Asset Pricing Models with

L´evy Processes Xuecan Cui &

Jang Schiltz Introduction Asset Pricing Model with Time-varying L´evy Processes Decomposing S&P500 index Summary

### Asset Pricing Models with Underlying

### Time-varying L´

### evy Processes

### Bachelier World Congress 2016, New York

### Xuecan CUI

### Jang SCHILTZ

### University of Luxembourg

### July 15th, 2016

Asset Pricing Models with

L´evy Processes Xuecan Cui &

Jang Schiltz Introduction Asset Pricing Model with Time-varying L´evy Processes Decomposing S&P500 index Summary 1

### Introduction

2

### Asset Pricing Model with Time-varying L´

### evy Processes

3

### Decomposing S&P500 index

Asset Pricing Models with

L´evy Processes Xuecan Cui &

Jang Schiltz Introduction Asset Pricing Model with Time-varying L´evy Processes Decomposing S&P500 index Summary

### Literature

### Time-varying Jump Diffusion Framework

Time-varying volatility: Empirical studies on the statistical properties of realized and/or implied volatilities have given rise to various stochastic volatility models in the literature, such as the Heston model, CEV models and also stochastic volatility models with jumps etc.

Existence of jumps is empirically supported: Carr and Wu (2003), Pan (2002).

Asset Pricing Models with

L´evy Processes Xuecan Cui &

Jang Schiltz Introduction Asset Pricing Model with Time-varying L´evy Processes Decomposing S&P500 index Summary

### Literature

### Time-varying Jump Diffusion Framework

Time-varying volatility: Empirical studies on the statistical properties of realized and/or implied volatilities have given rise to various stochastic volatility models in the literature, such as the Heston model, CEV models and also stochastic volatility models with jumps etc.

Existence of jumps is empirically supported: Carr and Wu (2003), Pan (2002).

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Literature

### Time-varying Jump Diffusion Framework

Time-varying volatility: Empirical studies on the statistical properties of realized and/or implied volatilities have given rise to various stochastic volatility models in the literature, such as the Heston model, CEV models and also stochastic volatility models with jumps etc.

Existence of jumps is empirically supported: Carr and Wu (2003), Pan (2002).

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Motivation & Contribution

Previous studies rely on a specific model structure for volatility and jumps (e.g. Santa-Clara and Yan 2010).

We introduce a general non-parametric time-varying jump diffusion framework as a natural generalisation of the results from literature (Bollerslev, Todorov and Xu, 2015 JFE). Theoretical part: We assume a time-varying L´evy process, with time-varying drift, volatility and jump intensity parameters, to model the jump diffusion economy. We study an equilibrium asset and option pricing model in this economy.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Motivation & Contribution

Previous studies rely on a specific model structure for volatility and jumps (e.g. Santa-Clara and Yan 2010).

We introduce a general non-parametric time-varying jump diffusion framework as a natural generalisation of the results from literature (Bollerslev, Todorov and Xu, 2015 JFE).

Theoretical part: We assume a time-varying L´evy process, with time-varying drift, volatility and jump intensity parameters, to model the jump diffusion economy. We study an equilibrium asset and option pricing model in this economy.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Motivation & Contribution

Previous studies rely on a specific model structure for volatility and jumps (e.g. Santa-Clara and Yan 2010).

We introduce a general non-parametric time-varying jump diffusion framework as a natural generalisation of the results from literature (Bollerslev, Todorov and Xu, 2015 JFE). Theoretical part: We assume a time-varying L´evy process, with time-varying drift, volatility and jump intensity parameters, to model the jump diffusion economy. We study an equilibrium asset and option pricing model in this economy.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Motivation & Contribution

We introduce a general non-parametric time-varying jump diffusion framework as a natural generalisation of the results from literature (Bollerslev, Todorov and Xu, 2015 JFE). Theoretical part: We assume a time-varying L´evy process, with time-varying drift, volatility and jump intensity parameters, to model the jump diffusion economy. We study an equilibrium asset and option pricing model in this economy.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Model Build-up

### Stock Market

An investment of St in the stock market is governed by: d St

St−

= µ(t)dt + σ(t)d Bt+ (ex− 1)d Nt− λ(t)E (ex− 1)dt, (1)
where St− is the value of S_{t} before a possible jump occurs;

µ(t) and σ(t) are the rate of return and the volatility of the investment.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Model Build-up

