Risk and CVA for exotic derivatives: the universal modeling

Transcription

1 Risk and CVA for exotic derivatives: the universal modeling Alexandre Antonov, Serguei Issakov and Serguei Mechkov Numerix Quant Congress USA, New-York July 2011 A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 1/ 51

2 Outline Exposure: Scenarios vs. Modeling Future price and exposure for callable instruments in the Modeling Framework Backwards pricing using the Least Squares MC Aggregation of exercises into the instrument exposure Direct approach: cumbersome tracking of exercise indicators New approach: automatic recursion CVA Risk: measure dependence and the real-world measure as fictitious currency Examples and conclusion A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 2/ 51

3 The main subject calculation of exotic portfolio exposure What is exposure at time t? Possible values of the portfolio price at t (possibilities are related with different scenarios of the market evolution) Two approaches: Scenarios Modeling Selected books, reviews and articles of the subject: Canabarro and Duffie (2003), Pykhtin (2005), Cesari et al (2010), Jon Gregory (2010), Brigo-Capponi (2010), Pykhtin-Rosen (2010) Brigo-Capponi-Pallavicini-Papatheodorou (2011) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 3/ 51

4 Scenario based approach Algorithm Generate a set of markets (scenarios) M i (t) at time t (yield curves, implied vols etc.) For each market choose a model and calibrate it to the market Price the portfolio with the calibrated model for each market Drawbacks Not clear how to generate scenarios The period from today to the observation time t not taken into account Computationally (very) intensive A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 4/ 51

5 Modeling approach Arbitrage-free model properly calibrated to today implied market the best scenario generator Full time coverage: scenarios for all the time-steps Exposure is automatically consistent with the pricing Adaptation to indexes projections by the model measure change Numerical efficiency Key idea associate portfolio exposure with its future price given by an arbitrage-free model (Cesari et al) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 5/ 51

6 Two types of instruments in the exposure calculation: For vanillas which price is defined uniquely by the market one can use the scenario approach For exotic instruments one needs the modeling approach Below we will consider a portfolio of exotic instruments and will apply the modeling approach (a.k.a. one-step MC). A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 6/ 51

7 Measures The model calibrated to today market still has an extra degree of freedom the model measure. Its properties: Instrument PV does not depend on the measure Distribution of underlying depends on the measure We address a real world measure at the end. Now consider our model evolution under any fixed measure. A Risk (VaR etc) is related to the distributions measure dependent A credit value adjustment (CVA) is linked with PV s of default dependent payments measure independent A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 7/ 51

8 Modeling approach: swap example Consider a swap paying an index α j at a payment date τ j for j = 1,,M. The instrument price PV (discounted expectation) of the payments M S(0) = E α j N(τ j ) where N(t) is the model numeraire and E[ ] is the pricing expectation in the model measure. The swap future price S F (t) for some observation date t = t obs is a conditional expectation M S F (t) = N(t) E α j N(τ j ) F t j=1 j=1 A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 8/ 51

9 The swap future price S F (t) consists of two parts M N(t) M S F (t) = α j N(τ j ) + N(t) E α j N(τ j ) F t j=1,τ j <t j=1,τ j t Payments before t are discounted forward to t α j N(t) N(τ j ) Payments after t are replaced by discounted expectations [ ] αj N(t) E F t N(τ j ) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 9/ 51

10 Conditional expectation and numerical methods Conditional expectation (CE) E [ F t ] averaging over the stochastic evolution after time t (the life before or on t is fixed). Example. For diffusion processes we fix the Brownian increments dw (τ) for τ t and average over dw (τ) for τ > t; the CE is thus a certain function of dw (τ) for τ t. For regular path-independent pay-offs P(T) fixed at time T, a (discounted) CE is a function of model states x i (t) [ E N(t) P(T) ] N(T) F t = f (t;x i (t)) The CE depends on the Brownian increments through the states. For example, the model states can be short rates and FX-rate A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 10/ 51

11 CE of an index fixed at τ before the time t coincides with the index itself nothing to average [ ] P(τ) N(t) E N(τ) F t = N(t) P(τ) N(τ) Below we consider MC simulations with a possibility of CE calculation typical numerical method of CE calculation is a regression to state variables, [ E N(t) P(T) ] N(T) F t ν j β j (x 1 (t),x 2 (t), ) j where ν j are regression coefficients and β j (x 1 (t),x 2 (t), ) are basis functions (their choice is the method key). Other known names of the method: Longstaff-Schwartz, Least Squares MC, American MC (see Andersen-Piterbarg (2010) for review) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 11/ 51

