Lecture 2: Rough Heston models: Pricing and hedging

Transcription

1 Lecture 2: Rough Heston models: Pricing and hedging Mathieu Rosenbaum École Polytechnique European Summer School in Financial Mathematics, Dresden August 217 Mathieu Rosenbaum Rough Heston models 1

2 Table of contents 1 Introduction Mathieu Rosenbaum Rough Heston models 2

3 Table of contents 1 Introduction Mathieu Rosenbaum Rough Heston models 3

4 A well-know stochastic volatility model The Heston model A very popular stochastic volatility model for a stock price is the Heston model : ds t = S t Vt dw t dv t = λ(θ V t )dt + λν V t db t, dw t, db t = ρdt. Popularity of the Heston model Reproduces several important features of low frequency price data : leverage effect, time-varying volatility, fat tails,... Provides quite reasonable dynamics for the volatility surface. Explicit formula for the characteristic function of the asset log-price very efficient model calibration procedures. Mathieu Rosenbaum Rough Heston models 4

5 But... Volatility is rough! In Heston model, volatility follows a Brownian diffusion. It is shown in Gatheral et al. that log-volatility time series behave in fact like a fractional Brownian motion, with Hurst parameter of order.1. More precisely, basically all the statistical stylized facts of volatility are retrieved when modeling it by a rough fractional Brownian motion. From Alos, Fukasawa and Bayer et al., we know that such model also enables us to reproduce very well the behavior of the implied volatility surface, in particular the at-the-money skew (without jumps). Mathieu Rosenbaum Rough Heston models 5

6 Fractional Brownian motion (I) Definition The fractional Brownian motion with Hurst parameter H is the only process W H to satisfy : Self-similarity : (W H at ) L = a H (W H t ). Stationary increments : (W H t+h W H t ) L = (W H h ). Gaussian process with E[W H 1 ] = and E[(W H 1 )2 ] = 1. Mathieu Rosenbaum Rough Heston models 6

7 Fractional Brownian motion (II) Proposition For all ε >, W H is (H ε)-hölder a.s. Proposition The absolute moments satisfy E[ W H t+h W H t q ] = K q h Hq. Mandelbrot-van Ness representation t Wt H = dw ( s (t s) H 1 (t s) H ( s) 1 2 H ) dw s. Mathieu Rosenbaum Rough Heston models 7

8 Modifying Heston model Rough Heston model It is natural to modify Heston model and consider its rough version : ds t = S t Vt dw t V t = V + 1 Γ(α) t (t s) α 1 λ(θ V s )ds + λν Γ(α) with dw t, db t = ρdt and α (1/2, 1). t (t s) α 1 V s db s, Mathieu Rosenbaum Rough Heston models 8

9 Pricing in Heston models Classical Heston model From simple arguments based on the Markovian structure of the model and Ito s formula, we get that in the classical Heston model, the characteristic function of the log-price X t = log(s t /S ) satisfies E[e iaxt ] = exp ( g(a, t) + V h(a, t) ), where h is solution of the following Riccati equation : t h = 1 2 ( a2 ia)+λ(iaρν 1)h(a, s)+ (λν)2 h 2 (a, s), h(a, ) =, 2 and t g(a, t) = θλ h(a, s)ds. Mathieu Rosenbaum Rough Heston models 9

10 Pricing in Heston models (2) Rough Heston models Pricing in rough models is much more intricate : Monte-Carlo for rough (non-heston) models : Bayer et al., Bennedsen et al., Horvath et al., McCrickerd and Pakkanen. Asymptotic formulas : Bayer et al., Forde et al., Jacquier et al. This work Goal : Deriving a Heston like formula in the rough case, together with hedging strategies. Tool : The microstructural foundations of rough volatility models based on Hawkes processes. We build a sequence of relevant high frequency models converging to our rough Heston process. We compute their characteristic function and pass to the limit. Mathieu Rosenbaum Rough Heston models 1

