Inverse Problems and Stochastic Volatility Models Jorge P . Zubelli IMPA, Rio de Janeiro, Brazil Thanks to A. Leit˜ ao and W. Muniz May 1, 2014 Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 1 / 69
Outline Intro and Background 1 Problem Statement and Results on Local Vol Models 2 Main Technical Results 3 Numerical Examples w/ Synthetic and w/ Real Data 4 Connections with Exponential Families and Risk Measures 5 Conclusions 6 Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 2 / 69
Historical Remarks Options and Derivatives 600BC Greek Philosopher and Geometer Thales from Miletus: Options and Futures of Olives (for Olive Oil) Call options on olive presses nine months ahead of the next harvest XVI century Dutch fisherman made forward contracts before departing XVII Netherlands - Options on prices of tulips XVII century future and option contracts on rice were negociated in Amsterdan and Osaka on rice. XIX century USA Contracts of to arrive type on flour started to be traded (CBOT) in 1849-1850, standard future contracts on grains were introduced in 1865. Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 3 / 69
Historical Remarks 1972, CME started to negociate the first future contracts on currencies (6 currencies) Since the mid 80’s the growth of derivative markets overcame the growth of the underlyings Theoretical Developments: 70’s F. Black - M. Scholes - R. Merton Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 4 / 69
Derivative Contracts European Call Option : a forward contract that gives the holder the right, but not the obligation, to buy one unit of an underlying asset for an agreed strike price K on the maturity date T . Its payoff is given by � X T − K if X T > K , h ( X T ) = if X T ≤ K . 0 Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 5 / 69
Derivative Contracts European Put Option : a forward contract that gives the holder the right to sell a unit of the asset for a strike price K at the maturity date T . Its payoff is � K − X T if X T < K , h ( X T ) = 0 if X T ≥ K . At other times, the contract has a value known as the derivative price . The option price at time t with stock price X t = x is denoted by P ( t , x ) . Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 6 / 69
Main Contributions Historical Remarks Thales (Miletus) L. Bachelier (Paris) P . Samuelson (Nobel 1970) F. Black M. Scholes (Nobel 1997) R. Merton (Nobel 1997) Figure: Thales Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 7 / 69
Main Contributions Thales (Miletus) L. Bachelier (Paris) P . Samuelson F. Black M. Scholes R. Merton Figure: L. Bachelier Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 8 / 69
Thales (Miletus) L. Bachelier (Paris) P . Samuelson F. Black M. Scholes R. Merton Figure: P . Samuelson Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 9 / 69
Some Background, Terminology, and Notation Classical Black-Scholes-Merton Under highly simplifying assumptions, the call option price C on an underlying X is given by C BS ( X , t ; K , T , r , σ ) = XN ( d + ) − Ke − r ( T − t ) N ( d − ) (1) where N is the cumulative normal distribution. ± σ √ d ± = log ( Xe r ( T − t ) / K ) T − t σ √ . (2) T − t 2 Some of the Assumptions: Non dividend paying (just for simplicity) Complete and Frictionless Markets Exponential Brownian motion dynamics Constant Volatility Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 10 / 69
Plot of the Black-Scholes Price of a Call Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 11 / 69
However... Volatility is not deterministic! It is even a multi-scale phenomena! It is not true that the underlying undergoes an Exponential Brownian Motion Even more so in high frequency contexts... Implied Volatility: The value of the volatility that should be used in the Black-Scholes formula to give the quote market price of a derivative. Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 12 / 69
The Concept of Implied Volatility Recall C BS ( X , t ; K , T , r , σ 0 ) = XN ( d + ) − Ke − r ( T − t ) N ( d − ) (3) where N is the cumulative normal distribution function and √ d ± = log ( Xe r ( T − t ) / K ) ± σ 0 T − t √ . (4) σ 0 T − t 2 Notion of Implied Volatility: Fix everything else and consider → C BS ( X , t ; K , T , r , σ ) σ �− The implied volatilty is the inverse to this map. IMPLIED VOL: ”wrong number that when plugged into the wrong equation gives the right price” Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 13 / 69
