Rough Volatility Modelling

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Rough Volatility Modelling

Xiaoyan Zhang

Master’s Thesis, Spring 2018

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This master’s thesis is submitted under the master’s programme Modelling and Data Analysis, with programme optionFinance, Insurance and Risk, at the Department of Mathematics, University of Oslo. The scope of the thesis is 60 credits.

The front page depicts a section of the root system of the exceptional Lie group E₈, projected into the plane. Lie groups were invented by the Norwegian mathematician Sophus Lie (1842–1899) to express symmetries in differential equations and today they play a central role in various parts of mathematics.

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Abstract

The objective of this thesis is to give a description of the rough stochastic volatility modelling based on fractional Brownian motion with Hurst exponentH < ¹₂, as well as the corresponding pricing model. Specifically we considered the Rough Fractional Stochastic Volatility (RFSV) model that is proposed in Gatheral et al. (2014) [16], we presented and verified its findings. The thesis starts with a review of some significant volatility and pricing models, for some of them we covered the details of the mathematical derivations.

The stochastic calculus with respect to fractional Brownian motion is also provided.

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Acknowledgements

At first, I would like to thank my supervisor Prof. Salvador Ortiz-Latorre for inspiring me to get to know this topic and for the opportunity to do a project on this topic. The discussion with him and the instructions I got from him were always helpful through this challenging but indeed valuable journey.

Further, I would like to thank all of my friends for their encouragements and company when I was writing this thesis. I feel lucky to have their friendship and trust.

Moreover, I would like to thank my parents for everything they did for me. I feel deeply grateful for all of the supports I got from them, both emotionally and financially. Thank them for that they did their best to raise and educate me to be the best version of myself, and most importantly, for being my role models.

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Introduction

Estimating volatility from high frequency data, Gatheral et. al. [16] recently found that volatility behaves essentially as a fractional Brownian motion with Hurst parameter H of order 0.1 at any reasonable time scale. In this paper a remarkable scaling property of financial time series is discovered and this scaling property gives rise to the Rough Frac- tional Stochastic Volatility (RFSV) model. The RFSV model is claimed to be remarkably consistent with financial time series data. The structure of RFSV model is as

dS(t)

S(t) =µ(t)dt+σ(t)dZ(t), σ(t) = exp(X(t)), t∈[0, T],

Where µ(t) is a drift term, Z(t) is a standard Brownian motion, X(t) is a Ornstein- Uhlenbeck process driven by fractional Brownian motion W^H(t) given by

dX(t) = −α(X(t)−m)dt+νdW^H(t),

where m∈R and ν, α are positive constants, with Z and W^H correlated in general.

In the subsequent paper by Bayer et al. (2016) [2], it shows how the RFSV model can be used to price options. One example given in this paper is the Rough Bergomi (rBergomi) model, this model is claimed to be superior when fitting the SPX volatility than the conventional Markovian stochastic volatility models, and with fewer parameters.

In this master thesis we aim to survey the literatures on rough volatility modelling and present the construction of the volatility and pricing models. Furthermore, we will also present the theory on fractional Brownian motion and its stochastic calculus. We proceed as follows: In chapter 1 we give a review on four generations of volatility and asset pricing models: Black & Scholes constant volatility; time-dependent volatility where volatility is a deterministic function of time; local volatility where volatility is a deterministic function of both time and stock price; and stochastic volatility, where volatility itself is governed by a stochastic differential equation.

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In chapter 2 we present the theory and stochastic calculus with respect to fractional Brownian motion (fBm). Including definitions and elementary properties, the different representations of fBm. We also give a method to simulate fBm.

In chapter 3 and 4 we present the details of the constructions of rough volatility and the pricing model based on [16] and [2]. We show how the remarkable scaling property is discovered and how it is used to establish the RFSV model. In chapter 3 we also tested the scaling property using five different indices. In chapter 4 we show how the rBergomi model is derived by generalise the Lorenzo Bergomi’s stochastic volatility model. We also present the change of measure with respect to fractional Brownian motion in rBergomi model.

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List of Figures

1.1 DAX daily log-returns from 2016 to 2018. . . 11

1.2 The empirical autocorrelation of the daily DAX from 01. Jan. 2016 with lag up to 100. . . 13

1.3 The empirical autocorrelation of the absolute daily DAX from 01. Jan. 2016 with lags up to 100. . . 13

1.4 Implied volatility surface in reality. . . 15

1.5 B&S implied volatility surface. . . 15

2.1 Paths of fBm whenH = 0.1, 0.2 and 0.3. . . 35

3.1 Autocorrelation function of fractional Brownian motion with H = 0.142 and lags up to 0.10 (100 days). The blue dashed line is the significance level. . . . 60

3.2 log m(q,∆) against log ∆, S&P. . . 63

3.3 ζ_q against q, S&P. . . 63

3.4 The distribution of increments of log σ_t for different lags ∆ of S&P indices, normal fit in yellow, normal fit of lag = 1 day rescaled by ∆^H in dashed line in green. . . 64

3.5 log m(q,∆) against log ∆, FTSE. . . 65

3.6 ζ_q against q, FTSE. . . 65

3.7 The distribution of increments of log σ_t for different lags ∆ of FTSE indices, normal fit in yellow, normal fit of lag = 1 day rescaled by ∆^H in dashed line in green. . . 66

3.8 log m(q,∆) against log ∆, AEX. . . 67

3.9 ζ_q against q, AEX. . . 67

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3.10 The distribution of increments of log σ_t for different lags ∆ of AEX indices, normal fit in yellow, normal fit of lag = 1 day rescaled by ∆^H in dashed line in green. . . 68 3.11 log m(q,∆) against log ∆, HSI. . . 69 3.12 ζ_q against q, HSI. . . 69 3.13 The distribution of increments of log σ_t for different lags ∆ of HSI indices,

normal fit in yellow, normal fit of lag = 1 day rescaled by ∆^H in dashed line in green. . . 70 3.14 log m(q,∆) against log ∆, MXX. . . 71 3.15 ζ_q against q, MXX. . . 71 3.16 The distribution of increments of log σ_t for different lags ∆ of MXX indices,

normal fit in yellow, normal fit of lag = 1 day rescaled by ∆^H in dashed line in green. . . 72

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Chapter 1 A review of asset pricing models and volatility dynamics

Since Bachelier first used Brownian motion to value stock options in 1900, various models are proposed for both asset price and volatility. In the first chapter of this master thesis, we are going to give a description of the development of asset and volatility dynamics modelling, focusing on the significant and well known continuous time models. We will also summarise the stylised facts of financial time series and empirical characteristics of volatility, as they are the features of financial assets that these models are designed to capture. We are trying to formulate the description in a progressive and logical way, meaning that the subsequent models are designed to overcome the shortcomings of the previous models.