12 Modeling approach: Bermudan swaption example Consider a Bermudan swaption giving a right to enter into the swap defined above on exercise dates T i. The swaption PV discounted non-conditional expectation of the payments subjected to the exercise conditions M V (0) = E α j I(τ j ) N(τ j ) j=1 where indicator I(τ j ) equals to one if we have entered into the swap before the payment date τ j and zero otherwise. The swaption future price for some observation date t = t obs discounted conditional expectation M V F (t) = N(t) E α j I(τ j ) N(τ j ) F t j=1 A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 12/ 51

13 Future price decomposition The obvious property of the discounted future price [ ] VF (t) V (0) = E N(t) The future price observed at t obs V F (t obs ) = V (t obs )(1 I(t obs )) + V e (t obs ) I(t obs ) can be split in two parts: continuation value V (t obs ) (the option was not exercised till t obs ) exercise value V e (t obs ) (the option was exercised before t obs ) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 13/ 51

14 Continuation value It can be shown that the continuation value satisfies recursive relation on exercise dates ( [ ] ) V (Tj+1 ) V (T j ) = max N Tj E N(T j+1 ) F T j, S(T j ) where S(t) is the swap as seen at time t with payments after t. S(t) = [ ] αj N(t)E N(τ j ) F t j,τ j t The continuation value coincide with option underlying calculated by a backwards induction. Note that the total option future value contains also exercise part. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 14/ 51

15 Introduce the instrument days T = {τ 1,τ 2, } {T 1,T 2, }, a union of exercise dates T j and payment dates τ k. Backward induction: Update underlyings on the instrument dates (in pseudo-code notations) max(v (T j ),S(T j )) V (T j ) on exercise dates T j S(τ k ) + α k S(τ k ) on payment dates τ k Calculate discounted conditional expectation between the instrument dates [ ] [ ] V (T) V (t) = N(t)E S(T) N(T) F t, S(t) = N(t)E N(T) F t for t,t T. Typical numerical method regression A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 15/ 51

16 Indicators The indicator I(t) participating in the future price formula can be constructed from conditional exercise indicators C j = 1 S(Tj )>V(T j ) which equals to 1 if we exercise at T j provided that we did not exercise before and 0 otherwise I(t) = I j for T j t < T j+1 where j I j = 1 (1 C i ) i=1 A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 16/ 51

17 Exercise value The exercise value consists of the payments after the observation date (swap S(t obs )) and payments before the observation date V e (t obs ) = S(t obs )I(t obs ) + N tobs j,τ j <t obs I(τ j ) α j N(τ j ) Remark. The second part of the exercise value is sum for forward discounted payments occurred before the observation date. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 17/ 51

18 Instrument exposure The exposure O(t obs ) is closely related with the option future price. There are two choices available: Exposure includes all payments (coincides with the future price) O(t obs ) = V F (t obs ) = V (t obs )(1 I(t obs )) + V e (t obs ) I(t obs ) Exposure only includes the future payments with respect to the observation date O(t obs ) = V (t obs )(1 I(t obs )) + S(t obs ) I(t obs ) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 18/ 51

19 Exposure path-dependence Exposure is a path-dependent product (for callable deals). For example, the continuation part V (t obs )(1 I(t obs )) consists of two contributions Continuation value V (t obs ) is a state Underlying i.e. a certain function of the model states on the observation date The indicator I(t obs ) is path-dependent (a combination of Underlyings for exercise dates before the observation) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 19/ 51

20 The exposure calculation Two approaches for the exposure calculation Direct Calculate all components in the backwards pricing procedure and assemble them in the forward pass Automatic Change Underlying operations to obtain the exposure as by-product of the pricing procedure Remark. A good quality backward induction algorithm is essential. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 20/ 51

21 Direct approach: algorithm Backward induction Calculate and store the underlying swap, the option continuation value and conditional exercise indicators by backward induction using the least-square MC Forward induction Calculate unconditional exercises and roll forward index payments to the observation date Results aggregation A similar procedure was proposed in Cesari et al. Modulo some difference (for example, the exercise time instead of the exercise indicators) it is equivalent to ours. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 21/ 51

22 Drawbacks of the direct exposure calculations 1. Requires modification of backward pricing procedure 2. Includes both backward and forward inductions 3. Includes the cumbersome logic of exercise indicators calculation and aggregation 4. Can be very complicated for exotic instruments with different types of exercises Alternative automatic procedure A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 22/ 51