11 Bibliography Main references El Euch and Rosenbaum : The characteristic function of rough Heston models (16). El Euch and Rosenbaum : Perfect hedging in rough Heston models (17). Advertisement Eduardo Abi Jaber s talk. Abi Jaber, Larsson, Pulido and Gatheral, Keller-Ressel forthcoming works. Mathieu Rosenbaum Rough Heston models 11

12 Table of contents 1 Introduction Mathieu Rosenbaum Rough Heston models 12

13 Building the model Necessary conditions for a good microscopic price model We want : A tick-by-tick model. A model reproducing the stylized facts of modern electronic markets in the context of high frequency trading. A model helping us to understand the rough dynamics of the volatility from the high frequency behavior of market participants. A model helping us to understand leverage effect. A model helping us to derive a Heston like formula and hedging strategies. Mathieu Rosenbaum Rough Heston models 13

14 Digression : How is leverage effect generated? Traditional macroscopic explanations for leverage effect Asset price declines company becomes automatically more leveraged since the ratio of its debt with respect to the equity value becomes larger risk of the asset (the volatility) should become more important. Forecast of an increase of the volatility should be compensated by a higher rate of return, which can only be obtained through a decrease in the asset value. Microstructural component for leverage effect? We want to address the following question : Can leverage effect be partly generated from high frequency features of the asset? Mathieu Rosenbaum Rough Heston models 14

15 Building the model Stylized facts 1-2 Markets are highly endogenous, meaning that most of the orders have no real economic motivations but are rather sent by algorithms in reaction to other orders, see Bouchaud et al., Filimonov and Sornette. Mechanisms preventing statistical arbitrages take place on high frequency markets, meaning that at the high frequency scale, building strategies that are on average profitable is hardly possible. Mathieu Rosenbaum Rough Heston models 15

16 Building the model Stylized facts 3-4 There is some asymmetry in the liquidity on the bid and ask sides of the order book. In particular, a market maker is likely to raise the price by less following a buy order than to lower the price following the same size sell order, see Brennan et al., Brunnermeier and Pedersen, Hendershott and Seasholes. A large proportion of transactions is due to large orders, called metaorders, which are not executed at once but split in time. Mathieu Rosenbaum Rough Heston models 16

17 Building the model Hawkes processes Our tick-by-tick price model is based on Hawkes processes in dimension two, very much inspired by the approaches in Bacry et al. and Jaisson and R. A two-dimensional Hawkes process is a bivariate point process (N + t, N t ) t taking values in (R + ) 2 and with intensity (λ + t, λ t ) of the form : ( ) ( λ + t µ + = λ t µ ) t + ( ) ( ϕ1 (t s) ϕ 3 (t s) dn +. s ϕ 2 (t s) ϕ 4 (t s) dns ). Mathieu Rosenbaum Rough Heston models 17

18 Building the model The microscopic price model Our model is simply given by P t = N + t N t. N t + corresponds to the number of upward jumps of the asset in the time interval [, t] and Nt to the number of downward jumps. Hence, the instantaneous probability to get an upward (downward) jump depends on the location in time of the past upward and downward jumps. By construction, the price process lives on a discrete grid. Statistical properties of this model have been studied in details. Mathieu Rosenbaum Rough Heston models 18

19 Encoding the stylized facts The right parametrization of the model Recall that ( ) ( λ + t µ + = λ t µ ) t + ( ) ( ϕ1 (t s) ϕ 3 (t s) dn +. s ϕ 2 (t s) ϕ 4 (t s) dns High degree of endogeneity of the market L 1 norm of the largest eigenvalue of the kernel matrix close to one. No arbitrage ϕ 1 + ϕ 3 = ϕ 2 + ϕ 4. Liquidity asymmetry ϕ 3 = βϕ 2, with β > 1. Metaorders splitting ϕ 1 (x), ϕ 2 (x) ). K/x 1+α, α.6. x Mathieu Rosenbaum Rough Heston models 19