It is common to categorise the development of volatility modelling into four phases:

constant volatility, time-dependent volatility, local volatility, and stochastic volatility. We will therefore introducing them in the corresponding order. This chapter is based on [21].

1.1 Some preliminaries and stylised facts.

Different types of volatility

1. Volatility as the diffusion term

Ever since Louis Bachelier first proposed to use Brownian motion to value stock price in his doctoral thesis in 1900, it has been popular to model asset price using

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stochastic differential equations. A stochastic processS(t)(e.g. stock price process) can be defined as a diffusion if it approximately follows the difference equation

S(t+ ∆t)−S(t) = µ(t, S(t))∆t+σ(t, S(t))∆W.

Where W is a Wiener process and ∆W(t) =W(t+ ∆t)−W(t), µ(t, S(t)) denotes the drift of S(t), and σ(t, S(t))is the diffusion term which denotes the volatility of S(t). In this case volatilityσ(t, S(t))measures the randomness in asset return.

2. Realised volatility (historical volatility)

This is a measure of volatility obtained by using empirical asset price data and is calculated as follows: Given the (discrete) stock price time series S(t₀), . . . , S(t_n), the realised volatility σˆ in discrete time is obtained by calculating the standard deviation of the stock’s continuously compound return per unit time (sampling interval) ∆t=t_i−ti−1:

ˆ σ =

pVˆ

∆t ,

where Vˆ denotes the sample variance of the log-returns of the stock, i.e., Vˆ = 1

n−1

n

X

i=1

(x_i−x)²,

with log-returns x_i and the sample mean x is given by x_i = ln X(ti)

X(ti−1)

x= 1 n

n

X

i=1

x_i.

3. Implied volatility

This is the volatility obtained by using empirical option price data and the Black and Scholes model.

4. Forward implied volatility

This is the volatility obtained using some forward instruments.

Note in theory the above four measurements of volatility should be consistent if the Black

& Scholes model is correct, however in reality it is not the case.

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Stylised facts of financial time series

• Stylised fact 1: Volatility Clustering

Volatility clustering can be explained by a positive autocorrelation of the absolute log-returns over few days. Figure 1.1 shows the daily log-returns of DAX from 01. Jan.

2016 to 01. Jan. 2018. The volatility clustering can be discovered from the phenomenon that large price changes are often followed by large price changes , and small price changes are often followed by small price changes.

Time[Days]

Logreturns

0 100 200 300 400 500

−0.10−0.050.000.05

Figure 1.1 – DAX daily log-returns from 2016 to 2018.

• Stylised fact 2: Leverage Effect

The leverage effect is another empirical characteristic of volatility. This effect is first observed by Black and describes the negative correlation with stock prices and volatility.

Black gives the following expression of leverage:

Leverage = v M KT,

where v is the the total debt of a firm and M KT is the market capitalisation which is the number of shares times the share price. The leverage effect also describes the phenomenon that large loss or a huge drop in the share prices will have stronger influence over the perception of risk than the large gains. To explain the effect, we see that either a drop in the share price or an increase in debt will increase the leverage, means a riskier situation, hence leads to increase in volatility.

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• Stylised fact 3: Long Memory of Volatility

It is also a stylised fact that volatility has long memory, i.e. the autocorrelation function decays as power law. In figure 1.2 and figure 1.3, we plotted the autocorrelation of the daily log-returns and absolute daily log-returns of DAX from 01. Jan. 2016 with lags up to 100 . The log-returns seems independent because the autocorrelation fluctuating around zero. But if we take out the direction of the fluctuation and plot the absolute log-returns we see that for lags up to 100 days the autocorrelation function is diminishing as the lags increase but still positive. This is an indication that the volatility has long range dependency. Also, the autocorrelation function ρ(n) of lag n is said to decay as a power law if:

ρ(n)∝ 1

|n|^α,as |n| → ∞. (1.1.1) When 0< α <1, the autocorrelation function is not integrable, i.e.,

∞

X

n=0

ρ(n)dn= +∞. (1.1.2)

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0 20 40 60 80 100 0.00.20.40.60.81.0 DAX, autocorrelation

Figure 1.2 – The empirical autocorrelation of the daily DAX from 01. Jan. 2016 with lag up to 100.

0 20 40 60 80 100

0.00.20.40.60.81.0 DAX, autocorrelation(absolute)

Figure 1.3 – The empirical autocorrelation of the absolute daily DAX from 01. Jan. 2016 with lags up to 100.

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Implied volatility and volatility surface

As we will show in the following section, the Black & Scholes formula gives the call option price P(t) =C_BS(t, S(t), σ, r, K, T) a closed form of solution. Since in reality the quoted option price p(t) and all other variables in C_BS(t) except volatility are observable, the implied volatility can be defined as the volatility needed for the solution C_BS(t) to give a price which is consistent with the market quoted call option price p(t). The plot of implied volatility against strike priceK and the time to maturityT is called the volatility surface. As we mentioned, the Black & Scholes model assumes a constant volatility, thus will have a volatility surface which is completely flat. But in reality this is not the case.

This is also a motivation of various models which are designed later on, for example, the time-dependent volatility, local volatility, or stochastic volatility models. Figure 1.4 is the volatility surface in reality and in Figure 1.5 we plotted the flat volatility surface in B&S model.

The plot of implied volatility against strike price K for a fixed maturity T is often symmetric and "u-shape" looking, hence called volatility smile. The lowest point on the plot is usually around the at the money point where S(t) = K. The implied volatility becomes higher when the option goes towards into the money or out of the money situation. The plot which gives an opposite shape is called a volatility frown, and when the plot is downward sloping, it is known as a volatility skew.

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0.4

3 0.6

2 2 2.5

0.8

1 1.5

1 0.5

Figure 1.4 – Implied volatility surface in reality.