23 Automatic exposure calculation Recall that the backward pricing reduces to recursive iterations the update of underlyings max(v (T j ),S(T j )) V (T j ), S(τ k ) + α k S(τ k ) the discounted conditional expectation [ ] [ ] V (T) V (t) = N(t)E S(T) N(T) F t, S(t) = N(t)E N(T) F t For the conditional exercise indicator C j = 1 S(Tj )>V(T j ) we can rewrite the option update as follows V (T j )(1 C j ) + S(T j ) C j V (T j ) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 23/ 51

24 Introduce exposure underlyings with the hat symbol ˆ for our products in hand: swaption exposure underlying ˆV and swap Ŝ. In parallel with the main pricing procedure execute the following: For dates after or on the observation date the exposure underlyings coincide with the pricing ones ˆV(t) = V (t) and Ŝ(t) = S(t) for t t obs For dates before the observation date proceed as below A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 24/ 51

25 For dates before the observation date t < t obs modify the update and conditioning procedures The same update rules for the swaption ˆV(T j )(1 C j ) + Ŝ(T j) C j ˆV(T j ) Modified update rules for the swap all payments in the exposure Ŝ(τ k ) + α k Ŝ(τ k) future payments in the exposure Ŝ(τ k ) Ŝ(τ k) No conditional expectation (regression) before the observation date (t < t obs ) ˆV(t) = N(t) ˆV (T) Ŝ(T), Ŝ(t) = N(t) N(T) N(T) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 25/ 51

26 Exposure O(t obs ) = ˆV(0)N(t obs ) Why so? Compare the option price M V (0) = E α j I(τ j ) N(τ j ) j=1 with its exposure underlying M ˆV(0) = E α j I(τ j ) N(τ j ) F t obs j=1 The only difference is the conditional expectation in the future price that is why we have stopped the regression in the exposure calculation before the observation date. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 26/ 51

27 Recursion example Future payments in the exposure no swap update on the payment dates before the observation date The swaption update on the exercise dates ˆV(T j )(1 C j ) + Ŝ(T j) C j ˆV(T j ) The discounting between the exercise dates (no events on the payment dates) ˆV(T j 1 ) = N(T j 1 ) ˆV (T j ) N(T j ), Ŝ(T j 1 ) = N(T j 1 ) Ŝ(T j) N(T j ) It can be shown that the recursion ˆV (T j 1 ) N(T j 1 ) = ˆV(T j ) N(T j ) (1 C j) + Ŝ(t obs) N(t obs ) C j leads to the desired O(t obs ) = ˆV (0)N(t obs ) = V (t obs )(1 I(t obs )) + S(t obs ) I(t obs ) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 27/ 51

28 Alternative algorithm For dates before the observation date t < t obs modify the update and conditioning procedures The same update rules for the swaption ˆV(T j )(1 C j ) + Ŝ(T j ) C j ˆV(T j ) Modified update rules for the swap all payments in the exposure: Ŝ(τ k ) + α k N(t obs ) N(τ k ) Ŝ(τ k) future payments in the exposure Ŝ(τ k ) Ŝ(τ k ) No conditional expectation (regression) and discounting after the observation Exposure O(t obs ) = ˆV(0) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 28/ 51

29 Similar recursion logic was applied in Egloff et al (2007) and Andersen-Piterbarg (2010) Sect in a context of pricing of Callable Libor Exotics. Our contribution: We target explicitly the future price (exposure) We generalize it for arbitrarily complex instrument Generalization. Imagine our (complicated) instrument containing multiple underlyings U m (T) (legs, swaps, options etc. having currency dimensions) and possibly multiple exercise conditions. To calculate exposures O m (t obs ) for all underlings define corresponding exposure underlyings Û m (T). They are identical to the underlyings for t > t obs Û m (t) = U m (t) for t > t obs and obey the following update rules for t < t obs. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 29/ 51

30 Linear update rules of the underlyings lead to the same relation with the exposure underlyings γ m,k U m (T) U k (T) m γ m,k Û m (T) Ûk(T) m where γ m,k are dimensionless values (for example, numbers or barrier exercise indicators) Non-linear update rules of the underlyings (optimal exercise) max(u m (T),U n (T)) U k (T) Û m (T)θ(U m (T) U n (T))+Ûn(T)θ(U n (T) U m (T)) Ûk(T) where θ(x) = 1 for positive x and zero otherwise A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 30/ 51

31 Update rules for pure payments α n,k having currency dimensions U n (τ k ) + α n,k U n (τ k ) lead to the following updates on the level of the exposure underlyings all payments in the exposure Û n (τ k ) + α n,k Ûn(τ k ) future payments only in the exposure Û n (τ k ) Û n (τ k ) No conditional expectation (regression) before the observation date (t < t obs ) Û n (t) = N(t)Ûn(T) N(T) Exposures O m (t obs ) = Ûm(0)N(t obs ) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 31/ 51

