Vega Maps: Predicting Premium Change from Movements of the Whole Volatility Surface

Transcription

1 Vega Maps: Predicting Premium Change from Movements of the Whole Volatility Surface Ignacio Hoyos Senior Quantitative Analyst Equity Model Validation Group Risk Methodology Santander Alberto Elices Head of Equity Model Validation Group Risk Methodology Santander Joint work with: Philippe Dérimay (Credit Swiss) Ignacio Ariño (Santander) Quantitative Equity Methodology and Analysis London, 30th November 2nd December, 2011

2 Outline Introduction General description of the method Application to different models: SVI, Heston, splines local volatility Case studies: vanilla, callable and barrier products Conclusions 2

3 Introduction Vega: sensitivity of the option price to the volatility of the underlying. Total Vega: sensitivity to parallel movements of the volatility surface. Vega map: sensitivity by buckets (maturities and strikes). Sensitivity to the value of the volatility surface for each maturity and strike. Allows predicting the P&L change for any movement in the volatility surface, therefore, hedging more than parallel movements. Particular interest for single underlying options, especially for equity indexes (due to liquidity). 3

4 General Description of the Method Vega map: sensitivity to the volatility by buckets (ne maturities and ns strikes). It allows estimating the change in P&L due to any change in the volatility surface. ( ) ( ) 0 P & L Nominal V V 0 VegaMap ( ' ) ( P & L) = σ σ σ = σ real ' nb VegaMap Motivation 1: to obtain a Vega map, we would need as many evaluations as the number of buckets considered for the volatility surface. Very time-consuming process % 70.00% 60.00% 50.00% Vol 40.00% 30.00% 20.00% 50% Motivation 2: difficult to calculate sensitivity of a single bucket (calibration) % 0.00% 1M Maturity 9M 3Y 10Y 200% 125% 150% 175% 75% 100% Strike 4

5 General Description of the Method Idea of the method Define a set of base movements so that any movement could be decomposed as a linear combination of them. Evaluate only with the surfaces given by the predefined movements. * Idea of the method and implementation for parametric models (SVI, Heston) developed by Philippe Dérimay. ** Extension to non-parametric model (splines local volatility) by Ignacio Ariño. 5

6 General Description of the Method Rearranging volatility surfaces (σ) or volatility movements ( σ) as σ M σ 1,1 1,ne ns,1 K K ne σ σ M ns,ne ns σ1,1 M σ ns,1 M σ1,ne M σ ns,ne ns x ne = nb (number of buckets) Every volatility movement ( σ) can be approximated by a linear combination (with coefficients ω 1,, ω nm ) of the nm predefined ( base ) movements p 1,, p nm. nm σ = ω p = P w j= 1 j j σ = p 1 K p nm nb 1 nb nm ω 1 M ω nm nm 1 6

7 General Description of the Method The price of the option (V) depends on the volatility surface. Assuming that any movement ( σ) is a combination of nm predefined movements, applying the chain rule: Vega map Evaluations with σ j = σ 0 + P j 7 unknown

8 General Description of the Method 0 σ = P w d dω dσ j i i i ij j ij j= 1,..., nm dω j i = 1, K, nb σ σ = p ω = p = σ = σ + t w P W σ We only need to evaluate the pseudo-inverse of P (equivalent to doing a linear regression of w with respect to σ). We use the singular value decomposition: 8

9 General Description of the Method Practical issues: Filtering: small singular values turn the pseudo-inverse computation unstable, as the inverse of a small singular value may be very large. Criterion: if s j <α s max (e.g. α = 3%) substitute s j -1 =0. Only maturities up to the pillar following to expiration of the product are needed. Calculation of the pseudo-inverse is only carried out for strikes 50% to 200% (lower strikes introduce noise and worsen the performance). The movements of the implied volatility surface are left for the whole range of strikes. The more movements considered the more exact the Vega map will be but the more evaluations will be needed (more time of calculation). 9

10 Application to splines local volatility Movements used: Set of components defined for all the moneyness levels and applied maturity by maturity to define each movement (e.g. 5 types of movements nm = 5 ne). Components: 10

11 Application to splines local volatility Movements: More robust than applying the component maturity by maturity individually nc number of components ne number of expiries ns number of strikes 11