### Stock Market

An investment of St in the stock market is governed by: d St

St−

= µ(t)dt + σ(t)d Bt+ (ex− 1)d Nt− λ(t)E (ex− 1)dt, (1)
where St− is the value of S_{t} before a possible jump occurs;

µ(t) and σ(t) are the rate of return and the volatility of the investment.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Money Market Account

We further assume that there is a market for instantaneous borrowing and lending at a risk-free rate r (t). The money market account, Mt, follows

d Mt Mt

= r (t)dt. (2)

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Representative Investor

Maximize the expected utility function of life time consumption

max ct Et Z T t p(t)U(ct)dt, where ct is the rate of consumption at time t, U(c) the utility function with U0> 0, U00< 0, and p(t) ≥ 0, 0 ≤ t ≤ T the time preference function.

Assume constant relative risk aversion (CRRA) utility function.

U(c) = (

c1−γ

1−γ γ > 0, γ 6= 1, ln c, γ = 1,

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Representative Investor

Maximize the expected utility function of life time consumption

max ct Et Z T t p(t)U(ct)dt, where ct is the rate of consumption at time t, U(c) the utility function with U0> 0, U00< 0, and p(t) ≥ 0, 0 ≤ t ≤ T the time preference function.

Assume constant relative risk aversion (CRRA) utility function.

U(c) = (

c1−γ

1−γ γ > 0, γ 6= 1, ln c, γ = 1,

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Total Wealth

The total wealth of the representative investor at time t: Wt = W1t+ W2t

where W1t= ωWt is invested in the stock market, and W2t= (1 − ω)Wt is invested in the money market.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Total Wealth

The total wealth of the representative investor at time t: Wt = W1t+ W2t

where W1t= ωWt is invested in the stock market, and W2t= (1 − ω)Wt is invested in the money market.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

Representative Investor’s Optimal Control Problem:

max ct,ω Et Z T t p(t)U(ct)dt, (3) subject to d Wt Wt = ωd St St− +(1−ω)d Mt Mt − ct Wt dt = [r (t) + ωµ(t) − ωr (t) − ωλ(t)E (ex− 1) − ct Wt ]dt + ωσ(t)d Bt + ω(ex− 1)dNt,

where φ(t) = µ(t) − r (t) is the equity premium.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

Representative Investor’s Optimal Control Problem:

max ct,ω Et Z T t p(t)U(ct)dt, (3) subject to d Wt Wt = ωd St St− +(1−ω)d Mt Mt − ct Wt dt = [r (t) + ωµ(t) − ωr (t) − ωλ(t)E (ex− 1) − ct Wt ]dt + ωσ(t)d Bt + ω(ex− 1)dNt,

where φ(t) = µ(t) − r (t) is the equity premium.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Equity Premium

### Proposition

In the production economy with jump diffusion and one representative investor with CRRA utility function, the equilibrium equity premium is given by

φ(t) = φσ(t) + φJ(t),

where φσ(t) = γσ(t)2 -diffusion risk premium

φJ(t) = λ(t)E [(1 − e −γx

)(ex− 1)] -jump risk premium

The risk-free rate is a time-varying function:

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Equity Premium

### Proposition

In the production economy with jump diffusion and one representative investor with CRRA utility function, the equilibrium equity premium is given by

φ(t) = φσ(t) + φJ(t),

where φσ(t) = γσ(t)2 -diffusion risk premium

φJ(t) = λ(t)E [(1 − e−γx)(ex− 1)] -jump risk premium

The risk-free rate is a time-varying function:

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Equity Premium

### Proposition

In the production economy with jump diffusion and one representative investor with CRRA utility function, the equilibrium equity premium is given by

φ(t) = φσ(t) + φJ(t),

where φσ(t) = γσ(t)2 -diffusion risk premium

φJ(t) = λ(t)E [(1 − e−γx)(ex− 1)] -jump risk premium

The risk-free rate is a time-varying function:

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### General Pricing Kernel

Proposition

The pricing kernel is given by

d πt

πt

= −r (t)dt − γσ(t)dBt+ (ey− 1)dNt− λ(t)E (ey− 1)dt,

or equivalently, after integration

πT
πt
= exp{−
ZT
t
γσ(s)dBs−
ZT
t
[r (s) +1
2γ
2_{σ}2_{(s)]ds − E (e}y_{− 1)}ZT
t
λ(s)ds +
Nt,T
X
i =1
yi}.

