Interest rate sensitivities and hedging under the G2++ modelrationnel-finance.com/Files/rakoto_g2___slides_x.pdf · 2013-06-20 · shock1 shock2 ZCpricechange 102 errorapprox 102

Interest rate sensitivities and hedgingunder the G2++ model

Yves Rakotondratsimba

The full paper ( including the practical implementation in Matlab ) is available fromwww.ssrn.com

17 juin 2013

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is avww.ssrn.com )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 1 / 10

The purpose

Our main purpose here is to provide some clarifications about theinterest rate sensitivities to make use for hedging basic financialinstruments as bonds and swaps.

When compared with equity options, for interest rate derivatives,there are at least two difficulties :

there is a need to simultaneously consider various types of interestrates,a given interest rate itself does not appear to be a tradable asset.

The literature is often focused on the hedging of a given singleposition by one or few instruments, though practitioners are ratherfaced with covering a portfolio position by another portfolio. There isno clearly identified generic result especially devoted to the analysis ofthis situation. We contribute here by filling this lack.

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) isrn.com )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 2 / 10

G2++ model

Under the G2++ model, the instantaneous short rate rt is given by

rt = ϕ(t) + xt;1 + xt;2 (1)

where t 7−→ ϕ(t) is a ( deterministic ) function which allows the model tofit the current observed interest rates. Here x;1 and x;2 may be viewed asstate variables governed by

dxt;1(·) = κ1{0− xt;1}dt + σ1dWt;1(·) (2)

anddxt;2(·) = κ2{0− xt;2}dt + σ2dWt;2(·). (3)

All of these dynamics are given under the risk-adjusted risk-neutralmeasure Q. Here W;1 and W;2 are two correlated standard Brownianmotions with a ( constant ) correlation ρ ≡ ρx1,x2 , with −1 < ρ < 1.Therefore κ1, κ2, σ1, σ2, ρ represent the model parameters.

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is www.ssrn.com )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 3 / 10

Sensitivities for the zero-coupon bond

Given a zero-coupon P(0,T ) maturing at time T , then under the G2++model, we define the corresponding sensitivity of order k , with k ≥ 1,computed at time 0 and for the horizon t, 0 < t < T , with respect to theshock factors ε1, ε2, as the (k + 1)-th dimensional vector given by

Sens_ZC(k; t,T ) ≡(c(l , k)×Θ× λl

1 × λk−l2

)l∈{0,...,k}

(4)

where c(l , k) = k!l!(k−l)!

Θ ≡ Θ

(t,T ; Pmkt(0, t),Pmkt(0,T ); x0;1, x0;2;κ1, κ2, σ1, σ2, ρ

)λ1 ≡ λ1(t,T , κ1, κ2, σ1, σ2, ρ) (5)

andλ2 ≡ λ2(t,T , κ1, κ2, σ1, σ2, ρ.)

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is avaiw.ssrn.com )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 4 / 10

The calibrated model

The interest rate is assumed to be driven by the G2++ model,calibrated at the initial time 0 with the parameters

κ1 = 12.09%, κ2 = 10.58%

σ1 = 16.42%, σ2 = 15.50% and ρ = −1.2%.

For shortness, the numerical illustrations considered are just related toa zero-coupon with a one-year maturity.

The hedging horizon is taken to be 90 days after the present time 0.

We will restrict to the shock scenarios spanned by ε1 and ε2 with

ε1, ε2 ∈ {−1,−0.5, 0, 0.5, 1}

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Mww.ssrn.com )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 5 / 10

Order 1 approximation error for the ZC change

shock1 shock2 ZC price change × 102 error approx ×102

−1.0000 −1.0000 11.8630 0.6409−1.0000 −0.5000 8.8938 0.3618−1.0000 0 6.0053 0.1634−1.0000 0.5000 3.1953 0.0436−1.0000 1.0000 0.4618 0.0002−0.5000 −1.0000 8.7780 0.3524−0.5000 −0.5000 5.8926 0.1572−0.5000 0 3.0857 0.0405−0.5000 0.5000 0.3552 0.0001−0.5000 1.0000 −2.3011 0.0339