32 CVA Consider now the Credit Value Adjustment (CVA) calculation 1, see also Cesari et al. Assume that the Counterparty survival process Λ(t) a Poisson process with stochastic intensity (hazard rate) h(t) and independent jumps Extend our initial pricing model with (possibly correlated) hazard rate process calibrated to the corresponding credit market simulate all the components Calculate the portfolio future values for a fine set of dates Π(t) using the algorithm above (do not include the credit hazard rates into the regression variables) Assume also that we have calculated the collateral, C(t), which consists of a certain number Φ(t) of units of some asset A(t) with known (simulated) evolution, C(t) = Φ(t)A(t). 1 Our portfolio does not have an explicit default risk. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 32/ 51

33 The Self ( our ) exposure at time t is O(t) = (Π(t) C(t)) + T [ CVA = (1 RR C ) E dλ(t) O(t) ] 0 N(t) where RR C is the Counterparty recovery rate which is assumed to be constant. Explanation. If the Counterparty defaults (dλ(t) = 1) on the interval [t,t + dt], our loss will be equal to (Π(t) C(t)) +, and this infinitesimal payment, as seen at the origin, is equal to the discounted expectation. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 33/ 51

34 Averaging over jumps (independent from the hazard rate) T CVA = (1 RR C ) = (1 RR C ) 0 T 0 [ E dh(t) O(t) N(t) ] [ E dt h(t)h(t) O(t) N(t) effective replacement of the survival processes by the hazard discount factor H(t) = e t 0 dτ h(τ) Our extended model, equipped with the Least Squares MC, is able to compute the above averages The future portfolio value is calculated as above (the credit hazard rates are not included into the regression variables) The collateral and the hazard discount factor are simulated The integral element is estimated by averaging Remark. DVA and Bilateral CVA can be computed in a similar way. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 34/ 51 ]

35 Risk The risk measure is a general term for statistical characteristics of the instrument exposure non-discounting (simple) averages E[f (O(t))] A simple average is measure-dependent (contrary to the discounted one E [f (O(t))/N(t)]) Examples. Potential Future Exposure (PFE) for a confidence level α q α (t) = inf{x : E[1 O(t)<x ] α} Expected Shortfall or Expected Tail Loss E[O(t) O(t) > q α (t)] Expected Positive Exposure (EPE) E[O(t) + ] A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 35/ 51

36 Right/wrong way exposure When the model evolution is highly correlated with the default process the exposure distribution conditional to the default is important, i.e. instead of the CDF of the exposure CDF(t,x) = E[1 O(t)<x ] we need a conditional CDF to default happening at time t CDF D (t,x) = E[1 O(t)<x τ = t] where τ is a stochastic default time. Following our CVA considerations we have CDF D (t,x) = E[1 O(t)<x dλ(t)] E[dΛ(t)] = E[1 O(t)<x h(t)h(t)] E[h(t) H(t)] where h(t) is the stochastic hazard rate and H(t) = e t 0 dτ h(τ) is the hazard discount factor A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 36/ 51

37 Real world measure A model calibrated to today s market still has an extra degree of freedom the model probability measure. We want to fix this measure to be the real-world one for the exposure distribution. The real-world (or physical) probability measure appeared early in Quantitative Finance has not been widely used due to its loose definition, contrary to the risk-neutral measure. The risk-neutral approach assumes that tradable securities have drift coinciding with the short rate r(t) a zero-bond SDE dp(t,t) = P(t,T)(r(t)dt + σ(t)dw (t)) In the real world, this drift is supposed to be different but its time-series estimations are vague. We need to link our model to the real-world in a more rigorous way by requiring that some non-discounting (simple) averages hold. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 37/ 51

38 Fixing projections Suppose that we have a set of indices I j fixed at some times t j and their projections in the future p j p j = E RW [I j ] as expectation in the real-world (RW) measure. Our initial arbitrage-free model in its risk-neutral measure cannot return a priori such averages p j E[I j ] So we should modify the model measure in order to meet the projections. Remark. Note that the model is calibrated to the implied market (initial rates, implied vols, etc.) and the only freedom to reproduce the projection is in the measure choice. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 38/ 51

39 Cross-currency analogy Set our initial model as foreign one w.r.t. some FX-rate process and a domestic currency model. Our initial model states will get a drift adjustment depending on FX vol and correlations. Result. The initial model in such cross-currency (CC) setup will have a different measure w.r.t. to its risk-neutral one. Realization. Domestic model (factitious ccy RW ) Trivial model with zero rates unit numeraire: N RW (t) = 1 FX-rate model (between the initial ccy and the factitious one) Black-Scholes model Foreign model (initial ccy) Out initial model Remark. The FX volatility permits to step up from the risk neutral measure towards the RW one. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 39/ 51

