Recent Advances in Fractional Stochastic Volatility Models
Transcript of Recent Advances in Fractional Stochastic Volatility Models
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Recent Advances inFractional Stochastic Volatility Models
Alexandra ChronopoulouIndustrial & Enterprise Systems EngineeringUniversity of Illinois at Urbana-Champaign
IPAM National Meetingof Women in Financial Mathematics
April 28, 2017
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Stochastic Volatility
– {St; t ∈ [0, T ]}: Stock Price Process
dSt = µStdt+ σ(Xt) St dWt,
σ(·) is a deterministic function and Xt is a stochastic process.
– Assumption: Xt is modeled by a diffusion driven by noise other thanW .
– Log-Normal : dXt = c1 Xt dt+ c2 Xt dZt,
– Mean Reverting OU: dXt = α (m−Xt) dt+ β dZt,
– Feller/Cox–Ingersoll–Ross (CIR): dXt = k (ν −Xt) dt+ v√Xt dZt.
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Stylized Fact: Volatility Persistence
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ACF of S&P 500 Returns2
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Stochastic Process with Long Memory
Long-memoryIf {Xt}t∈N is a stationary process and there exists H ∈ ( 1
2 , 1) such that
limt→∞
Corr(Xt, X1)c t2−2H = 1
then {Xt} has long memory (long-range dependence).
Equivalently,{Long Memory:
∑∞t=1 Corr(Xt, X1) =∞, when H > 1/2
Antipersistence:∑∞t=1 Corr(Xt, X1) <∞, when H < 1/2
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Stochastic Process with Long Memory
Long-memoryIf {Xt}t∈N is a stationary process and there exists H ∈ ( 1
2 , 1) such that
limt→∞
Corr(Xt, X1)c t2−2H = 1
then {Xt} has long memory (long-range dependence).
Equivalently,{Long Memory:
∑∞t=1 Corr(Xt, X1) =∞, when H > 1/2
Antipersistence:∑∞t=1 Corr(Xt, X1) <∞, when H < 1/2
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Fractional Stochastic Volatility Model
Log-Returns: {Yt, t ∈ [0, T ]}
dYt =(r − σt
2
2
)dt+ σt dWt.
where σt = σ(Xt) and Xt is described by:
dXt = α (m−Xt) dt+ β dBHt
– BHt is a fractional Brownian motion with Hurst parameter
H ∈ (0, 1).– Xt is a fractional Ornstein-Uhlenbeck process (fOU).
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Fractional Stochastic Volatility Model
Log-Returns: {Yt, t ∈ [0, T ]}
dYt =(r − σt
2
2
)dt+ σt dWt.
where σt = σ(Xt) and Xt is described by:
dXt = α (m−Xt) dt+ β dBHt
– BHt is a fractional Brownian motion with Hurst parameter
H ∈ (0, 1).– Xt is a fractional Ornstein-Uhlenbeck process (fOU).
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Fractional Brownian Motion
DefinitionA centered Gaussian process BH = {BHt , t ≥ 0} is called fractionalBrownian motion (fBm) with selfsimilarity parameter H ∈ (0, 1), if it hasthe following covariance function
E(BHt B
Hs
)= 1
2
{t2H + s2H − |t− s|2H
}.
and a.s. continuous paths.
– When H = 12 , BH is a standard Brownian motion.
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Fractional Brownian Motion
DefinitionA centered Gaussian process BH = {BHt , t ≥ 0} is called fractionalBrownian motion (fBm) with selfsimilarity parameter H ∈ (0, 1), if it hasthe following covariance function
E(BHt B
Hs
)= 1
2
{t2H + s2H − |t− s|2H
}.
and a.s. continuous paths.
– When H = 12 , BH is a standard Brownian motion.
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0 200 400 600 800 1000
-1.0
-0.5
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circFBM Path - N=1000, H=0.3
time
ans
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-0.8
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circFBM Path - N=1000, H=0.5
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-0.2
0.00.2
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circFBM Path - N=1000, H=0.7
time
ans
0 200 400 600 800 1000
0.00.2
0.40.6
0.81.0
1.2
circFBM Path - N=1000, H=0.95
time
ans
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Properties of FBM
The increments of fBm, {Bn −Bn−1}n∈N, are– stationary , i.e. E[(Bn −Bn−1) (Bn+h −Bn+h−1)] = γ(h).
