is given. For the time-horizon t, with 0 < t < T , the future price is unknown and may be viewed as a random variable F (t, T )(·) described by the model (4) and driven by the normal standard Gaussian εt (·). So it appears to be natural to investigate the sensitivity of the initial forward price F (0, T ) with respect to the change of this underlying uncertainty factor and for this horizon t. For this purpose, we introduce n h i o Sens fwd(0; 0, t, T ) ≡ exp −g1 (T − t, T ; α, σ) − 1 F (0, T ) (10) and
h i Sens fwd(l; 0, t, T ) ≡ g2l (T − t, T ; α, σ) exp −g1 (T − t, T ; α, σ) F (0, T )
(11)
where l is a nonnegative integer. Expressions in (10) and (11) may be seen respectively as the zero and high order sensitivities of the forward price with the maturity T , measured at time 0 and for the horizon t. We introduce the (conservative) view for some nonnegative real numbers ε• and ε•• then −ε• ≤ εt (·) ≤ ε•• .
(12)
Since εt (·) is a standard random Gaussian, then the view (12) is seen to be true with ε• = ε•• = 4 with a probability more than 99.9937%. Proposition 1 Let p be a nonnegative integer. Under the forward model (4),then we have the decomposition F (t, T )(·) − F (0, T ) = Sens fwd(0; 0, t, T ) +
p X 1 Sens fwd(l; 0, t, T )εlt (·) l! l=1
1 + Sens fwd0 (p + 1; 0, t, T, ε)εp+1 (·) t (p + 1)!
(13)
where Sens fwd0 (p + 1; 0, t, T, ε) is some random term which depends on εt (·). Under the view (12), deterministic estimates of this term are given by c•f wd t, T, ε• , α, σ Sens fwd(p + 1; 0, t, T ) < Sens fwd0 (p + 1; 0, t, T, ε) •• t, T, ε , α, σ Sens fwd(p + 1; 0, t, T ) < c•• f wd with and
(14)
n h io c•f wd t, T, ε• , α, σ ≡ min 1; exp −ε• g2 (T − t, T, α, σ)
(15)
n h io •• •• c•• t, T, ε , α, σ ≡ max 1; exp ε g (T − t, T, α, σ) . 2 f wd
(16)
Identity (13) may be seen as the right meaning of the approximation p X 1 Sens fwd(l; 0, t, T )εlt (·). F (t, T )(·) − F (0, T ) ≈ Sens fwd(0; 0, t, T ) + l!
(17)
l=1
This means that the change F (t, T )(·) − F (0, T ) may be approximated by a p order polynomial expression with constant coefficients defined by the forward price sensitivities measured at time 0 but
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for the horizon time t. As it is observed in our numerical illustration in Section 18, more the order decomposition p is chosen high, more the error approximation error fwd change(0, t, T )(·) = p n o X 1 l F (t, T )(·) − F (0, T ) − Sens fwd(0; 0, t, T ) + Sens fwd(l; 0, t, T )εt (·) l! l=1
(18) is small. Deterministic bound of this error term may be easily obtained under the view (12), since we have error fwd change(0, t, T )(·) 1 cf wd t, T, ε• , ε•• , α, σ Sens fwd(p + 1; 0, t, T ) max{ε• ; ε•• }p+1 (19) ≤ (p + 1)! where
n o cf wd t, T, ε• , ε•• , α, σ = max c• t, T, ε• , α, σ ; c•• t, T, ε•• , α, σ .
As a consequence to fulfill a requirement as error fwd change(0, t, T )(·) ≤ ρ with ρ = 10−12 it is sufficient to choose p as the first nonnegative integer for which 1 cf wd t, T, ε• , ε•• , α, σ Sens fwd(p + 1; 0, t, T ) max{ε• ; ε•• }p+1 ≤ ρ. (p + 1)!
(20)
(21)
(22)
It should be noted that the approximation (17) is an accurate alternative of a discrete version of the stochastic differential equation (1). Since it may arise that ε(·) = 3, then there is no reason to think about ε(·) to have a small size, as is commonly considered in literature when considering a sensitivity effect. By the way, the notion of infinitesimal shock ε(·) remains only a relative appreciation. Our approach covers the case ε(·) with an arbitrary size, and this is not a real drawback since we are able to have an accurate control of the error approximation as is displayed in (19). The approximation (17) appears to be useful in the hedging perspective, whose the idea relies on benefiting from the cancellation between the sensitivities of the position to hedge and the various hedging instruments. Deep analyses on this part will be performed in Section ???. The requirement of small error as ρ = 10−12 appears to be useful, since in practice the market participants have to deal not on a single contract but rather on a possibly large number N , as 10 000, of contracts such that the really important thing is the size of N error fwd change(0, t, T )(·) as perceived and tolerated by the hedger.
2.3
Sensitivities related to the spot price
Consistently with the model (9), it appears that the spot price at the time-horizon t depends essentially on the shock εt (·) which arises at this time. So it makes sense to introduce and consider the sensitivity of the present spot price with respect to the possible shock. This is performed in this subsection. Assuming that at the present time-0 the spot S(0) is known. For the time-horizon t, with 0 < t, the future price is unknown and may be viewed as a random variable S(t)(·) described by the model (9) and driven by the normal standard Gaussian εt (·). The suitable sensitivities of the initial spot price S(0) with respect to the shock at the horizon t are defined by h i (23) Sens spot(0; 0, t) ≡ exp −g1 (0, t; α, σ) F (0, t) − S(0) 7
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h i Sens spot(l; 0, t) ≡ g2l (0, t; α, σ) exp −g1 (0, t; α, σ) F (0, t)
(24)
where l is a nonnegative integer. Expressions in (23) and (24) may be seen respectively as the zero and high order sensitivities of the spot price S(0) measured at time 0 and for the horizon t. Proposition 2 Let p be a nonnegative integer. Under the forward model (9), then we have the decomposition S(t)(·) − S(0) = Sens spot(0; 0, t) +
p X 1 Sens spot(l; 0, t)εlt (·) l! l=1
1 + Sens spot0 (p + 1; 0, t, ε)εp+1 (·) t (p + 1)!
(25)
where Sens spot0 (p + 1; 0, t, ε) is some random term which depends on εt (·). Under the view (12), deterministic estimates of this term are given by c•f wd t, t, ε• , α, σ Sens spot(p + 1; 0, t) ≤ Sens spot0 (p + 1; 0, t, ε) •• t, t, ε , α, σ Sens spot(p + 1; 0, t) ≤ c•• f wd
(26)
•• with c•f wd t, t, ε• , α, σ and c•• f wd t, t, ε , α, σ are defined respectively in (15) and (16). Identity (25) yields the right meaning of the approximation S(t)(·) − S(0) ≈ Sens spot(0; 0, t) +
p X 1 Sens spot(l; 0, t)εlt (·). l!
(27)
l=1
Therefore the change S(t)(·) − S(0) may be approximated by a p order polynomial expression with constant coefficients defined by the spot price sensitivities measured at time 0 but for the horizon time t. As in the case of forward price, the error approximation is defined by error spot change(0, t)(·) = p n o X 1 l S(t)(·) − S(0) − Sens spot(0; 0, t) + Sens spot(l; 0, t)εt (·) . l! l=1
(28) Deterministic bound of this error term may be obtained under the view (12), since we have error spot change(0, t)(·) 1 ≤ c t, t, ε• , ε•• , α, σ Sens spot(p + 1; 0, t) max{ε• ; ε•• }p+1 (p + 1)! where
n o cf wd t, t, ε• , ε•• , α, σ = max c• t, t, ε• , α, σ ; c•• t, t, ε•• , α, σ .
As a consequence to fulfill the requirement error spot change(0, t)(·) ≤ ρ it is sufficient to choose p as the first nonnegative integer for which 1 cf wd t, t, ε• , ε•• , α, σ Sens spot(p + 1; 0, t) max{ε• ; ε•• }p+1 ≤ ρ. (p + 1)!
(29)
(30)
(31)
(32)
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Sensitivities related to a swap contract
Swaps in commodities are similar to swaps in financial markets and are essentially suited for hedging applications. They are flexible OTC, easily customizable transactions, typically financial settled, offbalance sheet and non-regulated. Most frequently one encounters fixed-price swap in which for a period of several months or years one counterparty pays another a fixed payment and in exchange receives a payment linked to a certain floating index. In this section we consider a swap contract such that at present time-0, the remaining payment cash-flow dates are t1 , . . . , tM with t1 < t2 < . . . < tk < . . . < tM such that tM = T may be seen as the maturity of the considered product. At each date tk , the cash-flow exchange is assumed to be based on some commodity volume ϑk , with 0 < ϑk , and the fixed price is denoted as K, with 0 < K. The present time-0 of the interest rates curve is given by the zero-coupon prices P (0, t1 ), . . . , P (0, tk ), . . . , P (0, tM ).
(33)
The time-0 value of such a contract(in the buyer viewpoint) is given by M n o X value swap 0, SW(0) ≡ P (0, tk )ϑk F (0, tk ) − K
(34)
k=1
where SW(0) = t1 , . . . , tk , . . . , tM ; ϑ1 , . . . , ϑk , . . . , ϑM ; K; P (0, t1 ), . . . , P (0, tk ), . . . , P (0, tM ); F (0, t1 ), . . . , F (0, tk ), . . . , F (0, tM ) . It means that the value of the considered swap at time 0 depends on: - the remaining payment dates t1 , . . . , tk , . . . , tM , - the volumes ϑ1 , . . . , ϑk , . . . , ϑM on which the exchange are based on, - the fixed payment K, - the zero-coupon prices P (0, t1 ), . . . , P (0, tk ), . . . , P (0, tM ), - the forward prices F (0, t1 ), . . . , F (0, tk ), . . . , F (0, tM ). It may be noted that if 0 is the time-issuance of the swap then the price K is always chosen such that value swap 0, SW(0) = 0. Consider a time-horizon t with 0 < t < t1 < t2 < . . . < tk < . . . < tM .
