Market models for the smile Local volatility, local-stochastic volatility

Transcription

1 Market models for the smile Local volatility, local-stochastic volatility Lorenzo Bergomi Global Markets Quantitative Research European Summer School in Financial Mathematics Le Mans, September / 48

2 Outline Usable models? The local volatility model The carry P&L of the LV model The delta the delta of a vanilla option Break-even levels for vols of implied vols / covariance of spot and implied volatilities SSR and volatilities of volatilities in the local volatility model Local-stochastic volatility models A criterion for admissibility Examples Conclusion 2 / 48

3 Intro a practically usable model? Imagine we have traded an option of maturity T on an asset S, whose payoff is f (S T ). The pricing library supplies a pricing function P(t, S). We have no idea of what s been implemented. How do we assess whether it s OK to use P(t, S)? Sanity check 1 Set t = T ; check that P(t = T, S) = f (S), S. If OK, then sanity check 2 Compute delta: = dp. P&L of a short delta-hegded position during [t, t + δt] is: ( ) ( ) P&L = P(t + δt, S + δs) (1 + rδt)p(t, S) + δs (r q)sδt Expand at order 2 in δs, 1 in δt: ( P&L = rp + dp dt + (r q)s dp ) δt 1 2 d 2 P 2 δs2 3 / 48

4 Intro a practically usable model? 2 P&L during δt is: P&L = A(t, S)δt B(t, S)δS 2 if A(t, S) 0, B(t, S) 0 Always loosing money: no good. if A(t, S) 0, B(t, S) 0 Always making money: no good either. OK to use P(t, S) only if signs of A and B different, S, t. P&L = Reasonable ansatz, if S is an equity: of A and B: rp + dp dt BS 2 ( ( δs S + (r q)s dp ) ) 2 + A BS 2 δt A BS 2 = cst = σ 2. Using expressions = σ S2 d2 P 2 This is in fact the BS equation. Carry P&L acquires simple form: P&L = 1 2 S2 d2 P 2 ( ( ) δs 2 σ δt) 2 S 4 / 48

5 Intro a practically usable model? 3 Simple form of P&L simple break-even criterion. Only reason why BS equation used in banks. No assumption that equities are lognormal they are not. No assumption that volatility is constant it is not. Not even the assumption of a process for S. Criterion for breakeven of P&L at order 2 in δs P solves parabolic equation probabilistic interpretation & P interpreted as an expectation. What if there are multiple hedge instruments? Carry P&L reads: P&L = 1 ( ) 2 S d 2 P δsi δs j is j C ij δt i j S i S j Criterion for P&L to be nonsensical: C must be positive matrix. There is exist breakeven covariance levels S, t that are payoff-independent. Important thing: only involves hedge instruments not model s state variables. S i underliers or 1 underlying & associated vanilla options. 5 / 48

6 What s left to do? Once option is delta-hedged, we are left with gamma/theta P&L. Total P&L incurred on [0, T ]: P&L T = Σ i e r(t t i ) Si 2 d 2 P 2 (ri 2 σ 2 δt), r i = δs i ti,s i S i Is this P&L sizeable? If S follows a lognormal process with volatility σ and δt 0, then P&L = 0. Returns of real undelyings (a) do not have exhibit volatility, (b) have non-gaussian conditional distributions. Set: Then: P&L T = Σ i e r(t t i ) S 2 i r i = σ i Z i, E[Z 2 i ] = 1 d 2 P 2 (σi 2 Z i 2 σ 2 δt) ti,s i Z i non-gaussian impacts short-maturity options. σ i random AND correlated impacts longer-maturity options. 6 / 48

7 What s left to do? 2 Use typical parameters. Stdev(P&L) as fraction of price for an ATM option, as a function of maturity: Real case without kurtosis term Lognormal case For 1y maturity: Black-Scholes: 5%, while 30% in the real case. Delta hedging better than nothing but remaining gamma/theta still too large. Gamma needs to be cancelled as well options are hedged with options. 7 / 48

8 What s left to do? conclusion P becomes a function of t, S and other derivative prices For example vanilla options: P(t, S, O KT ). This is called calibration. Admissible models are such that the P&L of a delta/vega-hedged option reads: P&L = 1 2 S i S j d 2 P i j ( δsi S i δs j S j ) C ij δt with C positive (implied) break-even covariance matrix of hedge instruments S i. C is payoff-independent. Ideally we would like to be able to choose the C ij. We call market models models satisfying this condition. Usually not able to write down SDEs for hedge instruments directly, so condition needs to be checked a posteriori. 2 examples: Local volatility Local-stochastic volatility 8 / 48