40 The CC model calibration: The foreign component (initial model) is calibrated to its implied market The FX-vol and correlations are free they are tools to calibrate the indexes projections The resulting CC model capabilities: Pricing of the initial instrument (identical to the initial pricing up to numerical errors) Exposure simulation in the real-world measure Remark. The BS process for the FX-rate makes a deterministic drift change for the foreign (initial) model. More complicated FX-rates evolution, say Heston, gives richer family of the measure change more degrees of freedom for the calibration to real-world projections. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 40/ 51

41 Numerical experiments Instrument (cancelable swap) swap we receive semi-annually a 6M Libor and pay annually a fixed rate (= 2.57%, a swap rate at origin) on 1 EUR notional exercise we have a right to cancel the swap annually from 4Y Output 6M Libors expectation in different measures distribution (CDF) of 6Y instrument exposure (including the future payments only) in different measures exposure profile in the risk-neutral measure PFE 97.5% in different measures A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 41/ 51

42 Model CC Model Domestic model (RW currency) Trivial model with zero interest rates Foreign model (EUR) Hull-White IR model with 3% rate, 5% mean-reversion and 1.5% volatility FX rate (RW currency/eur) BS model with correlation with HW Brownian motion ρ = 100% and a set of FX-volatilities: 0%,25%,50% Different FX-volatilities correspond to different measures. Zero FX-volatility gives the risk-neutral measure. For simplicity we consider the Counterparty default uncorrelated with the CC model factors. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 42/ 51

43 Libor average, % fx vol 0% fx vol 25% fx vol 50% date, years Figure: 6M Libor averages for different FX-vols (different measures). A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 43/ 51

44 1 0.8 CDF fx vol 0% fx vol 25% fx vol 50% Exposure Value Figure: CDF of 6Y exposure for different FX-vols (different measures). A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 44/ 51

45 value, EUR PFE 2.5% EPE PFE 97.5% date, years Figure: Risk profile in the risk-neutral measure. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 45/ 51

46 value, EUR date, years fx vol 0% fx vol 25% fx vol 50% Figure: PFE 97.5% for different FX-vols (different measures). A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 46/ 51

47 Observations Typical shape of the risk profile Measure change leads to a positive drift in rates: the bigger FX-vol, the bigger Libor average Significant differences is the exposure distributions in different measures It is important to use the pricing model in the real-world measure (the factitious CC model should be calibrated to the rates projections) to get the correct exposure distribution. A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 47/ 51

48 Conclusion We presented: Calculation of the portfolio exposure in a self-consistent way using arbitrage-free model calibrated to both implied market and real-world projections A new automatic method of exposure calculations especially attractive for exotic portfolios avoiding cumbersome exercise aggregation Efficient CVA calculation using the simulated information A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 48/ 51

49 Lief Andersen and Vladimir Piterbarg (2010) Interest Rate Modeling, Atlantic Financial Press Damiano Brigo and Agostino Capponi, (2010) Bilateral counterparty risk with application to CDSs, Risk Magazine, March 2010 Damiano Brigo, Agostino Capponi, Andrea Pallavicini, and Vasileios Papatheodorou (2011), Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment including Re-Hypotecation and Netting, Available at SSRN Eduardo Canabarro and Darrell Duffie (2003) Measuring and Marking Counterparty Risk, DefaultRisk Giovanni Cesari, John Aquilina, Niels Charpillon, Zlatko Filipovic, Gordon Lee, Ion Manda (2010) Modelling, Pricing, and Hedging Counterparty Credit Exposure: A Technical Guide, Springer Finance, Berlin A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 49/ 51

50 Daniel Egloff, Michael Kohler, and Nebojsa Todorovic (2007) A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options, Ann. Appl. Probab. Volume 17, Number 4, Jon Gregory (2010) Counterparty Credit Risk: The new challenge for global financial markets, Wiley Finance Michael Pykhtin (2005) Counterparty Credit Risk Modelling, Risk books Michael Pykhtin and Dan Rosen (2010) Pricing Counterparty Risk at the Trade Level and CVA Allocations, Journal of Credit Risk, vol. 6, pp A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 50/ 51

51 Contact information UK Numerix Software Ltd, 2nd floor 41 Eastcheap London EC3 Tel +44 (0) , Fax +44 (0) A. Antonov, S. Issakov and S. Mechkov; Numerix Risk and CVA for exotic derivatives: the universal modeling 51/ 51