(35)
It means that we assume that no cash-flow payment occurs until time t. As a consequence on has SW(t)(·) = t1 , . . . , tk , . . . , tM ; ϑ1 , . . . , ϑk , . . . , ϑM ; K; P (t, t1 ), . . . , P (t, tk ), . . . , P (t, tM ); F (t, t1 )(·), . . . , F (t, tk )(·), . . . , F (t, tM )(·) . Under the deterministic assumption of interest rates, the zero-coupon term structure P (t, t1 ), . . . , P (t, tk ), . . . , P (t, tM ) is assumed to be known from the present time 0. However the forward prices F (t, t1 )(·), . . . , F (t, tk )(·), . . . , F (t, tM )(·)
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remain unknown at the present time 0 and their values are governed by the model introduced in (4). As an alternative to the deterministic term SW(0) and the stochastic term SW(t)(·), we have to make use of the deterministic term SW(0, t) = t1 , . . . , tk , . . . , tM ; ϑ1 , . . . , ϑk , . . . , ϑM ; K; P (0, t1 ), . . . , P (0, tk ), . . . , P (0, tM ); P (t, t1 ), . . . , P (t, tk ), . . . , P (t, tM ); F (0, t1 ), . . . , F (0, tk ), . . . , F (0, tM ) . Similarly to the case of forward price, it may be desirable to introduce the sensitivities of the value of such a contract, measured at time zero, with respect to the change of the uncertainty factor εt (·) and for the time horizon t. They are defined as follows Sens value swap 0; SW(0, t) ≡ M n M on o X X P (t, tk ) − P (0, tk ) F (0, tk ) − K ϑk + P (t, tk )ϑk Sens fwd(0; 0, t, tk ) k=1
k=1
(36) and
M X P (t, tk )ϑk Sens fwd(l; 0, t, tk ) Sens value swap l; SW(0, t) ≡
(37)
k=1
where l is a nonnegative integer. Proposition 3 Let p be a nonnegative integer. Under the forward model (4), then we have the decomposition value swap t; SW(t) (·) − value swap 0; SW(0) = Sens value swap 0; SW(0, t) p X 1 + Sens value swap l; SW(0, t) εlt (·) l! l=1 1 (·) (38) + Sens value swap0 p + 1; SW(0, t), ε εp+1 t (p + 1)! where Sens value swap0 p + 1; SW(0, t), ε εp+1 is some random term which depends on εt (·). t Under the view (12), deterministic estimates of this term are given by c•swap t, T, ε• , α, σ Sens value swap p + 1; SW(0, t) 0 ≤ Sens value swap p + 1; SW(0, t), ε •• ≤ c•• t, T, ε , α, σ Sens value swap p + 1; SW(0, t) (39) swap
•• , α, σ are defined as where c•swap t, T, ε• , α, σ and c•• t, T, ε swap
and
n c•swap t, T, ε• , α, σ ≡ min c•f wd t, tk , ε• , α, σ
o k ∈ {1, . . . , M }
(40)
n •• • •• t, t , ε , α, σ c•• t, T, ε , α, σ ≡ max c k swap f wd
k ∈ {1, . . . , M }
o
(41)
.
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Identity (38) may viewed as value swap t; SW(t) (·) − value swap 0; SW(0) ≈ Sens value swap 0; SW(0, t) p X 1 + Sens value swap l; SW(0, t) εlt (·). l!
(42)
l=1
Under the view (12), the error approximation error swap change value 0, t, T ; SW(t) (·) ≡ value swap t; SW(t) (·) − value swap 0; SW(0) − Sens value swap 0; SW(0, t) p X 1 l + Sens value swap l; SW(0, t) εt (·) l!
(43)
l=1
satisfies error swap change value 0, t, T ; SW(t) (·) ≤ 1 cswap t, T, ε• , ε•• , α, σ Sens value swap p + 1; SW(0, t) max{ε• ; ε•• }p+1 (p + 1)! where
3
n o •• cswap t, T, ε• , ε•• , α, σ = max c•swap t, T, ε• , α, σ ; c•• . swap t, T, ε , α, σ
(44)
Hedging mechanism
Roll-over the position. We would like to clarify more the hedging corresponding to a one-period of time ?????
3.1
Profit& loss for a covered portfolio
Let us denote by V0 the present time 0-value of a portfolio made by J ∗∗ types of financial instruments ∗∗ in long positions and J ∗ types of instruments V ∗ in short positions as defined by V·;j ∗∗ ·;j ∗ ∗
∗∗
V0 =
J X j ∗∗ =1
∗∗ ∗∗ V0;j ∗∗ mj ∗∗
−
J X
∗ ∗ V0;j ∗ mj ∗ .
(45)
j ∗ =1
∗ Here the m∗∗ j ∗∗ , mj ∗ ’s are positive integer numbers which represent respectively the quantities of in∗∗ ∗ . These last are assumed to be non-negative real numbers. Moreover J ∗∗ struments Vt;j ∗∗ and Vt;j ∗ ∗ and J are nonnegative integer numbers. ∗∗ (·) At a future time horizon t, with t is some nonnegative real number, the unknown values of Vt;j ∗∗ ∗ ∗∗ ∗ and Vt;j ∗ (·) may be very different than V0;j ∗∗ and V0;j ∗ such that the portfolio may suffer from a loss, in the sense that Vt (·) < V0 . To try to maintain the (future) value Vt (·) to be close to V0 the portfolio manager has to put in place a hedging technique. ???? Various approaches are known in theory and used in practice ( see for instance [Ca-Da; 2010] and the references therein ). Here we slightly differ with the standard asset-liability management for which the liability and asset are taken to have the same value at the starting time immunization ?????.
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The hedging idea relies on using another portfolio, referred in the sequel as a hedging instrument, such that this last would lead to a nonnegative profit compensating the loss for the portfolio in case of adverse movements of the component instruments V;j∗∗∗∗ (·) and V;j∗∗ (·). Therefore instead of the absolute change Vt (·) − V0 ≡ P&L naked portfolio 0,t (·) associated with the naked portfolio, at the horizon t the change for the covered portfolio is given by P&L covered portfolio
0,t (·)
≡ {Vt (·) − V0 } + P&L hedging instrument
0,t (·).
(46)
As in (45), the (portfolio) hedging instrument has a time-0 value given by ∗∗
H0 =
∗
I X
∗∗ ∗∗ H0;i ∗∗ ni∗∗
−
i∗∗ =1
I X
∗ ∗ H0;i ∗ n i∗ .
(47)
i∗ =1
It means that the hedging portfolio H is made by I ∗∗ types of instruments H;i∗∗∗∗ in long positions and I ∗ types of instruments H;i∗∗ in short positions. For a given type i∗∗ ( resp. i∗ ), we make use of n∗∗ i∗∗ ( resp. n∗i∗ ) number of instruments H;i∗∗∗∗ ( resp. H;i∗∗ ). Building and benefiting from the use of such a hedging portfolio requires in counterpart some initial cost which is given by I ∗∗ n I∗ n o X o X ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ C≡ λ H0;i∗∗ ni∗∗ + Φ H0;i∗∗ ni∗∗ + λ∗ H0;i . ∗ ni∗ + Φ H0;i∗ ni∗ i∗∗ =1
(48)
i∗ =1
The transaction cost function Φ is a positive function defined in [0, ∞) and satisfying the property Φ(0) = 0 (which means that no fee if there is no transaction cost) and for x > 0 the expression Φ(x) is in general defined by piecewise affine function as considered in [Ra; 2009]. Depending on the considered financial product and the market practice, Φ(x) may be given by a proportional cost as Φ(x) = (0.1%) ∗ x. Here λ∗∗ and λ∗ are some constants such that 0 < λ∗ , λ∗∗ ≤ 1 reflecting the market rules and practices. For instance, when the instrument H;i∗∗∗∗ is an equity or bond then we take λ∗∗ = 1, which means that the hedger needs to buy the corresponding securities. If instead H;i∗∗∗∗ is a forward contract which does not required to spend an initial amount then 0 < λ∗∗ < 1, as for instance λ∗∗ = 25% corresponding to an initial deposit in order to secure the operation. Similarly for a short position in equity or bond λ∗ may take the form as λ∗ = λdeposit + rate lend ∗ t where for instance λdeposit = 20% and rate lend = 0.5%. This last term corresponds to the rate applied in counterpart of borrowing the securities required in the short position. Remind that s represents the hedging operation lifetime under consideration. It often happens that the hedger do not have the whole of the amount C required for the hedging operation. He needs to borrow money in the market. It means that the real amount initially required is reduced by the corresponding interest interest hedging cost ≡
n
o 1 −1 C P (0; t)
(49)
which is assumed in this paper to be payable in advance when starting the hedging operation. Here 1 P (0; t) denotes the time-0 price of the zero-coupon maturing at t, such that P (0;t) represents the amount capitalized when investing one unit of currency.
12
Hedging a commodity-linked portfolio
Therefore the profit&loss P&L hedging instrument ing operation during the time-period (0, 0 + t) is given by P&L hedging instrument
0,t (·)
ECE Paris 2012
corresponding to the use of the hedg-
0,t (·)
= {Ht (·) − H0 } − interest hedging cost X I ∗∗ I∗ X ∗∗ ∗∗ ∗ ∗ − Φ Ht;i∗∗ (·)ni∗∗ + Φ Ht;i∗ (·)ni∗ . i∗∗ =1
(50)
i∗ =1
This last expression corresponds to the fact that when undoing the hedging operation ( by selling securities previously in long positions and buying those in short positions ) then some transaction costs are required. To simplify, we have not included in (50) the possibility that some interest amount corresponding to the result of selling operation at the hedge initiation may be paid by the financial intermediary, since it is not the rule for all existing markets. Moreover we do not take into consideration margin payments possibly associated with the instruments positions if these last are for instance future contract products. Therefore plugging (50) and (49) inside (46) then one has P&L covered portfolio
0,t (·)
J ∗∗
J∗ n o o Xn X ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ = Vt;j ∗∗ (·) − V0;j ∗∗ mj ∗∗ − Vt;j ∗ (·) − V0;j ∗ mj ∗ j ∗∗ =1
j ∗ =1
I ∗∗
I∗ n o o Xn X ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ + Ht;i∗∗ (·) − H0;i∗∗ ni∗∗ − Ht;i ∗ (·) − H0;i∗ ni∗ i∗∗ =1
n −
i∗ =1
1 P (0; t)
I ∗∗ n I∗ n o o X o X ∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗ ∗∗ ∗∗ ∗∗ λ H0;i∗ ni∗ + Φ H0;i∗ ni∗ λ H0;i∗∗ ni∗∗ + Φ H0;i∗∗ ni∗∗ + −1 i∗ =1
i∗∗ =1
X I ∗∗ I∗ X ∗∗ ∗∗ ∗ ∗ Φ Ht;i∗ (·)ni∗ . − Φ Ht;i∗∗ (·)ni∗∗ + i∗∗ =1
(51)
i∗ =1
To avoid technical complexity we will focus in this paper to the case where the transaction cost function is proportional in the sense that for some constant φ, with 0 ≤ φ < 1, one has for all 0 ≤ x.
Φ(x) = φx
(52)
It means that under this assumption, the identity (51) becomes P&L covered portfolio
0,t (·)
J ∗∗
J∗ n o o Xn X ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ = Vt;j ∗∗ (·) − V0;j ∗∗ mj ∗∗ − Vt;j ∗ (·) − V0;j ∗ mj ∗ j ∗∗ =1
j ∗ =1 I ∗∗
∗
I n o o Xn X ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ +(1 − φ) Ht;i Ht;i ∗∗ (·) − H0;i∗∗ ni∗∗ − (1 + φ) ∗ (·) − H0;i∗ ni∗ i∗∗ =1 ∗∗ I X ∗∗ ∗∗ −λ∗∗ H0;i ∗∗ ni∗∗ φ i∗∗ =1
where λ∗∗ φ =φ
i∗ =1
− λ∗φ
I∗ X
∗ ∗ H0;i ∗ n i∗ .
(53)
i∗ =1
n 1 o n 1 o 1 1 + λ∗∗ −1 and λ∗φ = φ + λ∗ −1 P (0; t) P (0; t) P (0; t) P (0; t)
The problem of hedging the initial portfolio V ( as defined in (45) ) is reduced to be able to choose suitably the instruments ∗∗ H;1 , . . . , H;i∗∗∗∗ , . . . , H;I∗∗∗∗
∗ and H;1 , . . . , H;i∗∗ , . . . , H;I∗ ∗
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Hedging a commodity-linked portfolio
ECE Paris 2012
and the corresponding security numbers ∗∗ ∗∗ n∗∗ 1 , . . . , n;i∗∗ , . . . , n;I ∗∗
such that the value of
and n∗1 , . . . , n∗;i∗ , . . . , n∗;I ∗
P&L covered portfolio
0,t (·)
should be small possible. The difficulty here is linked to the fact that the future values of the hedging instruments at time t are unknown at the present time 0 where the hedge strategy is built. The choice of the hedging instruments is dictated by the willing that the resultant effect of their change variations would roughly compensate ( i.e. going in the opposite direction ) the change variation of the portfolio V to hedge. Once we are able to control all of the corresponding compensation then the problem should be reduced to a minimization problem in order to find the suitable allocation of ∗∗ ∗∗ ∗ ∗ ∗ the security numbers n∗∗ 1 , . . . , n;i∗∗ , . . . , n;I ∗∗ and n1 , . . . , n;i∗ , . . . , n;I ∗ .