9 The local volatility model 9 / 48

10 Local volatility intro: things heard on the street LV model used inconsistently: local vol surface is calibrated today; only to be recalibrated tomorrow. violates model s assumption of fixed LV surface. Trading practice: don t use LV delta instead compute sticky-strike delta: move S, keep implied vols unchanged, recalibrate local vol surface. Rationale: so that vanilla options have BS delta. On a scale from dirty to downright ugly, where do we stand? What is the carry P&L of an option position? By the way, what s the delta of a vanilla option? 10 / 48

11 Local volatility 1 Local volatility: simplest model that is able to take as inputs vanilla option prices. Provided: no time arbitrage: if zero int. rate: no strike arbitrage: d 2 C KT dk 2 0 dc KT dt 0 T 1 T 2 T 1 σ 2 KT 1 T 2 σ 2 KT 2 there exists a (single) local volatility function σ(t, S), given by the Dupire formula: such that, by using: σ (t, S) 2 = 2 t vanilla option prices are recovered. dc dt + qc + (r q) K dc dk K 2 d 2 C dk 2 = (r q)s t dt + σ(t, S t )S t dw t K=S T =t Pricing function of LV model reads: P(t, S, O KT ) or P(t, S, σ KT ) no parameter beside time & values of hedge instruments. Model assumes fixed σ(t, S) while, in practice, local volatility function is recalibrated every day. Does this make any sense? What are the deltas (vegas)? 11 / 48

12 Local volatility 2 Pricing equation of the local volatility model reads: dp LV dt + (r q)s dplv σ2 (t, S)S 2 d2 P LV 2 = rp LV Just like BS equation except σ(t, S) instead of cst volatility σ. Solution of PDE is P LV (t, S, σ) In LV model all instruments have 1-d Markov representation as a function of t, S: σ KT (t, S) Σ LV KT (t, S,σ) Imagine trading the LV delta: LV = dplv σ P&L during δt of delta-hedged option is: P&L LV = 1 2 S2 d2 P LV 2 ( ( ) δs 2 σ 2 (t, S)δt) S P&L LV actual P&L only if market implied vols move as prescribed by Σ LV KT (t, S,σ). LV useless 12 / 48

13 Local volatility carry P&L Let s compute the carry P&L in the LV model. Use (black-box) pricing function P(t, S, σ KT ) given by: P (t, S, σ KT ) P LV (t, S, σ [t, S, σ KT ]) P LV (t, S, σ) = P ( t, S, Σ LV KT (t, S, σ) ) Start with P&L of naked option position: P&L = [ ] P(t + δt, S + δs, σ KT + δ σ KT ) (1 + rδt)p(t, S, σ KT ) Expand at order 1 in δt, 2 in δs and δ σ KT : P&L = rpδt dp dt ( 1 2 δt dp δs d 2 P 2 δs2 + d2 P dp d σ KT δ σ KT d σ KT δ σ KT δs d 2 ) P δ σ KT δ σ K d σ KT d σ T K T Notation stands for: df d σ KT δ σ KT dkdt δf df δ σ KT Σ ij δ σ KT d σ Ki T j δ σ Ki T j 13 / 48

14 Local volatility carry P&L 2 dp, dp dt are computed keeping the σ KT fixed the LV function is not fixed. Define sticky-strike delta SS : SS = dp P is not solution of the LV pricing PDE P LV is: σkt P LV (t, S, σ) = P ( t, S, σ KT = Σ LV KT (t, S, σ) ) Express derivatives of P LV in terms of derivatives of P: dp LV dt = dp dt + dp dσlv d σ KT dt KT d 2 P LV 2 = ( d 2 P d 2 P d σ KT dp LV dσlv KT = dp + + dp dσlv d σ KT d 2 P d σ KT d σ K T KT dσlv KT dσ LV K T ) KT + dp d 2 Σ LV d σ KT 2 Now insert in LV pricing equation: dp LV dt + (r q)s dplv σ2 (t, S) S 2 d2 P LV... to generate relationship involving derivatives of P. 2 = rp LV 14 / 48

