Interest rate models enhanced with local volatility

Transcription

1 1/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Ineres rae models enhanced wih local volailiy Lingling Cao Join work wih Pierre Henry-Labordère Global Markes Quaniaive Research, SOCIÉTÉ GÉNÉRALE

2 2/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor 1 Inroducion 2 Maching a rolling mauriy swapion 3 An example: Cheyee s model wih LV 4 Exensions o muli-d Cheyee and Libor Marke Model

3 3/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Inroducion Backgound: In fixed income world, Dupire-like local volailiy does no exis. In [3], Gaarek e el have considered a one-dimensional Cheyee process enhanced wih a local vol, and derived an (approximae) Dupire-like local vol. Dimensional curse for markov funcionals.

4 4/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Inroducion Our work: A general equaion which impose a generic (muli-facors) ineres rae model is calibraed o a srip of rolling mauriy swapions. (Consan enor, diagonals, ec.) As an example: Cheyee 1-d enhanced wih local vol. Exension o muli-dimensional models (Cheyee, BGM).

5 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Maching a rolling mauriy swapion: consan enor Rolling Swap rae s,+θ wih mauriy and enor θ, is dynamics is: ds,+θ = σ,+θ }{{} arge local vol dw,+θ + d Proposiion (Local vol calibraion condiion) C(, K ) = C mk (, K ) for all (, K ) if and only if E Q [(σ,+θ ) 2 s,+θ = K ] = 2 Cmk (,K ) KC mk (,K )+K 2 K C mk (,K ) K 2 Cmk (,K ) +2 EQ [ 1,+θ s >K B (f, f,+θ P,+θ )] K 2 Cmk (,K ) (1) 5/13 f,α : forward insananeous rae a α. P,α : zero coupon of mauriy α.

6 6/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor An example: Cheyee s model wih LV Markovian processes: zero-coupon bond: dx = (Y λ X )d + σ dw dy = (σ 2 2λY )d P T = P 0T e G T X 1 2 G 2 T Y, G T = e λ(t ) 1 P 0 λ f T T ln P T = f 0T + e λ(t ) (X G T Y ) r = f 0 + X Local volailiy specificaion: σ = Swap rae dynamics is hen: ds,+θ = X s,+θ() σ(,s,+θ() ) ( X s,+θ() )(,X,Y ) (, X, Y ) dx = σ(, s,+θ() ) dw,+θ

7 7/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Maching marginals From Equaion (1), C(, K ) C mk (, K ) for all (, K ) [0, T ] R if and only if σ(, K ) is given by wih σ(, K ) 2 = σ loc (, K ) 2 Ξ(, K ) +2 }{{} K 2 Cmk (, K ) can be read from marke daas Ξ(, K ) E Q [e 0 rsds ξ ] ξ 1 s,+θ >K (f, f,+θ P,+θ ) Numerical soluions paricle mehod: Approximae Ξ(, K ) by empirical disribuionaeachsep

8 8/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Numerical examples 6.5 5y5y y5y Marke Model Marke Model bps/day bps/day Sdev Sdev Figure: Swapion smile wih mauriies 5Y/10Y and enor of 5Y compared o implied volailiies (EUR, 15-April-2016). N = 2 12 paricles, 2 15 simulaions.

9 9/13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor Marke Model 5y10y Marke Model 10y10y bps/day bps/day Sdev Sdev Figure: Swapion smile wih mauriies 5Y/10Y and enor of 10Y compared o implied volailiies (EUR, 15-April-2016). N = 2 12 paricles, 2 15 simulaions.

10 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor 10/13 Exensions o muli-dimensional Cheyee A muli-dimensional Cheyee model: dx = d + Σ(, x ) dw Σ(, x ) is N N, W a N-dimensional Brownian moion. Swap rae dyanmics: ds α,β = ( x s,+θ ) Σ(, x ) dw α,β Take Σ(, x ) = ( x s,+θ ) 1 σ(, s,+θ )Φ() where σ(, s,+θ ) is a scaling funcion and Φ() a deerminisic N N marix. Calibraion condiion is: σ(, K ) 2 Tr ( Φ() Φ() ) = σ loc (, K ) 2 Ξ(, K ) + 2 K 2 Cmk (, K )

11 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor 11/13 Exensions o Libor marke models Swap rae dynamics under LLM: ds α,β = β i=α+1 s α,β L i Σ i dw α,β, Σ i is he insananeous volailiy of Libor L i L(, T i, T i+1 ). Take Σ i = ( ) s,+θ 1 L i σ(, s,+θ )Φ i ()

12 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor 12/13 Open quesions Exisence of local vol: exisence of soluion of Mckean-Vlasov ype SDE. Calibraion on Vol cube: how o paramerize?

13 Inroducion Maching a rolling mauriy swapion An example: Cheyee s model wih LV Exensions o muli-d Cheyee and Libor 13/13 Cheyee, O. : Markov represenaion of he Heah-Jarrow-Moron model, Available a SSRN: hp://ssrn.com/absrac=6073. Chibane, M., Law, D. : A quadraic volailiy Cheyee model, Risk magazine, Jul Gaarek, D., Jablecki, J., Qu, D. : Non-parameric local volailiy formula for ineres rae swapions, Risk magazine (2016). Guyon J., Henry-Labordère, P. : Being paricular abou calibraion, Risk magazine, Jan Guyon J., Henry-Labordère, P. : Non Linear opion pricing, Chapman and Hall/CRC Financial Mahemaics Series (2014). Hun, P., Kennedy, J., Pelsser, A. : Markov-funcional ineres rae models, Finance and Sochasics, Augus 2000, Volume 4, Issue 4, pp Qu, D., Zhu, D. : Libor Local Volailiy Model: A New Ineres Rae Smile Model, Wilmo Magazine, March 2016, pp