Numerics for SLV models in FX markets

Transcription

1 Numerics for SLV models in FX markets Christoph Reisinger Joint with Andrei Cozma, Ben Hambly, & Matthieu Mariapragassam Mathematical Institute & Oxford-Man Institute University of Oxford Project partially funded by OMI and BNPP London Models and Numerics in Financial Markets, 27 May 2015

2 OUTLINE Introduction Derivative Pricing Models Markovian Projections Dupire-Type Formula for Barrier Options Main Result Dupire-type Formula Forward PIDE for Barrier Options Numerical Approximation Monte Carlo Exotics Pricing Under SLV

3 FRAMEWORK In derivative pricing we try to accurately estimate the prices of OTC exotic contracts (accumulator, cliquet) hedge ratios with contracts available on the listed market (calls, barriers). Underlying spot processes are calibrated to data. It is an inverse problem: Given quoted vanillas, what can one say about the spot diffusion. It is ill-posed: stability of calibrated parameters problematic. Regularization (mathematically!) well-posed problem

4 MODELS FOR EXOTICS PRICING LOCAL VOLATILITY ds t S t = (r q) dt + σ (S t, t) dw t Widely used for OTC contracts with barriers (even if not perfect) Exact repricing of calls Forward smile flattens out with time bad for cliquets Application to long-dated FX, e.g., Deelstra and Rayeé (2012) sigma T K Figure: S&P500: Left: Implied volatility. Right: Local volatility.

5 LOCAL VOLATILITY AND DUPIRE Recall the local volatility model: ds t S t = (r q) dt + σ (S t, t) dw t The parameter σ (S t, t) has to be calibrated such that for any call of strike K, maturity T: C (K, T) = E Q [ exp ( rt) (S T K) +] = C Market (K, T) Theorem (Dupire) The market-consistent local volatility parameter σ is given by: ( C(K,T) (r (T) q (T)) σ 2 T (K, T) = 1 2 K2 2 C(K,T) K 2 C (K, T) K C(K,T) K )

6 LOCAL VOLATILITY AND DUPIRE Recall the local volatility model: ds t S t = (r q) dt + σ (S t, t) dw t The parameter σ (S t, t) has to be calibrated such that for any call of strike K, maturity T: C (K, T) = E Q [ exp ( rt) (S T K) +] = C Market (K, T) Theorem (Dupire) The market-consistent local volatility parameter σ is given by: ( C(K,T) (r (T) q (T)) σ 2 T (K, T) = 1 2 K2 2 C(K,T) K 2 C (K, T) K C(K,T) K Dupire formula very sensitive to approximation/interpolation: Small change in input market prices big change in local volatility Extremely bad for adapted Greeks (adapted Gamma particularly) )

7 DUPIRE FORWARD PDE AND REGULARISATION Corollary The call price C (K, T) is the solution to a Forward PDE ( ) C(K,T) + r (T) C (K, T) + K C(K,T) 1 T K 2 σ2 (K, T) K 2 2 C(K,T) = 0, K 2 C (K, 0) = (S 0 K) + C (0, T) = S 0 D(T) Opens the possibility to regularize the problem well-posedness. Idea: Minimize over σ at each maturity T: N K ( f T (σ) = i=1 C LocVol T,K i where P a regularization term. (σ) C Market T,K i ) 2 + λ P (σ), Penalise non-desirable local vol shapes: non-convex, non-smooth,... Theoretical results by Berestycki, Busca and Florent (2002), Crepey (2003), Egger & Engl (2005), Achdou & Pironneau (2005),...

8 MODELS FOR EXOTICS PRICING HESTON ds t S t = (r q) dt + V tdw t, dv t = κ (θ V t) dt + ξ V tdw v t Good smile and spot-vol dynamics (with correlation ρ) Good model for forward start features (cliquet/compound) No exact calibration to calls drawback for pricing Heston Calibrated Volatility Surface 0.42 sigma T T K K Figure: S&P500: Left: Implied volatility. Right: Fitted model implied vol.

