Simple Smiles For The Mixing Setup Joint work with D. Sloth Elisa Nicolato Department of Economics and Business, Aarhus University Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 1 / 26
Few Words of Introduction Analytical approximations of implied volatility have been and continue to be proposed, even for solvable models, for the need of Transparency Robustness Speed In this work, we propose an approximation of the implied volatility which can be used for a large variety of well-established models and is Transparent, as it decomposes the /smile into meaningful quantities associated with higher-order option risks. Simple, fast and easy to implement. Quite accurate where it matters. Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 2 / 26
The Mixing Setup The Mixing Setup Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 3 / 26
The Mixing Setup The Mixing Setup The mixing setup is a natural generalization of the Black-Scholes model � � − 1 2 σ 2 t + W σ 2 t S t = S 0 exp . obtained by randomizing the spot S 0 and the total variance σ 2 t via their stochastic counterparts. The risk-neutral dynamics of the asset price are given by � � − 1 S t = S eff 2 V eff V eff 0 = 0 , S eff t exp + W V eff , 0 = S 0 , t t where S eff is a positive martingale, V eff is an increasing process and W is a Brownian motion, independent of ( S eff , V eff ). The price process S is a conditionally log-normal martingale. Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 4 / 26
The Mixing Setup Embedded subclasses: SV models The mixing setup contains stochastic volatility models of the following type �� � S t √ v t 1 − ρ 2 dW + ρ dB dS t = = µ ( v t , t ) dt + σ ( v t , t ) dB t , v 0 > 0 , dv t where W ⊥ B are Brownian motions and ρ is the correlation parameter. The Heston model, the 3/2 model, and the quadratic class specification are examples of solvable specifications. The mixing representation follows by setting � t (1 − ρ 2 ) V eff = 0 v s ds , t � � � t � t √ v s dB s − ρ 2 S eff = S 0 exp 0 v s ds + ρ , t 2 0 Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 5 / 26
The Mixing Setup Purely jumping models Purely jumping models are obtained by setting V eff as an increasing and purely jumping semimartingale. The price dynamics are then specified as � � − 1 S t = S eff t exp 2 V eff + W V eff , t t S eff S 0 exp ( − K t ( c ) + cV eff = t ) , t where the real parameter c allows for correlation between S and V eff , and K ( c ) is the cumulant exponent process. evy models are obtained by modeling V eff as a drift-less L´ Exponential L´ evy subordinator. Models of this class include, for instance, the VG model, the NIG model, and the CGMY model. Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 6 / 26
The Two Series Expansions The Two Series Expansions Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 7 / 26
The Two Series Expansions The � S , V � -expansion By conditional log-normality, the price C ( S 0 , K , τ ) of a call with strike K and expiry τ > 0 admits the mixing representation C ( S 0 , K , τ ) = E [( S τ − K ) + ] = E [ C BS ( S eff τ , V eff τ )] , where C BS ( S , V ) denotes the BS call-price in terms of total variance V = στ . First, we apply a 2-dimensional Taylor series expansion around the point ( E S eff τ , E V eff τ ) = ( S 0 , E V eff τ ) τ − S 0 ) 2 ] ∂ 2 C BS τ ) + 1 E [ C BS ( S eff τ , V eff τ )] = C BS ( S 0 , E V eff 2! E [( S eff ∂ S 2 τ ) 2 ] ∂ 2 C BS + 1 2! E [( V eff τ − E V eff ∂ V 2 τ )] ∂ 2 C BS + E [( S eff τ − S 0 )( V eff τ − E V eff ∂ S ∂ V + · · · Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 8 / 26
The Two Series Expansions The � Σ � -expansion Next, recall that by definition C ( S 0 , K , τ ) ≡ C BS ( S 0 , Σ) . where Σ = τ I 2 denotes the implied total variance . Then expand this expression in the second variable Σ around E V eff using a τ one-dimensional Taylor series. � � ∂ C BS C ( S 0 , K , τ ) = C BS ( S 0 , E V eff Σ − E V eff τ ) + τ ∂ V � � 2 ∂ 2 C BS + 1 Σ − E V eff + · · · τ ∂ V 2 2! Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 9 / 26