Figure: Implied Volatility Surface- (From Bruno Dupire - IMPA talk) Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 14 / 69
Remarks Implied Vol is only a way of renaming the price (change of variables). 1 Crucial to find the connection with the underlying dynamics. 2 Crucial to introduce the concept of LOCAL VOLATILITY 3 Concept of Stochastic Volatility: dX t = µ t dt + σ t dW t X t How to model the process σ t ? Fundamental Issue: One Possibility: Following Dupire... (Local Volatility Model) assume that σ t = σ ( t , X t ) Assume that the process σ t = f ( Y t ) undergoes its own Another Possibility: dynamics according to a SDE. Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 15 / 69
Figure: Example of Data from IBOVESPA Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 16 / 69
Limitations of Classical Black-Scholes log-normality of asset prices is not verified by statistical tests option prices are subjet to the smile effects volatility tends to fluctuate with time Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 17 / 69
Local Volatility Models Idea: Assume that the volatility is given by σ = σ ( t , x ) i.e.: it depends on time and the asset price. Easy to check that the Black-Scholes eq. holds. � � 2 σ ( t , x ) 2 x 2 ∂ 2 P ∂ P x ∂ P ∂ t + 1 ∂ x 2 + r ∂ x − P = 0 (5) P ( T , x ) = h ( x ) (6) or in the case you have dividends: 2 σ ( t , x ) 2 x 2 ∂ 2 P ∂ P ∂ x 2 +( r − d ) x ∂ P ∂ t + 1 ∂ x − rP = 0 P ( T , x ) = h ( x ) Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 18 / 69
Motivation and Goals Focus Dupire [Dup94] local volatility models Goal: Present a unified framework for the calibration of local volatility models Use recent tools of convex regularization of ill-posed Inverse Problems. Present convergence results that include convergence rates w.r.t. noise level in fairly general contexts Go beyond the classical quadratic regularization. Applications risk management hedging evaluation of exotic derivatives. Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 19 / 69
The Direct and the Inverse Problem The Direct Problem Given σ = σ ( t , x ) and the payoff information, determine P = P ( t , x , T , K ; σ ) The Inverse Problem Given a set of observed prices { P = P ( t , x , T , K ; σ ) } ( T , K ) ∈ S find the volatility σ = σ ( t , x ) . The set S is taken typically as [ T 1 , T 2 ] × [ K 1 , K 2 ] . In Practice: Very limited and scarce data Note: To price in a consistent way the so-called exotic derivatives one has to know σ and not only the transition probabilities Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 20 / 69
The Smile Curve and Dupire’s Equation Assuming that there exists a local volatility function σ = σ ( x , t ) for which (5) holds Dupire(1994) showed that the call price satisfies � ∂ T C − 1 2 σ 2 ( K , T ) K 2 ∂ 2 K C + rS ∂ K C = 0 , S > 0 , t < T (7) C ( K , T = 0 ) = ( X − K ) + , Theoretical: way of evaluating the local volatility � � ∂ T C + rK ∂ K C � σ ( K , T ) = 2 (8) K 2 ∂ 2 K C In practice To estimate σ from (7), limited amount of discrete data and thus interpolate. Numerical instabilities! Even to keep the argument positive is hard. Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 21 / 69
Literature Very vast!!! Abken et al. (1996) Avellaneda et al. Ait Sahalia, Y & Lo, A (1998) [ABF + 00, Ave98c, Ave98b, Berestycki et al. (2000) Ave98a, AFHS97] Buchen & Kelly (1996) Bouchev & Isakov [BI97] Coleman et al. (1999) Crepey [Cr´ e03] Cont, Cont & Da Fonseca (2001) Derman et al. [DKZ96] Jackson et al. (1999) Egger & Engl [EE05] Jackwerth & Rubinstein (1998) Hofmann et al. [HKPS07, HK05] Jourdain & Nguyen (2001) Jermakyan [BJ99] Lagnado & Osher (1997) Achdou & Pironneau (2004) Samperi (2001) Roger Lee (2001,2005) Stutzer (1997) Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 22 / 69
Problem Statement Starting Point: Dupire forward equation [Dup94] − ∂ T U + 1 2 σ 2 ( T , K ) K 2 ∂ 2 K U − ( r − q ) K ∂ K U − qU = 0 , T > 0 , (9) K = X 0 e y , τ = T − t , b = q − r , u ( τ , y ) = e q τ U t , X ( T , K ) (10) and a ( τ , y ) = 1 2 σ 2 ( T − τ ; X 0 e y ) , (11) Set q = r = 0 for simplicity to get: u τ = a ( τ , y )( ∂ 2 y u − ∂ y u ) (12) and initial condition u ( 0 , y ) = X 0 ( 1 − e y ) + (13) Stochastic Volatility Models � J.P c .Zubelli (IMPA) May 1, 2014 23 / 69
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