Pricing and hedging with rough-heston models

Transcription

1 Pricing and hedging with rough-heston models Omar El Euch, Mathieu Rosenbaum Ecole Polytechnique 1 January 216 El Euch, Rosenbaum Pricing and hedging with rough-heston models 1

2 Table of contents Introduction to rough-heston models 1 Introduction to rough-heston models 2 3 El Euch, Rosenbaum Pricing and hedging with rough-heston models 2

3 Table of contents Introduction to rough-heston models 1 Introduction to rough-heston models 2 3 El Euch, Rosenbaum Pricing and hedging with 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. El Euch, Rosenbaum Pricing and hedging with rough-heston models 4

5 But... Introduction to rough-heston models 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. 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). El Euch, Rosenbaum Pricing and hedging with rough-heston models 5

6 Rough-Heston model Mandelbrot-van Ness representation Wt H = Rough-Heston model dw ( s (t s) H 1 (t s) H ( s) 1 2 H 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 s) α 1 λ(θ V s )ds + λν Γ(α) with dw t, db t = ρdt and α (1/2, 1). ) dw s. (t s) α 1 V s db s, El Euch, Rosenbaum Pricing and hedging with rough-heston models 6

7 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 g(a, t) = θλ h(a, s)ds. El Euch, Rosenbaum Pricing and hedging with rough-heston models 7

8 Pricing in rough-heston models This work Goal : Deriving a Heston like formula in the rough case. 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. Application : Pricing and. El Euch, Rosenbaum Pricing and hedging with rough-heston models 8

9 Table of contents Introduction to rough-heston models 1 Introduction to rough-heston models 2 3 El Euch, Rosenbaum Pricing and hedging with rough-heston models 9

10 Price model Introduction to rough-heston models A tick by tick price model based on a two-dimensional Hawkes process (N + t, N t ) with intensity : ( ) ( λ + t µ + = λ t µ ) + ( ) ( ϕ1 (t s) ϕ 3 (t s) dn +. s ϕ 2 (t s) ϕ 4 (t s) dns where µ +, µ R + and ϕ 1, ϕ 2, ϕ 3, ϕ 4 : R + R + ), The microscopic price model The price model : P t = N + t N t El Euch, Rosenbaum Pricing and hedging with rough-heston models 1

11 Encoding microscopic stylized facts ( λ + t λ t ) ( µ + = µ ) + ( ) ( ϕ1 (t s) ϕ 3 (t s) dn +. s ϕ 2 (t s) ϕ 4 (t s) dns ), Encoding microscopic stylized facts No arbitrage µ + = µ and ϕ 1 + ϕ 3 = ϕ 2 + ϕ 4. Liquidity asymmetry on the bid and ask sides of the order book ϕ 3 = βϕ 2 ; β > 1. High endogeneity of markets ρ( φ) 1. Effect of metaorders ϕ 1 (x), ϕ 2 (x) K/x 1+α. El Euch, Rosenbaum Pricing and hedging with rough-heston models 11

12 The scaling limit of the price model Limit theorem Under previous assumptions, and after suitable scaling in time and space, the long term limit of our price model satisfies the following rough-heston log-price : V t = V + 1 Γ(α) with X t = Vs dw s 1 V s ds, 2 (t s) α 1 λ(θ V s )ds + λν Γ(α) d W, B t = (t s) α 1 V s db s, 1 β 2(1 + β 2 ) dt. El Euch, Rosenbaum Pricing and hedging with rough-heston models 12

13 Characteristic function of multidimensional Hawkes processes Using a population interpretation of Hawkes processes : Theorem The characteristic function of d-dimensional Hawkes process (N t ) t with intensity : λ t = µ(t) + φ(t s).dn s R d. is given by : E[exp(ia.N t )] = exp ( ( ) ) 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 ( ia + φ (s).(c(a, t s) 1)ds ). El Euch, Rosenbaum Pricing and hedging with rough-heston models 13

14 Deriving the characteristic function 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. El Euch, Rosenbaum Pricing and hedging with rough-heston models 14

15 Characteristic function of rough-heston models 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 E[exp ( ia log(s t /S ) ) ( ) ] = exp θλ h(a, s)ds + V I 1 α h(a, t), with h(a, ) the unique solution of the fractional Riccati equation : D α h(a, s) = 1 2 ( a2 ia) + λ(iaρν 1)h(a, s) + (λν)2 h 2 (a, s). 2 El Euch, Rosenbaum Pricing and hedging with rough-heston models 15

16 Table of contents Introduction to rough-heston models 1 Introduction to rough-heston models 2 3 El Euch, Rosenbaum Pricing and hedging with rough-heston models 16

17 Characteristic function of rough-heston models Theorem The characteristic function at time t for the rough-heston model could be also written as E[exp ( ia log(s t /S ) ) ( ) ] = exp g(a, t s)e[v s ]ds, with : g(a, t) = 1 2 ( a2 ia) + iaρλνh(a, s) + (λν)2 h 2 (a, s). 2 This expression holds even for an extended rough-heston model with time dependent parameter θ : Extended rough-heston model V t = V + 1 Γ(α) ds t = S t Vt dw t (t s) α 1 λ(θ(s) V s )ds+ λν Γ(α) (t s) α 1 V s db s. El Euch, Rosenbaum Pricing and hedging with rough-heston models 17

18 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 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 s) α 1 λ ( θ t (s) V t ) λν s ds+ Γ(α) (t s) α 1 V t s db t with (W t t, B t t ) = (W t +t W t, B t +t B t ) is a correlated Brownian motion independent of F t and θ t is a F t measurable process. s, El Euch, Rosenbaum Pricing and hedging with rough-heston models 18

19 Dynamics of the characteristic function process Define : P T t (a) = E[exp(ia log(s T )) F t ] Dynamics of the characteristic function process and with : 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) = S P T t (a)ds t + V P T t (a).(de[v T s F t ]) s T t S P T t (a) = ia PT t (a) S t ; V P T t (a) = P T t (a)(g(a, s)) s T t We can hedge the option with the spot price and the forward variance curve! El Euch, Rosenbaum Pricing and hedging with rough-heston models 19

20 Conclusion Introduction to rough-heston models About the rough-heston model Consistent with historical data. Consistent with implied volatility surface data (In particular for the skew ATM). With a time dependent θ, the model is consistent with the forward variance curve of the market. Explicit formula of the characteristic function fast calibration. Hedge formula using the underlying and the forward variance curve. But... Negative skew for the VIX : work in progress... El Euch, Rosenbaum Pricing and hedging with rough-heston models 2