The random variable y modeling the jump size in the logarithm of
the pricing kernel, satisfies E [(ey_{− e}−γx

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### General Pricing Kernel

Proposition

The pricing kernel is given by

d πt

πt

= −r (t)dt − γσ(t)dBt+ (ey − 1)dNt− λ(t)E (ey− 1)dt,

or equivalently, after integration

πT
πt
= exp{−
ZT
t
γσ(s)dBs−
ZT
t
[r (s) +1
2γ
2_{σ}2_{(s)]ds − E (e}y_{− 1)}ZT
t
λ(s)ds +
Nt,T
X
i =1
yi}.

The random variable y modeling the jump size in the logarithm of
the pricing kernel, satisfies E [(ey_{− e}−γx

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### General Pricing Kernel

Proposition

The pricing kernel is given by

d πt

πt

= −r (t)dt − γσ(t)dBt+ (ey − 1)dNt− λ(t)E (ey− 1)dt,

or equivalently, after integration

πT
πt
= exp{−
ZT
t
γσ(s)dBs−
ZT
t
[r (s) +1
2γ
2_{σ}2_{(s)]ds − E (e}y_{− 1)}ZT
t
λ(s)ds +
Nt,T
X
i =1
yi}.

The random variable y modeling the jump size in the logarithm of
the pricing kernel, satisfies E [(ey_{− e}−γx

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### General Pricing Kernel

Proposition

The pricing kernel is given by

d πt

πt

= −r (t)dt − γσ(t)dBt+ (ey − 1)dNt− λ(t)E (ey− 1)dt,

or equivalently, after integration

_{σ}2_{(s)]ds − E (e}y_{− 1)}ZT
t
λ(s)ds +
Nt,T
X
i =1
yi}.

_{− e}−γx

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### European Call

### Proposition

### The price of a European call, c(S

t### , t), in the jump diffusion

### economy satisfies

∂c(St, t) ∂t + 1 2σ 2_{(t)S}2 t ∂2

_{c(S}t, t) ∂S2 + [r (t) − λ Q

_{(t)E}Q

_{(e}x

_{− 1)]S}t ∂c(St, t) ∂S − r (t)c(St, t) + λQ(t){EQ[c(Stex, t)] − c(St, t)} = 0,

with final condition

c(ST, T ) = max (ST− K , 0),

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### European Call

### Proposition

### Pricing formula of a European call option:

c(St, t) =
+∞
X
n=0
e−RtTλ
Q_{(s)ds}(R_{t}TλQ(s)ds)n
n! E
Q
n[cBS(SeXe−E
Q_{(e}x_{−1)}RT
t λ
Q_{(s)ds}
, t)],

where cBS_{(S , t) is the Black-Scholes formula price for the European}
call option and

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Empirical Part

### Decompose the S&P500 Index into time-varying components,

### using

### the Hodrick-Prescott Filter

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Hodrick-Prescott Filter

### The Hodrick-Prescott filters was first proposed in

### Whittaker (1923), then popularized in economics by

### Hodrick and Prescott (1997).

### The method serves to decompose the time series

### y

t### = ln(S

t### ) into a trend component τ

t### , and a cyclical

### component c

t### :

### y

t### = τ

t### + c

t### , for t = 1, . . . , T .

### Condition: For a given a, τ

t### satisfies

### min

τ### (

T### X

t=1### (y

t### − τ

t### )

2### + a

T −1### X

t=2### [(τ

t+1### − τ

t### ) − (τ

t### − τ

t−1### )]

2### ),

### where a = 129600 for monthly data

1### .

1

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Hodrick-Prescott Filter

### The Hodrick-Prescott filters was first proposed in

### Whittaker (1923), then popularized in economics by

### Hodrick and Prescott (1997).

### The method serves to decompose the time series

### y

t### = ln(S

t### ) into a trend component τ

t### , and a cyclical

### component c

t### :

### y

t### = τ

t### + c

t### , for t = 1, . . . , T .

### Condition: For a given a, τ

t### satisfies

### min

τ### (

T### X

t=1### (y

t### − τ

t### )

2### + a

T −1### X

t=2### [(τ

t+1### − τ

t### ) − (τ

t### − τ

t−1### )]

2### ),

### where a = 129600 for monthly data

1### .

1

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Hodrick-Prescott Filter

### The Hodrick-Prescott filters was first proposed in

### Whittaker (1923), then popularized in economics by

### Hodrick and Prescott (1997).

### The method serves to decompose the time series

### y

t### = ln(S

t### ) into a trend component τ

t### , and a cyclical

### component c

t### :

### y

t### = τ

t### + c

t### , for t = 1, . . . , T .