Constant Implied Volatility Surface

0

0.5 1

1.5 2

2.5 3

Time to Maturity T 0

1 2 3

Momeyness M=^S/_K -0.5

0 0.5 1 1.5

Implied Volatility (T,M)

Figure 1.5 – B&S implied volatility surface.

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1.2 Constant volatility : Black & Scholes framework

In 1973 Black and Scholes (B&S) proposed to model stock price as a geometric Brownian motion with a constant volatility assumption. This is usually considered as the first generation model. In this section we will present the model structure under the B&S framework and will also present the details of the derivation of the pricing formula according to [4].

Although the constant volatility is obviously inconsistent with the stylised facts we listed above, it is worth to know because it will help us to understand how advanced mathematics is applied in finance. Moreover, as we will see, B&S model is very practical in the sense that it provides a closed form expressions for the price of basic options like calls and puts.

Consider a market consisting of one risky assetS and one interest-based risk-less asset Z. LetS(t) and Z(t) be the price processes of the risky and risk-less assets respectively.

Under Black and Scholes (B&S) framework S(t) is modelled as

S(t) = S(0)exp(µt+σB(t)). (1.2.1)

Where B(t) is Brownian motion defined in definition (A.9). Since S(t) is modelled as the exponential of Brownian motion, it is a geometric Brownian motion, and thus is a lognormal process. The structure of the model is given by

dS(t) = S(t) (µdt+σdB(t)), S(0) >0, (1.2.2)

dZ(t) = Z(t)rtdt, Z(0) = 1, (1.2.3)

where the interest rate r, the drift µ and the volatility σ are assumed to be constant, with σ > 0. One huge advantage of B&S model is that it gives a closed form solution to call and put options. The solution is obtained by finding the value of a self-financing, arbitrage-free hedging strategy of the option. We will in the following formulate how the pricing formula for European call option is derived based on [4].

Black & Scholes partial differential equation

Let P(t) be the price of a contingent claim at time t, (a(t), b(t), c(t)) be the amount of risky asset, risk-less asset and contingent claim at time t respectively. The value V(t)of the portfolio (a, b, c)at time t is then

V(t) = a(t)S(t) +b(t)Z(t) +c(t)P(t).

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Definition 1.2.1. self-financing portfolio A portfolio strategy (a, b, c) is called self- financing if

dV(t) =a(t)dS(t) +b(t)dZ(t) +c(t)dP(t).

Intuitively, for a self-financing portfolio there is no new investment or any withdraw.

Definition 1.2.2. arbitrage opportunity A self-financing portfolio strategy is called an arbitrage opportunity if V(0) ≤0, V(T)>0 and E[V(T)]>0.

Assumption: Assume the market is arbitrage-free and complete. Complete means every contingent claim in the market can be hedged (replicated).

Let H = (a^H, b^H) denote the hedging strategy and H(t) denote its value at time t.

Let P(t) denote the value of the claim that we aim to hedge, andX denote its payoff at exercise time T. Under the above assumption, the following equalities should hold:

H(t) =P(t), (1.2.4)

H(T) =X. (1.2.5)

We omit the proof of why when the above equalities do not hold, there are arbitrage opportunities. For details see [4]. Let P(T) = g(S(T)) and P(t) = C(t, S(t)), where the payoff function g and the function C(t, x) are both square integrable. Assume further thatC(t, x)is twice differentiable in xand once differentialble int, applying Itô’s formula yields

dP(t) = d(C(t, S(t))) =n∂C(t, S(t))

∂t +µ∂C(t, S(t))

∂t S(t) + 1

2σ²C(t, S(t))

∂x² S²(t)o dt +σ∂C(t, S(t))

∂x S(t)dB(t). (1.2.6)

On the other hand we have

dP(t) =dH(t) = a^H(t)dS(t) +b^H(t)dZ(t)

= µa^H(t)S(t)) +rb(t)Z(t)

dt+σa^H(t)S(t)dB(t). (1.2.7) It is obvious that equation (1.2.6) and equation (1.2.7) should be equal and this implies

∂C(t, x)

∂t =rxC(t, x)

∂x + 1

2σ²x²∂C(t, x)

∂x² =rC(t, x), (1.2.8) for allx≥0.Equation 1.2.8 is known as the Black & Scholes partial differential equation.

Thus to find the priceP(t)for a claim is equivalent to solve the B&S PDE. We will in the

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following show that the solution of equation (1.2.8) can be represented as the expectation of a random variable.

Theorem 1.2.1. Let Y^t,x(s) for s≥t be a stochastic process defined by Y^t,x(s) =x+

Z s t

rY^t,x(u)du+ Z s

t

σY^t,x(u)dB(u). (1.2.9) then

C(t, x) = e^−r(T^−t)E[f(Y^t,x)(T)] (1.2.10) is the solution to the Black & Scholes partial differential equation (1.2.8).

Proof. Note first the process Y^t,x(s) in (1.2.9) is the same as Y^t,x(t) =xexp

r− 1 2σ²

(s−t) +σ B(s)−B(t)

. (1.2.11)

This can be seen by following arguments.

Define first a function

f(s, y) =xexp

r− 1 2σ²

(s−t) +σy)

,

with y =B^t,0(s), i.e., the Brownian motion in state 0 at time t. Applying Itô’s formula to f(s, y) yields

df(s, B^t,0(s)) = rf(s, B^t,0(s))ds+σf(s, B^t,0(s))dB^t,0(s).

Integrating the above differential equation from time t to time s with initial valuef(t,0) yields

f(s, B^t,0(s))−f(t,0) = Z s

t

rf(u, B^t,0(u))du+ Z s

t

σf(u, B^t,0(u))dB(u).

Let B^t,0(s) :=B(s)−B(t), then Z^t,x(u) = f(u, B^t,0(u)) and f(t,0) = x for all u∈ [t, s], and it follows

Y^t,x(s) =x+ Z s

t

rY^t,x(u)du+ Z s

t

σY^t,x(u)dB(u).

We see Y^t,x(s) is a geometric Brownian motion, and lnY^t,x(T) = lnx+

r− 1 2σ²

(T −t) +σ(B(T)−B(t)), which is a Brownian motion with

E[lnY^t,x(T)] = lnx+ r−σ²

2

(T −t) Var[lnY^t,x(T)] =σ(T −t).