– H–selfsimilar , i.e. c−H (Bn −Bn−1) ∼D (Bc n −Bc(n−1))
– dependent– When H > 1
2 , the increments exhibit long-memory , i.e.∑E [B1 (Bn −Bn−1)] = +∞.
– When H < 12 , the increments exhibit antipersistence, i.e.∑
E [B1 (Bn −Bn−1)] < +∞.
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Properties of FBM
The increments of fBm, {Bn −Bn−1}n∈N, are– stationary , i.e. E[(Bn −Bn−1) (Bn+h −Bn+h−1)] = γ(h).
– H–selfsimilar , i.e. c−H (Bn −Bn−1) ∼D (Bc n −Bc(n−1))
– dependent– When H > 1
2 , the increments exhibit long-memory , i.e.∑E [B1 (Bn −Bn−1)] = +∞.
– When H < 12 , the increments exhibit antipersistence, i.e.∑
E [B1 (Bn −Bn−1)] < +∞.
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Properties of FBM
The increments of fBm, {Bn −Bn−1}n∈N, are– stationary , i.e. E[(Bn −Bn−1) (Bn+h −Bn+h−1)] = γ(h).
– H–selfsimilar , i.e. c−H (Bn −Bn−1) ∼D (Bc n −Bc(n−1))
– dependent– When H > 1
2 , the increments exhibit long-memory , i.e.∑E [B1 (Bn −Bn−1)] = +∞.
– When H < 12 , the increments exhibit antipersistence, i.e.∑
E [B1 (Bn −Bn−1)] < +∞.
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Properties of FBM
The increments of fBm, {Bn −Bn−1}n∈N, are– stationary , i.e. E[(Bn −Bn−1) (Bn+h −Bn+h−1)] = γ(h).
– H–selfsimilar , i.e. c−H (Bn −Bn−1) ∼D (Bc n −Bc(n−1))
– dependent– When H > 1
2 , the increments exhibit long-memory , i.e.∑E [B1 (Bn −Bn−1)] = +∞.
– When H < 12 , the increments exhibit antipersistence, i.e.∑
E [B1 (Bn −Bn−1)] < +∞.
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Properties of FBM
The increments of fBm, {Bn −Bn−1}n∈N, are– stationary , i.e. E[(Bn −Bn−1) (Bn+h −Bn+h−1)] = γ(h).
– H–selfsimilar , i.e. c−H (Bn −Bn−1) ∼D (Bc n −Bc(n−1))
– dependent– When H > 1
2 , the increments exhibit long-memory , i.e.∑E [B1 (Bn −Bn−1)] = +∞.
– When H < 12 , the increments exhibit antipersistence, i.e.∑
E [B1 (Bn −Bn−1)] < +∞.
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Properties of FBM
– It has a.s. Holder-continuous sample paths of any order γ < H.
– Its 1H -variation on [0, t] is finite. In particular, fBm has an infinite
quadratic variation for H < 1/2.
– When H 6= 12 , BHt is not a semimartingale.
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Properties of FBM
– It has a.s. Holder-continuous sample paths of any order γ < H.
– Its 1H -variation on [0, t] is finite. In particular, fBm has an infinite
quadratic variation for H < 1/2.
– When H 6= 12 , BHt is not a semimartingale.
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Properties of FBM
– It has a.s. Holder-continuous sample paths of any order γ < H.
– Its 1H -variation on [0, t] is finite. In particular, fBm has an infinite
quadratic variation for H < 1/2.
– When H 6= 12 , BHt is not a semimartingale.
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Stochastic Integral wrt fBm
– Pathwise Riemann-Stieltjes integral, when H > 1/2. (Lin; Dai andHeyde).
– Stochastic calculus of variations with respect to a Gaussian process.(Decreusefond and Ustunel; Carmona and Coutin; Alos, Mazet andNualart; Duncan, Hu and Pasik-Duncan and Hu and Oksendal).
– Pathwise stochastic integral interpreted in the Young sense, whenH > 1/2 (Young; Gubinelli).
– Rough path-theoretic approach by T. Lyons.
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Fractional Stochastic Volatility Model
{dYt =
(r − σ2(Xt)
2
)dt+ σ(Xt) dWt,
dXt = α (m−Xt) dt+ β dBHt
Some Properties– Holder continuity : Holder continuous of order γ, for all γ < H.
