Stochastic Volatility and Jump Modeling in Finance

Transcription

1 Stochastic Volatility and Jump Modeling in Finance HPCFinance 1st kick-off meeting Elisa Nicolato Aarhus University Department of Economics and Business January 21, 2013 Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

2 Outline The aim of this talk is to give an informal, strictly non-rigorous and basically conversational introduction to advanced modelling techniques in Finance. Motivate why advanced modelling is often (although not always) necessary. Hint briefly at SV models and Lévy models Attempt to highlight common traits in these two seemingly very different model classes Touch upon modelling frameworks combining the best features of SV and Jump models. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

3 The seminal Black-Scholes model In the BS model, the price-process of an asset S is described by a GBM ds t = µs t dt +σs t dz t d(logs t ) = dx t = (µ 1 2 σ2 )dt +σdz t Implications: The volatility is constant and log-returns are i.i.d normally distributed X t+ t X t (µ 1 2 σ2 ) t +σ tn(0,1) The risk-neutral price process follows a GMB, albeit with µ = r and the price of a call stroke at K and expiring at T is given by with C BS denoting the BS formula. e rt E Q [(S T K) + ] = C BS (S 0,K,σ,T) Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

4 Model vs Empirics - laughing off constant vol -1 This plot shows SPX daily log returns from 02/01/2001 to 02/31/ Volatility doesn t seem to be constant. Large moves follow large moves and small moves follow small moves- volatility clustering Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

5 Model vs Empirics - laughing off constant vol -2 This plot shows SPX daily volatility estimates for the same period Although naive estimates (60 days-rolling window), clear indication of non-constant vol. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

6 Model vs Empirics - laughing off normality -1 Frequency distribution of (10 years of) SPX daily returns compared with the normal distribution Descriptive statistics: mean=0.0136% (annualized=3.42%), STD=1.37% (annualized=21.81%), skewness=0.0846, kurtosis= Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

7 Model vs Empirics - laughing off normality -2 Empirical distribution (kernel density estimator) vs fitted normal. 60 Empirical vs Normal log scale Stylized features: High central peak (top panel) and fat tails (bottom panel in log-scale). Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

8 Mean-Variance Mixtures Distributions: a first look High central peak/fat tails and random variance are in fact two sides of the same coin. The so-called mean-variance mixtures distributions embed the two features and are constructed as follows: X µ+θv +σ V N(0,1), V N(0,1) (1) where µ, θ and σ are real parameters with σ > 0; V is a distribution on the positive real line (representing the random variance); V is independent of the standard normal N(0, 1). µ is a location parameter, θ is a skewness parameter, while σ regulates the level of the overall random variance Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

9 Variance Gamma Distribution A notorious choice for V is the gamma distribution Γ(µ, ν) with density f V (x) = ( µ µ ν) 2 ν x µ2 ν 1 exp ( µ ν x) Γ( µ2 ν ) x > 0 (2) The resulting mixed distribution X µ+θv +σ V N(0,1) is known as the Variance-Gamma distribution. Denoting with K η ( ) is a modified Bessel, the density is f X (x) = 2exp( θ(x µ)/σ 2) ( ) 1/ν 1/2 σ x µ (3) 2πν 1/ν Γ(1/ν) 2θ2 /ν +σ 2 ( x µ ) 2σ K 2 /ν +θ 2 1/ν 1/2 < x + The (much simpler) c.f. φ X (u) = E[exp(iuX)] is given by σ 2 φ X (u) = e iµu( 1 iθνu +(σ 2 νu 2 )/2 ) 1 ν (4) Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

10 VG vs Normal Fixing θ = 0, comparison between VG and Normal distribution, with the same mean=0.0136% and STD=1.37% 40 Normal vs VG 5 log scale Normal vs VG 5 log scale Top panel ν = 0.5, bottom panel ν = 3. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

11 So what have we found out? Constant volatility is an absurd assumption. The normal distribution is not a good fit to observed returns due to fat tails. Mean-variance mixture distributions, such as VG; obtained by randomizing variance in a normal display such a property and therefore have the potential to fit the empirical distribution of returns Now all this is information coming from historical/time-series preliminary statistical analysis. How about empirical evidence against BS assumptions coming from liquid derivative prices? In other words, is the normality assumption valid/acceptable in the risk-neutral world? Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