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Hence the density function of Y^t,x(T)is p(t, y) = 1

p2πσ²(T −t)exp

− y−lnx−(r−σ²/2)(T −t))² 2σ²(T −t)

. We claim we can express C(t, x) as

C(t, x) =e^−r(T^−t) Z ∞

−∞

f(e^y)p(t, y;x)dy

In order to differentiate C(t, x) we need to differentiate the densityp, observe that

∂p(t, y)

∂t =p(t, y)n 1

2(T −t) − (r−σ²/2)(y−lnx−(r−σ²/2)(T −t)) σ²(T −t)

− (y−lnx−(r−σ²/2)(T −t))² 2σ²(T −t)²

o ,

∂p(t, y)

∂x =p(t, y)n(y−lnx−(r−σ²/2)(T −t) xσ²(T −t)

o ,

∂p(t, y)

∂x² =p(t, y)

n(y−lnx−(r−σ²/2)(T −t))²

x²σ⁴(T −t)² −(y−lnx−(r−σ²/2)(T −t)) x²σ²(T −t)

o . Hence the we have following equality

∂p(t, y)

∂t +rx∂p(t, y)

∂x + 1

2σ²x²∂p(t, y)

∂x² = 0.

The last step is to differentiate C(t, x) with respect to t, and with the help of the above equality we have

∂C(t, x)

∂t =rC(t, x) +e^−r(T^−t)∂

∂t Z

R

f(e^y)p(t, y)dy

=rC(t, x) +e^−r(T^−t) Z

R

f(e^y)∂p(t, y)

∂t dy

=rC(t, x) +e^−r(T^−t) Z

R

f(e^y)−rx∂p(t, y)

∂x −1

2σ²x²∂²p(t, y)

∂x² dy

=rC(t, x)−rx∂C(t, x)

∂x − 1

2σ²x²∂²C(t, x)

∂x² .

We arrived at the desired partial differential equation and thus proved Theorem 1.2.1.

Black & Scholes formula for European call option

As we mentioned, Black & Scholes gives a closed form expressions for the price of call and put options, in this section, we will show how the Black & Scholes formula for pricing European call option is derived. A European call option written on a stock S with strike K, maturityT has the payoff function

C(T) = max{S(T)−K,0}.

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Then the payoff function g(S(T))becomesg(x) = max(0, x−K).Observe thatσ(B(T)− B(t))is a normal random variable with mean zero and varianceσ²(T −t).It follows then

σ(B(T)−B(t))∼σ√

T −t·Z where Z = (B(T)−B(t))/√

T −t∼ N(0,1). Then the payoff

E[g(Y^t,x)(T)] =E[max(0, Y^t,x(T)−K)] =E[max(0, e^ln^Y^t,x^(T⁾−K)]

=E[max(0, e^lnx+(r−σ²^/2)(T^−t)σ

√T−t·Z−K)]. (1.2.12)

Note when Z <−d2, where

d₂ := ln(x/K) + (r−σ²/2)(T −t) σ√

T −t ,

the random variable inside expectation 1.2.12 becomes zero. Then it follows E[max(0, Y^t,x(T)−K)] =

Z ∞

−d2

e^ln^x+(r−σ²^/2)(T^−t)σ

√

T−t·z−K

φ(z)dz

=xe^r(T^−t) Z ∞

−d₂

e^−σ²^(T^−t)/2+σ

√T−t·z

φ(z)dz (1.2.13)

−K Z ∞

−d2

φ(z)dz, (1.2.14)

where φ is the density for a standard normal random variable, and satisfies P(Z >−d₂) =

Z ∞

−d2

φ(z)dz =P(Z < d₂), Hence the integral (1.2.14) becomes

K Z ∞

−d₂

φ(z)dz =KΦ(d2),

where Φ is the cumulative distribution function for a standard normal random variable.

Making change of variables v =z−σ√

T −t to the integral ( 1.2.13) gives Z ∞

−d2

e^−σ²^(T^−t)/2+σ

√T−t·z

φ(z)dz = Z ∞

−d2

√1

2πe^−(z−σ

√T−t)²/2dz

= Z ∞

−d2−σ√ T−t

√1

2πe^−v²^/2dv= Φ

d₂+σ√

T −t . We conclude our derivations in the following theorem.

Theorem 1.2.2. The price of European call option written on a stock S with strike K and exercise time T is

C(t, x, T, K, σ)_BS =S_tΦ(d₁)−Ke^−r(T^−t)Φ(d₂), (1.2.15)

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where

d₁ =d₂+σ√

T −t (1.2.16)

d₂ = ln(S(t)/K) + (r−σ²/2)(T −t) σ√

T −t . (1.2.17)

The implied volatility in Black & Scholes model

We have seen in Black & Scholes model, the price process of the risky asset is a geometric Brownian motion of which the volatility is constant. However, empirical data shows that options written on the same underlying asset with different strike price K and exercise time T have different implied volatilities. In addition to this fact, we have listed some stylised facts on volatility (e.g. volatility clustering), it altogether indicates that the constant volatility assumption in Black & Scholes framework is erroneous.

1.3 Time-dependent volatility model

We have seen the simplest way to model volatility is to assign a constant value to it, however, we have also seen from the volatility smile that volatility does vary with the exercise timeT.Hence, a simple improvement can be to model volatility as a deterministic function of time. Consider the dynamic of the stock price

dS(t)

S(t) =µdt+σ(t)dW(t). (1.3.1)

Merton(1973) proposed the pricing formula using time dependent volatility. The option price can still be calculated by the Black & Scholes formula, but the volatility is calculated from the annualised variance, and is given by

σ_a = s

1 T −t

Z T t

σ²(τ)dτ , (1.3.2)

whereσ²(t)is the instantaneous variance,RT

t σ²(τ)dτis the variance of log-returnlog(S_T/S_t) from time t to T. We see σa is still a constant but now is dependent of t and T. The distribution of log-return log(S_T/S_t) is then

log(S(T) S(t))∼ N

(µ− 1

2σ_a²)(T −t), σ_a²(T −t)

. (1.3.3)

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The d₁ and d₂ terms appeared in B&S formula becomes d₁ = log_K^S +µ(T −t) + ¹₂RT

t σ²(τ)dτ q

RT

t σ²(τ)dτ

, and (1.3.4)

d₂ = log_K^S +µ(T −t)−¹₂RT

t σ²(τ)dτ qRT

t σ²(τ)dτ

. (1.3.5)

1.4 Local volatility models

Since time dependent volatility can only vary with time deterministically, it neither can capture the empirical characteristics like the leverage effect nor can fit the smile. Hence motivates the next generation model: the local volatility models where volatility is a deterministic function of time and stock price. The structure of local volatility models is as

dS(t)

s(t) =µdt+σdW(t), (1.4.1)

σ=f(t, S(t)). (1.4.2)

These kind of models are known as local volatility models. Note in a local volatility model, the volatility is known once the stock price is known. One simple example can be the instantaneous volatility computed by log-returns of the underlying assets, i.e.,

σ(t, S(t)) = 1

√∆tlnS(t+ ∆t) S(t) .