15 Local volatility carry P&L 3 dp = rp (r q)s dp dt dp µ KT d σ KT ( 1 2 σ2 (t, S) S 2 d 2 P d2 P dσlv KT d σ KT + d 2 P d σ KT d σ K T dσlv KT dσ LV K T ) with µ KT given by: µ KT = dσlv KT dt σ2 (t, S) S 2 d2 Σ LV KT 2 + (r q)s dσlv KT Now use this expression of dp dt to rewrite P&L of naked option position: P&L = dp (δs (r q)sδt) dp d σ KT (δ σ KT µ KT δt) ( σ2 (t, S) S 2 d 2 P d2 P dσlv KT d σ KT ( 1 2 d 2 P 2 δs2 + d2 P δ σ KT δs + 1 d σ KT 2 + d 2 P d σ KT d σ K T dσlv KT d 2 ) P δ σ KT δ σ K d σ KT d σ T K T dσ LV K T ) δt 15 / 48

16 Local volatility carry P&L 4 Introduce implied (log-normal) vol of vol of σ KT : Rewrite P&L as: ν KT = 1 Σ LV KT dσ LV KT Sσ(t, S) P&L = dp 1 2 S2 d2 P (δs (r q)sδt) dp d σ KT (δ σ KT µ KT δt) [ δs 2 S 2 d2 P d σ KT S σ KT ] σ2 (t, S) δt [ δs S δ σ KT σ KT d 2 P d σ KT d σ K T σ KT σ K T ] σ(t, S) ν KT δt [ δ σkt δ σ K T σ KT σ K T ] ν KT ν K T δt Only uses market observables: P(t, S, σ KT ) no LV function involved. P&L expression is that of market model. Variance/covariance breakeven levels are well-defined, payoff-independent, and make up a positive covariance matrix. Delta is sticky-strike delta dp, vegas simple vegas. 16 / 48

17 Local volatility carry P&L 5 σ KT implied vol plays no special role. Use instead price O KT : P(t, S, O KT ). P(t, S, σ KT ) = P ( t, S, O KT = P BS KT (t, S, σ KT ) ) P LV (t, S, σ) = P ( t, S, Ω LV KT (t, S, σ)) Ω LV KT (t, S, σ) price in LV model with LV function σ. Everything same as before, except σ KT O KT, Σ LV KT Ω LV KT. Drift µ KT simplifies: µ KT = dωlv KT dt σ2 (t, S) S 2 d 2 Ω LV KT 2 + (r q)s dωlv KT = rω LV KT = ro KT OK P&L of naked option position using only asset prices no LV function involved: P&L = dp (δs (r q)sδt) dp do KT (δo KT ro KT δt) 1 d 2 P [ δs σ 2 (t, S) S 2 δt ] [ ] d2 P δsδo KT σ 2 (t, S) S 2 dωlv KT do KT δt [ 1 d 2 P δo KT δo K 2 do KT do T σ2 (t, S) S 2 dωlv KT K T dω LV K T δt ] 17 / 48

18 Local volatility carry P&L 6 Expression of carry P&L inclusive of recalibration of local volatility function has typical form of market models. Hedge instruments all treated on equal footing. Implied break-even levels of cross-gammas are payoff-independent are determined by market smile prevailing at time t. spot/vol correl = 100% vol/vol correl = 100% vol of σ KT is ν KT = 1 Σ LV KT Hedge ratios simply dp dσ LV KT Sσ(t, S) OKT and dp do KT S Delta of the local volatility model is market model delta: MM = dp Delta of vanilla option irrelevant notion. OKT akin to asking model to generate a hedge ratio of one hedging instrument on another hedging instrument. Result seems natural; looks like any P that s the solution of a parabolic PDE will do the job but see pathologies in local/stoch vol models. 18 / 48

19 Consistency of sticky-strike and market-model deltas Use S, O KT P(t, S, O KT ). Hedge ratios MM = dp Use S, σ KT P (t, S, σ KT ). Hedge ratios SS = dp dp d σ KT offset by trading BS-delta-hedged vanilla options OKT, σkt, dp S do KT dp S d σ KT Hedge portfolio is: Π = dp S + Rewrite in terms of delta-hedged vanillas: Π = Spot hedge ratio? [ dp + dp dpbs KT do KT ] S + dp do KT O KT dp do KT [O KT dpbs KT Move spot + move vanilla prices by their Black-Scholes deltas akin to: move vanilla prices keeping implied vols fixed sticky strike delta SS = dp + dp dpbs KT do KT Once hedge portfolio broken down into underlying + naked vanilla options, delta always equal to MM = dp OKT. Nothing fundamental about SS tied to a particular representation of vanilla option prices. 19 / 48 ] S