12 The Implied Volatility In the BS model, the price of a call option is given by C BS (S 0,K,σ,T) = e rt E Q [(S T K) + ] = S 0 N(d 1 ) Ke rt N(d 2 ) (5) with d 1/2 = log(s0ert /K) σ T ± σ T 2. Given an observed option price C(S 0,K,T), the Implied Volatility is the volatility value σ BS (S 0,K,T) to be plugged in the BS formula to match C(S 0,K,T), i.e. such that C(S 0,K,T) = C BS (S 0,K,σ BS (S 0,K,T),T) IV has been described as the wrong number in the wrong formula to get the right price (R. Rebonato). By its very definition, for prices coming from the BS model with volatility σ σ BS (S 0,K,T) σ for all K,T. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

13 Implied volatilities are in one-to-one correspondence with option prices. They can be seen as a useful alternative way to quote options instead of raw prices: option prices can be compared across different strikes, maturities and underlyings. Given a surface of option prices C(S 0,K,T), the volatility surface σ BS (S 0,K,T) is the associated surface of implied vols. The volatility surface associated with BS model with volatility σ is a flat surface of height σ. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

14 Observed Implied Vol Surface from JG, 15/09/2005 Here s a 3D plot pf the (smoothed) SPX volatility surface as of September 15,2005. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

15 Less smooth but more real pics Raw SPX implied vols observed on March 15, 2010, for two different expiries. S 0 = 1150 Expiry:17/04/2010 0,25 0,2 0,15 0,1 0,05 0 0,9 0,95 1 1,05 1,1 1,15 1,2 K/S Expiry 19/06/2010 0,4 0,35 0,3 0,25 0, ,15 0,1 0,05 0 0,8 0,85 0,9 0,95 1 1,05 1,1 1,15 1,2 K/S Typical behavior across moneyness K/S known as smile (hokey-stick) effect. Notice negative slope around the at-the-money point, the skew. If BS was a good model for market prices, we should observe a flat vol surface! Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

16 How to generate a smile? A naive experiment. By risk-neutral evaluation, we have that the price of a call is given by C(S 0,K,T) = E Q [(e X T K) + ] (6) where X T is the distribution of the log-price at maturity under the simplifying assumption r = d = 0. In the BS model X T = logs T = logs 0 +r 1/2σ 2 T + σ 2 TN(0,1). Inspired by the return analysis, we could consider σ 2 as a random variable and model X T as a mean-variance mixture as introduced in (1). We could fix T = 1 and choose σ 2 σ 2 Γ(1,ν), σ 2 R (as in the VG model introduced by Carr& Madan & Chang (1998). Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

17 VG prices VG Prices can then be computed as the integral C VG (S 0,K,T) = E Q [(e X T K) + ] = (e x K)f X (x)dx (7) where f X (x) is the VG density given in (3): Although known explicitly, it is definitely a mess! Due to the numerical complexity of the integral above, one typically plugs the integrand through the Fourier transform machinery to obtain the equivalent expression in terms of the c.f. S0 K C(S 0,K,T) = S 0 π 0 R du u 2 +1/4 Re[ e iuk φ 0 (u i/2) ] (8) where k := log(k/s 0 ) and φ 0 ( ) denotes the c.f. of the standardized log-price X T X 0 = log S T S0 at maturity. For the VG distribution (and in many other cases) φ 0 ( ) is way more tractable than the density at the point that the integral (8) is know as closed-form solution. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

18 The VG implied vol function And now the experiment goes as follows: For the fixed maturity (e.g. T = 1), and for some fixed parameters σ and ν, compute the call prices C VG (S 0,K j,t) for a set of strikes K j, j = 1,...,m. Numerically invert C VG (S 0,K j,t) = C BS (S 0,K j,σ BS (S 0,K j,t),t) to find the implied volatilities plot them against K. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

19 A smile! And the result is as follows (green line BS prices corresponding to σ = 26%) 0.35 Prices 0.29 Implied Vols We do manage to obtain a smile!!! We can conclude that stochastic volatility injects curvature in an otherwise flat implied vol function. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