20 The scaling limit of the price model Limit theorem After suitable scaling in time and space, the long term limit of our price model satisfies the following rough Heston dynamics : V t = V + 1 Γ(α) with t P t = t Vs dw s 1 t V s ds, 2 (t s) α 1 λ(θ V s )ds + λν Γ(α) d W, B t = t (t s) α 1 V s db s, 1 β 2(1 + β 2 ) dt. Mathieu Rosenbaum Rough Heston models 2

21 The scaling limit of the price model Comments on the theorem The Hurst parameter H = α 1/2. Hence stylized facts of modern market microstructure naturally give rise to fractional dynamics and leverage effect. One of the only cases of scaling limit of a non ad hoc micro model where leverage effect appears in the limit. Compare with Nelson s limit of GARCH models for example. Uniqueness of the limiting solution is a difficult result. The proof requires the use of recent results in SPDEs theory by Mytnik and Salisbury. Obtaining a non-zero starting value for the volatility is a tricky point. To do so, we in fact consider a time-dependent µ. Mathieu Rosenbaum Rough Heston models 21

22 Table of contents 1 Introduction Mathieu Rosenbaum Rough Heston models 22

23 A general case Multidimensional Hawkes process To obtain the characteristic function of our microscopic price process, we derive the characteristic function of multidimensional Hawkes processes. Let us consider a d-dimensional Hawkes process N = (N 1,..., N d ) with intensity λ 1 t λ t =. λ d t = µ(t) + t φ(t s).dn s. Mathieu Rosenbaum Rough Heston models 23

24 Multidimensional Hawkes process Population interpretation Migrants of type k {1,.., d} arrive as a non-homogenous Poisson process with rate µ k (t). Each migrant of type k {1,.., d} gives birth to children of type j {1,.., d} following a non-homogenous Poisson process with rate φ j,k (t). Each child of type k {1,.., d} also gives birth to other children of type j {1,.., d} following a non-homogenous Poisson process with rate φ j,k (t). Mathieu Rosenbaum Rough Heston models 24

25 Multidimensional Hawkes process Towards the characteristic function Let (Ñ k,j ) 1 j d be d multivariate Hawkes processes with migrant rate (φ j,k ) 1 j d (for given k) and kernel matrix φ. Let Nt,k be the number of migrants of type k arrived up to time t of the initial Hawkes process. Let T1 k <... < T k N,k t type k. We have [, t] the arrival times of migrants of Nt k = Nt,k + law 1 j d 1 l N,j t Ñ j,k,(l) t T j l where the (Ñ j,k,(l) ) are independent copies of (Ñ j,k )., Mathieu Rosenbaum Rough Heston models 25

26 Characteristic function of multidimensional Hawkes processes Theorem We have E[exp(ia.N t )] = exp ( t ( ) ) C(a, t s) 1.µ(s)ds, where C : R d R + C d is solution of the following integral equation : C(a, t) = exp ( t ia + φ (s).(c(a, t s) 1)ds ). Mathieu Rosenbaum Rough Heston models 26

27 Deriving the characteristic function of the rough Heston model Strategy From our last theorem, we are able to derive the characteristic function of our high frequency price model. We then pass to the limit. Mathieu Rosenbaum Rough Heston models 27

28 We write : I 1 α f (x) = 1 x Γ(1 α) f (t) (x t) α dt, Dα f (x) = d dx I 1 α f (x). Theorem The characteristic function at time t for the rough Heston model is given by ( t exp g(a, s)ds + V ) θλ I 1 α g(a, t), with g(a, ) the unique solution of the fractional Riccati equation : D α g(a, s) = λθ 2 ( a2 ia) + λ(iaρν 1)g(a, s) + λν2 2θ g 2 (a, s). Mathieu Rosenbaum Rough Heston models 28