20 So, what is the LV model? The LV model is a usable model. It is a market model for the underlying and vanilla options... that happens to have a 1-d Markov representation in terms of (t, S). This is a mathematical technicality of which the LV function is a by-product that facilitates pricing. Nothing fundamental. Daily recalibration of LV function is exactly how it has to be used. Consequences of 1-d Markov representation: The break-even covariance matrix is of rank 1 correls = 100%. No control on break-even levels of volatilities of implied volatilities. They are set by the configuration of S, σ KT and will vary unpredictably. Like them, use model don t like them, don t use model. LV model completely specified by feeding in the values of the hedge instruments no parameters whatsoever. This is how much we can get in a model with a 1-d Markov representation. 20 / 48

21 Using the LV model What s left before we can use LV model? Output the ν KT, see if we like them. More practical to look at implied vols for floating strike fixed moneyness. Look at vols of vols and spot/vol covariances. For ATMF vol σ FT T equivalently look at SSR R T R T = 1 S T S T = d σ KT d ln K d σft T dlns (dlns) 2 FT = 1 d σ FT T S T dlns Vol of vol: Thus: d σ FT T σ FT T = 1 σ FT T d σ FT T dlns dlns t = d σ F T T dlns vol( σ FT T ) = R T S T ( σf0 0 σ FT T ) σ(t, S) dw t σ FT T Assume following expression for LV function: σ(t, S) = σ(t) + α(t)x + β(t) ( S ) 2 x2, x = ln F t and calculate S T, R T at order 1 in α (t), β (t). 21 / 48

22 Expansion of Implied volatilities Consider an LV model Model 1: LV function σ 1 (t, S), pricing function P 1 (t, S). dp 1 dt + (r q)s dp σ2 1 (t, S) S2 d2 P 1 2 = rp 1 Now consider arbitrary diffusive model Model 2: instantaneous volatility σ 2t. t = (r q) S t dt + σ 2t S t dw t Consider process Q t defined by: Q t = e rt P 1 (t, S t ) At t = 0, Q t=0 = P 1 (0, S 0 ). At t = T, Q t=t = e rt P 1 (T, S 2T ) = e rt f (S 2T ), that is the final payoff. dq t = e rt [ ( rp 1 + dp 1 dt = e rt [ ( rp 1 + dp 1 dt ) dt + dp 1 t ) dt + dp 1 t S2 t = e rt [ dp1 ( t (r q)s t dt) S2 t d 2 ] P t d 2 ] P 1 2 σ2 2t dt d 2 P 1 2 (σ2 2t σ2 1 (t, S t))dt ] E 2 [dq t t, S t ] = e rt S2 t 2 d 2 P 1 2 ( σ 2 2t σ 1 (t, S t ) 2) dt 22 / 48

23 Expansion of Implied volatilities 2 E 2 [Q T ] = Q 0 + T 0 E 2 [dq t ] = P 1 (0, S 0 ) + E 2 [ T 0 e rt S2 t 2 d 2 P 1 2 ( σ 2 2t σ 1 (t, S t ) 2) ] dt So, price in Model 2 given by: P 2 (0, S 0, ) = P 1 (0, S 0 ) + E 2 [ T 0 e rt S2 t 2 d 2 P 1 2 ( σ 2 2t σ 1 (t, S t ) 2) ] dt where other state variables of Model 2. Price(Model 2) = Price(Model 1) + gamma/theta P&L, incurred by hedging payoff using Model 1 with dynamics of S t generated by Model 2. Efficient numerical algorithm for generating vanilla smiles of stochastic volatility models see book. Imagine Model 1 is BS model with implied vol = σ KT P 2 (0, S 0, ) = P σkt (0, S 0 ) 0 = E 2 [ T 0 e rt S2 t 2 d 2 P σkt 2 ] ( σ 2 2t σ KT 2 ) dt 23 / 48