3.2
Hedging and the sensitivities
∗∗ (·) − V ∗∗ is to make use of some sensitivities A standard idea to deal with each change value as Vt;j ∗∗ 0;j ∗∗ with respect to the one or more underlying factor(s). Our presentation is focused on the single uncertainty factor modelling framework as is introduced in Section 2. Therefore we assume that for any non-negative integer p there exists a real number ε(·) satisfying ∗∗ ∗∗ ∗∗ Vt;j ∗∗ (·) − V0;j ∗∗ = Sens(0, V;j ∗∗ ) p X (−1)l
+
l=1
Sens(l, V;j∗∗∗∗ )εl (·) +
l!
(−1)p+1 Sens0 p + 1, V;j∗∗∗∗ ; ε ε(·)p+1 (p + 1)!
(54)
(−1)p+1 Sens0 p + 1, V;j∗∗ ; ε εp+1 (·) (p + 1)!
(55)
(−1)p+1 Sens0 p + 1, H;i∗∗∗∗ ; ε εp+1 (·) (p + 1)!
(56)
(−1)p+1 Sens0 p + 1, H;i∗∗ ; ε εp+1 (·). (p + 1)!
(57)
∗ ∗ ∗ Vt;j ∗ (·) − V0;j ∗ = Sens(0, V;j ∗ )
+
p X (−1)l l=1
l!
Sens(l, V;j∗∗ )εl (·) +
∗∗ ∗∗ ∗∗ Ht;i ∗∗ (·) − H0;i∗∗ = Sens(0, H;i∗∗ )
+
p X (−1)l l=1
l!
Sens(l, H;i∗∗∗∗ )εl (·) +
and ∗ ∗ ∗ Ht;i ∗ (·) − H0;i∗ = Sens(0, H;i∗ )
+
p X (−1)l l=1
l!
Sens(l, H;i∗∗ )εl (·) +
For convenience the short notation Sens(l, V;j∗∗∗∗ ) is used instead of Sens(l, 0, t; V;j∗∗∗∗ ) by which we mean the l-th order sensitivity ( computed at time 0 and and which applies for the the horizon t ) of the instrument V;j∗∗∗∗ with respect to the uncertainty factor change underlying the considered model. It should be emphasized that the real ε(·), representing the factor shock value, is the same for all of the considered instruments and depends on p and t. It may be important to note that there is no reason that ε(·) is a very small number, since values as ε = 3 and ε = −3 may be allowed. The change value balance is ensured by the sensitivities which are expected to decrease as soon as the considered order increases. The term Sens0 p + 1, V;j∗∗∗∗ ; ε may be seen as an unknown remainder term we also assume to be controllable in the sense that for each shock ε(·) lying on some interval [−ε• , ε•• ], with 0 < ε• , ε•• , then some constant c(ε• , ε•• , V;j∗∗∗∗ ) exists such that (58) Sens0 p + 1, V;j∗∗∗∗ ; ε ≤ c(ε• , ε•• , V;j∗∗∗∗ )Sens(p + 1, V;j∗∗∗∗ ). 14
Hedging a commodity-linked portfolio
ECE Paris 2012
Similarly we suppose that 0 ∗ Sens p + 1, V ; ε ≤ c(ε• , ε•• , V;j∗∗ )Sens(p + 1, V;j∗∗ ) ;j ∗
and
(59)
0 ∗∗ Sens p + 1, H ; ε ∗∗ ≤ c(ε• , ε•• , H;i∗∗∗∗ )Sens(p + 1, H;i∗∗∗∗ ) ;i
(60)
Sens0 p + 1, H;i∗∗ ; ε ≤ c(ε• , ε•• , H;i∗∗ )Sens(p + 1, H;i∗∗ )
(61)
respectively under the assumption that ε belong to the interval [−ε• , ε•• ]. With decomposition (54) we can write that J ∗∗ n o X ∗∗ ∗∗ ∗∗ Vt;j (·) − V ∗∗ 0;j ∗∗ mj ∗∗ j ∗∗ =1 ∗∗
∗∗
J X
=
Sens(0, V;j∗∗∗∗ )m∗∗ j ∗∗
+
j ∗∗ =1
+
J p X (−1)l X l!
l=1
(−1)p+1 (p + 1)!
J ∗∗
X
l Sens(l, V;j∗∗∗∗ )m∗∗ j ∗∗ ε (·)
j ∗∗ =1
p+1 Sens0 p + 1, V;j∗∗∗∗ ; ε m∗∗ (·). j ∗∗ ε
(62)
j ∗∗ =1
Similarly using decomposition (57) we obtain ∗
J n o X ∗ ∗ ∗ Vt;j ∗ (·) − V0;j ∗ mj ∗ j ∗ =1 ∗
=
J X
∗
Sens(0, V;j∗∗ )m∗j ∗
j ∗ =1
+
J p X (−1)l X l!
l=1
Sens(l, V;j∗∗ )m∗j ∗
εl (·)
j ∗ =1
J∗ (−1)p+1 X 0 ∗ ∗ + Sens p + 1, V;j ∗ ; ε mj ∗ εp+1 (·). (p + 1)! ∗
(63)
j =1
With (62) and (63) we can write that the portfolio change Vt+s (·) − Vt is given by J∗ n J ∗∗ n o o X X ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ Vt+s;j Vt+s (·) − Vt = Vt+s;j ∗∗ (·) − Vt;j ∗∗ mj ∗∗ − ∗ (·) − Vt;j ∗ mj ∗ j ∗∗ =1
=
j ∗ =1
J ∗∗
J∗
X
X
Sens(0, V;j∗∗∗∗ )m∗∗ j ∗∗ −
j ∗∗ =1
! Sens(0, V;j∗∗ )m∗j ∗
j ∗ =1
# J∗ X (−1)l X + Sens(l, V;j∗∗∗∗ )m∗∗ Sens(l, V;j∗∗ )m∗j ∗ εl (·) j ∗∗ − l! j ∗∗ =1 j ∗ =1 l=1 " J ∗∗ # J∗ X (−1)p+1 X Sens0 p + 1, V;j∗∗∗∗ ; ε m∗∗ + Sens0 p + 1, V;j∗∗ ; ε m∗j ∗ εp+1 (·). j ∗∗ − (p + 1)! ∗∗ ∗ p X
"
J ∗∗
j
=1
j =1
(64) For convenience, we feel the need to introduce the following notations ∗∗
ΘV0
≡
J X j ∗∗ =1
∗
Sens(0, V;j∗∗∗∗ )m∗∗ j ∗∗
−
J X
Sens(0, V;j∗∗ )m∗j ∗
(65)
j ∗ =1
∗∗ ∗∗ ∗∗ Θ∗∗ 0;i∗∗ ≡ (1 − φ)Sens(0, H;i∗∗ ) − λφ H;i∗∗
for i∗∗ ∈ {1, . . . , I ∗∗ }
(66) 15
Hedging a commodity-linked portfolio
Θ∗0;i∗ ≡ (1 + φ)Sens(0, H;i∗∗ ) + λ∗φ H;i∗∗ ∗∗
ΘVl
≡
J X
for i∗ ∈ {1, . . . , I ∗ }
(67)
∗
Sens(l, V;j∗∗∗∗ )m∗∗ j ∗∗
−
j ∗∗ =1
J X
Sens(l, V;j∗∗ )m∗j ∗
for l ∈ {1, . . . , p}
(68)
j ∗ =1
∗∗ Θ∗∗ l;i∗∗ ≡ (1 − φ)Sens(l, H;i∗∗ )
Θ∗l;i∗ ≡ (1 + φ)Sens(l, H;i∗∗ )
for l ∈ {1, . . . , p} and i∗∗ ∈ {1, . . . , I ∗∗ }
(69)
for l ∈ {1, . . . , p} and i∗ ∈ {1, . . . , I ∗ }
(70)
∗∗
ΘVp+1 (ε)
ECE Paris 2012
J X
≡
∗
0
Sens p +
1, V;j∗∗∗∗ ; ε
m∗∗ j ∗∗
J X
−
j ∗∗ =1
Sens0 p + 1, V;j∗∗ ; ε m∗j ∗
(71)
j ∗ =1
0 ∗∗ Θ∗∗ p+1;i∗∗ (ε) ≡ (1 − φ)Sens p + 1, H;i∗∗ ; ε
for i∗∗ ∈ {1, . . . , I ∗∗ }
(72)
Θ∗p+1;i∗ (ε) ≡ (1 + φ)Sens0 p + 1, H;i∗∗ ; ε
for i∗ ∈ {1, . . . , I ∗ } .
(73)
and
Using the above notations (65) to (73) and decompositions of change values as (64) then the covered portfolio P&L may be written as P&L covered portfolio
0,t (·)
I ∗∗
=
ΘV0
+
X
∗
∗∗ Θ∗∗ 0;i∗∗ ni∗∗
−
i∗∗ =1
I X
Θ∗0;i∗ n∗i∗
i∗ =1
I∗ I ∗∗ X X (−1)l V ∗ ∗∗ ∗ ∗∗ Θl;i∗ ni∗ εl (·) Θl;i∗∗ ni∗∗ − Θl + + l! i∗ =1 i∗∗ =1 l=1 ∗∗ I I∗ X X (−1)p+1 V ∗∗ ∗∗ ∗ ∗ + Θp+1 (ε) + Θp+1;i∗∗ (ε)ni∗∗ − Θp+1;i∗ (ε)ni∗ εp+1 (·). (p + 1)! ∗∗ ∗ p X
i
=1
(74)
i =1
At this stage it may be useful to recall that our hedging problem is reduced to find the number of securities ∗∗ ∗∗ n∗∗ and n∗1 , . . . , n∗i∗ , . . . , n∗I ∗ 1 , . . . , ni∗∗ , . . . , nI ∗∗ such that the quantity P&L covered portfolio 0,t (·) should be small as possible for all ε(·) ∈ [−ε• , ε•• ]. The last expression in (53) may be considered as a remainder, which with the help of (58) to (61) can be estimated as I ∗∗ I∗ (−1)p+1 X X V ∗∗ ∗∗ ∗ ∗ p+1 Θp+1 (ε) + Θp+1;i∗∗ (ε)ni∗∗ − Θp+1;i∗ (ε)ni∗ ε (·) (p + 1)! i∗∗ =1 i∗ =1 I ∗∗ I∗ X X 1 V ∗∗ ∗∗ ∗ ∗ Υp+1 + Υp+1;i∗∗ ni∗∗ + Υp+1;i∗ ni∗ max{ε• , ε•• }p+1 (75) ≤ (p + 1)! ∗∗ ∗ i
=1
i =1
where ΥVp+1 ≡ ΥVp+1 (ε• , ε•• ) ∗∗
=
J X j ∗∗ =1
∗
•
••
c(ε , ε
, V;j∗∗∗∗ )Sens(p
+
1, V;j∗∗∗∗ )m∗∗ j ∗∗
+
J X
c(ε• , ε•• , V;j∗∗ )Sens(p + 1, V;j∗∗ )m∗j ∗
j ∗ =1
(76) ∗∗ • •• • •• ∗∗ ∗∗ Υ∗∗ p+1;i∗∗ ≡ Υp+1;i∗∗ (ε , ε ) = c(ε , ε , H;i∗∗ )Sens(p + 1, H;i∗∗ )
for all i∗∗ ∈ {1, . . . , I ∗∗ }
(77)
and Υ∗p+1;i∗ ≡ Υ∗p+1;i∗ (ε• , ε•• ) = c(ε• , ε•• , H;i∗∗ )Sens(p + 1, H;i∗∗ )
for all i∗ ∈ {1, . . . , I ∗ } .