9 MODELS FOR EXOTIC PRICING LSV Local-stochastic volatility: ds t S t = (r q) dt + σ (S t, t) V t dw t dv t = κ (θ V t) dt + ξ V t dwt v d[w t, Wt V ] = ρ dt. Spot-vol dynamics and smile regeneration Better model for most exotics Smile dynamics is good and can be controlled (trough the vol-of-vol)

10 MODELS FOR EXOTIC PRICING LSV Local-stochastic volatility: ds t S t = (r q) dt + σ (S t, t) V t dw t dv t = κ (θ V t) dt + ξ V t dwt v d[w t, Wt V ] = ρ dt. Spot-vol dynamics and smile regeneration Better model for most exotics Smile dynamics is good and can be controlled (trough the vol-of-vol) Calibration can be involved and time consuming. Bootstrapping with Kolmogorov forward equation, see Ren, Madan, & Qian (2007). Approximations, e.g., Bompis & Gobet (2015), Lorig, Pagliarini, & Pascucci (2014)

11 MODEL CLASS We consider Itô process models of the form ds t S t = (r q) dt + α tdw t, where α a continuous semi-martingale such that [ T ] E αt 2 S 2 t dt <. 0 Example: Local-Stochastic Volatility (LSV) model of Heston-type: α t = σ (S t, t) V t dv t = κ (θ V t) dt + ξ V tdwt v dw tdwt V = ρdt We are ultimately interested in situations such as κ = κ(v t, t) (Dai, Tang, & Yue 2012), ξ = ξ(s t, t) etc. [NB: Moment conditions on model parameters apply.]

12 MIMICKING AN ITÔ PROCESS Recall ds t S t = (r q) dt + α t dw t. Definition The Markovian projection of α 2 t onto S t is [ ] σ 2 S (K, t) = E Q αt 2 S t = K. Theorem (Gyöngy 1986) There exists a weak solution X t to the SDE dx t X t = (r q) dt + σ S (X t, t) dw t, such that the distributions of S t and X t are the same for all t > 0.

13 SMILE CALIBRATION PROBLEM SOLVED? Corollary For a given strike K and maturity T, the LSV replicates exactly call option prices if and only if σ 2 (S T, T) E Q [V T S T = K] = σ 2 Dupire (K, T) 1. Pre-calibrate stochastic volatility parameters (κ, θ, ρ, ξ). 2. The calibration problem for vanillas is solved if we can compute E Q [V T S T = K] efficiently. 3. Iterative procedure to find σ. 4. Decoupled timestep-wise in Ren, Madan, & Qian (2007).

14 THE BARRIER OPTION CASE The price of an up-and-out call with strike K, barrier B and maturity T is where M t = max 0 u t S u. E Q [ (S T K) + 1 MT <B], Hence, market quotes contain information about the joint density of (S t, M t). Simultaneous calibration to both barriers and vanillas should give: More accuracy for exotics with barrier features (accumulators / auto-callables) A good estimation of the market implied spot-vol dynamics (excellent for cliquets / compounds) and vol-of-vol (ξ) Accurate sticky model adapted Greeks (from the pricing PDE grid)

15 MIMICKING AN ITÔ PROCESS Recall ds t S t = (r q) dt + α t dw t. Definition The Markovian projection of α 2 t onto (S t, M t) is [ ] σ 2 S,M (K, B, t) = E Q αt 2 S t = K M t = B. Theorem (Brunick and Shreve (2013)) There exists a weak solution X t to the SDE { dxt X t M X t = (r q) dt + σ S,M ( Xt, M X t, t ) dw t = max 0 u t X u such that the joint distributions of (S t, M t) and ( ) X t, M X t are the same for all t > 0.

16 PROBLEM Assume: ds t S t = (r q) dt + α t dw t Gyöngy σ 2 S (K, t) = E Q [ α 2 t S t = K ] = Dupire C (r q)(c K C T K ) 1 2 K2 2 C K 2 σ 2 S,M (K, B, t) = E [ Q αt 2 S t = K M t = B ] =? Brunick& Shreve Is there a Dupire-type formula for Barrier Options?