– Self-similarity : Self-similar in the sense that
{BHct ; t ∈ R} ∼D {cHBHt ; t ∈ R}, ∀c ∈ R
This property is approximately inherited by the fOU, for scalessmaller than 1/α.
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Fractional Stochastic Volatility Model
Hurst Index Model
H > 1/2 Long-Memory SV: persistence
ACF Decay ∼ dt2H−2
H < 1/2 Rough Volatility: anti-persistence
ACF Decay ∼ dtH
H = 1/2 Classical SV
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Long Memory Stochastic Volatility Models
– Comte and Renault (1998)
– Comte, Coutin and Renault (2010)
– C. and Viens (2010, 2012)
– Gulisashvili, Viens and Zhang (2015)
– Guennoun, Jacquier, and Roome (2015)
– Garnier & Solna (2015, 2016)
– Bezborodov, Di Persio & Mishura (20176)
– Fouque & Hu (2017)
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Rough Stochastic Volatility Models
– Gatheral, Jaisson, and Rosenbaum (2014)
– Bayer, Friz, and Gatheral (2015)
– Forde, Zhang (2015)
– El Euch, Rosenbaum (2016, 2017)
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Related Literature
– Willinger, Taqqu, Teverovsky (1999): LRD in the stock market
– Bayraktar, Poor, Sircar (2003): Estimation of fractal dimension of S&P500.
– Bjork, Hult (2005): Fractional Black-Scholes market.
– Cheriditio (2003): Arbitrage in fractional BS market.
– Guasoni (2006): No arbitral under transaction cost with fBm.
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Research Questions
1 Option Pricing
2 Statistical Inference
3 Hedging
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Option Pricing
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Option Pricing
– In practice, we have access to discrete-time observations of historicalstock prices, while the volatility is unobserved.
Two-steps1 Estimate the empirical stochastic volatility distribution.
– Through adjusted particle filtering algorithms.
2 Construct a multinomial recombining tree to compute the optionprices.
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Option Pricing
– In practice, we have access to discrete-time observations of historicalstock prices, while the volatility is unobserved.
Two-steps1 Estimate the empirical stochastic volatility distribution.
– Through adjusted particle filtering algorithms.
2 Construct a multinomial recombining tree to compute the optionprices.
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Option Pricing
– In practice, we have access to discrete-time observations of historicalstock prices, while the volatility is unobserved.
Two-steps1 Estimate the empirical stochastic volatility distribution.
– Through adjusted particle filtering algorithms.
2 Construct a multinomial recombining tree to compute the optionprices.
![Page 31: Recent Advances in Fractional Stochastic Volatility Models](https://reader035.fdocument.org/reader035/viewer/2022071000/62c81f21b1ba5712677e222d/html5/thumbnails/31.jpg)
Option Pricing
– In practice, we have access to discrete-time observations of historicalstock prices, while the volatility is unobserved.
Two-steps1 Estimate the empirical stochastic volatility distribution.
– Through adjusted particle filtering algorithms.
2 Construct a multinomial recombining tree to compute the optionprices.
![Page 32: Recent Advances in Fractional Stochastic Volatility Models](https://reader035.fdocument.org/reader035/viewer/2022071000/62c81f21b1ba5712677e222d/html5/thumbnails/32.jpg)
Option Pricing
– In practice, we have access to discrete-time observations of historicalstock prices, while the volatility is unobserved.
Two-steps1 Estimate the empirical stochastic volatility distribution.
– Through adjusted particle filtering algorithms.
2 Construct a multinomial recombining tree to compute the optionprices.
![Page 33: Recent Advances in Fractional Stochastic Volatility Models](https://reader035.fdocument.org/reader035/viewer/2022071000/62c81f21b1ba5712677e222d/html5/thumbnails/33.jpg)
Multinomial Recombining Tree
(i) At each step, a value for the volatility is drawn from the volatility filter.(ii) A standard pricing technique using backward induction can be used to compute the option
price.(iii) We iterate this procedure by using N repeated volatility samples, constructing a different
tree with each sample and averaging over all prices.