(78) 16
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Using (74) and (75) we get the estimate P&L covered portfolio 0,t (·) ≤ ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ F n1 , . . . , ni∗∗ , . . . , nI ∗∗ , n1 , . . . , ni∗ , . . . , nI ∗ ;
(79)
where = max{ε• , ε•• }
(80)
and ∗∗ ∗∗ ∗ ∗ ∗ F n∗∗ 1 , . . . , ni∗∗ , . . . , nI ∗∗ , n1 , . . . , ni∗ , . . . , nI ∗ ; I ∗∗ I∗ X X V ∗∗ ∗∗ ∗ ∗ = Θ0 + Θ0;i∗∗ ni∗∗ − Θ0;i∗ ni∗ i∗∗ =1
i∗ =1
p I ∗∗ I∗ X X X 1 V ∗∗ ∗∗ ∗ ∗ l + Θl + Θl;i∗∗ ni∗∗ − Θl;i∗ ni∗ l! i∗∗ =1 i∗ =1 l=1 ∗∗ I I∗ X X 1 V ∗∗ ∗∗ ∗ ∗ Υp+1 + Υp+1;i∗∗ ni∗∗ + Υp+1;i∗ ni∗ p+1 . + (p + 1)! ∗∗ ∗ i
=1
(81)
i =1
∗∗ ∗∗ Various approaches may be used to find the I ∗∗ +I ∗ positive integer numbers unknown n∗∗ 1 , . . . , ni∗∗ , . . . , nI ∗∗ ∗ ∗ ∗ and n1 , . . . , ni∗ , . . . , nI ∗ such that
∗∗ ∗∗ ∗ ∗ ∗ F n∗∗ 1 , . . . , ni∗∗ , . . . , nI ∗∗ , n1 , . . . , ni∗ , . . . , nI ∗ ; should be small as possible.
For instance if the development order p is previously chosen such that p + 1 = I ∗∗ + I ∗ then the idea more and less followed by authors on hedging problems is to solve the linear system of p + 1 equations and unknowns I∗ I ∗∗ X X ∗ ∗ ∗∗ V Θ0;i∗ ni∗ − Θ∗∗ (82) 0;i∗∗ ni∗∗ = Θ0 i∗ =1
and
∗
I X i∗ =1
i∗∗ =1 ∗∗
Θ∗l;i∗ n∗i∗
−
I X
∗∗ V Θ∗∗ l;i∗∗ ni∗∗ = Θl
for l ∈ {1, . . . , p} .
(83)
i∗∗ =1
If such a system admits an unique solution and may be solved then it is expected that the remainder term ∗∗ ∗∗ ∗ ∗ ∗ Rem n∗∗ , . . . , n , . . . , n , n , . . . , n , . . . , n ; ∗∗ ∗∗ ∗ ∗ 1 i I 1 i I ∗∗ I I∗ X X 1 V ∗∗ ∗∗ ∗ ∗ Υp+1 + Υp+1;i∗∗ ni∗∗ + Υp+1;i∗ ni∗ p+1 (84) = (p + 1)! ∗∗ ∗ i
=1
i =1
appears to be a small term in regard of the hedger view point. It may be noted that solving system of equations (82) and (83) does not lead in general to integer num∗∗ ∗∗ ∗ ∗ , . . . , x∗ bers and consequently we have to round the real solutions x∗∗ 1 , . . . , xi∗∗ , . . . , xI ∗∗ and x1 , . . . , x i∗ I∗
∗∗ ∗∗ founded. However by so doing it introduces new hedging error and it is not obvious that Rem x∗∗ 1 , . . . , xi∗∗ , . . . , xI ∗∗ has a small size as it is expected.
The reasonable situation we have to face on is that For instance if the order development p is previously chosen such that I ∗∗ + I ∗ < p + 1. Indeed in general it may be seen empirically that one has to choose p large enough to ensure that the approximation errors, arising from decompositions (54) to (57), to be small enough. Moreover due to the few instruments, which may be available in the market and for the hedger, most of the case where are in the case that I ∗∗ + I ∗ remains a small
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Hedging a commodity-linked portfolio
ECE Paris 2012
quantity. Consequently we think that the suitable approach to the hedging problem is to consider the minimization problem n ∗∗ ∗∗ ∗ ∗ ∗ min F n∗∗ , . . . , n , . . . , n , n , . . . , n , . . . , n ; ∗∗ ∗∗ ∗ ∗ 1 i I 1 i I o ∗∗ ∗∗ ∗ ∗ ∗ (n∗∗ (85) 1 , . . . , ni∗∗ , . . . , nI ∗∗ , n1 , . . . , ni∗ , . . . , nI ∗ ) ∈ C . ∗∗ ∗∗ ∗ ∗ ∗ Here C is the constraint made by the set of positive integers n∗∗ 1 , . . . , ni∗∗ , . . . , nI ∗∗ , n1 , . . . , ni∗ , . . . , nI ∗ satisfying I ∗∗ I∗ X X ∗∗ ∗∗ ϑ;i∗∗ ni∗∗ + ϑ∗;i∗ n∗i∗ ≤ D (86) i∗∗ =1
i∗ =1
where ∗∗ ϑ∗∗ ;i∗∗ = λ
n
and ϑ∗;i∗ = λ∗
n
o 1 − 1 H;i∗∗∗∗ P (0; t)
o 1 − 1 H;i∗∗ . P (0; t)
(87)
(88)
Here D is a given nonnegative real number which corresponds to the maximal amount the hedger is allowed to make use in the hedging operation. The first member of expression (86) is nothing else than the interest hedging cost as introduced in (49). Therefore our solution to the hedging problem is ∗∗ ∗∗ ∗ ∗ ∗ n∗∗ , . . . , n , . . . , n , n , . . . , n , . . . , n ∗∗ ∗∗ ∗ ∗ 1 i I 1 i I n ∗∗ ∗∗ ∗ ∗ ∗ = argmin F n∗∗ , . . . , n , . . . , n , n , . . . , n , . . . , n ; ∗∗ ∗∗ ∗ ∗ 1 i I 1 i I o ∗∗ ∗∗ ∗ ∗ ∗ (n∗∗ , . . . , n , . . . , n , n , . . . , n , . . . , n ) ∈ C (89) ∗∗ ∗∗ ∗ ∗ 1 i I 1 i I and the corresponding deterministic estimates of the hedging error is given by ∗∗ ∗∗ ∗ ∗ ∗ F n∗∗ , . . . , n , . . . , n , n , . . . , n , . . . , n ; . ∗∗ ∗∗ ∗ ∗ 1 i I 1 i I It may be observed that (89) is an integer multi-dimensional optimization problem and related to a non-linear and non-convex function. So, in general, there is no immediately available and common approach to solve this problem. Actually it can be proved ( see for instance [Ja-Ya-Ra; 2012] for details in the case of bond hedging ) that the given problem is equivalent to a Mixed-Integer-Linear-Problem, which can be handled by common solvers as the commercial application CPLEX. When the number of hedging instruments used is very low, then the problem may be used by direct enumerative method.
4 4.1
Numerical Illustrations Experiments related to the order of approximation
The framework we introdyced implies several complex equations in which multiple variables play specific roles. This part permits to study the influence some variables on the system’s performance. Alternatively, it aims to find the values for which the system displays the best performances. Concerning sensitivities, different tests are performed. The objective is to analyze the influence of the variables: first in 4.1.1 the order into which the sensitivity equation is extended, namely p, then the absolute error accepted ρ in 4.1.2 and finally the maturities T for futures and swaps in . These tests are conducted for the three financial instruments under consideration (futures, spots and swaps) whenever it is possible. To observe these features we calculate the absolute error between the exact variation and the approximated variation calculated with the sensitivity decomposition. That will allow us to see how close the approximation is from the reality (the basic dynamic process with the uncertainty factor). The goal is to show that the stochastic process is fairly well explained by the 18
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sensitivities. For the following calculations, we decided to fix some variables as follows : • The horizon studied as half a month : t = 1/24 • The maturity as 3 months for calculation where it is needed to be fixed : T = 3/12 • The initial future price as 100 : F0 = 100 • The initial spot price as 100 : S0 = 100 • The initial swap price as 100 : SW0 = 100 • The speed of mean reversion ; α = 0.34 • The level price returns volatility : σ = 0.31 • The time between each observation : δt = 7/360 4.1.1
Influence of the order on the approximated variations
Futures In this subpart, the estimation of the future change given by the p order polynomial equation (11) is studied. The goal is to evaluate this equation with different orders and, with different shock value within the interval [-4;4]. This evaluation is relevant as it permits to determine the order to use as we mentioned earlier (for which the results are the most accurate). To do so, we evaluate the absolute error between the real future change F (t, T )(·) − F (0, T ) and the estimated future change calculated using the sensitivities: Sens fwd(0; 0, t, T ) +
p X 1 Sens fwd(l; 0, t, T )εlt (·) l!
(90)
l=1
. The calculation is repeated for several orders. The future change estimation is considered for the order 1, 2, 3, 5, 7, 9, 11 and 13, and the results of the absolute error are detailed in the tables 1 and 2 in the appendix B. These results and the plots on figure 4 clearly show that the higher the order is, the more accurate the estimation of the future change is. Additionally the errors stabilize around 10−15 when the order of 12 is crossed for every shock considered, so we decided to fix p = 12 for which the estimation seems to be optimized for the future. Let us just point out that when the shock is equal to 0, there is no uncertainty in the process so that is why no error is observed. Spots In the same way, we assess the estimation of the spot change given by the p order polynomial equation (24). The absolute error between the real spot change S(t)(·) − S(0) and the estimated spot change calculated is computed using the sensitivities: Sens spot(0; 0, t) +
p X 1 Sens spot(l; 0, t)εlt (·) l!
(91)
l=1
. The spot change estimation is also considered for the order 1, 2, 3, 5, 7, 9, 11 and 13, and the results are detailed in tables 3 and 4 in the appendix B. These results and the plots on figure 4 show also that the higher the order, the most accurate the approximation. The error stabilizes around 10−15 with an order of 12 as well. Therefore, p is fixed to 12 for the spot.