17 OUTLINE Introduction Derivative Pricing Models Markovian Projections Dupire-Type Formula for Barrier Options Main Result Dupire-type Formula Forward PIDE for Barrier Options Numerical Approximation Monte Carlo Exotics Pricing Under SLV

18 MAIN RESULT Recall the Dupire formula for European calls: C (K, T) T = 1 2 σ2 S (K, T) K 2 2 C (K, T) K 2 (r = q = 0 is assumed for readability. Two extra terms appear otherwise.) Theorem (Hambly, Mariapragassam and R. (2014)) For any strike K, barrier B and maturity T 2 C(K,B,T) B T = 1 2 ( 1 2 σ2 S,M (K, B, T) K 2 3 C(K,B,T) K 2 B (B K) + B 2 σ S,M 2 (B,B,T) 3 C(K,b,T) K 2 b K=B,b=B. B ) Proof. Mainly relies on Trotter-Meyer theorem and properties of local times. Alternatively, use PDE techniques.

19 WHAT THE EQUATION DOES AND DOES NOT BRING It links the Markovian projection to the market prices of Barrier options. It gives existence and uniqueness of the mimicking coefficient given a continuum of quotes. Problems to solve: Markovian projection involved at two strike levels, σ 2 S,M (K, B, T) and σ 2 S,M (B, B, T) How to retrieve σ 2 S,M from market quotes? There is a fourth order derivative involved potentially strong numerical instabilities. C ( B, B, T) might be hard to retrieve from the market.

20 OUTLINE Introduction Derivative Pricing Models Markovian Projections Dupire-Type Formula for Barrier Options Main Result Dupire-type Formula Forward PIDE for Barrier Options Numerical Approximation Monte Carlo Exotics Pricing Under SLV

21 DUPIRE-TYPE FORMULA Corollary The unique Brunick-Shreve mimicking coefficient for up-and-out call options is: σ Brunick (K, B, T) = 2 C(K,B,T) B T K B 2 C(0,B,T) T 1 2 K2 3 C(K,B,T) K 2 B (B K) 2 C(0,B,T) B T B where C (0, B, T) = E Dom [S T 1 MT <B] = S 0Q For (M T < B) is the price of the foreign no-touch (quoted on FX market).

22 HOW TO USE THE MODEL We do not believe in the projection á la Brunick-Shreve as a model, i.e., we do not postulate a dynamics of the form ds t S t = r (t) dt + σ Brunick (S t, M t, t) dw t (but see, e.g., Guyon 2014). Rather, we see σ Brunick as a code book for barrier prices, in the spirit of the dynamic local vol of Carmona and Nadtochiy (2009).

23 HOW TO USE THE MODEL We do not believe in the projection á la Brunick-Shreve as a model, i.e., we do not postulate a dynamics of the form ds t S t = r (t) dt + σ Brunick (S t, M t, t) dw t (but see, e.g., Guyon 2014). Rather, we see σ Brunick as a code book for barrier prices, in the spirit of the dynamic local vol of Carmona and Nadtochiy (2009). The LSV model as defined is under parametrized : We can only play with (κ, θ, ρ, ξ) since σ is fully determined by the vanilla prices. Considerations: Local stochastic vol parameters, parametric, as: Quotes of Barrier/Touch options on the market are scarce. For Touches: 5 Barrier levels, 10 maturities up to 3 years. Existence and uniqueness of the solution?