### Condition: For a given a, τ

t### satisfies

### min

τ### (

T### X

t=1### (y

t### − τ

t### )

2### + a

T −1### X

t=2### [(τ

t+1### − τ

t### ) − (τ

t### − τ

t−1### )]

2### ),

### where a = 129600 for monthly data

1### .

1

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Extract Drift

### Data:

### S&P500 index, daily, 1985 - 2014.

### In each month, we use the 5% to 95% quantile of ln(S

t### ),

### compute the mean as a monthly data input for HP filter.

### As a result, we decompose the stock index into a time-varying

### trend component T and a component C :

### ln(S

t### ) = T + C .

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Extract Drift

### Data:

### S&P500 index, daily, 1985 - 2014.

### In each month, we use the 5% to 95% quantile of ln(S

t### ),

### compute the mean as a monthly data input for HP filter.

### As a result, we decompose the stock index into a time-varying

### trend component T and a component C :

### ln(S

t### ) = T + C .

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Time-varying Drift

Figure 1.mean (×10−5) volatility skewness kurtosis ∆ ln(S) 31.9 0.0115 -1.3044 31.8

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### By taking the difference of the time-varying trend (∆T ), we

### can observe that:

### Regime Switching: Before 2000, stock return was positive.

### However, after 2000 we can observe that it fluctuates

### around zero.

### Volatility/Jump Clustering: In negative return periods,

### there exists jumps and volatility clustering. By contrast, in

### positive return period, volatility/jump process is much less

### volatile.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtering Problems

### For filtering problem, the data is generated by the state

### space model, which consists of the observation and state

### evolution equations,

Observation equation: yt = f (xt, y t) State evolution: xt+1= g (xt, xt+1),

### where

y_{t+1}

### is the observation error or “noise”, and

x_{t+1}

### are state shocks.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtering Problems

### For filtering problem, the data is generated by the state

### space model, which consists of the observation and state

### evolution equations,

Observation equation: yt = f (xt, y t) State evolution: xt+1= g (xt, xt+1),

### where

y_{t+1}

### is the observation error or “noise”, and

x_{t+1}

### are state shocks.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Particle Filters

### Particle filters use a sampling approach with a set of

### particles to represent the posterior density of a latent state

### space (Johannes, Polson and Stroud 2009, RFS).

### They are simulation-based estimation methods, which

### include a set of algorithms that estimate the posterior

### density by directly implementing the Bayesian recursion

### equations.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Particle Filters

### Particle filters use a sampling approach with a set of

### particles to represent the posterior density of a latent state

### space (Johannes, Polson and Stroud 2009, RFS).

### They are simulation-based estimation methods, which

### include a set of algorithms that estimate the posterior

### density by directly implementing the Bayesian recursion

### equations.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Particle Filters

### Particle filters use a sampling approach with a set of

### particles to represent the posterior density of a latent state

### space (Johannes, Polson and Stroud 2009, RFS).

### They are simulation-based estimation methods, which

### include a set of algorithms that estimate the posterior

### density by directly implementing the Bayesian recursion

### equations.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### SIR Algorithm

### The Sampling Importance Resampling (SIR) algorithm is a

### classical particle filtering algorithm developed by Gordon,

### Salmond, and Smith (1993).

### SIR includes two steps: given samples from p

N### (x

t### |y

t### ),

S1. Propagation: for i = 1, ..., N, draw
x_{t+1}(i ) ∼ p(xt+1|x
(i )
t ).
S2. Resampling: for i = 1, ..., N,
draw z(i )_{∼ Mult}
N(w
(1)
t+1, . . . , w
(N)
t+1),

with importance sampling weights w_{t+1}(i ) = p(yt+1|xt+1(i ))

PN

l =1p(yt+1|xt+1(l ))

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### SIR Algorithm

### The Sampling Importance Resampling (SIR) algorithm is a

### classical particle filtering algorithm developed by Gordon,

### Salmond, and Smith (1993).

### SIR includes two steps: given samples from p

N### (x

t### |y

t### ),

S1. Propagation: for i = 1, ..., N, draw
x_{t+1}(i ) ∼ p(xt+1|x
(i )
t ).
S2. Resampling: for i = 1, ..., N,
draw z(i )_{∼ Mult}
N(w
(1)
t+1, . . . , w
(N)
t+1),

with importance sampling weights w_{t+1}(i ) = p(yt+1|x(i )t+1)

PN

l =1p(yt+1|xt+1(l ))