The Two Series Expansions From Call Prices to Implied Vols Finally truncate the � Σ � -expansion to the first order and the � S , V � -expansion to the q -th order, to approximate I 2 as q k � � I 2 ≈ E V eff + 1 D s l v k − l τ τ − S 0 ) l ( V eff τ ) k − l ] , l !( k − l )! E [( S eff τ − E V eff τ τ k =2 l =0 � ∂ C BS � − 1 ∂ m + n C BS where D s m v n ≡ ∂ S m ∂ V n denote the Vega-normalized Black-Scholes ∂ V derivatives. � t ) n � t ) m ( S eff ( V eff Application demands that E are easy to compute. � � e uX eff + wV eff This is a simple task whenever L eff t ( u , w ) = E with t t X eff = log S eff , is available in (semi) closed-form. In this case � = ∂ m L eff �� � m � � n � t ( u , w ) � V eff S eff E � u = n , w =0 , t t ∂ w m Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 10 / 26
The Two Series Expansions A Simple Quadratic Approximation The 2nd-order expansion yields a simple approximation I 2 Q ( x , τ ) of the implied variance which is quadratic in x = log K / S . Specifically Q ( x , τ ) = E V eff + V ar [ S eff τ ] D ss + C ov [ S eff τ , V eff τ ] D sv + V ar [ V eff τ ] I 2 ≈ I 2 τ D vv , τ 2 τ τ 2 τ where the normalized Gamma D ss , Vanna D sv and Volga D vv are D vv = x 2 D ss = 2 SV + 1 x 2 V 2 − 1 8 − 1 S 2 , D sv = 2 S , 2 V . We see that - the Gamma risk D ss only contributes to the level of smile, - the Vanna term D sv determines the slope - the Volga term D vv introduces convexity. Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 11 / 26
The Two Series Expansions A first look at ATM term-structure Re-arranging the terms we obtain I 2 Q ( x , τ ) = I 0 ( τ ) + I 1 ( τ ) x + I 2 ( τ ) x 2 , where I . ( τ ) describe the term structure of the approximate smile: - ATM Variance: � � I 0 ( τ ) = E V eff + V ar [ S eff τ ] + C ov [ S eff τ , V eff τ ] − V ar [ V eff τ ] 1 + 1 τ 4 E V eff τ S 2 τ 0 τ 2 S 0 τ 4 τ E V eff τ - ATM Skew: I 1 ( τ ) = 1 C ov [ S eff τ , V eff τ ] τ S 0 E V eff τ - ATM Curvature: V ar [ V eff I 2 ( τ ) = 1 τ ] τ 4( E V eff τ ) 2 Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 12 / 26
Does it work? Does it work? Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 13 / 26
Does it work? Illustration for SV Models Naturally, we consider the Heston (1993) model dv t = κ ( θ − v t ) dt + ε v 1 / 2 dB t . t We also consider the 3/2 model with the instantaneous variance dv t = v t κ ( θ − v t ) dt + ε v 3 / 2 dB t . t Both models are solvable, as the joint Laplace transform L XV of X = log S � · t and V = 0 v s ds has closed-form. Also the relevant moments are computable, since it holds that � � u , (1 − ρ 2 )( w + 1 2 u − 1 t ( u , w ) = L XV 2 u 2 ) L eff , t between the ”standard” and the ”effective” transforms. However, Fourier inversion is numerically not trivial in the 3/2 model, due to complex evaluations of the confluent hypergeometric function. Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 14 / 26
Does it work? ATM Implied Vols and Skews ATM Vols. Heston ATM skews Heston 20.1% 0% Fourier Quadratic 20% −0.5% 19.9% −1% 19.8% −1.5% ATM vols skews 19.7% −2% 19.6% −2.5% 19.5% −3% 19.4% −3.5% Fourier Quadratic 19.3% −4% 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 maturity maturity log−scale ATM Vols 3/2 ATM skews 3/2 20% 0% Fourier Quadratic −0.5% 19.5% −1% 19% −1.5% ATM vols 18.5% skews −2% 18% −2.5% 17.5% −3% 17% −3.5% Fourier Quadratic 16.5% −4% 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 maturity log−scale maturity log−scale ATM vols (left) and skews (right) for the Heston (top) and the 3/2 (bottom). The maturity ranges from τ = 0 . 05 up to τ = 18 years. Parameters are as in Forde et al. (2012). Elisa Nicolato (Department of Economics and Business, Aarhus University) Simple Smiles For The Mixing Setup 15 / 26
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