René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27
Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding Market Models for Credit Portfolios (Shoenbucher or SPA?) 2 Choosing Time Evolution for a NonStationary Markov Process 3 Let s do it for Equity Markets 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton Lecture Notes in Mathematical Finance. Lecture Notes in Math #1919 R.C. & S. Nadtochy, submitted
Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding Market Models for Credit Portfolios (Shoenbucher or SPA?) 2 Choosing Time Evolution for a NonStationary Markov Process 3 Let s do it for Equity Markets 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton Lecture Notes in Mathematical Finance. Lecture Notes in Math #1919 R.C. & S. Nadtochy, submitted
Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding Market Models for Credit Portfolios (Shoenbucher or SPA?) 2 Choosing Time Evolution for a NonStationary Markov Process 3 Let s do it for Equity Markets 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton Lecture Notes in Mathematical Finance. Lecture Notes in Math #1919 R.C. & S. Nadtochy, submitted
Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding Market Models for Credit Portfolios (Shoenbucher or SPA?) 2 Choosing Time Evolution for a NonStationary Markov Process 3 Let s do it for Equity Markets 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton Lecture Notes in Mathematical Finance. Lecture Notes in Math #1919 R.C. & S. Nadtochy, submitted
Equity Market Model {S t } t price process interest rate (discount factor β t 1) No dividend Classical Approach Specify dynamics for S t, e.g. GBM in Black Scholes case ds t = S t σ t dw t Compute prices of derivatives by expectation, e.g. C (T, K ) = E{(S T K ) + }
Actively/Liquidly Traded Instrument Main Assumptions At each time t we observe C t (T, K ) the market price at time t of European call options of strike K and maturity T > t. Market prices by expectation C t (T, K ) = E{(S T K ) + F t } for some measure (not necessarily unique) P Empirical Fact Many observed option price movements cannot be attributed to changes in S t Fundamental market data: Surface {C t (T, K )} T,K instead of S t
Remarks No arbitrage implies C (T, K ) increasing in T C (T, K ) non-increasing and convex in K lim K C (T, K ) = lim K C (T, K ) = S Realistic Set-Up We actually observe C (T i, K ij ) i = 1,, m, j = 1,, n i Davis-Hobson & references therein.
More Remarks Switch to notation τ = T t for time to maturity (Musiela) Call surface { C t (τ, K )} of prices C t (T, K ) parameterized by τ and K. C t (τ, K ) = E{(S t+τ K ) + F t } = E Pt {(S t+τ K ) + }. C t (τ, K ) = (x K ) + dµ t,t+τ (dx) Crucial Fact (Breeden-Litzenberger) For each τ >, the knowledge of all the prices C t (τ, K ) completely determines the marginal distribution µ t,t+τ of S t+τ w.r.t. P t.
Black-Scholes Formula Dynamics of the underlying asset Wiener process {W t} t, σ >. ds t = S tσdw t, S = s Price of a call option C t(τ, K ) = S tφ(d 1 ) K Φ(d 2 ) with d 1 = log Mt + τσ2 /2 σ, d 1 = log Mt τσ2 /2 τ σ τ M t = K /S t moneyness of the option Φ error function Φ(x) = 1 2π x e y2 /2 dy, x R.
