1. Stochastic Volatility Modelling Bruno Dupire Nice 14/02/03

2. Bruno Dupire Structure of the talk 1.Model review 2.Forward equations 3.Forward volatility 4.Arbitrageable & admissible smile dynamics

3. Bruno Dupire Model Requirements Has to fit static/current data: Spot Price Interest Rate Structure Implied Volatility Surface Should fit dynamics of: Spot Price (Realistic Dynamics) Volatility surface when prices move Interest Rates (possibly) Has to be Understandable In line with the actual hedge Easy to implement

4. Bruno Dupire : Bachelier 1900 : Black-Scholes 1973 : Merton 1973 : Merton 1976 : Hull&White 1987 A Brief History of Volatility (1)

5. Bruno Dupire Dupire 1992, arbitrage model which fits term structure of volatility given by log contracts. Dupire 1993, minimal model to fit current volatility surface. A Brief History of Volatility (2)

6. Bruno Dupire Heston 1993, semi-analytical formulae. Dupire 1996 (UTV), Derman 1997, stochastic volatility model which fits current volatility surface HJM treatment. A Brief History of Volatility (3)

7. Bruno Dupire Bates 1996, Heston + Jumps: Local volatility + stochastic volatility: –Markov specification of UTV –Reech Capital Model: f is quadratic –SABR: f is a power function A Brief History of Volatility (4)

8. Bruno Dupire Lévy Processes Stochastic clock: –VG (Variance Gamma) Model: BM taken at random time  (t) –CGMY model: same, with integrated square root process Jumps in volatility Path dependent volatility Implied volatility modelling Incorporate stochastic interest rates n dimensional dynamics of  n assets stochastic correlation A Brief History of Volatility (5)

9. Bruno Dupire BWD Equation: price of one option C K 0,T 0 for different (S,t) FWD Equation: price of all options C K,T for current (S 0,t 0 ) Advantage of FWD equation: If local volatilities known, fast computation of implied volatility surface, If current implied volatility surface known, extraction of local volatilities, Understanding of forward volatilities and how to lock them. Forward Equations (1)

10. Bruno Dupire Several ways to obtain them: Fokker-Planck equation: –Integrate twice Kolmogorov Forward Equation Tanaka formula: –Expectation of local time Replication –Replication portfolio gives a much more financial insight Forward Equations (2)

11. Bruno Dupire FWD Equation: dS/S =  (S,t) dW Equating prices at t 0 : STST STST STST

12. Bruno Dupire FWD Equation: dS/S = r dt +  (S,t) dW Time Value + Intrinsic Value (Strike Convexity) (Interest on Strike) STST TVIV Equating prices at t 0 : TV IV STST STST

13. Bruno Dupire TV + Interests on K – Dividends on S STST TV IV Equating prices at t 0 : STST STST FWD Equation: dS/S = (r-d) dt +  (S,t) dW

14. Bruno Dupire If known, quick computation of all today, If all known: Local volatilities extracted from vanilla prices and used to price exotics. Stripping Formula

15. Bruno Dupire FWD Equation for Americans Local Volatility model in the continuation region if, In Black Scholes : in the continuation region (Carr 2002) Same FWD Eq. than in the European case (+Boundary Condition)

16. Bruno Dupire FWD Equation for Americans in LVM If do we have for Americans? BSLVM BWD Eq.Same for Eur/Am FWD Eq.Same for Eur/Am ?

17. Bruno Dupire Counterexample Assume for, FWD Eq. for Europeans: When indeed

18. Bruno Dupire Risk Neutral Dynamics Smile Calibrated 1D Diffusion Local Volatility Model : Simplest model which fits the smile Models of interest Local Volatility Model

19. Bruno Dupire When, gives at T: Equating Prices at t 0 : from local vol model Any stochastic volatility model which matches the initial smiles has to satisfy: FWD Equation when dS/S =  t dW

20. Bruno Dupire Convexity Bias likely to be high if K

21. Bruno Dupire Impact on Models Risk Neutral drift for instantaneous forward variance Markov Model: fits initial smile with local vols