One advantage of the local volatility model is that, unlike stochastic volatility, it captures the dependency between volatility and time and stock price without adding any extra randomness. Hence theoretically the models are complete and all contingent claims can be perfectly hedged. Local volatility models are also the one-dimensional diffusion process that can fit the smile. We will in the following present some common local volatility models.

Mixture distribution models

One advantage of the mixed distribution model is that they can capture different distributions of the stock price thus is able to capture different empirical characteristics. e.g Bingham and Kiesel (2002) [8] showed that volatility smiles can be generated through

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such models. In this subsection we will look at one application of mixture distribution model in mathematical finance provided by Brigo and Mercurio (2000) [7], they were the first to establish the theoretical connection between volatility and a weighted sum of lognormal distributions. Let P(t, S)be the risk neutral density of a stock price S(t)at time t, and p_i(t, S) be lognormal density for each i∈N, and denotes the i-th component i of the weighted sum. Let w_i(t) be the weight coefficient for each componenti. This mixed distribution model is give by

P(t, S(t)) =

N

X

i=1

wi(t)pi(t, S(t)), where (1.4.3)

N

X

i=1

w_i(t) = 1, w_i(t)≥0 ∀t. (1.4.4)

Assume for each i p_i(t, S(t))has the same mean µbut different varianceσ_t²(t).The stock price is specified by the following stochastic differential equation

dS(t)

S(t) =µdt+σ(t, S(t))dW(t), where (1.4.5) σ²(t, S(t)) =

N

X

i=1

wiσ_i²(t). (1.4.6)

Implied local volatility: Dupire’s formula

Before we move on to introduce Dupire’s implied volatility model, two assumptions should be made first.

Assumptions:

• The set of market prices of European call options of all strikes and maturities is continuous.

• There are no dividends paid to the stock.

Consider the stock price modelled by a risk-neutral diffusion process dS(t) = S(t) rdt+σ(t, S(t))dW^Q(t)

. (1.4.7)

Dupire (1994) [13], Derman and Kani (1994) [14] have showed that for local volatility there exists a unique such risk-neutral process that is consistent with option data, and

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the price of the European call option by risk neutral valuation is given by

C =e^−rTE^Q[S(T)−K]⁺, (1.4.8)

=e^−rT Z ∞

K

(S(T)−K)p(S(T))dS(T), (1.4.9) where P(S(T)) is the risk neutral probability density function for S(T), and K is the strike.

Later on, Breeden and Litzenberger (1978) [3] showed that the risk neutral cumulative distribution function F(·) is given by

∂C

∂K =−e^−rTF(S(T)≥K), (1.4.10)

hence the risk neutral density p(S(T) = K)is given by

∂²C

∂K²e^rT =p(S(T) = K). (1.4.11) Therefore with option data one can retrieve the density p(S(t)).

Theorem 1.4.1. Fokker-Planck equation For a diffusion process defined in 1.4.7, with drift r and diffusion coefficient D(t, S_t) = ^σ²^(t,S₂ ^t⁾, the Fokker-Planck equation for the probability density p(t, s) of the random variable S_t is

∂

∂tp(t, s) =− ∂

∂s[rp(t, s)] + ∂²

∂s²[D(t, s)p(t, s)]. (1.4.12) Dupire (1994) applied equation (1.4.11) to the Fokker-Planck equation (1.4.12) and obtained Dupire’s equation:

∂C

∂T =σ²(S, T)· 1

2 ·S²· ∂²C

∂S² −rS· ∂C

∂S, (1.4.13)

hence

σ(S, T) =

r

_∂C

∂T +rS^∂C_∂S

1

2S²^∂²^C

∂S2

. (1.4.14)

Note that once we have the option price data, the volatility can be obtained by using the above formula. However the shortcoming of this model is that, one needs to differentiate option price with respect to the maturity T and strike K to calculate the volatility, but this requires a continuous set of the option price data. However, in reality the empirical option data is discrete. Hence there is no closed form solution to this model and one solution to this problem is to convert these discrete data into a continuous modification by some interpolation method.

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Constant elasticity of variance model

Another well known example of local volatility model is constant elasticity of variance (CEV) model. Let S(t), t∈ [0, T] be the stock price, the structure of the model is given by

dS(t)

S(t) =µdt+σ(S(t))dW(t), (1.4.15) σ(S(t)) =α S(t)^n/2−1

, (1.4.16)

where n >0, α >0

Equation (1.4.16) implies the variance of return is

v(S(t)) =α²S(t)ⁿ⁻² (1.4.17)

The elasticity of variance _v(S(t)) with respect to the stock price is defined by the relative change in variance divided by the relative change in price:

_v(S(t)) = dv(S(t))/v(S(t))

dS(t)/S(t) = dv(S(t))

dS(t) · S(t) v(S(t))

=α²(n−2)S(t)ⁿ⁻³· S(t) α²S(t)ⁿ⁻²

=n−2. (1.4.18)

From (1.4.18) we know the elasticity of variance in this model is always a constant, hence gives rise to the name of the model. Note when n= 2, S(t)becomes geometric Brownian motion and we retrieve constant volatility. Hence B&S model can be considered as one special case of the CEV model. When n = 1, _v(S(t)) =−1, this is a model proposed by Cox and Ross (1976) [12]. When n <2volatility and stock price is negatively correlated, we will in the following focus on this situation since the leverage effect is automatically captured. When n > 2volatility increases as the stock price increases.