24 Expansion of Implied volatilities 3 Thus: σ 2 KT = E 2 [ T 0 e rt S 2 t E 2 [ T 0 e rt S 2 t d 2 P σkt 2 σ 2 2t dt ] d 2 P σkt 2 ] dt Work with variances u = σ 2. Set u 0 = σ 2 0 and σ 2 2t = u 0 + δu(t, S). Expand at order 1 in δu: σ 2 KT = σ δ( σ 2 KT ). σ 2 KT = σ2 0 + δ( σ2 KT ) = E u 0 +δu[ (u 0 + δu)] E u0 +δu[ ] = u 0 + E u 0 +δu[ δu] E u0 +δu[ ] Thus: σ 2 KT = E σ0 = u 0 + E u 0 [ δu] E u0 [ ] = E u 0 [ (u 0 + δu)] E u0 [ ] [ T 0 e rt u (t, S) S 2 d2 P σ0 E σ0 [ T 0 e rt S 2 d2 P σ0 2 2 ] dt ] dt Density and gamma available in closed form in BS model. 24 / 48

25 Dynamics in LV model 2 Calculation can be done with deterministic u 0 (t) = σ 2 0(t). At order 1 in δu: σ 2 KT = 1 T T 0 dt + y dy e 2 2 2π u ( t, F t e ωt ω T x K + (ωt ωt )ωt ) ωt y where F t forward for maturity t, x K = ln( K F T ) and ω t = t 0 σ2 0(τ)dτ. Expanding around a cst σ(t) = σ 0 : u 0 = σ 2 0 σ KT = 1 T T 0 dt + y dy e 2 2 2π σ ( ) t, F t e T t x (T t)t K + σ 0 y T Implied vol average of local vol around straight line in ln S from S to K. S T = d σ KT d ln K Cst α, β: d 2 σ KT d ln K 2 d σ KT d ln S d σ KT d ln K = 1 K=FT T = 1 K=FT T = 1 K=FT T T 0 T 0 T 0 t T α(t)dt ( t T ( 1 t T ) 2β(t)dt ) α(t)dt skew averaging see also V. Piterbarg K=FT = α 2, d 2 σ KT d ln K 2 K=FT = β 3 25 / 48

26 Dynamics in LV model 3 From 1st equation: α (t) = d dt (ts t) + S t. d σ FT (S)T dlns = ( d σkt d ln K + d σ KT K=FT d ln S ) K=FT = 1 T T 0 α(t)dt = S T + 1 T T 0 S t dt Get expression of SSR: R T = 1 d σ FT T dlns S T (dlns) 2 = 1 S T d σ FT (S)T dlns : R T = T T 0 S t S T dt For typical equity smiles, S t decreases with t R T 2. Limiting behavior Short maturities: lim R T = 2 T 0 Lognormal vol of short ATMF vol = twice the skew. Long maturities take S T 1 T γ : lim R T = 2 γ T 1 γ For typical value γ = 1 2, lim T R T = / 48

27 Dynamics in LV model 4 Check approx of SSR on 2 smiles of Eurostoxx50 50% 50% 40% 40% 30% 30% 20% Initial smile 10% Spot = 90% Spot = 110% 0% 50% 70% 90% 110% 130% 150% 20% Initial smile 10% Spot = 90% Spot = 110% 0% 50% 70% 90% 110% 130% 150% Actual Benchmark Actual Benchmark Figure: Top: smiles of the Eurostoxx50 index for a maturity 1 year observed on October 4, 2010 (left) and May 16, 2013 (right). Bottom: term structures of ATMF skew and power-law fits with γ = 0.37 (left), γ = 0.52 (right), as a function of T (years). 27 / 48

28 Dynamics in LV model 5 Real versus approximate SSR Actual Approx Actual Approx Figure: R T as a function of T (years) computed: (a) in FD (actual), (b) using expression R T = T S t T 0 dt (approx). S T What about smile with S T 1 T? Approx fomula gives lim T R T = (logarithmic divergence of R T ): Approx slightly overestimates SSR Actual Benchmark 2.0 Actual Approx / 48

29 Conclusion LV model is a genuine market model for underlying + vanilla options The only diffusive market model that possesses a 1-d Markov representation in terms of (t,s) Generates well-defined break-even levels for spot/vol and vol/vol covariances in the carry P&L. Daily recalibration of LV function an ancillary object is exactly how model should be used and deltas calculated. Spot/vol break-even correlations = 100%, vol/vol break-even correlations = 100%. Volatilities of implied volatilities given by: vol( σ KT ) = 1 Σ LV KT Delta is well-defined: MM = dp dσ LV KT Sσ(t, S). OKT. Delta of vanilla option irrelevant notion. When vega-hedging with (BS) delta-hedged vanilla options, sticky-strike delta should be used. Good approximate formulae for sizing up break-even vols of ATMF vols or equivalently SSR: T S t R T = dt T 0 S T ( σf0 ) 0 vol( σ FT T ) = R T S T σ FT T 29 / 48