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is aww.ssrn.com )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 6 / 10

shock1 shock2 ZC price change × 102 error approx ×102

0 −1.0000 5.7801 0.15110 −0.5000 2.9762 0.03740 0 0.2487 00 0.5000 −2.4047 0.03670 1.0000 −4.9860 0.1457

0.5000 −1.0000 2.8669 0.03450.5000 −0.5000 0.1423 0.00010.5000 0 −2.5082 0.03970.5000 0.5000 −5.0867 0.15140.5000 1.0000 −7.5950 0.33331.0000 −1.0000 0.0360 0.00021.0000 −0.5000 −2.6116 0.04281.0000 0 −5.1872 0.15731.0000 0.5000 −7.6928 0.34191.0000 1.0000 −10.1302 0.5946

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is available fro)Interest rate sensitivities and hedging under the G2++ model17 juin 2013 7 / 10

High order approximation error for the ZC change

shock1 shock2 er2 ×103 er3 ×106 er5 ×109 er7 ×1013

−1.0000 −1.0000 2.3800 6.6540 2.7874 6.2732−1.0000 −0.5000 1.0165 2.1483 0.5136 0.6596−1.0000 0 0.3108 0.4441 0.0485 0.0294−1.0000 0.5000 0.0432 0.0321 0.0009 −0.0007−1.0000 1.0000 0.0000 0.0000 0.0000 0.0011−0.5000 −1.0000 0.9776 2.0395 0.4752 0.5954−0.5000 −0.5000 0.2933 0.4112 0.0432 0.0237−0.5000 0 0.0386 0.0276 0.0008 −0.0004−0.5000 0.5000 0.0000 0.0000 −0.0000 −0.0001−0.5000 1.0000 −0.0300 0.0199 0.0005 −0.0001

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is available froInterest rate sensitivities and hedging under the G2++ model17 juin 2013 8 / 10

shock1 shock2 er2 ×103 er3 ×106 er5 ×109 er7 ×1013

0 −1.0000 0.2765 0.3801 0.0384 0.02220 −0.5000 0.0343 0.0236 0.0006 −0.00150 0 0 0 0 00 0.5000 −0.0339 0.0234 0.0006 0.00020 1.0000 −0.2690 0.3718 0.0378 0.0211

0.5000 −1.0000 0.0304 0.0201 0.0005 0.00130.5000 −0.5000 −0.0000 0.0000 0.0000 0.00060.5000 0 −0.0380 0.0273 0.0007 0.00090.5000 0.5000 −0.2852 0.4020 0.0425 0.02390.5000 1.0000 −0.9375 1.9723 0.4640 0.58341.0000 −1.0000 −0.0000 0.0000 0.0000 0.00041.0000 −0.5000 −0.0425 0.0317 0.0009 −0.00031.0000 0 −0.3020 0.4340 0.0477 0.02801.0000 0.5000 −0.9743 2.0766 0.5013 0.64741.0000 1.0000 −2.2499 6.3612 2.6993 6.1186

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Comments about these numerical results

These tables show how are the error approximation sizes resultingfrom our approach under some scenarios of shocks ε1 and ε2.

Monte-Carlo simulations are not useful to appreciate the accuratenessof our approach. Actually we can derive analytic control of the errorwhen imposing some views on shocks.

Observe that for ε1 = ε2 = 0 then the error is equal to zero since inthis case the approximation becomes exact.

In these results, sensitivities of high orders are considered as they leadto reduce the error approximations. This is particularly useful whenlarge size positions are considered.

Numerical results for various times horizon are not reported here forshortness. Just it may be stressed here that trends going in the samedirection as the above are also available, showing the robustness of ourapproach with respect to the time horizon.

Yves Rakotondratsimba ( The full paper ( including the practical implementation in Matlab ) is avai.sscom )Interest rate sensitivities and hedging under the G2++ model17 juin 2013 10 / 10