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Multinomial Recombining Tree
(i) At each step, a value for the volatility is drawn from the volatility filter.(ii) A standard pricing technique using backward induction can be used to compute the option
price.(iii) We iterate this procedure by using N repeated volatility samples, constructing a different
tree with each sample and averaging over all prices.
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Multinomial Recombining Tree
(i) At each step, a value for the volatility is drawn from the volatility filter.(ii) A standard pricing technique using backward induction can be used to compute the option
price.(iii) We iterate this procedure by using N repeated volatility samples, constructing a different
tree with each sample and averaging over all prices.
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S&P 500 Data
800 1000 1200 1400 1600 1800
0100
200
300
400
500
Strike Price
Optio
n Pric
e
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Statistical Inference
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Inference for FSV Models
{dYt =
(µ− σ2(Xt)
2
)dt+ σ(Xt) dWt,
dXt = α (m−Xt) dt+ β dBHt
– We can also assume that Corr(BHt ,Wt) = ρ
(leverage effects).
– Parameters to estimate: θ = (α,m, β, µ, ρ) and H.
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Inference for FSV Models
Remark– The estimation of H is decoupled from the estimation of the drift
components, but not from the estimation of the “diffusion” terms.
Framework– Observations:
Historical stock prices → Discrete, even when in high-frequency.
– Unobserved State:Stochastic Volatility with non-Markovian structure is hidden.
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Inference for FSV Models
Extension of classical statistical methods– For H known:
µ, m, α and β can be estimated with standard techniques (Fouque,Papanicolaou and Sircar, 2000) using high-frequency data.
– For H unknown:
Traditional volatility proxies: Squared returns, logarithm of squaredreturns.
Use a non-parametric method to estimate H through the squaredreturns and then go back to classical techniques and estimate theremaining parameters.
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Inference for FSV Models
Extension of classical statistical methods– For H known:
µ, m, α and β can be estimated with standard techniques (Fouque,Papanicolaou and Sircar, 2000) using high-frequency data.
– For H unknown:
Traditional volatility proxies: Squared returns, logarithm of squaredreturns.
Use a non-parametric method to estimate H through the squaredreturns and then go back to classical techniques and estimate theremaining parameters.
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Inference for FSV Models
Extension of classical statistical methods– For H known:
µ, m, α and β can be estimated with standard techniques (Fouque,Papanicolaou and Sircar, 2000) using high-frequency data.
– For H unknown:
Traditional volatility proxies: Squared returns, logarithm of squaredreturns.
Use a non-parametric method to estimate H through the squaredreturns and then go back to classical techniques and estimate theremaining parameters.
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Inference for FSV Models
Extension of classical statistical methods– For H known:
µ, m, α and β can be estimated with standard techniques (Fouque,Papanicolaou and Sircar, 2000) using high-frequency data.
– For H unknown:
Traditional volatility proxies: Squared returns, logarithm of squaredreturns.
Use a non-parametric method to estimate H through the squaredreturns and then go back to classical techniques and estimate theremaining parameters.
![Page 44: Recent Advances in Fractional Stochastic Volatility Models](https://reader035.fdocument.org/reader035/viewer/2022071000/62c81f21b1ba5712677e222d/html5/thumbnails/44.jpg)
Inference for FSV Models
Simulation Based Methods– Employ a Sequential Monte Carlo (SMC) method to estimate the
unobserved state along with the unknown parameters.– Denote θ the vector of all parameters, except for H.
– Observation equation: f(Yt|Xt; θ)– State equation: f(Xt|Xt−1, . . . , X1; θ)
– Key Idea: Sequentially compute
f(X1:t; θ|Y1:t) ∝ f(X1) · f(X2|X1; θ) · . . . · f(Xn|Xn−1, . . . , X1; θ)
·t∏
i=1
f(Yi|Xi; θ) · f(θ|Yt),
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Inference for FSV Models
Simulation Based Methods– Employ a Sequential Monte Carlo (SMC) method to estimate the
unobserved state along with the unknown parameters.– Denote θ the vector of all parameters, except for H.
– Observation equation: f(Yt|Xt; θ)– State equation: f(Xt|Xt−1, . . . , X1; θ)
– Key Idea: Sequentially compute
f(X1:t; θ|Y1:t) ∝ f(X1) · f(X2|X1; θ) · . . . · f(Xn|Xn−1, . . . , X1; θ)
·t∏
i=1
f(Yi|Xi; θ) · f(θ|Yt),
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Inference for FSV Models
Simulation Based Methods– Employ a Sequential Monte Carlo (SMC) method to estimate the
unobserved state along with the unknown parameters.– Denote θ the vector of all parameters, except for H.