30 Local-stochastic volatility models and non-models 30 / 48

31 Motivation In LV model, nothing to enter beside values of hedge instruments zero parameter. Break-even covariances are set by prevailing smile. If smile is flat, implied vols of vols = 0. Can we regain some leverage on the model-implied dynamics of hedge instruments? Poor man s fix: Pick your favourite stochastic volatility model. Decorate SV instantaneous volatility with local volatility component. Is it a (usable) model? Provided answer is positive What is the delta? What are the vegas? What kind of model is it? 31 / 48

32 SV models Which SV model should we use? Unlike LV model, SV models have parameters that we can use to drive the dynamics of the σ KT. First generation of SV models: based on instantaneous variance V t, e.g. the Heston model: { t = (r q)s t dt + V t S t dw t dv t = k(v t V 0 )dt + ν V t dz t Pbm: V t not an asset no way to generate P&L (V t2 V t1 ) dynamics of σ KT needs to be checked a posteriori. Better to model dynamics of hedge instruments directly, for example forward variances ξt T : ξ T t = E t [ ( T S T ) 2 ] = E t [V T ] Can be bought/sold by trading variance swaps (VS) at no cost. VS volatility for maturity T at time t, σ T (t) given by: σ 2 T (t) = 1 T t T t ξ T t dt ξ T t is driftless: dξ T t = dw T t 32 / 48

33 Forward variance models Need to specify a dynamics for the curve ξ T t such that: Low-dimensional Markov representation Able to generate flexible patterns for volatilities of VS volatilities σ T. Typically: vol( σ T ) 1, α [0.3, 0.6] T α In practice using two Brownian motions with exponential weightings is sufficient: dξt T ξt T = (2ν)N [ ] (1 θ)e k 1(T t) dwt 1 + θe k 2(T t) dwt 2 with ν: volatility of a volatility with vanishing maturity and N normalization factor. ξ T t = f T (t, X 1 t, X 2 t ) with Xt 1, Xt 2 two OU processes easily simulated exactly. Process for S t is: t = (r q)s t dt + ξ t t S tdw S t Also able to generate decay of ATMF skew S t 1 T γ with γ typically 1 2. see papers Smile Dynamics II, III, IV. 33 / 48

34 Models used as examples in presentation Mixed Heston model { t = (r q)stdt + σ(t, St) Vt StdWt dv t = k(v t V 0 )dt + ν V t dz t Mixed two-factor model t = (r q)s t dt + σ(t, S t ) ζt t S tdwt S dζt T ζt T where α θ = 1/ = 2ν N [ ] ((1 θ) e k 1(T t) dwt 1 + θe k 2(T t) dwt 2 (1 θ) 2 + θ 2 + 2ρθ (1 θ); ν vol of short vol. LV component σ(t, S) calibrated on vanilla smile. Pricing function in mixed model: in Heston model P M (t, S, σ, V ), V number. in two-factor model P M (t, S, σ, ζ u ), ζ u curve. 34 / 48

35 Usage of mixed models Choose model parameters & initial values of state variables: In Heston: ( k, σ, ρ, V 0), V. In two-factor model: (k 1, k 2, θ, ν, ρ 12, ρ S1, ρ S2 ), ζ. Calibrate local volatility function σ(t, S) to market smile. In 1-factor model like Heston: solve fwd PDE for density. General technique: particle method of P. Henry-Labordère/J. Guyon (2009). Then Shift+F9 produces a (real) number. Is it a price? What about deltas? Typically, move spot, recalibrate local vol and reprice. Is it right delta? What kind of carry P&L does this materialize? Let s assume this is a model. Can we have an approximate way of sizing up: volatilities of implied vols covariances of spot and implied vols equivalently SSR? 35 / 48

36 Two pricing functionals P M (t, x): takes as inputs t, S, LV function + state variables λ of SV model: P M (t, S, σ(, ), λ) In Heston: λ = V number In two-factor model: λ = ζ u curve P(t, x) takes as inputs t, S, implied vols + state variables of SV model: P (t, S, σ KT, λ) Could include in x, x model parameters as well ( state variables with zero drift/vol). Will use P( ) rather than P M ( ) to do P&L accounting. Could use prices rather than implied vols. 36 / 48