Closed-form solution for European call options using Non-central Chi-square Approach

One advantage of the (CEV) model is that it gives a closed form solution to value option. Here we follow Hsu et al. (2008) [17] and Schroder (1989) [27] to present the approach using non-central Chi-square distribution. Consider the following CEV model

dS_t =µS_tdt+σS_t^n/2dW(t), n <2, (1.4.19)

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where µ=r−a,σ is a constant, we proceed by defining a new process

X(t, S) =S²⁻ⁿ. (1.4.20)

Applying Itô’s formula to X(t, S)with

∂X

∂S = (2−n)S¹⁻ⁿ,∂X

∂t = 0,and, ∂²X

∂S² = (2−n)(1−n)S⁻ⁿ yields dX =

(r−a)(2−n)X+1

2σ²(n−1)(n−2)

dt+σ²(2−n)²XdW. (1.4.21) Applying Kolmogorov’s forward equation to process X gives

∂P

∂t =1 2

∂²

∂X²[σ²(2−n)XP]

− ∂

∂X

[(r−a)(2−n)X+ 1

2σ²(n−1)(n−2)]P . (1.4.22) Hence f(S_T|S_t, T > t) = f(X_T|x_t, T > t)× |J|with J = ^∂X_∂S = (2−n)S¹⁻ⁿ.

Consider a parabolic equation in the following form:

∂P

∂t = ∂²

∂²x(αxP) + ∂

∂x[(βx+h)P], 0< x <∞, (1.4.23) where P = P(t, x), and α > 0, β, h are constants. This parabolic equation can be inter- preted as the Fokker - Planck equation of a diffusion problem with drift term βx+hand the diffusion coefficient αx.

Lemma 1.4.1. Feller’s lemmaLet f(x, t|x₀)be the conditional probability density function for x and t with condition x₀. The explicit form of the fundamental solution to the above parabolic equation is given by

f(t, x|x0) = β α(e^βt)−1

e^−βtx x₀

^(h−α)/2α

×expn−β(x+x₀e^βt) α(e^βt−1)

o I₁₋h

α

2β

α(1−e^−βt)(e^−βtxx₀)^1/2

, (1.4.24) where Iv(x) is the modified Bessel function of the first kind of order v and is defined as

Iv(x) =

∞

X

i=0

(x/2)^2i+v

i!Γ(i+ 1 +v), (1.4.25)

where Γ(y) =R∞

0 x^y−1e^−xdx denotes the Gamma function.

Applying Feller’s lemma to equation (1.4.22) with

• α= ¹₂σ²(2−n)², β=µ(2−n), h= ¹₂σ²(n−2)(1−n), τ = (T −t);

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• x= _T¹, x₀ = ¹_t yields

f(S_T|S_t, T > t) =f_S_T|St=z(x) = (2−n)k^01/(2−n)(xz¹⁻ⁿ)^1/(2(2−n))e^−x−zI ¹

2−n(2(xz)^1/2), (1.4.26)

where k⁰ = ^2µ

σ²(2−n)[e^µ(2−n)τ−1], x = k⁰S_t²⁻ⁿe^µ(2−n)τ, and z = k⁰S_T²⁻ⁿ. Hence we have obtained the transition probability density from S_t to S_T for T > t. Now it is handy to look at some definitions that are necessary for further derivation.

Definition 1.4.1. (central) X²(Chi-square)-distribution Let X₁, X₂. . . X_k be i.i.d. N(0,1)−distributed random variables. Then the random variable Z = Pk

i=1X_i² is said to be X²-distributed with k degrees of freedom (denoted X_(k)² −distribution). The probability density function of Z is given by

f(x;k) =







x^k/2−1e^−x/2

2^k/2Γ(^k₂) , x >0;

0, x <0,

where for x >0,Γ(x) =R∞

0 y^x−1e^−ydy denotes the gamma function.

Definition 1.4.2. Non-central X²-distribution Let X₁, X₂. . . X_k be i.i.d. N(µ_i,1)- distributed random variables. Then the random variable Z = Pk

i=1X_i² is said to be non-central X²-distributed with k degrees of freedom (denoted X_(k,m)² −distribution) and non-central parameter m=Pk

i=1µ_i. The probability density function of Z is given by

f(x;k, m) =







x^k/2−1e^−(x+m)/2 2^k/2

P

∞

i=0

(

^m₄

)

ⁱ ^xⁱ

i!Γ(^k₂+i), x >0;

0, x <0,

where for x >0,Γ(x) =R∞

0 y^x−1e^−ydy denotes the gamma function.

Thus the probability density function of non-central chi-square random variableX_k,m² can be considered as the mixed distribution of central Chi-square probability density functions.

The cumulative distribution of X_(k,m)² is then

F(x;m, k) = P(X_k,m² ≤x) (1.4.27)

=e^−m/2

∞

X

i=0

(m/2)ⁱ

i!2^k/2 Γ(k/2 +i) Z x

0

y^k/2+i−1e^−y/2dy, x >0. (1.4.28)

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Rearranging the terms in (1.4.28) we obtain an alternative expression for F(x;k, m) F^∗(x;k, m) =

∞

X

i=0

(m/2)ⁱe^−m/2 i!

P(X_k+2i² ≤x). (1.4.29) The complementary distribution function of X_k,m² is then

Q(x;k, m) = 1−F(x;k, m). (1.4.30) Rearranging the terms in (1.4.2) leads to an alternative expression fro the probability density function of X_(k,m)² forx >0

f^∗(x;k, m) = 1 2(x

m)^(k−2)/4exp

− (m+x) 2 I^k−2

2 (√

mx), x > 0, (1.4.31) where

I_v(y) = (1 2y)^v

∞

X

i=0

(y²/4)ⁱ

i!Γ(v+i+ 1) (1.4.32)

is the modified Bessel function of the first kind of order v. And the complementary distribution function obtained from this alternative expression of probability density is then

Q^∗(x;k, m) = Z ∞

x

f^∗(x;k, m)(y)dy. (1.4.33) Using the transition probability function in (1.4.26) gives the pricing formula of call option:

C=E[max(0, S_T −K)]

=e^−rτ Z ∞

K

f(S_T|S_t, T > t)(S_T −K)dS_T

=e^−rτ Z ∞

K

S_Tf(S_T|S_t, T > t)dS_T −e^−rτK Z ∞

K

f(S_T|S_t, T > t)dS_T

=A₁−A₂, (1.4.34)

where τ =T −t. Making change of variable with z =k⁰S_T²⁻ⁿ and u=k⁰K²⁻ⁿ yields dS_T = (2−n)⁻¹k^{0−1/(2−n)}z(n−1)/(2−n)

dz, (1.4.35)