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Swaps Finally, the estimation of the swap change given by the p order polynomial equation (42) is assessed. The method is the same as before. We look after the absolute of the difference between the stochastic swap change value swap t; SW(t) (·) − value swap 0; SW(0) and the estimated variation calculated using the sensitivities: p X 1 Sens value swap l; SW(0, t) εlt (·) Sens value swap 0; SW(0, t) + l! l=1
(92) . The swap change estimation is also considered for multiple orders: 1, 2, 3, 5, 7, 9, 11 and 13. The results of the absolute error are presented in tables 6 and 7 in the appendix B. Together with the plots on figure 5 they imply that the higher the order is, the more accurate the estimation of the future change is. The errors also stabilize when the order cross 12 for every shock considered. They are around 10−13 so we decided to set p = 12 for all instrument estimations in the end. 4.1.2
Influence of the absolute error
Futures This subsection presents the influence of ρ on the order and on the quality of the estimation for the future change given by the equation (11) page 6. In the same time, it gives the error variation defined by the formula (18) page 7. Test are done for shock values within the interval [-4;4]. The evaluation of the influence of ρ is also important as it is a fixed variable which permits to determine the best order to use in the calculation of the sensitivity. The maturity 3, 6, 12 and 24 months are evaluated, and for each maturity the values 1e − 02, 1e − 06, 1e − 08, 1e − 10 and 1e − 13 for ρ are tested. The different tables 10, 11, 12 and 13 present the evolution of the order and the error according to the variation of ρ enumerated before. One table is made for each maturity introduced: The results on the tables 10, 11, 12 and 13 globally show that the value of the error get smaller with the decrease of ρ. This observation is confirmed by the plots displayed in tables 8, 9, 10 and 11. Indeed, we observed that for ρ = 1e − 02 the error is about 1e-05, that for ρ = 1e − 08 it is about 1e-11 and that for ρ = 1e − 13 about 1e − 14. Furthermore we also noticed that the maturity doesn’t have much impact on the variation of the error. This is clearly visible on the plots and thiss confirm the results we obtain in the previous section. According to these figures, ρ = 1e − 12 seems to be adequate to obtain the best approximation of the variation of the futures, for the different maturities.
Spots Let us have a look on the influence of ρ for the spot case. As we did for the future, the values 1e − 02, 1e − 06, 1e − 08, 1e − 10 and 1e − 13 for ρ are evaluated and the table 14 presents the evolution of the order and the error according to the variation of ρ enumerated before: The results presented on the table 14 show that the value of the error decreases with ρ. This observation is confirmed by the plots displayed in Table ??. Indeed, we can see that for ρ = 1e − 02 the error takes values around 1e-05, for ρ = 1e − 08 around 1e-11 and for ρ = 1e − 13 1e − 14. Plus we noticed that for ρ = 1e − 12 the values of the error with the different shock values are fine enough such that we obtain accurate assessments of the spot change for a horizon t.
Swaps The estimation of the swap change value swap t; SW(t) (·) − value swap 0; SW(0) given by the p-order polynomial equation (42) page 11 is studied. The aim here is to evaluate how the 20
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threshold ρ influences the estimated change with several shocks within the interval [-4;4]. Indeed, as before it allows to set the order to its most fitted value. 1e − 02, 1e − 06, 1e − 08, 1e − 10 and 1e − 13 for ρ are tested again and the table 14 draws the evolution of the order and the estimated swap change according to each level ρ. The results presented in tables (15) and the curves (?? ) show that the smaller the ρ is, the better the estimation of the change is. In the same time, the order of approximation gets higher which is consistent with previous conclusions. According to these observations, we choose to set ρ = 1e − 12.
4.1.3
Influence of the maturity
Futures Now we are interested in the error variation for futures from the formula (18) given page 7 when we run the calculations with different maturities. They are set to 3, 6, 12 and 24 months. As well as we did for previous tests, observations are made for different shocks within the interval [-4;4]. It is decided to fix the order l of the sensitivity from equation (11) page 6 to 12. This test should emphasize the robustness of the framework to the maturity. The figures in table 8 illustrate the fact that the maturity doesn’t have a significant influence on the calculations. This is confirmed by the evolution of the error displayed on Figure ??. Indeed for the order l = 12, the error found for the maturities vary between 1e − 14 and 1e − 15. These results prove that an arbitrary fixed order can be chosen since that for different maturities, we have sufficiently good performances.
Swaps This subpart followsthe same principle as beforebut for the swaps. It presents the estimation value swap t; SW(t) (·) − value swap 0; SW(0) of the swap change given by the p-order polynomial equation (42) page 11. The table 9 and the plots on figure ?? show the same conclusion as for futures. The maturity does not influence on the estimations. That proves that we can set a value for the order l without worries with regards to the performances of the framework.
4.2
Portfolio hedging : scopes and applications
In this part, we are interested in setting simple hedging cases that relate to reality. The good results obtained for sensitivity tests allow us to integrate them on the general framework testing. Here comes the main focus. We apply the theoretical strategy to an initial portfolio. The objective is to see how well the cover works for basic situations. To do so, fictive specifications for instruments are given based on real figures. Information were found on the ”U.S Energy Information Administration” website (www.eia.gov). For convenience, we reduce the numerical applications to the use of futures. Then conclusions would be derived immediately. It is also interesting to consider the agent’s case that need to get provided with commodities for its personal use and that must have future agreements. Further investigations should be undertaken by including other derivatives and more realistic situations. For the simulation, we consider an initial portfolio composed by one future (fictive financial instrument we generated). Let us recall that the numbers of contracts needed to cover the portfolio is what we search. To find them, we minimize the final portfolio value. This corresponds to the problem derived in 85. The solution gives the numbers of instruments that reduce at most this objective function. There are obviously some constraints defined as the cost accepted for the operation as defined in 86. Finally, the portfolio-covered value obtained helps to assess the quality of the operation. For the illustrative portfolios, it should be mentioned that we do not calibrate the model. Therefore, values for α and σ are observed on the market. These values are inserted into Clewlow and Strickland future process. 21
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The cost of benefits from obtaining a hedging portfolio are introduced in formula 48. The function Φ(x) is defined as Φ(x) = Φ × x. In the same way, we set the constants λ∗∗ = 25% and λ∗ = 20% taking into account real world situations (for futures). Let us recall that these components represent the market rules and practices, especially deposits that are imposed by Clearhouses. Besides, the horizon t interval for which we consider the hedging operation is also fixed. Since all elements that appear in the valorization formula are all settled, we may proceed to simulations. 4.2.1
Futures’ specifications
To build our portfolios, we need to define the set of Futures under consideration. For simplicity, complicated situations are avoided. The reader needs to see the ”main position” of the portfolio holder immediatly. The list of information is drawn up hereafter. It should be pointed out that the evaluations are made at time to t = 0 to the hedging lifetime t. This information is crucial when the future sensitivities are computed. To make the ”scope of the experiments” perfectly clear, the position related to the financial instruments are also given. Data for crude oil come from the website of the ”ICE : InterContinentalExchange”. They correspond to Brent prices from the january 2nd, 2007. Maturity 5 years 5 months 5 months 5 months 4.2.2
Price 63.470$ 51.000$ 128.500$ 148.675$
Contract index 1 2 3 4
Holding position Long Short Short Short
α 0.500 0.230 0.300 0.450
σ 0.346 O.3OO 0.380 0.200
Product type Brent crude oil Uranium Non Fat Dry Milk Feeder Cattle
Brief summary of the environment
Two principal cases are studied in this paper. The first consists in hedging a naked portfolio constituted by crude oil futures in long position (index 1). In the counterpart, the agent covers his position with the use of 5-month maturity instruments (index 3). This is the Non Fat Dry Milk. Crude oil could have just been considered but these situation assumptions do not impact numerical results. The other case is more concerned with real life situations. Futures on Brent crude oil (index 1) composes the naked portfolio and to find a good hedging strategy, the agent is furnished with 3 types of commodities indexes 2, 3 and 4. All these commodity processes are generated with their respective α and σ plugged into the onefactor model. The order chosen for approximated the derivative values is p = 12. It is the smallest order from which we obtain satisfying approximations as seen in the previous part. 4.2.3
Simulations and results
By considering the actual market trend (April 2012) and the values introduced before, the process generates a strategy of 131 Milk short position future contracts for 1000 Crude oil long position future contracts. Then the covered portfolio is equal to 17.22$. This is very satisfying when we recall that the naked portfolio is set to 1000*63.470 = 63470$. The same simulation is run on this initial portfolio when the second hedging scenario is assumed. The strategy generated by the hedging process is constituted by: • 45 Uranium future contracts • 31 Non Fat Dry Milk future contracts • 153 Feeder Cattle contracts 22
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The covered portfolio value that comes out from this strategy is 19.86$ which is once again very small. These results show the efficiency of the general framework. In both cases, the agent disposes of two realistic and profitable strategies. Alternatively, some additionnal tests should be defined to test the system performances. The first one will be about the influence of the horizon of cover t (the horizon is defined by the difference between time t and time t = 0). The curve produced allow us to observe how well the cover is performed since we draw the covered portfolio value F (the objective function) with respect to the cost allowed to perform the operation D. The cost allowed for the operation varies from 0.011% to 0.11% of the initial portfolio value. Both situations as described in the simulation environment ?? are used on these tests but for convenience we only refer to the one that uses 3 instruments in the hedging portfolio. This test gives us a real insight of the model’s consistency. As a matter of fact, the figure ?? in the appendix B shows that with a growing D, that is the amount of money the agent uses, we get better performances. Consequently, the objective function gets lower and lower at each step. From this curve, it is clearly visible that there is a threshold value from which the hedging efficiency remains constant. In this simulation, this threshold appears to be very close to 0.070% of the naked portfolio value. This test allows the agent to save money, investing more than the threshold value would be a waste because the optimal solution would be already reached. The other interesting test is to see the performance with respect to the time considered in the cover. The figure ?? in the appendix B draws the corresponding curve. It shows that as the horizon grows, the efficiency of the hedging process is wearing away. At time t = 0 it is seen that the strategy has no chance to fail, which is logical because no uncertainties are introduced in the assessment as all data are given. Then as when we move forward in time it is harder to get an accurate estimation of the market trend. That is why the efficiency of the strategy decreases proportionately, and inversely the covered portfolio value is increasing in the same way.
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5
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Conclusion • The results of the tests for the sensitivities obtained in section 4 are conclusive. With these observations, we give the proof that it is coherent to make use of linear approximations of the stochastic process under consideration to work on hedging strategies. • Now the difficulties appear for the minimization problem and its initialization conditions. The algorithm used for the covered portfolio minimization determines the numbers of contracts considered. It is working in a continuous universe when the solution has to be discrete. To overcome this fact, the best strategy may be selected by a upper/lower bound study. As a consequence if the agent is furnished with n types of commodities, then 2n possibilities are taking into account. Nevertheless, the solution happens not to be immediatly found. Besides, the minimization function is not concave by definition. Then the minimum found for the contract numbers are not global. This recalls that the initialization of the optimization problem’s numbers of contrats is a very important deal. In the same time, it gives the opportunity for further studies that may be concern with the best algorithm to use in this kind of problem. • Since we use linear approximations, the calculus that must be done are not costly in terms of time. Therefore, a trader or any person that gives a lot of attention to the computation speed can be satisfyied by such an approach.
Through this paper, it is seen that even if many important points have been studied, there are still some that may conduct to multiple researches. Some could also be looked into more precisely. There are several interesting derivatives to study such as options (even if the mathematical expression might be complicated). Many strategies and more real ones may be defined to test the different variables of the general framework. It is possible to add suppositions, maybe more pertinent according to the evolution of the commodity market. We defined in this paper a solid basis by setting the stochastic process for futures, but now it is important to consider more pertinent models that might turn out to be relevant. The main difficulty in this remains mathematical. Finally all the tools necessary to develop the proposed general framework for hedging a commoditylinked portfolio are presented. The opportunities that come from this method clearly show its quality and hence they should convince the scientific community of its worthiness.