,
and set x_{t+1}(i ) = xz(i )

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### State Variable of Volatility

### As a state space model (for volatility) is necessary to

### implement particle filters, we assume that following

### dynamics for the stochastic variance:

### d ν

t### = k(θ − ν

t### )dt + σ

ν### √

### ν

t### dB

tν### ,

### where ν

t### is a mean-reverting stochastic process. B

tν### is a

### Brownian motion correlated with B

t### , with correlation

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filter out Volatility

### Based on the result from HP filter, we further apply SIR to

### decompose the time-varying volatility and jump

### (component C ).

### We start with particle filters under stochastic volatility

### (SV) model without jumps, then we apply particle filters

### under stochastic volatility and jump (SVJ) model.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filter out Volatility

### Based on the result from HP filter, we further apply SIR to

### decompose the time-varying volatility and jump

### (component C ).

### We start with particle filters under stochastic volatility

### (SV) model without jumps, then we apply particle filters

### under stochastic volatility and jump (SVJ) model.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filter out Volatility

### Based on the result from HP filter, we further apply SIR to

### decompose the time-varying volatility and jump

### (component C ).

### We start with particle filters under stochastic volatility

### (SV) model without jumps, then we apply particle filters

### under stochastic volatility and jump (SVJ) model.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtered Volatility Processes I - SV model

We run particle filters under SV model three times. Estimated volatilities stay around 0.34-0.35. However, the pattern of the volatility processes varies each time.

Note that the hump shape on the left sides are caused by an adaptation period (around 200 initial data points) needed by the algorithm.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtered Volatility Processes II - SVJ model

Following Eraker, Johannes and Polson (2003), we assume a jump intensity of λ = 0.006, meaning 1 to 2 jumps per year; the jump size follows a normal distribution.

With the SVJ model, filtered volatilities decrease to 0.2-0.25, as jumps account for some of the excess variance.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtered Volatility Processes II - SVJ model

Following Eraker, Johannes and Polson (2003), we assume a jump intensity of λ = 0.006, meaning 1 to 2 jumps per year; the jump size follows a normal distribution.

With the SVJ model, filtered volatilities decrease to 0.2-0.25, as jumps account for some of the excess variance.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtered Volatility Processes II - SVJ model

Following Eraker, Johannes and Polson (2003), we assume a jump intensity of λ = 0.006, meaning 1 to 2 jumps per year; the jump size follows a normal distribution.

With the SVJ model, filtered volatilities decrease to 0.2-0.25, as jumps account for some of the excess variance.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Filtered Volatility Processes II - SVJ model

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Future Research

### The decomposition of the time-varying component of

### drift, volatility and jumps from S&P500 index using HP

### filter and particle filter is still a preliminary result.

### Here we studied the SVJ model only with fixed jump

### intensity; another possibility is to consider the jump

### intensity as a time-varying function, for example a

### function of time-varying drift or volatility, or some other

### possible exogenous determinant.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Future Research

### The decomposition of the time-varying component of

### drift, volatility and jumps from S&P500 index using HP

### filter and particle filter is still a preliminary result.

### Here we studied the SVJ model only with fixed jump

### intensity; another possibility is to consider the jump

### intensity as a time-varying function, for example a

### function of time-varying drift or volatility, or some other

### possible exogenous determinant.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Future Research

### The decomposition of the time-varying component of

### drift, volatility and jumps from S&P500 index using HP

### filter and particle filter is still a preliminary result.

### Here we studied the SVJ model only with fixed jump

### intensity; another possibility is to consider the jump

### intensity as a time-varying function, for example a

### function of time-varying drift or volatility, or some other

### possible exogenous determinant.

Asset Pricing Models with

L´evy Processes Xuecan Cui &

### Main References

### Bjørn Eraker, Micheal S. Johannes, Nicolas G. Polson

### (2003). The Impact of Jumps in Volatility and Returns,

### The Journal of Finance, Vol. LVIII, No.3 (June 2003).

### Robert J. Hodrick, Edward C. Prescott (1997). Postwar

### U.S. Business Cycles: An Empirical Investigation, Journal

### of Money, Credit and Banking, Vol. 29, No.1 (Feb 1997).

### Micheal S. Johannes, Nicolas G. Polson, and Jonathan R.

### Stroud (2009). Optimal Filtering of Jump Diffusions:

### Extracting Latent States from Asset Prices, The Review of

### Financial Studies, Vol. 22, No. 7.

Asset Pricing Models with

L´evy Processes Xuecan Cui &