A₁ =e^−rτ Z ∞

u

e^−x−z(x/z)^1/(4−2n)I ¹

2−n(2√

xz)(z/k⁰)^1/(2−n)dz

=e^−rτ(x/k⁰)^1/(2−n) Z ∞

u

e^−x−z(x/z)^{−1/(4−2n)}I ¹

2−n(2√ xz)dz

=e^−rτS_te^µτ Z ∞

u

e^−x−z(z/x)^1/(4−2n)I ¹

2−n(2√ xz)dz

=e^−aτS_t Z ∞

u

e^−x−z(z/x)^1/(4−2n)I ¹

2−n(2√

xz)dz, (1.4.36)

A₂ =Ke^−rτ Z ∞

u

x⁴⁻²ⁿ¹ z(1−2n+2n−2)/(4−2n)

e^−x−nI ¹

2−n(2√ xz)dz

=Ke^−rτ Z ∞

u

e^−x−z(x/z)^1/(4−n)I ¹

2−n(2√

xz)dz. (1.4.37)

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Using the complementary distribution function given in (1.4.33), and making change of variables with z⁰ = 2z and x⁰ = 2x gives

A₁ =e^−aτS_t Z ∞

u

e^(−x−z)/2(z/x)^1/(4−2n)I ¹

2−n(2√ xz)dz

=e^−aτS_t Z ∞

2u

e^−(x⁰^+z⁰^)/2(z⁰

x⁰)^1/(4−2n)I ¹

2−n(2√ x⁰z⁰)1

2dz⁰

=e^−aτStQ(2u;k, x⁰)

=e^−aτS_tQ(2u; 2 + 2

2−n,2x), (1.4.38)

A2 =Ke^−rτ Z ∞

u

e^−x−z(x/z)^1/(4−n)I ¹

2−n(2√ xz)dz

=Ke^−rτ Z ∞

2u

e^−(x⁰^+z⁰^)/2(x⁰

z⁰)^1/(4−2n)I ¹

2−n(s√ x⁰z⁰)1

2dz⁰

=Ke^−rτQ(2u; 2− 2

2−n,2x). (1.4.39)

Hence we have

C=A₁−A₂

=e^−aτS_tQ(2u; 2− 2

2−n,2x) +Ke^−rτQ(2u; 2− 2

2−n,2x). (1.4.40) For the case when k = 2− _2−n² is negative when n <2, see [17].

It is worth noting that this formula and the B&S formula are essentially in the same form. Instead of using the cumulative normal distribution in B&S formula, now we use the complementary non-central Chi-sqaure distribution.

1.5 Stochastic volatility models

Although the local volatility models allow volatility vary with strike K and maturity T, they captures only few empirical characteristics of volatility and still have some disadvantages. For example, since volatility is a deterministic function of time and stock prices, it is perfectly correlated with stock price, that is, the absolute value of the correlation between volatility and the stock price is 1. However, there is no perfect correlation empirically observed. Moreover, volatility clustering is not captured in local volatility models. These defects motivated the next generation of the volatility modelling, which is the stochastic volatility.

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Consider the following stochastic volatility model dS(t)

S(t) =µ_tdt+σ(t)dW(t)₁. (1.5.1) The stock price is driven by the Wiener process W₁ and µ_t is the drift term. However the volatility σ(t) now is also stochastic. There are lots of different suggestions for σ(t).

A typical stochastic volatility process is governed by a stochastic differential equation, which is driven by another Wiener process W₂ that is correlated with W₁. i.e.,

corr(W₁(t), W₂(t)) =ρ. (1.5.2) Note that ρ can take any value in [-1, 1] for different purpose. For example, in currency markets, ρis close to 0, but in order to capture the leverage effect, ρmust be negative. In some models ρ can be a function of time. Compare to the previous three generations of volatility models, the stochastic volatility models have a lot more advantages. Firstly, they capture more empirical characteristics such as heavy tail, skewness as well as volatility clustering and leverage effect by choosing a negative value for ρ. Secondly stochastic volatility has higher variability and is able to capture extreme events like the dramatic changes of volatility during financial crisis. The disadvantages of stochastic volatility are mainly due to the extra randomness from the second stochastic process W₂. Stochastic volatility serves less practical purposes since the market is no longer complete, therefore not all contingent claims can be perfectly hedged and the price can not be determined uniquely. Furthermore, it is common that stochastic models do not have closed form of solution for option price, so the prices must be calculated by simulation.

In this subsection, we will give a survey on some well known stochastic volatility models.

• mean-reverting models

The driving process of many stochastic volatility models is mean-reverting. Mean- reversion means that the volatility tends to return to its mean over time. The Ornstein-Uhlenbeck process is a typical mean-reverting process and its dynamics is specified as

dY(t) =α(m−Y(t))dt+βdW₂(t), (1.5.3)

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where β ≥ 0, and β, µ₂ are constants, α is the rate of reversion which controls the degree of volatility clustering. Using Itô’s lemma gives the solution

Y(t) = m+ (Y(0)−m)e^−αt+β Z t

0

e^−αtdW₂(t). (1.5.4) Then a large class of stochastic volatility can be represented as a function of Y,i.e.,

σ(t) = f(Y). (1.5.5)

• Jonson and Shanno (1987)

dS(t) =µ₁S(t)dt+σ(t)S(t)^κdW₁(t), (1.5.6) dσ(t) =µ₂σ(t)dt+δσ^β(t)dW₂(t), (1.5.7) where κ, β are nonnegative constants.

• Scott (1987)

dS(t)

S(t) =µdt+σ(t)dW₁(t), (1.5.8)

dσ(t) =α(m−σ(t))dt+βdW₂(t). (1.5.9) In this model, Scott assumes ρ= 0 to compute the price of options.

• Hull-White model (1987) dS(t)

S(t) =µ1dt+σ(t)dW1(t), (1.5.10) dσ²(t)

σ²(t) =µ2dt+δdW2(t). (1.5.11) Hull-White model provides a closed form solution to European option prices when ρ= 0.The solution is obtained by using Black & Scholes formula (1.2.15) with

σ(t) = s

1 T −t

Z T t

σ²(s)ds.