12 Application to parametric models (SVI, Heston) Same idea but the movements used are defined by the change of one parameter at a time (not any movement can be considered). By only moving one of SVI or Heston parameters it is not possible to reproduce movements similar to the gaussian impulses. This may restrict the precision of the Vega maps for parametric models. σ1,1 K σ1,ne σ M M = σ K σ ns,1 ns,ne ns ne = nb calibration parametrization θ1,1 K θ 1,ne θ = M M θp1 θ K p,ne p ne = np 12

13 Application to parametric models (SVI, Heston) We have dv dv dθ = dσ dθ dσ known (model) Target: obtain A θ dv θ = A σ A = Vega map = A σ d θ [ nb 1 ] = [ nb np] [ np 1] Random θ movements (nr) Θ = Λ A rand rand ( ) A = Λ rand + Θ rand nr [ Θ ] [ Λ ] 1 1 M M [ Θ k ] = nr [ Λ k ] A M M [ Θ nr ] [ Λ nr ] np nb np nb Pseudo-inverse (SVD, filtering) Θ = θ Λ = σ K θ k 1,k np,k K σ k 1,k nb,k 13

14 Application to parametric models (SVI, Heston) Practical issues: Better to use normally distributed increments with the same variance but no correlation, instead of varying one parameter at a time. Many random movements increase precision while filtering small singular values keep stability σ σ 0 x x x x x x Simplified representation of the linear regression as if nb=1 and np=1 θ 0 θ 14

15 Outline Introduction General description of the method Application to different models: SVI, Heston, splines local volatility Case studies: vanilla, callable and barrier products Conclusions 15

16 Case studies: vanilla, callable and barrier products Qualitative analysis: expected Vega map shape. Tenors where the Vega must be concentrated: expiry date, cancellation dates Given the payoff function, strikes where the Vega must be positive, null or negative. Quantitative analysis: prediction capacity of Vega map Define a set of volatility movements to test the Vega Map. Obtain premiums for the original and modified volatility (real P&L variation). Evaluate the Vega map predictions, multiplying bucket by bucket the Vega map by the volatility movement and adding all. Compare the real P&L variation with the P&L variation estimated with the Vega map. 16

17 Case studies: vanilla, callable and barrier products Movements to test the Vega map: 6 pure movements (parallel/skew/smile) and 2 combined ones. * Movements calculated with a parametrization different from SVI and applied to the all maturities. 17

18 Case studies: vanilla, callable and barrier products Case studies: Deal characteristics: 4-year maturity. Underlying: Eurostoxx index. Spot values close to reference fixings (ATM). Products: Vanilla cases: out-of-the-money and ATM call and put, risk reversal, butterfly. Callable cases: yearly cancellations. Barrier cases: non-already-touched barriers applied over the whole life of the option. 18

19 Case studies: vanilla, callable and barrier products Linearity checks: the more linear the better Vega map prediction: Linearity in products: when applying certain movement ( σ), if a product is a combination of other more simple ones, the P&L experimented by this product is the sum of the P&L experimented by those simple products? e.g. σ in a collar, butterfly or a risk reversal give the same P&L that the sum of P&L of their corresponding vanillas. Linearity in movements: a movement which is a combination of two movements give the same P&L variation than the sum of the P&L variations of those two? 19

20 Case studies: vanilla products Qualitative analysis Vega concentrated at maturity (4Y tenor), with positive sign around the strike level and equal to zero (with a bit of noise) far from the strike. Put 70% Call ATM Call 130% 20

21 Case studies: vanilla products Qualitative analysis Risk reversal Butterfly Sensitive to skew Sensitive to smile 21

22 Case studies: vanilla products Quantitative analysis Vanilla call at 130%: out-of-the-money option, sensitive to all pure movements. Pure movements Combined movements Linearity in movements (quite) Maximum errors Lower sensitivity (relatively high errors) 22

23 Case studies: vanilla products Quantitative analysis Vanilla put at 70%: out-of-the-money option, sensitive to all pure movements. Pure movements Combined movements Linearity in movements (high) Lower sensitivity (relatively high errors for SVI) Maximum errors 23

24 Case studies: vanilla products Quantitative analysis Vanilla call ATM: only sensitive to parallel movements. Pure movements Combined movements Sensitivity concentrated in flat movements Maximum errors 24