20 Summing up From the distributional point of view we have seen that mean-variance mixture laws have the potential to fit the empirical distributions of returns as well as to capture the implied volatility smile. From now on we focus on derivatives and we are interested in finding dynamical models in continuous time capable to match the properties of empirically observed IV surfaces. In other words: how do we bend the seminal BS model so that the flat volatility surface turns into a smile? Roughly speaking there are two main ways to score this goal: SV Models Bend it like Heston Lévy Models Bend it like Madan Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

21 Summing up From the distributional point of view we have seen that mean-variance mixture laws have the potential to fit the empirical distributions of returns as well as to capture the implied volatility smile. From now on we focus on derivatives and we are interested in finding dynamical models in continuous time capable to match the properties of empirically observed IV surfaces. In other words: how do we bend the seminal BS model so that the flat volatility surface turns into a smile? Roughly speaking there are two main ways to score this goal: SV Models Bend it like Heston Lévy Models Bend it like Madan Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

22 Summing up From the distributional point of view we have seen that mean-variance mixture laws have the potential to fit the empirical distributions of returns as well as to capture the implied volatility smile. From now on we focus on derivatives and we are interested in finding dynamical models in continuous time capable to match the properties of empirically observed IV surfaces. In other words: how do we bend the seminal BS model so that the flat volatility surface turns into a smile? Roughly speaking there are two main ways to score this goal: SV Models Bend it like Heston Lévy Models Bend it like Madan Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

23 Summing up From the distributional point of view we have seen that mean-variance mixture laws have the potential to fit the empirical distributions of returns as well as to capture the implied volatility smile. From now on we focus on derivatives and we are interested in finding dynamical models in continuous time capable to match the properties of empirically observed IV surfaces. In other words: how do we bend the seminal BS model so that the flat volatility surface turns into a smile? Roughly speaking there are two main ways to score this goal: SV Models Bend it like Heston Lévy Models Bend it like Madan Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

24 Summing up From the distributional point of view we have seen that mean-variance mixture laws have the potential to fit the empirical distributions of returns as well as to capture the implied volatility smile. From now on we focus on derivatives and we are interested in finding dynamical models in continuous time capable to match the properties of empirically observed IV surfaces. In other words: how do we bend the seminal BS model so that the flat volatility surface turns into a smile? Roughly speaking there are two main ways to score this goal: SV Models Bend it like Heston Lévy Models Bend it like Madan Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

25 Summing up From the distributional point of view we have seen that mean-variance mixture laws have the potential to fit the empirical distributions of returns as well as to capture the implied volatility smile. From now on we focus on derivatives and we are interested in finding dynamical models in continuous time capable to match the properties of empirically observed IV surfaces. In other words: how do we bend the seminal BS model so that the flat volatility surface turns into a smile? Roughly speaking there are two main ways to score this goal: SV Models Bend it like Heston Lévy Models Bend it like Madan Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

26 SV models: diffusion formulation Stochastic volatility models are models generalizing the seminal BS, allowing the volatility parameter itself to be a stochastic process with its own source of randomness. In a quite (but not the most) general SV model, the price S of a non dividend paying underlying asset is given by: ds t = rs t dt + v t S t db t (9) dv t = a(v t )dt +b(v t )dw t (10) where B and W are two Brownian Motions, possibly correlated d B,W = ρdt, while v t is a process representing the random instantaneous variance. Here the model is specified under the risk-neutral measure Q. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

27 Examples of SV models The spot is always described by ds t = rs t dt + v t S t db t The Good: Hull and White (1987): dv t = αv t dt +ηv t dw t GBM implying log-normal v t, no mean reversion. The Bad: Heston (1993): dv t = λ(v t v)dt +η v t dw t Stationary, Gamma invariant law. The Ugly: 3/2 (???): v = 1/x with x described by square root process as in Heston: dv t = λv t (v t v)dt +ηv 3/2 t dw t Level dependent speed of mean reversion, stationary, Inverse Gamma invariant law. Common to all these models (and many others) is the fact that the instantaneous variance process v t is Markovian in itself. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