• Heston model (1993) dS(t)

S(t) =µdt+σ(t)dW₁(t) (1.5.12)

dσ²(t) = α(m−σ²(t))dt+βσ(t)dW₂(t). (1.5.13) The dynamics of varianceσ² in (1.5.13) is a CIR interest model (1985) [10]. Heston model is a significant model because it provides an analytical solution for European options, but due to the incompleteness of the market, the risk neutral measure has to be specified first.

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Chapter 2 Fractional Brownian motion

In the previous chapter we surveyed some asset pricing models and volatility dynamics which are based on Brownian motion. However, in the next chapter, we are going to see that the rough volatility in Gatheral et al. (2014) [16] is governed by a stochastic differential equation driven by fractional Brownian motion (fBm). Hence in this chapter we will describe some mathematical preliminaries of fBm, and introduce the Mandelbrot- Van Ness representation of fBm that is used for a pricing model which is based on rough volatility in chapter 4. Furthermore, we will also introduce representation of fBm on an interval. At the end of this chapter, we will present one method to simulate fBm, together with the corresponding algorithm. The core knowledge in this chapter follows closely [24]

and [28] .

2.1 Elementary properties of fBm

Definition 2.1.1. fractional Brownian motion A centered Gaussian process B^H = {B_t^H, t∈R}is called a two-sided fractional Brownian motion (fBm) with Hurst parameter H ∈(0,1) if it has the covariance function

R_H(t, s) =E[B_t^HB_s^H] = 1

2(s^2H +t^2H − |t−s|^2H) s, t ∈R. (2.1.1) For a Gaussian process, its distribution is uniquely determined by its mean and covariance function. Therefore, the distribution of a fBm is uniquely specified by the above definition.

Definition 2.1.2. homogeneous function A homogeneous function f of variables x

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and y is a real-valued function that satisfies

f(tx, ty) =t^kf(x, y),

for some constant k and all real numbers t. The constant k is called the degree of homo- geneity.

The covariance function of fBm is then homogeneous of order2H. From this property it can be deduced that fBm is H self-similar, that is, forα >0, {B_αt^H, t ∈R}has the same distribution as {α^HB_t^H, t∈R}. It can be derived from equation (2.1.1) that

E[|B_t^H −B_s^H|²] =|t−s|^2H, s, t ∈R. (2.1.2) This implies that fBm has stationary increments.

From equality (2.1.2) one can establish the continuity of fBm with the help of Kolmogorov- Chentsov continuity theorem.

Theorem 2.1.1. Kolmogorov-Chentsov Continuity Theorem. Assume that for a stochastic process {Xt, t ≥0} such that for s, t ∈[0, T] and T ≥0,

E[|X_t−X_s|^p]≤D|t−s|^1+β,

with positive constants p, β, D. Then there exists a continuous modification Xˆ of X, i. e.

1. Xˆ has continuous paths.

2. P(X_t = ˆX_t) = 1 for t∈[0, T].

Furthermore, the process Xˆ is κ-Hölder continuous, i. e., for any 0< κ < ^β_p, sup

0≤s<t≤T

|Xˆt−Xˆs| (t−s)^κ <∞.

Corollary 2.1.1. The fractional Brownian motion {B_t^H, t ∈R} has a modification with continuous trajectories. Moreover, for any κ ∈ (0, H) this modification is κ-Hölder continuous on each finite interval.

Proof. We have E[|B_t^H −B_s^H|^p] =Dp|t−s|^pH since the increment(B_t^H −B_s^H)is centered Gaussian variable with variance|t−s|^H.Then the continuous modification is obtained by taking p > _H¹. Moreover, we have the Hölder continuity with exponent κ ∈(0, H −1/p), the desired statement is achieved by choosing psufficiently large.

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Note that whenH = ¹₂, we retrieve the standard Brownian motionB

1 2

t with covariance functionE[B

1 2

t B

1

s2] = min(|t|,|s|)fort, s∈Rand independent increments. However, when H ∈ (0,¹₂)∪(¹₂,1), the increments of the fBm on disjoint intervals are not independent.

We will show this in the following way.

From (2.1.1), the covariance between two increments (Bt+h −Bt) and (Bs+h −Bs) where s+h≤t, and t−s =nh is

R_H(n) =E[(B_t+h−B_t)((B_s+h−B_s))] = 1

2h^2H((n+ 1)^2H + (n−1)^2H −2n^2H) (2.1.3)

∼h^2HH(2H−1)n^2H−2 →0, as n→ ∞. (2.1.4)

It follows immediately that:

1. WhenH ∈(0,¹₂), R_H(n)<0 and P∞

n=1|R_H(n)|<∞.

2. WhenH ∈(¹₂,1), R_H(n)>0 and P∞

n=1R_H(n) =∞.

Hence in both cases, the increments of fBm are not independent.

Another way to understand the dependence of increments of fBm is as follows: Let s1 < t1 < s1 < t2, so that[s1, t1],[s2, t2]are two disjoint intervals. Consider the covariance of fBm on these two time intervals

E[(B_t^H₁ −B_s^H₁)(B_t^H₂ −B_s^H₂)]

= 1

2 |t₁−s₂|^2H +|t₂−s₁|^2H − |t₂−t₁|^2H − |s₂−s₁|^2H

. (2.1.5)

Let g(x) = x^2H, ξ₁ =t₂−s₁, ξ₂ =t₂−t₁, ζ₁ = s₂−s₁, ζ₂ = s₂−t₁, then the right-hand side of (2.1.5) can be written as

1

2((g(ξ₁)−g(ξ₂))−(g(ζ₁)−g(ζ₂))).

Now it is easily to see that the following holds:

1. WhenH ∈(0,¹₂), g(x) is concave, andE[(B_t^H₁ −B_s^H₁)(B_t^H₂ −B_s^H₂)]<0.

2. WhenH ∈(¹₂,1), g(x) is convex, and E[(B_t^H

1 −B_s^H

1)(B_t^H

2 −B^H_s

2)]>0.

From both of the above two analyses, we can see that in the first case the increments are negatively correlated, and the process present counter-persistence, that is, if the increment is increasing in the past, it is more likely to decrease in the future. Hence the paths are

Rough Volatility Modelling