25 Case studies: vanilla products Quantitative analysis Risk reversal (strikes 70% & 130%): mainly sensitive to skew movements. Pure movements Combined movements Linearity in movements (lower) Maximum errors Sensitivity concentrated in skew * Linearity in products: difference <0.5 bps in real P&L (<0.1 in estimated P&L) 25

26 Case studies: vanilla products Quantitative analysis Butterfly (strikes 70%, 100% & 130%): the most sensitive to convexity movements. Pure movements Combined movements Linearity in movements (lower) Maximum errors Higher smile sensitivity (higher errors for SVI) * Linearity in products: difference <0.5 bps in real P&L (<0.1 in estimated P&L) 26

27 Case studies: callable products Description Callable1: base case (digital jump at maturity). - 4Y maturity - Yearly callable - Tr = 102, 104, 106, 108% - C = 102, 104, 106, 108% - MatPay = 100% - Put(108%) - CS = 2% 100% 108% at maturity Callable2: without digital jump at maturity. - Tr = 102, 104, 106, 108% - C = 102, 104, 106, 108% - MatPay = 108% - Put(108%) 100% 108% at maturity Callable3: with higher digital jump, lower probability of early cancellation. - Tr = 105, 110, 115, 120% - C = 105, 110, 115, 120% - MatPay = 100% - Put(120%) 100% 120% at maturity 27

28 Case studies: callable products Qualitative analysis Sensitivity to several tenors Callable1 Callable2-4Y maturity - Yearly callable - Tr = 102, 104, 106, 108% - C = 102, 104, 106, 108% - MatPay = 100% - Put(108%) - CS = 2% Digital jump on reaching maturity No digital jump at maturity (MatPay equal to vanilla) 4Y = vanilla Vega maps (noise in the tails) - 4Y maturity - Yearly callable - Tr = 102, 104, 106, 108% - C = 102, 104, 106, 108% - MatPay = 108% - Put(108%) - CS = 2% 28

29 Case studies: callable products Qualitative analysis Callable1-4Y maturity - Yearly callable - Tr = 102, 104, 106, 108% - C = 102, 104, 106, 108% - MatPay = 100% - Put(108%) - CS = 2% Higher Tr and Coupons higher effect of the digital at maturity, rarely cancellations in 1 st tenors Callable3-4Y maturity - Yearly callable - Tr = 105, 110, 115, 120% - C = 105, 110, 115, 120% - MatPay = 100% - Put(120%) - CS = 2% 29

30 Case studies: callable products Quantitative analysis Callable3 Pure movements Combined movements - 4Y maturity - Yearly callable - Tr = 105, 110, 115, 120% - C = 105, 110, 115, 120% - MatPay = 100% - Put(120%) - CS = 2% Linearity in movements (normally high) Maximum errors Low sensitivity to smile 30

31 Case studies: barrier products Description Barrier1: downand-out barrier at 70% with a call 100% at maturity. - 4Y maturity - Barrier: Down-and-out 70% - MatPay = Call(100%) barrier strike at maturity Barrier2: up-andout barrier at 130% with a call 100% at maturity. - 4Y maturity - Barrier: Up-and-out 130% - MatPay = Call(100%) strike barrier at maturity 31

32 Case studies: barrier products Qualitative analysis - 4Y maturity - Barrier: Down-and-out 70% - MatPay = Call(100%) strike effect - 4Y maturity - Barrier: Up-and-out 130% - MatPay = Call(100%) barrier effect Barrier level where the MatPay is worth 0 touching makes no difference similar to vanilla case Touching makes a difference (from getting ~30% to 0). Negative Vega for barrier level 4Y Positive Vega at strike level 32

33 Case studies: barrier products Quantitative analysis Up-and-out barrier at 130% (with ATM call at maturity) Pure movements Combined movements - 4Y maturity - Barrier: Up-and-out 130% - MatPay = Call(100%) Linearity in movements (lower) Maximum errors Low sensitivity 33

34 Conclusions A method to efficiently obtain a Vega map for both parametric and non-parametric models has been presented. Performance of the method has been tested for different products (vanillas, callables, barriers) with rather accurate P&L predictions. Linear behaviour respect to volatility movements is a key point for accurate predictions. For non-parametric models, a trade-off between precision and speed must be reached (consider more or less base movements). For parametric models, the parametrization may limit the precision of the Vega map if it does not well replicate actual volatility changes. 34

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