Implied Volatility Classical Black-Scholes framework On any given day t fix maturity T (or time to maturity τ) strike K price is an increasing function of the parameter σ σ C (BS) t (τ, K ) one-to-one In general case, given an option price C quoted on the market, its implied volatiltiy is the unique number σ = Σ t (τ, K ) for which C t (τ, K ) = C. Used by ALL market participants as a currency for options the wrong number to put in the wrong formula to get the right price. (Black)
Implied Volatility Code-Book { C t (τ, K ); τ >, K > } {Σ t (τ, K ); τ >, K > } Static (t = ) No arbitrage conditions difficult to formulate (B. Dupire, Derman-Kani, P.Carr,...) Dynammic No arbitrage conditions difficult to check in a dynamic framework (Derman-Kani for tree models)
Search for another Option Code-Book ds t = S t σ t dw t, S = s If t > is fixed, for any τ 1 and τ 2 such that < τ 1 < τ 2, then for any convex function φ on [, ) we have (Jensen) Or φ(x)µ t,t+τ1 (dx) µ t,t+τ1 µ t,t+τ2 φ(x)µ t,t+τ2 (dx) {µ t,t+τ } τ> non-decreasing in the balayage order (Kellerer) Existence of a Markov martingale {Y τ } τ with marginal distributions {µ t,t+τ } τ>. NB{Y τ } τ contains more information than the mere marginal distributions {µ t,t+τ } τ>
Local Volatility Code-Book On Wiener space (in Brownian filtration) Martingale Property implies Markov Property implies Y τ = Y + τ Y s a(s) db s a(s, ω) = a t (s, Y s (ω)) At each time t, I choose surface {a t (τ, K )} τ>,k > as an alternative code-book for { C(τ, K )} τ>,k >. {a t (τ, K )} τ>,k > was introduced in a static framework (i.e. for t = ) simultaneously by Dupire and Derman and Kani called local volatility surface
PDE Code, I Assume with initial condition and µ t,t+τ has density g t (τ, x). dy τ = Y τ a t (τ, Y τ )d B τ, τ > Y = S t Breeden-Litzenberger argument (specific to the hockey-stick pay-off function) C t (τ, K ) = (x K ) + g t (τ, x)dx Differentiate both sides twice with respect to K 2 K 2 C t (τ, K ) = g t (τ, K ). (1)
PDE Code, II Tanaka s formula: (Y τ K ) + = (Y K ) + + τ 1 [K, ) (Y s )dy s + 1 2 τ δ K (Y s ) d[y, Y ] s and taking E t - expectations on both sides using the fact that Y is a martingale satisfying d[y, Y ] s = Y 2 s a t (s, Y s ) 2 ds, we get: C t (τ, K ) = (S t K ) + + 1 2 = (S t K ) + + 1 2 τ τ Take derivatives with respect to τ on both sides C(τ, K ) τ E t {δ K (Ys)Y 2 s a t (s, Y s ) 2 } ds K 2 a t (s, K ) 2 g t (s, K ) ds. = 1 2 K 2 a t (τ, K ) 2 g t (τ, K ).
PDE Code, III Equate both expressions of g t (τ, K ) a t (τ, K ) 2 = 2 τ C(τ, K ) K 2 2 KK C(τ, K ) Smooth Call Prices Local Volatilities
PDE Code IV From local volatility surface {a t (τ, K )} τ,k to call option prices { C t (τ, K )} τ,k solve PDE (Dupire s PDE) τ C(τ, K ) = 1 2 K 2 a 2 (τ, K ) 2 KK C(τ, K ), τ >, K > C(, K ) = (S t K ) + { C t (τ, K ); τ >, K > } {a t (τ, K ); τ >, K > } Why would this approach be better? NEED ONLY POSITIVITY for no arbitrage
Dupire Formula If ds t = S t σ t dw t for some Wiener process {W t } t and some adapted non-negaitve process {σ t } t, then a t (τ, K ) 2 = E t {σ 2 t+τ S t+τ = K }.
HJM Prescription Proposed by Derman-Kani in 1998, but NEVER developed! Compute a (τ, K ) from market call prices (Initial condition) Define a dynamic model by defining the dynamics of the local volatility surface da t (τ, K ) = α t (τ, K )dt + β t (τ, K )dw t
Consistency Question Under what conditions do the Call Prices computed from the dynamics of a t (τ, K ) come from a model of the form of the form ds t = S t σ t db 1 t with initial condition S = s the underlying instrument? Answer σ t = a t (, S t )