37 Carry P&L P(t, S, σ KT, λ) In mixed model for a set LV function σ KT is a function of t, S, σ(, ) + state variables: x = x(t, x): P M (t, x) = P (t, x(t, x)) Implied vols given by: σ KT Σ M KT (t, S, σ, λ) P M, P related through: P M (t, S, σ, λ) = P ( t, S, Σ M KT (t, S, σ, λ), λ ) Pricing equation for P M with set LV function zero rates: dp M dt + ( Σ k µ k d + 1 dx k 2 Σ d 2 kl a kl dx k dx l ) P M = 0 37 / 48

38 Carry P&L 2 Switch to variables x: with: dp dt + ( Σ i µ i d + 1 d x i 2 Σ d 2 ij â ij d x i d x j µ i = d x i dt + Σ kµ k d x i dx k Σ kl a kl ) P = 0 d 2 x i dx k dx l â ij = Σ kl a kl d x i dx k d x j dx l µ i drift of x i and â ij covariance matrix of x i and x j as generated by mixed model with fixed LV function. d x i involve derivatives of functional Σ dx M k KT (t, S, σ, λ) with respect to t, S, λ. Now consider P&L of short option position unhedged for now zero rates: P&L = P(t + δt, x + δ x) + P (t, x) 38 / 48

39 Carry P&L 3 Expand at order two in δ x, one in δt. P&L = dp dt δt Σ i dp d x i δ x i 1 2 Σ ij d 2 P d x i d x j δ x i δ x j = Σ i dp d x i ( δ xi µ i δt ) 1 2 Σ ij d 2 P ( δ xi δ x j â ij δt ) d x i d x j Among components of x: O i market observables: S, σ KT. λ k = state variables of SV model Rewrite P&L: P&L = Σ i dp do i ( δoi µ i δt ) 1 2 Σ ij Σ k dp dλ k ( δλk µ k δt ) 1 2 Σ kl d 2 P dλ k dλ l ( δλk δλ l â kl δt ) Σ ik d 2 P ( δoi δo j â ij δt ) do i do j d 2 P ( δoi δλ k â ik δt ) do i dλ k 39 / 48

40 P&L hedged portfolio Portfolio: option + hedges that offset sensitivities dp do i : P H = P + Σ i α i f i (t, S, O i ) P&L equation also holds for hedge instruments canceling δo i term cancels µ i δt contribution. P&L of hedged position: P&L H = 1 2 Σ ij d 2 P H ( δoi δo j â ij δt ) do i do j Σ k dp H dλ k ( δλk µ k δt ) 1 2 Σ kl d 2 P H dλ k dλ l ( δλk δλ l â kl δt ) Σ ik d 2 P H ( δoi δλ k â ik δt ) do i dλ k 1st piece OK: thetas matching gammas on market instruments. â ij positive covariance matrix: â ij = Σ kl a kl d x i dx k d x j dx l 2nd / 3d pieces no good. P&L leakage from variation (or not) of SV state variables. By construction value of hedges indpdt on λ k : dp H = dp, dλ k dλ k d 2 P H = d 2 P, dλ k dλ l dλ k dλ l df i dλ k = 0, so: d 2 P H = d 2 P do i dλ k do i dλ k 40 / 48

41 P&L hedged portfolio 2 δλ k are not market values are in our control. For example, take δλ k = µ k δt. Still leaves us with 3d piece in P&L: P&L leak H = 1 2 Σ kl d 2 P dλ k dλ l ( δλk δλ l â kl δt ) Σ ik d 2 P ( δoi δλ k â ik δt ) do i dλ k Is there a solution to P&L leakage? YES need condition on P(t, O, λ): dp dλ k S, σkt = 0, k Pricing functional P(t, S, σ KT, λ) must have zero sensitivity to SV state variables. 41 / 48

42 Conclusion: admissible (or gauge-invariant) models Criterion for models that can be used in trading: P (t, S, σ KT, λ): dp dλ = 0 k S, σkt P&L H of delta-hedged/vega-hedged position then has typical form of market models: P&L H = 1 2 Σ ij d 2 P H ( δoi δo j â ij δt ) do i do j Break-even covariance levels are given by covariances in model with fixed LV d x function: â ij = Σ kl a i d x j kl dx k dx l. Pbm: condition dp dλ S, σkt = 0 usually not satisfied. Ex: not satisfied in local/stoch vol model built on Heston model: d dv P (t, S, σ KT, V ) 0 P&L leakage Not usable in trading. Do admissible models exist at all? YES. 42 / 48