29 Table of contents 1 Introduction Mathieu Rosenbaum Rough Heston models 29

30 Dynamics of a European option price Consider a European option with payoff f (log(s T )). We study the dynamics of C T t = E[f (log(s T )) F t ]; t T. Define P T t (a) = E[exp(ia log(s T )) F t ]; a R. Fourier based hedging Writing ˆf for the Fourier transform of f, we have Ct T = 1 ˆf (a)pt T (a)da; dct T = 1 ˆf (a)dpt T (a)da. 2π 2π a R a R Mathieu Rosenbaum Rough Heston models 3

31 Conditional law of the rough Heston model Theorem The law of the process (S t t, V t t ) t = (S t+t, V t+t ) t is that of a rough Heston model with the following dynamics : ds t t = S t t V t t dw t t ; S t = S t, V t t =V t + 1 Γ(α) t t (t s) α 1 λ ( θ t (s) V t ) λν s ds+ (t s) α 1 V t s db t Γ(α) where (W t t, B t t ) = (W t +t W t, B t +t B t ) and θ t is an explicit F t -measurable process, depending on (V u ) u t. s, Mathieu Rosenbaum Rough Heston models 31

32 Generalized rough Heston model Generalized rough Heston So we naturally generalize the definition of the rough Heston model as follows : ds t = S t Vt dw t V t = V + 1 Γ(α) t (t s) α 1 λ(θ (s) V s )ds+ λν Γ(α) with dw t, db t = ρdt, α (1/2, 1). t (t s) α 1 V s db s, Mathieu Rosenbaum Rough Heston models 32

33 Characteristic function of the generalized rough-heston model Using the Hawkes framework, we get the following result : Theorem The characteristic function of log(s t /S ) in the generalized rough Heston model is given by exp ( t h(a, t s)(λθ s α (s) + V Γ(1 α) ds)), where h is the unique solution of the fractional Riccati equation D α h(a, t) = 1 2 ( a2 ia) + λ(iaρν 1)h(a, s) + (λν)2 h 2 (a, s). 2 Mathieu Rosenbaum Rough Heston models 33

34 Link between the characteristic function and the forward variance curve Link between θ and the forward variance curve θ. = D α (E[V. ] V ) + E[V. ]. Suitable expression for the characteristic function The characteristic function can be written as follows : with exp ( t g(a, t s)e[v s ]ds ), g(a, t) = 1 2 ( a2 ia) + λiaρνh(a, s) + (λν)2 h 2 (a, s). 2 Mathieu Rosenbaum Rough Heston models 34

35 Dynamics of a European option price Recall that P T t (a) = E[exp(ia log(s T )) F t ]. The conditional law of the rough Heston model being a generalized rough Heston, we deduce the following theorem : Theorem and P T t (a) = exp ( ia log(s t ) + T t g(a, s)e[v T s F t ]ds ) dp T t (a) = iap T t (a) ds t S t T t + Pt T (a) g(a, s)de[v T s F t ]ds. We can perfectly hedge the option with the underlying stock and the forward variance curve! (at least theoretically) Mathieu Rosenbaum Rough Heston models 35

36 Calibration We collect S&P implied volatility surface, from Bloomberg, for different maturities T j =.25,.5, 1, 1.5, 2 years, and different moneyness K/S =.8,.9,.95,.975, 1., 1.25, 1.5, 1.1, 1.2. Calibration results on data of 7 January 21 : ρ =.68; ν =.35; H =.9. Mathieu Rosenbaum Rough Heston models 36

37 Calibration results : Market vs model implied volatilities, 7 January 21 Mathieu Rosenbaum Rough Heston models 37

38 Calibration results : Market vs model implied volatilities, 7 January 21 Mathieu Rosenbaum Rough Heston models 38

39 Stability : Results on 8 February 21 (one month after calibration) Mathieu Rosenbaum Rough Heston models 39

40 Stability : Results on 8 February 21 (one month after calibration) Mathieu Rosenbaum Rough Heston models 4

41 Stability : Results on 7 April 21 (three months after calibration) Mathieu Rosenbaum Rough Heston models 41

42 Stability : Results on 7 April 21 (three months after calibration) Mathieu Rosenbaum Rough Heston models 42