– Observation equation: f(Yt|Xt; θ)– State equation: f(Xt|Xt−1, . . . , X1; θ)
– Key Idea: Sequentially compute
f(X1:t; θ|Y1:t) ∝ f(X1) · f(X2|X1; θ) · . . . · f(Xn|Xn−1, . . . , X1; θ)
·t∏
i=1
f(Yi|Xi; θ) · f(θ|Yt),
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Inference for FSV Models
Simulation Based Methods– Employ a Sequential Monte Carlo (SMC) method to estimate the
unobserved state along with the unknown parameters.– Denote θ the vector of all parameters, except for H.
– Observation equation: f(Yt|Xt; θ)– State equation: f(Xt|Xt−1, . . . , X1; θ)
– Key Idea: Sequentially compute
f(X1:t; θ|Y1:t) ∝ f(X1) · f(X2|X1; θ) · . . . · f(Xn|Xn−1, . . . , X1; θ)
·t∏
i=1
f(Yi|Xi; θ) · f(θ|Yt),
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Inference for FSV Models
Simulation Based Methods– Employ a Sequential Monte Carlo (SMC) method to estimate the
unobserved state along with the unknown parameters.– Denote θ the vector of all parameters, except for H.
– Observation equation: f(Yt|Xt; θ)– State equation: f(Xt|Xt−1, . . . , X1; θ)
– Key Idea: Sequentially compute
f(X1:t; θ|Y1:t) ∝ f(X1) · f(X2|X1; θ) · . . . · f(Xn|Xn−1, . . . , X1; θ)
·t∏
i=1
f(Yi|Xi; θ) · f(θ|Yt),
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Learning θ sequentially
Filtering for states and parameter(s): Learning Xt and θ sequentially.
Posterior at t : f(Xt|θ, Yt) f(θ|Yt)⇓
Prior at t+ 1 : f(Xt+1|θ, Yt) f(θ|Yt)⇓
Posterior at t+ 1 : f(Xt+1|θ, Yt+1) f(θ|Yt+1)
Advantages1 Sequential updates of f(θ|Yt), f(Xt|Yt) and f(θ,Xt|Yt).2 Sequential h-step ahead forecasts f(Yt+h|Yt)3 Sequential approximations for f(Yt|Yt−1).
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Learning θ sequentially
Filtering for states and parameter(s): Learning Xt and θ sequentially.
Posterior at t : f(Xt|θ, Yt) f(θ|Yt)⇓
Prior at t+ 1 : f(Xt+1|θ, Yt) f(θ|Yt)⇓
Posterior at t+ 1 : f(Xt+1|θ, Yt+1) f(θ|Yt+1)
Advantages1 Sequential updates of f(θ|Yt), f(Xt|Yt) and f(θ,Xt|Yt).2 Sequential h-step ahead forecasts f(Yt+h|Yt)3 Sequential approximations for f(Yt|Yt−1).
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Learning θ sequentially
Filtering for states and parameter(s): Learning Xt and θ sequentially.
Posterior at t : f(Xt|θ, Yt) f(θ|Yt)⇓
Prior at t+ 1 : f(Xt+1|θ, Yt) f(θ|Yt)⇓
Posterior at t+ 1 : f(Xt+1|θ, Yt+1) f(θ|Yt+1)
Advantages1 Sequential updates of f(θ|Yt), f(Xt|Yt) and f(θ,Xt|Yt).2 Sequential h-step ahead forecasts f(Yt+h|Yt)3 Sequential approximations for f(Yt|Yt−1).
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Artificial Evolution of θ
Draw θ from a mixture of Normals:– ∀t update f(θ|Yt)
– Compute a Monte Carlo approximation of f(θ|Yt), by using samplesθ
(j)t and weights w(j)
t .
– Smooth kernel density approximation
f(θ|Yt) ≈N∑
j=1
ω(j)t N (θ|m(j)
t , h2Vt)
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Artificial Evolution of θ
Draw θ from a mixture of Normals:– ∀t update f(θ|Yt)
– Compute a Monte Carlo approximation of f(θ|Yt), by using samplesθ
(j)t and weights w(j)
t .