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A A.1
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Appendix A : Proofs of Results Proof of Proposition 1
By the definition of Sens fwd(0; 0, t, T ) we can just consider the case 0 < ε ≡ εt (·) or ε ≡ εt (·) < 0. For shortness let us set the following notations g1 ≡ g1 (T − t, T ; α, σ),
g2 ≡ g2 (T − t, T ; α, σ)
and
A ≡ exp[−g1 ]F (0, T )
such that Sens fwd(0; 0, t, T ) ≡ A − F (0, T ),
and
Sens fwd(l; 0, t, T ) ≡ Ag2l (t, T ; α, σ)
for l ∈ {1, . . . , p + 1}. It may be noted that a real number ρ, satisfying 0 < ρ < ε or ε < ρ < 0, does exist and for which p h i h i X 1 l l 1 exp g2 ε = 1 + g2 ε + g2p+1 εp+1 exp g2 ρ . l! (p + 1)!
(93)
l=1
Since ρ depends on ε then we can introduce h i Sens fwd (p + 1; 0, t, T, ε) = exp g2 ρ Sens fwd(p + 1; 0, t, T ) 0
such that the double-estimates (39) holds under the view (12). Now the decomposition (13) arises from (93) as follows h i F (t, T )(·) − F (0, T ) = A exp g2 ε − F (0, T ) h i = Sens fwd(0; 0, t, T ) + A exp g2 ε − 1 X p h i 1 l l 1 p+1 p+1 exp g2 ρ g ε + g ε = Sens fwd(0; 0, t, T ) + A l! 2 (p + 1)! 2 l=1
= Sens fwd(0; 0, t, T ) +
p X l=1
1 Sens fwd(l; 0, t, T )εl l!
1 + Sens fwd0 (p + 1; 0, t, T, ε)εp+1 . (p + 1)! It may be observed that h i Sens fwd0 (p + 1; 0, t, T, ε) = exp g2 ρ Sens fwd(p + 1; 0, t, T ). Under the view (12) then it is clear that n h io h i n h io min 1; exp −ε• g2 < exp g2 ρ < max 1; exp ε•• g2 . With these last two inequalities then we get estimates (39).
A.2
Proof of Proposition 2
The proof is very closed to that of Proposition 1. By the definition of Sens spot(0; 0, t) we can just consider the case 0 < ε ≡ εt (·) or ε ≡ εt (·) < 0. For shortness let us set the following notations g1 ≡ g1 (0, t; α, σ),
g2 ≡ g2 (0, t; α, σ)
and
A ≡ exp[−g1 ]F (0, t)
such that Sens spot(0; 0, t) ≡ A − S(0),
and
Sens spot(l; 0, t) ≡ Ag2l (0, t; α, σ) 25
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for l ∈ {1, . . . , p + 1}. As for the proof of Proposition 1, a real number ρ, satisfying 0 < ρ < ε or ε < ρ < 0, does exist and for which the identity (93) is satisfied. Since ρ depends on ε then we can introduce h i Sens spot0 (p + 1; 0, t, ε) = exp g2 ρ Sens spot(p + 1; 0, t) such that the double-estimates (26) holds under the view (12). Now the decomposition (25) arises from (93) as follows h i S(t)(·) − S(0) = A exp g2 ε − S(0) h i = Sens spot(0; 0, t) + A exp g2 ε − 1 X p h i 1 l l 1 p+1 p+1 = Sens spot(0; 0, t) + A g ε + g ε exp g2 ρ l! 2 (p + 1)! 2 l=1
= Sens spot(0; 0, t) +
p X l=1
+
A.3
1 Sens spot(l; 0, t)εl l!
1 Sens spot0 (p + 1; 0, t, ε)εp+1 . (p + 1)!
Proof of Proposition 3
The swap change value stated in (38) can be readily derived from the forward change (13) found in Proposition 1 as follows value swap t; SW(t) (·) − value swap 0; SW(0) =
=
M n M on o n o X X P (t, tk ) − P (0, tk ) F (0, tk ) − K ϑk + P (t, tk )ϑk F (t, tk )(·) − F (0, tk ) k=1 M n X
k=1
P (t, tk ) − P (0, tk )
on o F (0, tk ) − K ϑk
k=1
p X 1 P (t, tk )ϑk Sens fwd(0; 0, t, tk ) + Sens fwd(l; 0, t, tk )εlt (·) l! k=1 l=1 1 p+1 0 Sens fwd (p + 1; 0, t, tk , ε)εt (·) + (p + 1)! = Sens value swap 0; SW(0, t) +
M X
+
p X 1 Sens value swap l; SW(0, t) εlt (·) l! l=1 1 + Sens value swap0 p + 1; SW(0, t), ε εp+1 (·). t (p + 1)!
Since Sens value swap0 p + 1; SW(0, t), ε =
M X
(94)
h i P (t, tk )ϑk exp g2 (tk − t, tk ; α, σ)ρ Sens fwd(p + 1; 0, t, tk )
k=1
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and under the view (12) n h io min 1; exp −ε• g2 (tk − t, tk ; α, σ) < h i exp g2 (tk − t, tk ; α, σ)ρ n h io < max 1; exp ε•• g2 (tk − t, tk ; α, σ) then it appears that c•swap t, T, ε• , α, σ Sens value swap p + 1; SW(0, t) < Sens value swap0 p + 1; SW(0, t), ε • < c•• swap t, T, ε , α, σ Sens value swap p + 1; SW(0, t) • , α, σ are defined as in (40) and (41). With these last where c•swap t, T, ε• , α, σ and c•• t, T, ε swap two inequalities then we get estimates (39).
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B
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Appendix B : Tables and Plots
B.1 B.1.1
Influence of the order on the approximation of the change Futures Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1
2
3
5
0.0021 7.038174e-03 3.820776e-03 1.853169e-03 7.634366e-04 2.429550e-04 4.827011e-05 3.034490e-06 5.773160e-15 3.070224e-06 4.941368e-05 2.516399e-04 8.000398e-04 1.964893e-03 4.098845e-03 7.639350e-03 1.311119e-02
1.9042e-06 9.961072e-06 3.966475e-06 1.333835e-06 3.511014e-07 6.274747e-08 5.531605e-09 8.679235e-11 5.773160e-15 8.751533e-11 5.624885e-09 6.434156e-08 3.630446e-07 1.390791e-06 4.170584e-06 1.056163e-05 2.363426e-05
Error 321.5027 1.959088e+00 1.452997e+00 1.018654e+00 6.581904e-01 3.738000e-01 1.677422e-01 4.234366e-02 5.773160e-15 4.317799e-02 1.744177e-01 3.963345e-01 7.116218e-01 1.123053e+00 1.633482e+00 2.245851e+00 2.963188e+00
1.3123 1.360430e-01 8.628282e-02 5.029012e-02 2.593393e-02 1.101999e-02 3.288900e-03 4.141118e-04 5.773160e-15 4.202165e-04 3.386584e-03 1.151459e-02 2.749740e-02 5.410818e-02 9.420244e-02 1.507205e-01 2.266901e-01
Table 1: Results of the evolution of the error with the order and with a maturity of 3 months for the future.
Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
7 2.179864e-08 7.514139e-09 2.196309e-09 5.124328e-10 8.623857e-11 8.652634e-12 3.499423e-13 2.220446e-15 5.773160e-15 3.996803e-15 3.410605e-13 8.819612e-12 8.851231e-11 5.293721e-10 2.283699e-09 7.864159e-09 2.296303e-08
9
11
Error 1.333689e-11 3.552714e-15 3.520739e-12 3.552714e-15 7.531753e-13 3.552714e-15 1.243450e-13 1.776357e-15 3.552714e-15 8.881784e-15 8.881784e-15 8.881784e-15 1.065814e-14 1.065814e-14 8.881784e-16 8.881784e-16 5.773160e-15 5.773160e-15 5.329071e-15 5.329071e-15 1.776357e-15 1.776357e-15 1.243450e-14 1.243450e-14 3.552714e-15 8.881784e-15 1.278977e-13 1.776357e-15 7.602807e-13 1.776357e-14 3.634426e-12 1.776357e-14 1.391243e-11 3.552714e-15
13 3.552714e-15 3.552714e-15 3.552714e-15 1.776357e-15 8.881784e-15 8.881784e-15 1.065814e-14 8.881784e-16 5.773160e-15 5.329071e-15 1.776357e-15 1.243450e-14 8.881784e-15 1.776357e-15 1.776357e-14 1.776357e-14 1.065814e-14
Table 2: Results of the evolution of the error with the order and with a maturity of 3 months for the future.
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Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
0
10
−2
10
−4
10
−6
Error
10
−8
10
−10
10
−12
10
−14
10
2
4
6
8
10
12
Order
Figure 1: Plots of the evolution of the error with the order of the approximation for the future.
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B.1.2
ECE Paris 2012
Spots Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1 2.903864e+00 2.245767e+00 1.666734e+00 1.169291e+00 7.560392e-01 4.296671e-01 1.929472e-01 4.874059e-02 0.000000e+00 4.977220e-02 2.012012e-01 4.575315e-01 8.221109e-01 1.298394e+00 1.889947e+00 2.600447e+00 3.433691e+00
2
3
Error 2.482858e-01 1.579247e-02 1.675980e-01 9.313801e-03 1.063498e-01 5.058195e-03 6.201788e-02 2.454355e-03 3.199826e-02 1.011522e-03 1.360396e-02 3.220398e-04 4.062214e-03 6.400958e-05 5.117522e-04 4.025657e-06 0.000000e+00 0.000000e+00 5.198545e-04 4.076565e-06 4.191862e-03 6.563876e-05 1.426042e-02 3.344128e-04 3.407346e-02 1.063670e-03 6.708577e-02 2.613533e-03 1.168624e-01 5.454382e-03 1.870822e-01 1.017038e-02 2.815414e-01 1.746310e-02
5 3.372149e-05 1.520043e-05 6.054556e-06 2.036612e-06 5.362512e-07 9.586548e-08 8.453710e-09 1.326836e-10 0.000000e+00 1.338623e-10 8.606848e-09 9.848211e-08 5.558557e-07 2.130106e-06 6.389610e-06 1.618630e-05 3.623252e-05
Table 3: Results of the evolution of the error with the order for the spot.
Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
7 3.833581e-08 1.321764e-08 3.864287e-09 9.017995e-10 1.518092e-10 1.525891e-11 5.941914e-13 5.773160e-15 0.000000e+00 1.199041e-14 6.048495e-13 1.557154e-11 1.561080e-10 9.338343e-10 4.029594e-09 1.387971e-08 4.053813e-08
9
11
Error 2.703970e-11 1.065814e-14 7.133849e-12 3.552714e-15 1.538325e-12 1.065814e-14 2.451372e-13 3.552714e-15 1.598721e-14 1.065814e-14 8.881784e-15 7.105427e-15 2.664535e-15 2.664535e-15 3.552714e-15 3.552714e-15 0.000000e+00 0.000000e+00 1.421085e-14 1.421085e-14 8.881784e-16 8.881784e-16 1.776357e-15 3.552714e-15 1.598721e-14 1.065814e-14 2.486900e-13 7.105427e-15 1.580958e-12 3.552714e-15 7.432277e-12 1.065814e-14 2.829026e-11 3.552714e-15
13 3.552714e-15 0.000000e+00 1.065814e-14 3.552714e-15 1.065814e-14 7.105427e-15 2.664535e-15 3.552714e-15 0.000000e+00 1.421085e-14 8.881784e-16 3.552714e-15 1.065814e-14 7.105427e-15 3.552714e-15 7.105427e-15 1.776357e-14
Table 4: Results of the evolution of the error with the order for the spot.