No-Arbitrage Condition Question Under what conditions on the dynamics of a t (τ, K ) are the call prices (local) martingales? Answer (α + β 2 2 ) 2 K 2 C + t a, 2 K 2 C t = T a 2 K 2 C Recall classical HJM drift condition T α(t, T ) = β(t, T ) β(t, s)ds = t d j=1 T β (j) (t, T ) β (j) (t, s)ds. t
Main Result Statement The dynamic model of the local volatility surface given by the system of equations dã t(τ, K ) = α t(τ, K )dt + β t(τ, K )dw t, t, (2) is consistent with a spot price model of the form ds t = S tσ tdb t for some Wiener process {B t} t, and does not allow for arbitrage if and only if a.s. for all t > : ã t(, S t) = σ t τ ã t(τ, K ) KK 2 C t(τ, K ) = (ã t(τ, K ) α t(τ, K ) + βt(τ, K ) 2 2 t quadratic covariation of two semi-martingales. (3) (4) ) KK 2 C t(τ, K ) + d dt ã (τ, K )2, KK 2 C (τ, K ) t
Practical Monte Carlo Implementation Start from a model for β t (τ, K ) (say a stochastic differential equation); Get S and C (τ, K ) from the market and compute 2 KK C, a and β from its model; Loop: for t =, t, 2 t, 1 Get α t(τ, K ) from the drift condition; 2 Use Euler to get a t+ t (τ, K ) from the dynamics of the local volatility; S t+ t from S t Dynamics; β t+ t from its own model;
Markovian Spot Models (β ) α t(τ, K ) = d ãt(τ, K ). dt Drift condition reads Hence τ ã t(τ, K ) = α t(τ, K ) τ ã t(τ, K ) = d dt ãt(τ, K ) which shows that for fixed K, ã t(τ, K ) is the solution of a transport equation whose solution is given by: ã t(τ, K ) = ã (τ + t, K ) and the consistency condition forces the special form of the spot volatility. Hence we proved: σ t = a (t, S t) The local volatility is a process of bounded variation for each τ and K fixed if and only if it is the deterministic shift of a constant shape and the underlying spot is a Markov process.
A First Parametric Family 2 a 2 i= (τ, x, Θ) = p iσ i e x 2 /(2τσ 2 i ) τσ 2 i /8 2 i= (p i/σ i )e x 2 /(2τσi 2 ) τσi 2 /8 for Θ = (σ, σ 1, σ 2, p 1, p 2 ) Mixture of Black-Scholes Call surfaces for 3 different volatilities Singularity when τ
Numerical Evidence of Singularity
A Second Parametric Family As in Brigo-Mercurio Still a mixture of Black-Scholes Call surfaces for 3 different volatilities Each volatility is time dependent t σ i (t) σ () = σ 1 () = σ 2 () a 2 (Θ, τ, x) = (1 (p 1 + p 2 )τ) σe d 2 (σ)/2 + p 1 τσ 1 e d2 (σ 1 )/2 + p 2 τσ 2 e d2 (σ 2 )/2 (1 (p 1 + p 2 )τ) 1 σ e d2 (σ)/2 + p 1 τ 1 e σ d2 (σ 1 )/2 + p 2 τ 1 e 1 σ d2 (σ 2 )/2 2 where d(σ) = s x + ( r + 1 2 σ2) τ σ τ Θ = (p 1, p 2, σ, σ 1, σ 2, s, r)
Fit to Real Data S&P5 April 3, 26.7.6.5 LVol.4.3.2 3 2 T 1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 log K
Stochastic Volatility Models with where Usually Special cases: dσ 2 t ds t = σ t S t dw t = b(σt 2 )dt + a(σt 2 )d W t d W, W t = ρdt. b(σ 2 ) = κ(σ 2 σ 2 ) a(σ 2 ) = γ, (Hull-White) a(σ 2 ) = γ σ 2 (Heston)
Local Volatility of SV Models a 2 (τ, K ) = { 2 τ C K 2 KK 2 C = E σ2 1 ρ2 { E S σ2 T σ T e d 2 1 2 where σ T = σ T 1 T, and σ σ T = T σ2 s ds ( ρσ S = s exp ˆσ ( στ 1) 1 2 σ2 ρ 2 σ ) τ 2 τ } } S σ T e d2 12 and d 1 = log(s ) log(k ) + ρσ ˆσ ( στ 1) + ( 1 2 ρ2 )σ 2 σ τ 2 τ 1 ρ2 σ σ τ τ
First Example: ρ =.5 Rho=.5.3.25 LVol.2.15.1.5 5 2 3 1 log K 5 T
Second Example: ρ =.1 rho=.1.4.3.2.1 5 1 2 3 5
Third Example: ρ =.75 Rho=.1, N=1,.4.35.3.25 LVol.2.15.1.5 5 log K 5 1 T 2 3
Comparing SV Models Rho=.75, N=1, Rho=.75, N=1,.2.2.15.15 LVol.1.5 5 log K 5 3 2.5 2 1.5 1.5 T LVol.1.5 5 log K/S 5 1 2 T 3