22. Bruno Dupire Locking FWD Variance | S T =K STST STST Constant curvature STST STST

23. Bruno Dupire Portfolio Initial Cost = 0 at T: if 0if Replicates the pay-off of a conditional variance swap Conditional Variance Swap Local Vol stripped from initial smile It is possible to lock this value

24. Bruno Dupire It is not possible to prescribe just any future smile If deterministic, one must have Not satisfied in general Deterministic future smiles S0S0 K T1T1 T2T2 t0

25. Bruno Dupire If stripped from Smile S.t Then, there exists a 2 step arbitrage: Define At t 0 : Sell At t: gives a premium = PL t at t, no loss at T Conclusion: independent of from initial smile Det. Fut. smiles & no jumps => = FWD smile t0t0 tT S0S0 S K

26. Bruno Dupire Sticky Strike assumption: Each (K,T) has a fixed  impl (K,T) independent of (S,t) Sticky Delta assumption:  impl (K,T) depends only on moneyness and residual maturity In the absence of jumps, - Sticky Strike is arbitrageable - Sticky  is (even more) arbitrageable Consequence of det. future smiles

27. Bruno Dupire Each C K,T lives in its Black-Scholes (  impl (K,T))world P&L of Delta hedge position over  t: If no jump Example of arbitrage with Sticky Strike !

28. Bruno Dupire average of  2 weighted by S 2  0   : model with local vols ,  0 with  0 Where  0 =  of C under  0, and  =RN density under   Implied vol solution of From Local Vols to implied Vols (equivalent to density of Brownian Bridge from (S,t 0 ) to (K,T))

29. Bruno Dupire Market Model of Implied Volatility Implied volatilities are directly observable Can we model directly their dynamics? where is the implied volatility of a given Condition on dynamics?

30. Bruno Dupire Drift Condition Apply Ito’s lemma to Cancel the drift term Rewrite derivatives of gives the condition that the drift of must satisfy. For short T, (Short Skew Condition :SSC) where

31. Bruno Dupire Local Volatility Model Case det. function of det. function of and :, SSC: solved by

32. Bruno Dupire “Dual” Equation The stripping formula can be expressed in terms of When solved by

33. Bruno Dupire Large Deviation Interpretation The important quantity is If then satisfies: and

34. Bruno Dupire Consider, for one maturity, the smiles associated to 3 initial spot values Skew case -ATM short term implied follows the local vols -Similar skews Smile dynamics: Local Vol Model 1 Local vols K

35. Bruno Dupire Pure Smile case -ATM short term implied follows the local vols -Skew can change sign Smile dynamics: Local Vol Model 2 K Local vols

36. Bruno Dupire Skew case (  <0) - ATM short term implied still follows the local vols - Similar skews as local vol model for short horizons - Common mistake when computing the smile for another spot: just change S 0 forgetting the conditioning on  : if S : S 0  S + where is the new  ? Smile dynamics: Stoch Vol Model 1 Local vols K 

37. Bruno Dupire Pure smile case  - ATM short term implied follows the local vols - Future skews quite flat, different from local vol model -Again, do not forget conditioning of vol by S Smile dynamics: Stoch Vol Model 2 Local vols K 

38. Bruno Dupire Skew case - ATM short term implied constant (does not follows the local vols) - Constant skew - Sticky Delta model Smile dynamics: Jump Model Local vols K

39. Bruno Dupire Pure smile case - ATM short term implied constant (does not follows the local vols) - Constant skew - Sticky Delta model Smile dynamics: Jump Model Local vols K

40. Bruno Dupire 2 ways to generate skew in a stochastic vol model -Mostly equivalent: similar (S t,  t  ) patterns, similar future evolutions -1) more flexible (and arbitrary!) than 2) -For short horizons: stoch vol model  local vol model + independent noise on vol. Spot dependency STST  STST 

41. Bruno Dupire Conclusion Local vols are conditional forward values that can be locked All stochastic volatility models have to respect the local vols in expectation Without jumps, the only non arbitrageable deterministic future smiles are the forward smile from the local vol model In particular, without jump, sticky strikes and sticky delta are arbitrageable Jumps needed to mimic market smile move