30
Hedging a commodity-linked portfolio
Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Variation -22.376935059245820 -19.899582994038088 -17.343165856359004 -14.705160272251074 -11.982962333888555 -9.173885029326648 -6.275155590220760 -3.283912754897802 -0.197203944077984 2.988017653540652 6.274896088711729 9.666675755052598 13.166704591502278 16.778437386986347 20.505439190550678 24.351388830330183 28.320082544825940
ECE Paris 2012
Approximation Variation -22.376935059245810 -19.899582994038084 -17.343165856358993 -14.705160272251078 -11.982962333888565 -9.173885029326641 -6.275155590220763 -3.283912754897798 -0.197203944077984 2.988017653540638 6.274896088711728 9.666675755052594 13.166704591502267 16.778437386986340 20.505439190550675 24.351388830330194 28.320082544825937
Error 1.065814e-14 3.552714e-15 1.065814e-14 3.552714e-15 1.065814e-14 7.105427e-15 2.664535e-15 3.552714e-15 0.000000e+00 1.421085e-14 8.881784e-16 3.552714e-15 1.065814e-14 7.105427e-15 3.552714e-15 1.065814e-14 3.552714e-15
Table 5: Results of the approximation and the error with the order p = 11 for the spot.
Shock = −4 Shock = −2 Shock = 2 Shock = 4
0
10
−2
10
−4
10
−6
Error
10
−8
10
−10
10
−12
10
−14
10
2
4
6
8
10
12
Order
Figure 2: Plots of the evolution of the error with the order of the approximation for the spot.
31
Hedging a commodity-linked portfolio
B.1.3
ECE Paris 2012
Swaps Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1 3.215027e+02 2.483172e+02 1.840505e+02 1.289484e+02 8.326362e+01 4.725578e+01 2.119178e+01 5.345894e+00 0.000000e+00 5.443780e+00 2.197495e+01 4.989948e+01 8.953183e+01 1.411952e+02 2.052216e+02 2.819526e+02 3.717389e+02
2
3
Error 2.374552e+01 1.312287e+00 1.601341e+01 7.733600e-01 1.015158e+01 4.196854e-01 5.914142e+00 2.034871e-01 3.048426e+00 8.379987e-02 1.294749e+00 2.665899e-02 3.862336e-01 5.294714e-03 4.860830e-02 3.327325e-04 0.000000e+00 0.000000e+00 4.927744e-02 3.364085e-04 3.969406e-01 5.412353e-03 1.348960e+00 2.755239e-02 3.219791e+00 8.756515e-02 6.332609e+00 2.149797e-01 1.101955e+01 4.482884e-01 1.762197e+01 8.351969e-01 2.649069e+01 1.432881e+00
5 2.147246e-03 9.674112e-04 3.851380e-04 1.294851e-04 3.407656e-05 6.088697e-06 5.366425e-07 8.419192e-09 0.000000e+00 8.486268e-09 5.452044e-07 6.235026e-06 3.517288e-05 1.347133e-04 4.038736e-04 1.022537e-03 2.287647e-03
Table 6: Results of the evolution of the error with the order for the swap.
Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
7 1.904151e-06 6.562805e-07 1.917974e-07 4.474305e-08 7.529479e-09 7.557901e-10 3.217338e-11 1.193712e-12 0.000000e+00 1.421085e-12 3.058176e-11 7.698873e-10 7.719564e-09 4.615094e-08 1.990725e-07 6.854179e-07 2.001081e-06
9
11
Error 1.065473e-09 1.364242e-12 2.823981e-10 1.818989e-12 6.184564e-11 1.364242e-12 1.023182e-11 6.821210e-13 9.094947e-13 2.273737e-13 2.273737e-13 2.273737e-13 2.614797e-12 2.614797e-12 1.080025e-12 1.080025e-12 0.000000e+00 0.000000e+00 1.307399e-12 1.307399e-12 5.684342e-13 5.684342e-13 2.273737e-13 2.273737e-13 1.818989e-12 6.821210e-13 8.640200e-12 1.364242e-12 6.184564e-11 4.547474e-13 2.905836e-10 0.000000e+00 1.110038e-09 1.364242e-12
13 9.094947e-13 1.818989e-12 1.364242e-12 6.821210e-13 2.273737e-13 2.273737e-13 2.614797e-12 1.080025e-12 0.000000e+00 1.307399e-12 5.684342e-13 2.273737e-13 6.821210e-13 1.364242e-12 4.547474e-13 0.000000e+00 9.094947e-13
Table 7: Results of the evolution of the error with the order for the swap.
32
Hedging a commodity-linked portfolio
ECE Paris 2012
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
0
10
−2
10
−4
10
−6
Error
10
−8
10
−10
10
−12
10
−14
10
2
4
6
8
10
12
14
Order
Figure 3: Plots of the evolution of the error with the order of the approximation for the swap.
B.2 B.2.1
Influence of the maturity on the approximation of the change Futures Order: Maturity: Shock Value -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
12 3 months 3.552714e-15 3.552714e-15 3.552714e-15 1.776357e-15 8.881784e-15 8.881784e-15 1.065814e-14 8.881784e-16 5.773160e-15 5.329071e-15 1.776357e-15 1.243450e-14 8.881784e-15 1.776357e-15 1.776357e-14 1.776357e-14 1.065814e-14
6 months
12 months Error 7.105427e-15 1.065814e-14 7.105427e-15 7.105427e-15 7.105427e-15 1.065814e-14 7.105427e-15 0.000000e+00 8.881784e-15 3.552714e-15 0.000000e+00 3.552714e-15 3.552714e-15 9.769963e-15 1.776357e-15 2.664535e-15 5.773160e-15 8.881784e-16 2.664535e-15 7.105427e-15 0.000000e+00 1.332268e-14 5.329071e-15 8.881784e-15 1.776357e-15 1.421085e-14 3.552714e-15 1.421085e-14 3.552714e-15 1.065814e-14 7.105427e-15 7.105427e-15 1.421085e-14 2.486900e-14
24 months 1.421085e-14 1.421085e-14 0.000000e+00 8.881784e-15 3.552714e-15 1.776357e-15 9.769963e-15 4.884981e-15 1.332268e-15 1.376677e-14 1.421085e-14 1.065814e-14 1.776357e-14 8.881784e-15 7.105427e-15 1.776357e-15 3.552714e-15
Table 8: Results of the evolution of the error with the maturity and with order p = 12 for the future.
33
Hedging a commodity-linked portfolio
ECE Paris 2012
−14
10
x 10
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
9 8 7
Error
6 5 4 3 2 1
4
6
8
10
12
14 Maturity
16
18
20
22
24
Figure 4: Plots of the evolution of the error with the maturity of the approximation for the future.
B.2.2
Swaps Order: Maturity: Shock Value -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
3 months 1.3642e-12 1.8190e-12 1.3642e-12 6.8212e-13 2.2737e-13 2.2737e-13 2.6148e-12 1.0800e-12 0.0000e+00 1.3074e-12 5.6843e-13 2.2737e-13 6.8212e-13 4.5475e-13 0.0000e+00 1.3642e-12 1.065814e-14
12 6 months 12 months Error 9.0949e-13 1.1369e-12 4.5475e-13 1.8190e-12 2.2737e-13 1.1369e-12 2.2737e-13 1.1369e-12 6.8212e-13 1.1369e-12 9.0949e-13 6.8212e-13 1.1369e-12 2.8422e-12 6.8212e-13 4.5475e-13 0.0000e+00 0.0000e+00 2.2737e-13 1.3642e-12 1.8190e-12 4.5475e-12 1.1369e-12 4.5475e-13 6.8212e-13 2.0464e-12 1.1369e-12 3.4106e-12 9.0949e-13 2.5011e-12 4.5475e-13 4.5475e-13 1.8190e-12 4.5475e-13
24 months 2.2737e-13 4.5475e-13 9.0949e-13 5.6843e-13 1.7053e-13 2.8422e-13 1.7053e-13 9.0949e-13 0.0000e+00 5.6843e-13 1.3074e-12 6.2528e-13 4.5475e-13 7.9581e-13 2.2737e-12 0.0000e+00 0.0000e+00
Table 9: Results of the evolution of the error with the maturity and with order p = 12 for the swap.
34
Hedging a commodity-linked portfolio
ECE Paris 2012
−11
1
x 10
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
0.9 0.8 0.7
Error
0.6 0.5 0.4 0.3 0.2 0.1 0
4
6
8
10
12
14 Maturity
16
18
20
22
24
Figure 5: Plots of the evolution of the error with the maturity of the approximation for the swap.
B.3
Influence of the absolute error on the approximation of the change
B.3.1
Futures 0
0
10
10
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4 −5
−5
10
error
error
10
−10
−10
10
10
−15
−15
10
10 0
−10
−5
−10
−10
−10 rho
Figure 6: Plots of the evolution of the error of the approximation with the absolute error and T = 3 months for the future.
0
−10
−5
−10
−10
−10 rho
Figure 7: Plots of the evolution of the error of the approximation with the absolute error and T = 6 months for the future.
35
Hedging a commodity-linked portfolio
Maturity ρ Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1e − 02 4
1e − 06 7
5.633556e-04 2.903267e-04 1.349660e-04 5.450001e-05 1.794453e-05 4.278892e-06 5.662070e-07 1.778004e-08 5.773160e-15 1.795435e-08 5.773635e-07 4.405981e-06 1.865868e-05 5.722464e-05 1.431031e-04 3.108495e-04 6.090946e-04
7.424193e-07 2.925911e-07 9.981448e-08 2.795691e-08 5.884214e-09 7.883063e-10 4.629541e-11 3.623768e-13 5.773160e-15 3.606004e-13 4.698464e-11 8.057786e-10 6.058965e-09 2.899871e-08 1.042945e-07 3.079694e-07 7.871810e-07
3 months 1e − 08 8 Error 5.685692e-10 1.714326e-10 4.293810e-11 8.343548e-12 1.133316e-12 9.414691e-14 8.881784e-15 8.881784e-16 5.773160e-15 5.329071e-15 0.000000e+00 7.283063e-14 1.140421e-12 8.595791e-12 4.445155e-11 1.785878e-10 5.958185e-10
ECE Paris 2012
1e − 10 9
1e − 13 13
1.333689e-11 3.520739e-12 7.531753e-13 1.243450e-13 3.552714e-15 8.881784e-15 1.065814e-14 8.881784e-16 5.773160e-15 5.329071e-15 1.776357e-15 1.243450e-14 3.552714e-15 1.278977e-13 7.602807e-13 3.634426e-12 1.391243e-11
3.552714e-15 3.552714e-15 3.552714e-15 1.776357e-15 8.881784e-15 8.881784e-15 1.065814e-14 8.881784e-16 5.773160e-15 5.329071e-15 1.776357e-15 1.243450e-14 8.881784e-15 1.776357e-15 1.776357e-14 1.776357e-14 3.552714e-15
Table 10: Results of the evolution of the error with the absolute error for maturity T = 3 months and order p = 12 for the future.