24 OUTLINE Introduction Derivative Pricing Models Markovian Projections Dupire-Type Formula for Barrier Options Main Result Dupire-type Formula Forward PIDE for Barrier Options Numerical Approximation Monte Carlo Exotics Pricing Under SLV

25 FORWARD PIDE Corollary The up-and-out call price follows a Volterra-type PIDE expressed as an IBVP: C (K, B, T) T = 1 2 σ2 S,M (K, B, T) K 2 2 C (K, B, T) K σ2 S,M (B, B, T) B 2 (B K) 3 C (B, B, T) K 2 B B 1 2 C (K, b, T) σ 2 S 0 K 2 K2 S,M (K, b, T) db. K 2 b C (K, B, 0) = (S 0 K) + 1 S0 <B T = 0 C (B, B, T) = 0 K = B C (K, S 0, T) = 0 B = S 0

26 OUTLINE Introduction Derivative Pricing Models Markovian Projections Dupire-Type Formula for Barrier Options Main Result Dupire-type Formula Forward PIDE for Barrier Options Numerical Approximation Monte Carlo Exotics Pricing Under SLV

27 A FULL-TRUNCATION EULER SCHEME Consider where dw t, dw V t = ρdt. ds t S t = (r q) dt + σ (S t, t) V t dw t, dv t = κ (θ v t) dt + ξ V t dw v t, 1. Approximate V t as in Lord et al (2010), by V tn+1 = V tn + κ(θ V + t n )(t n+1 t n) + ξ V + t n (Wt v n+1 Wt v n ), V t = V tn, t (t n, t n+1 ), where y + = max(y, 0). 2. Then define, for piecewise (in t) constant σ, t t S t = S 0 + (r q)s u du + σ(s u, u) V us u dw u. 0 0

28 CONVERGENCE Consider a (potentially path-dedependent) option pay-off f and U = E [exp( rt)f (S)], [ ] U = E exp( rt)f (S). Theorem (Cozma and R. (2015)) Assume σ Lipschitz with 0 sup S,t σ(s, t) σ max <. Then the approximations to the following option values converge as δt 0: 1. European puts, up-and-out calls, and any barrier puts; 2. if, additionally, k > ξ max(1, ξt/4), then also European calls, Asian options, down-and-in/out and up-and-in barrier calls etc. Extends to stochastic (CIR) rates. For variance reduction see A. Cozma poster.

29 CONCLUSION Used Markovian projection for calibration to barrier options; Derived Dupire-type formula for barriers, Forward PIDE for barrier / touch options; Monte Carlo scheme for exotics Next steps: Solution of Fokker-Planck equation for (St, V t, M t ); Forward boot-strapping algorithm by maturity.

30 NUMERICAL SCHEME ) u 0.,j = ((S 0 K i ) + 1 S0 <Bj 0 i N u 0.,0 = 0 f ṃ,0 = 0 for ( j = 0 ; j P ; j + +) * solve B j layer PDE for ( ) u ṃ,j : 0 m M δ T u m i,j ( 1 2 σ 2 S,M ( ) ) ( ) 1 σ 2 S,M Ki, B j, T m Ki, B j, T m B Ki 2 δ KK u m i,j 2 B σ2 S,M ( Bj, B j, T m ) B 2 j ( Bj K )+ δ KKBu m n j,j j 1 1 = 2 K2 i δ KK u m σ 2 S,M (K i, B n, T m) i,j B n=1 B * compute RHS ṃ,j+1 from RHS ṃ,j and u ṃ,j (for j < P) end for

31 VALIDATION: FRAMEWORK We compare: 1. The Forward PIDE with one numerical solution for the whole set of deal parameters. 2. The Backward Feynman-Kac pricing PDE with as many solutions as sets of deal parameters. We use: S 0 = 100, r d = 10% and the dividend yield is r f = 5% space steps / 1000 time steps for the PIDE and PDE solution Compared prices for (K, B, T) covering the set [0, 120] [100, 120] {1} 120 points in strike / 40 points in barrier levels.

32 VALIDATION: FRAMEWORK The Brunick volatility we use is arbitrary and defined using an SVI parametrisation (Gatheral (2006)). Figure: Brunick-Shreve Volatility Surface, a = 0.04, b = 0.2, σ = 0.2, ρ = m = 0 Brunick Volatility Surface Year 0.5 Year 1 Year Volatility Barrier Strike

33 VALIDATION Figure: Forward PIDE vs Backward PDE: S=100 and T=1Y

34 VALIDATION Strike Forward Backward Rel. PIDE PDE Diff e e e e e e e e e e e e e e