43 Admissible models 2 Consider mixed two-factor model. Pricing function P(t, S, σ KT, ζ u ). Model equivalently written as: t = (r q)s t dt + ζ t f (t, X 1 t, X 2 t ) σ(t, S t) S t dw S t dxt 1 = k 1 Xt 1 dt + dwt 1 dxt 2 = k 2 Xt 2 2 dt + dwt with X 1 0 = 0, X 2 0 = 0 and: f (t, x 1, x 2 ) = e ( 2να 2να θ[(1 θ)x 1 +θx θ ) 2 2 ] 2 χ(t) χ (t) = (1 θ) 2 1 e 2k 1 t 2k 1 + θ 2 1 e 2k2t 2k 2 + 2ρθ (1 θ) 1 e (k1+k2)t k 1 + k 2 Pick arbitrary ϕ u, do following transformation: ζ u ϕ u ζ u 1 σ(u, S) σ(u, S) ϕu SDEs for S t, Xt 1, Xt 2 unchanged: δp δζ u = 0 mixed two-factor model admissible. 43 / 48

44 Admissible models 3 Other admissible models: lognormal model for V t (SABR) smiled version of two-factor model (see SD III) Significance of condition dp S, σkt = 0 dp dλ S, σkt dλ 0: price depends on more state variables than hedge instruments. Ex. with Heston model: P (t, S, σ KT, V ). State variables λ are stochastic model allocates thetas proportional to d2 P P&L leakage, even if δλ = 0. dλ 2, d 2 P dλdo Does not happen with model parameters V 0, k, ν, ρ do not generate P&L leakage. Model allocates no theta to gammas on model params. Like making P a function of a non-financial state variable e.g. temperature. In admissible models, SV degrees of freedom do impact dynamics of assets, yet do not require extra hedges. 44 / 48

45 Now know which models are usable what s left to do? Size up break-even covariance levels for S/ σ KT, σ KT / σ K T. Like them, use model; don t like them, don t use model. In practice, look at dynamics of implied vols with floating strikes fixed moneyness, rather than fixed strikes. Approximate formulae for vols of vols and spot/vol covariances for ATMF vols? Consider in particular SSR: R T = 1 d σ T dln S S T (dln S) 2 Expand at order one in vol of vol ν and local vol function. 45 / 48

46 Example Pick as mkt smile smile generated by two-factor model. Parameters typical of Eurostoxx50 smile. VS vols flat at 20%. So that full SV situation attainable. Parameters so that vol( σ T ) 1 T 0.6. ρ SX 1, ρ SX 2 (calibrated on actual smile) so that S T 1 T /105 vol pts Model Power-law exp = 0.5 Model params nu 310.0% theta 13.9% k k rho XY 0% rho_sx -54.0% rho_sy -62.3% Mat - years 46 / 48

47 Example 2 Test 1: use same parameters for underlying SV model local vol flat = 1. MC: computed numerically other curves: order-1 formulae Everything as function of maturity (years) SSR Pure SV - MC Pure SV LV % % % 140% 120% 100% 80% 60% 40% 20% Vol of ATMF vol Pure SV - MC Pure SV LV 0% -20% -40% -60% -80% Spot/ATMF vol correl Pure SV - MC Pure SV LV % % Test 2: Now halve vol of vol of underlying SV model Pure SV LV Mixed SSR 200% 180% 160% 140% 120% 100% 80% 60% 40% 20% Vol of ATMF vol Pure SV LV Mixed Mixed - MC 0% -20% -40% -60% -80% Spot/ATMF vol correl Pure SV LV Mixed Mixed - MC 0.0 Mixed - MC 0% % 47 / 48

48 Conclusion Characterization of local/stoch vol models that can be used for trading. Pricing function P(t, S, σ KT, λ) has to be such that: dp dλ = 0 S, σkt Models not obeying this condition P&L leakage. Models obeying condition are genuine market models: thetas matching asset/asset cross-gammas with positive break-even covariance matrix. Delta and vega given simply by dp σkt recalibrated. and dp d σ KT S the LV function is Good approximate expressions for break-even covariances for ATMF vols & spot. 48 / 48