Stochastic Volatility Modelling Bruno Dupire Nice 140203 Bruno

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

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

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

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

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

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

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 n assets stochastic correlation Bruno Dupire

Forward Equations (1) BWD Equation: price of one option CK 0, T 0 for different (S, t) FWD Equation: price of all options CK, 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. Bruno Dupire

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

FWD Equation: d. S/S = (S, t) d. W ST ST Equating prices at t 0: Bruno Dupire ST

FWD Equation: d. S/S = r dt + (S, t) d. W TV IV Time Value + Intrinsic Value (Strike Convexity) (Interest on Strike) ST TV IV ST Equating prices at t 0: Bruno Dupire ST

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

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

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) Bruno Dupire

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

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

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

FWD Equation when d. S/S = t d. W 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: Bruno Dupire

Convexity Bias K likely to be high if Bruno Dupire

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

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

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

Deterministic future smiles It is not possible to prescribe just any future smile If deterministic, one must have Not satisfied in general K S 0 t 0 Bruno Dupire T 1 T 2

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

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

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: ! Bruno Dupire If no jump

From Local Vols to implied Vols S: model with local vols , S 0 with 0 Where G 0=G of C under S 0, and f=RN density under S Implied vol solution of average of 2 weighted by S 2 G 0 f (equivalent to density of Brownian Bridge from (S, t 0) to (K, T)) 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? 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 Bruno Dupire

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

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

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

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

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

Smile dynamics: Stoch Vol Model 1 Skew case (r<0) Local vols s K - 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 ? Bruno Dupire

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

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

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

Spot dependency 2 ways to generate skew in a stochastic vol model s s ST ST -Mostly equivalent: similar (St, 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. 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 Bruno Dupire