Stochastic Volatility Modelling

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

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

3. 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 Bruno Dupire

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

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

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

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

8. A Brief History of Volatility (5) • Lévy Processes • Stochastic clock: • VG (Variance Gamma) Model: BM taken at random time g(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 s • n assets stochastic correlation Bruno Dupire

9. Forward Equations (1) • BWD Equation: • price of one option CK0,T0 for different (S,t) • FWD Equation: • price of all options CK,T for current (S0,t0) • 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. Bruno Dupire

10. Forward Equations (2) • 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 Bruno Dupire

11. FWD Equation: dS/S = s(S,t) dW ST ST ST Equating prices at t0: Bruno Dupire

12. FWD Equation: dS/S = r dt + s(S,t) dW TV IV Time Value + Intrinsic Value (Strike Convexity) (Interest on Strike) ST IV TV ST ST Equating prices at t0: Bruno Dupire

13. FWD Equation: dS/S = (r-d) dt + s(S,t) dW TV + Interests on K – Dividends on S IV TV ST ST ST Equating prices at t0: Bruno Dupire

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

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

16. FWD Equation for Americans in LVM • Ifdo we have for Americans? Bruno Dupire

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

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

19. FWD Equation when dS/S = st dW • When , gives at T: • Equating Prices at t0: • from local vol model • Any stochastic volatility model which matches • the initial smiles has to satisfy: Bruno Dupire

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

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

22. Locking FWD Variance | ST=K ST Constant curvature ST ST ST Bruno Dupire

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

24. Deterministic future smiles It is not possible to prescribe just any future smile If deterministic, one must have Not satisfied in general K S0 t0 T1 T2 Bruno Dupire

25. Det. Fut. smiles & no jumps => = FWD smile If stripped from SmileS.t Then, there exists a 2 step arbitrage: Define At t0 : Sell At t: gives a premium = PLt at t, no loss at T Conclusion: independent of from initial smile K S0 S t0 t T Bruno Dupire

26. Consequence of det. future smiles • 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 D is (even more) arbitrageable Bruno Dupire

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

28. From Local Vols to implied Vols S: model with local vols s, S0 with s0 Where G0=G of C under S0, and f=RN density under S Implied vol solution of average of s2 weighted by S2G0f (equivalent to density of Brownian Bridge from (S,t0) to (K,T)) Bruno Dupire

29. 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? Bruno Dupire

30. 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 Bruno Dupire

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

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

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

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

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

36. Smile dynamics: Stoch Vol Model 1 • Skew case (r<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 S0 forgetting the conditioning on s : • if S : S0 S+ where is the new s ? Local vols s K Bruno Dupire

37. Smile dynamics: Stoch Vol Model 2 • Pure smile case (r=0) • 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 s Local vols K Bruno Dupire

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

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

40. Spot dependency • 2 ways to generate skew in a stochastic vol model • Mostly equivalent: similar (St,st) patterns, similar future • evolutions • 1) more flexible (and arbitrary!) than 2) • For short horizons: stoch vol model  local vol model + independent noise on vol. s s ST ST Bruno Dupire

41. 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 Bruno Dupire