28 Examples of SV models The spot is always described by ds t = rs t dt + v t S t db t The Good: Hull and White (1987): dv t = αv t dt +ηv t dw t GBM implying log-normal v t, no mean reversion. The Bad: Heston (1993): dv t = λ(v t v)dt +η v t dw t Stationary, Gamma invariant law. The Ugly: 3/2 (???): v = 1/x with x described by square root process as in Heston: dv t = λv t (v t v)dt +ηv 3/2 t dw t Level dependent speed of mean reversion, stationary, Inverse Gamma invariant law. Common to all these models (and many others) is the fact that the instantaneous variance process v t is Markovian in itself. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

29 Examples of SV models The spot is always described by ds t = rs t dt + v t S t db t The Good: Hull and White (1987): dv t = αv t dt +ηv t dw t GBM implying log-normal v t, no mean reversion. The Bad: Heston (1993): dv t = λ(v t v)dt +η v t dw t Stationary, Gamma invariant law. The Ugly: 3/2 (???): v = 1/x with x described by square root process as in Heston: dv t = λv t (v t v)dt +ηv 3/2 t dw t Level dependent speed of mean reversion, stationary, Inverse Gamma invariant law. Common to all these models (and many others) is the fact that the instantaneous variance process v t is Markovian in itself. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

30 Examples of SV models The spot is always described by ds t = rs t dt + v t S t db t The Good: Hull and White (1987): dv t = αv t dt +ηv t dw t GBM implying log-normal v t, no mean reversion. The Bad: Heston (1993): dv t = λ(v t v)dt +η v t dw t Stationary, Gamma invariant law. The Ugly: 3/2 (???): v = 1/x with x described by square root process as in Heston: dv t = λv t (v t v)dt +ηv 3/2 t dw t Level dependent speed of mean reversion, stationary, Inverse Gamma invariant law. Common to all these models (and many others) is the fact that the instantaneous variance process v t is Markovian in itself. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

31 Distribution of log-price X T The log price process is given by X t = X 0 +rt 1 t t v s ds + vs db s Let V t = t 0 v sds denote the integrated variance up to time t: By the Dambis-Dubins-Schwarz theorem, X t admits the following time-change representation X t = X 0 +rt 1 2 V t + B Vt where B is a Brownian motion (in a suitable filtration) Hence, the distribution of the log-price X t is given by a mean-variance mixture X t X 0 rt 1 2 V t + V t N(0,1) (11) However, V t and N(0,1) in this case are not necessarily independent, but they have correlation ρ. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

32 Given the discussion above, it is natural to expect that any SV model will be capable to generate a smile. And so it is. However, the Heston model, as any bad guy, is possibly the most popular both among practitioners and academics. Why? Quoting Gatheral The great difference...is the existence of a fast and easily implemented quasi-closed form solution for European options. That is, the c.f. φ 0 (u) of X T is available in terms of elementary functions and therefore we may use S0 K du [ ] C(S 0,K,T) = S 0 π u 2 +1/4 Re e iuk φ 0 (u i/2) introduced above to compute call option prices. 0 Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

33 The Heston c.f. Define a = 1 2 (z2 +iz) b = λ ρηiz d = b 2 2aη 2 r ± = b ±d η 2 g = r r + then the c.f. φ 0 (z) takes the form φ 0 (z) = e C(z,T)+D(z,T)v0 (12) where D(z,τ) = r 1 e dτ 1 ge dτ C(z,τ) = λ v {r τ 2η ( )} 1 ge dτ 2 log 1 g The exponential-affine form (12) is not a coincidence but a consequence of the fact that v is described by an affine diffusion Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

34 A couple of words about the good and the ugly In principle it is possible to compute the c.f. φ 0 also for the 3/2 model, but it is in terms of the Confluent Hypergeometric function (a series expansion) It is so ugly (and untractable) that I will not even begin to write it down. Still, this model is attracting attention (again) lately, due to the properties of its invariant law in connection with the pricing of VIX option. For the good old Hull and White model, no explicit or semi-explicit formula is available. Such a pity, as the log-normal law offers the best description of the empirical law of the instantaneous variance. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