Maturity ρ Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1e − 02 4
1e − 06 7
3.695461e-04 1.903745e-04 8.846691e-05 3.570974e-05 1.175316e-05 2.801465e-06 3.705607e-07 1.163177e-08 5.773160e-15 1.173647e-08 3.772618e-07 2.877801e-06 1.218210e-05 3.734620e-05 9.335406e-05 2.027002e-04 3.970153e-04
4.105574e-07 1.617564e-07 5.516567e-08 1.544682e-08 3.250218e-09 4.352980e-10 2.555822e-11 1.985079e-13 5.773160e-15 1.993961e-13 2.590816e-11 4.441567e-10 3.338773e-09 1.597472e-08 5.743581e-08 1.695489e-07 4.332384e-07
6 months 1e − 08 8 Error 2.651461e-10 7.992185e-11 2.000711e-11 3.895551e-12 5.329071e-13 3.907985e-14 3.552714e-15 1.776357e-15 5.773160e-15 2.664535e-15 1.776357e-15 3.375078e-14 5.364598e-13 3.991474e-12 2.066258e-11 8.299850e-11 2.768132e-10
1e − 10 9
1e − 13 13
5.709211e-12 1.513456e-12 3.304024e-13 4.618528e-14 3.552714e-15 0.000000e+00 3.552714e-15 1.776357e-15 5.773160e-15 2.664535e-15 0.000000e+00 5.329071e-15 7.105427e-15 4.973799e-14 3.268497e-13 1.563194e-12 5.957901e-12
3.552714e-15 3.552714e-15 7.105427e-15 7.105427e-15 8.881784e-15 0.000000e+00 3.552714e-15 1.776357e-15 5.773160e-15 2.664535e-15 0.000000e+00 5.329071e-15 1.776357e-15 3.552714e-15 3.552714e-15 7.105427e-15 1.776357e-14
Table 11: Results of the evolution of the error with the absolute error for maturity T = 6 months and order p = 12 for the future.
36
Hedging a commodity-linked portfolio
Maturity ρ Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1e − 02 4
1e − 06 7
4.352397e-03 2.562635e-03 1.389413e-03 6.730447e-04 2.769155e-04 8.801194e-05 1.746352e-05 1.096410e-06 8.881784e-16 1.106402e-06 1.778327e-05 9.044025e-05 2.871492e-04 7.042790e-04 1.467144e-03 2.730670e-03 4.680070e-03
1.254686e-07 4.940861e-08 1.684182e-08 4.713424e-09 9.912569e-10 1.326939e-10 7.797318e-12 6.394885e-14 8.881784e-16 5.417888e-14 7.863044e-12 1.349747e-10 1.013984e-09 4.848953e-09 1.742465e-08 5.140922e-08 1.312914e-07
12 months 1e − 08 8 Error 2.852737e-09 9.826557e-10 2.870095e-10 6.691536e-11 1.125144e-11 1.125322e-12 3.463896e-14 2.664535e-15 8.881784e-16 7.105427e-15 3.108624e-14 1.155520e-12 1.147171e-11 6.861001e-11 2.958203e-10 1.017959e-09 2.970104e-09
ECE Paris 2012
1e − 10 9
1e − 13 13
5.763567e-11 1.737277e-11 4.355627e-12 8.455459e-13 1.172396e-13 1.243450e-14 9.769963e-15 2.664535e-15 8.881784e-16 7.105427e-15 1.332268e-14 1.776357e-14 1.030287e-13 8.490986e-13 4.455103e-12 1.793055e-11 5.973178e-11
2.842171e-14 1.065814e-14 1.065814e-14 0.000000e+00 3.552714e-15 3.552714e-15 9.769963e-15 2.664535e-15 8.881784e-16 7.105427e-15 1.332268e-14 8.881784e-15 1.421085e-14 1.421085e-14 1.065814e-14 1.065814e-14 7.105427e-15
Table 12: Results of the evolution of the error with the absolute error for maturity T = 12 months and order p = 12 for the future.
Maturity ρ Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1e − 02 4
1e − 06 7
1.129167e-03 6.640032e-04 3.595553e-04 1.739508e-04 7.147844e-05 2.268881e-05 4.496157e-06 2.819156e-07 1.332268e-15 2.837419e-07 4.554599e-06 2.313263e-05 7.334875e-05 1.796588e-04 3.737597e-04 6.947069e-04 1.189035e-03
6.323532e-07 2.844440e-07 1.130593e-07 3.794993e-08 9.971137e-09 1.778723e-09 1.565059e-10 2.446265e-12 1.332268e-15 2.448930e-12 1.579519e-10 1.803492e-09 1.015679e-08 3.883530e-08 1.162319e-07 2.937779e-07 6.561232e-07
24 months 1e − 08 8 Error 1.890861e-10 6.508216e-11 1.899991e-11 4.417799e-12 7.407408e-13 7.638334e-14 6.661338e-15 4.884981e-15 1.332268e-15 1.376677e-14 1.110223e-14 6.394885e-14 7.371881e-13 4.497736e-12 1.941913e-11 6.674838e-11 1.946105e-10
1e − 10 9
1e − 13 13
2.732037e-12 8.277823e-13 2.042810e-13 4.796163e-14 8.881784e-15 1.776357e-15 9.769963e-15 4.884981e-15 1.332268e-15 1.376677e-14 1.421085e-14 1.065814e-14 1.243450e-14 3.197442e-14 2.149392e-13 8.384404e-13 2.792433e-12
2.131628e-14 0.000000e+00 1.776357e-15 8.881784e-15 3.552714e-15 1.776357e-15 9.769963e-15 4.884981e-15 1.332268e-15 1.376677e-14 1.421085e-14 1.065814e-14 1.776357e-14 8.881784e-15 8.881784e-15 1.065814e-14 3.907985e-14
Table 13: Results of the evolution of the error with the absolute error for maturity T = 24 months and order p = 12 for the future.
37
Hedging a commodity-linked portfolio
ECE Paris 2012
0
10
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
−2
10
−4
10
−6
error
10
−8
10
−10
10
−12
10
−14
10
−16
10
0
−10
−5
−10
−10
−10 rho
Figure 8: Plots of the evolution of the error of the approximation with the absolute error and T = 12 months for the future.
0
10
Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4 −5
error
10
−10
10
−15
10
0
−10
−5
−10
−10
−10 rho
Figure 9: Plots of the evolution of the error of the approximation with the absolute error and T = 24 months for the future.
38
Hedging a commodity-linked portfolio
B.3.2
ECE Paris 2012
Spots ρ Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1e − 02 4
1e − 06 7
8.003376e-04 4.125959e-04 1.918716e-04 7.750546e-05 2.552810e-05 6.089326e-06 8.060572e-07 2.532078e-08 0.000000e+00 2.558733e-08 8.231177e-07 6.283674e-06 2.662020e-05 8.167218e-05 2.043158e-04 4.439827e-04 8.702916e-04
3.833581e-08 1.321764e-08 3.864287e-09 9.017995e-10 1.518092e-10 1.525891e-11 5.941914e-13 5.773160e-15 0.000000e+00 1.199041e-14 6.048495e-13 1.557154e-11 1.561080e-10 9.338343e-10 4.029594e-09 1.387971e-08 4.053813e-08
1e − 08 8 Error 1.073495e-09 3.237517e-10 8.109424e-11 1.577050e-11 2.133405e-12 1.527667e-13 7.105427e-15 3.552714e-15 0.000000e+00 1.421085e-14 3.552714e-15 1.598721e-13 2.165379e-12 1.626432e-11 8.421353e-1 3.383178e-10 1.128825e-09
1e − 10 9
1e − 13 13
2.703970e-11 7.133849e-12 1.538325e-12 2.451372e-13 1.598721e-14 8.881784e-15 2.664535e-15 3.552714e-15 0.000000e+00 1.421085e-14 8.881784e-16 1.776357e-15 1.598721e-14 2.486900e-13 1.580958e-12 7.432277e-12 2.829026e-11
1.065814e-14 3.552714e-15 1.065814e-14 3.552714e-15 1.065814e-14 7.105427e-15 2.664535e-15 3.552714e-15 0.000000e+00 1.421085e-14 8.881784e-16 3.552714e-15 1.065814e-14 7.105427e-15 3.552714e-15 1.065814e-14 3.552714e-15
Table 14: Results of the evolution of the error with the absolute error for order p = 12 for the spot. 0
10
Shock = −4 Shock = −2 Shock = 2 Shock = 4
−5
error
10
−10
10
−15
10
0
−10
−2
−10
−4
−10
−6
−8
−10
−10
−10
−10
−12
−10
rho
Figure 10: Plots of the evolution of the error of the approximation with the absolute error for the spot.
39
Hedging a commodity-linked portfolio
B.3.3
ECE Paris 2012
Swaps ρ Order Shock -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1e − 02 4
1e − 06 7
5.807961e-02 2.992348e-02 1.390698e-02 5.614196e-03 1.848013e-03 4.405399e-04 5.827865e-05 1.829559e-06 0.000000e+00 1.846464e-06 5.936050e-05 4.528636e-04 1.917262e-03 5.878394e-03 1.469599e-02 3.191343e-02 6.251451e-02
1.904151e-06 6.562805e-07 1.917974e-07 4.474305e-08 7.529479e-09 7.557901e-10 3.217338e-11 1.193712e-12 0.000000e+00 1.421085e-12 3.058176e-11 7.698873e-10 7.719564e-09 4.615094e-08 1.990725e-07 6.854179e-07 2.001081e-06
1e − 08 8 Error 4.737694e-08 1.428225e-08 3.575678e-09 6.946266e-10 9.367795e-11 7.275958e-12 2.387424e-12 1.080025e-12 0.000000e+00 1.307399e-12 7.958079e-13 6.821210e-12 9.640644e-11 7.134986e-10 3.699370e-09 1.485523e-08 4.955245e-08
1e − 10 9
1e − 13 13
1.065473e-09 2.823981e-10 6.184564e-11 1.023182e-11 9.094947e-13 2.273737e-13 2.614797e-12 1.080025e-12 0.000000e+00 1.307399e-12 5.684342e-13 2.273737e-13 1.818989e-12 8.640200e-12 6.184564e-11 2.905836e-10 1.110038e-09
1.364242e-12 1.818989e-12 1.364242e-12 6.821210e-13 2.273737e-13 2.273737e-13 2.614797e-12 1.080025e-12 0.000000e+00 1.307399e-12 5.684342e-13 2.273737e-13 6.821210e-13 1.364242e-12 4.547474e-13 0.000000e+00 1.364242e-12
Table 15: Results of the evolution of the error with the absolute error for order p = 12 for the swap.
Variation of the error with the value of rho for the spot Shock = −4 Shock = −2 Shock = 0 Shock = 2 Shock = 4
0
10
−2
10
−4
10
−6
Error
10
−8
10
−10
10
−12
10
−14
10
0
−10
−2
−10
−4
−10
−6
−10 Rho
−8
−10
−10
−10
−12
−10
Figure 11: Plots of the evolution of the error of the approximation with the absolute error for the swap.
40
Hedging a commodity-linked portfolio
B.4
ECE Paris 2012
Figures related to hedging tests
Figure 12: Variation of the operation’s efficiency with the cost allowed
Figure 13: Variation of the operation’s efficiency with the horizon
41