35 Let s bend it like Heston Here s the 3D plot of a Heston imp. vol. surface obtained with parameters fitted to the observed surface on 15,September The values are λ = 0.54, ρ = 0.7,η = 0.30, v = 0.044, v 0 = implied vols log moneyness log(k/s) time to maturity 3 Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

36 Games you might play: parameter sensitivities Here s the graphical illustration of how the parameters affect the volatility shape for a maturity fixed at T = 0.3. Left panel: the correlation parameter ρ is taking values 0.7 (blue),0 (green) and 0.7 (red). Right panel: the vol-of-vol parameter η is taking values 0.1 (blue),0.6 (green) and 1.1 (red). The remaining parameters are same as from fit to 15, September, Sensitivity w.r.t. ρ 0.5 Sensitivity w.r.t. η Imp. Vols Imp. Vols log moneyness log(k/s) log moneyness log(k/s) Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

37 Heston: Limitations It seems that the Heston model is not quite capable to capture the behavior of the Skew σ2 BS (κ,t) κ for short maturities. For short maturities, the empirically observed skew seems to be much steeper than the skew generated by Heston model (or any other SV model, actually) Hence the need of introducing new models beyond/different than SV. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

38 Back to square one Let s get back to Black-Scholes model for which (risk-neutral) log-prices follow dx = (r 1/2σ 2 )dt +σdb (r 1/2σ 2 ) t +σn(0,dt) What is really annoying here in terms of the smile is the normality of the increments of the BM B, which transfers to the distributions of log-returns. The idea here is: can we substitute B with another continuous time process L which has independent and stationary increments but described by a distribution capable to bend a smile? The answer is yes and it is captured by the class of the exponential Lévy models. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

39 A brief introduction to Lévy processes Definition: A stochastic process (L t ) t 0 such that L 0 = 0 is called a Lévy process if 1 Independent increments: for every increasing sequence of times t 0 t n, the random variables L t0,l t1 L t0,...,l tn L tn 1 are independent. 2 Stationary increments: the law of L t+h L t does not depend on t, i.e. L t+h L t L h. 3 Stochastic continuity: ε > 0,lim h 0 Q( L t+h L t ε) = 0. - The third property does not imply in any way that the sample paths are continuous. It means that for a given t, the probability of a jump to occur exactly at t is zero. - The most prominent examples are the Brownian motion (continuous paths, the one and only) and the Poisson process (discontinuous paths). Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

40 Lévy models Definition: A probability distribution F on R is said to be infinitely divisible if for any integer n 2 there exists n i.i.d. random variables Y 1,...,Y n such that Y 1 + +Y n has distribution F. Given an infinitely divisible distribution F, then there exists a Lévy process (L t ) such that L 1 F, We can then define a risk-neutral log price with unit F distribution as follows dx t = rdt + drift dt +dl t (13) where the drift is determined by risk-neutrality and is given by where φ 0 (z) is the c.f. of F. drift = logφ 0 ( i) The VG distribution is infinitely divisible, implying that via (13) we can define a log price process with VG distributions as marginals: The VG model of Madan et al. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

41 The Seagull Volatility Surface Once again prices can be computed via Fourier transform. The typical volatility surface in the VG model looks like this implied vols log moneyness log(k/s) time to maturity 1 This time, the smile is not well behaved in the long-term end, i.e. for high maturities Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34

42 Injecting SV in Lévy models Lévy models can capture the IV smile for a single maturity, but do not accommodate the term structure of the surface of in the long-term end. On the other hand SV models are better at generating IVs across maturities but fail to reproduce strong skews at short maturities. To depict the volatility surface in the strike as well as in the maturity dimensions, Carr et al. propose the Lévy-Stochastic Volatility process. Here, the driving source of randomness X is constructed by subordinating a general Lévy process L to an absolutely continuous random clock Y (X t ) := (L Yt ), with Y t = t 0 y s ds, (14) where the instantaneous activity rate y is given by some positive process independent of L. Elisa Nicolato (Aarhus University DepartmentStochastic of Economics Volatility and Business and Jump ) Modeling in Finance January 21, / 34