– Smooth kernel density approximation
f(θ|Yt) ≈N∑
j=1
ω(j)t N (θ|m(j)
t , h2Vt)
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Artificial Evolution of θ
Draw θ from a mixture of Normals:– ∀t update f(θ|Yt)
– Compute a Monte Carlo approximation of f(θ|Yt), by using samplesθ
(j)t and weights w(j)
t .
– Smooth kernel density approximation
f(θ|Yt) ≈N∑
j=1
ω(j)t N (θ|m(j)
t , h2Vt)
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Artificial Evolution of θ
Draw θ from a mixture of Normals:– ∀t update f(θ|Yt)
– Compute a Monte Carlo approximation of f(θ|Yt), by using samplesθ
(j)t and weights w(j)
t .
– Smooth kernel density approximation
f(θ|Yt) ≈N∑
j=1
ω(j)t N (θ|m(j)
t , h2Vt)
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Filter convergence
Let φ : X 7→ R be an appropriate test function and assume that we wantestimate
φt =∫φt(x1:t)p(x1:t, θ(t)|Y1:t)dx1:tdθ(t).
The SISR algorithm provides us with the estimator
φNt =∫φt(x1:t)πN (dx1:t) =
N∑i=1
W itφt
(X
(i)1:t−1,t−1, X
(i)t,t
)
CLT for the filter (C. and Spiliopoulos)√N(φNt − φt
)⇒ N
(0, σ2(φt)
)as N →∞.
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Convergence of the Parameter
θN(t) =N∑i=1
W(i)t θ
(N,i)(t) , where θ
(N,i)(t) ∼ N (m(N,i)
t−1 |h2V Nt−1)
with mNt−1 = αθ(N,i)(t−1) + (1− α)θN(t−1), V Nt−1 = 1
N−1
∑N
i=1
(W
(i)t−1θ
(N,i)(t−1) − θ
N(t−1)
)2.
CLT for the parameterAssuming that Eπθt ‖Wt‖2+δ
<∞:
√N(θ
(N)(t) − θ(t)
)⇒ N
(0, σ2(θ(t))
), as N →∞ (1)
Moreover, if the model Pθ is identifiable, then the posterior mean θ(t)consistently estimates the true parameter value θ, as t→∞.
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S& P 500: Volatility Particle FilterDensity
0.31 0.32 0.33 0.34 0.35 0.36 0.37
010
2030
40
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S& P 500: Parameter Estimators
0 200 400 600 800 1000
24
68
10
alpha
0 200 400 600 800 1000
12
34
56
7
mu
(a) Estimator of µ (b) Estimator of α
0 200 400 600 800 1000
46
810
m
0 200 400 600 800 1000
24
68
sigma
(c) Estimator of m (d) Estimator of β
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Model Validation: 1-Step Ahead Prediction
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Model Validation: Residuals
(a) Residuals (b) ACF of Residuals
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Hedging
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Hedging
– Conditionally on the past and the entire volatility pathlnST /St ∼ N
(r − (1/2)
∫ Ttσ2sds,
∫ Ttσ2sds).
So,
Ct = St
{EQ[Φ(xtVt,T
+ Vt,T2
)]− e−xtEQ
[Φ(xtVt,T
− Vt,T2
)]},
where xt = ln(e−rTSt/K
)and Vt,T =
(∫ Ttσ2sds)1/2
.
– Imperfect Delta-Sigma hedging strategy
∆t(xt, σt) = EQ[Φ(xtVt,T
+ Vt,T2
)]
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Hedging
– Conditionally on the past and the entire volatility pathlnST /St ∼ N
(r − (1/2)
∫ Ttσ2sds,
∫ Ttσ2sds).
So,
Ct = St
{EQ[Φ(xtVt,T
+ Vt,T2
)]− e−xtEQ
[Φ(xtVt,T
− Vt,T2
)]},
where xt = ln(e−rTSt/K
)and Vt,T =
(∫ Ttσ2sds)1/2
.
– Imperfect Delta-Sigma hedging strategy
∆t(xt, σt) = EQ[Φ(xtVt,T
+ Vt,T2
)]
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Hedging
– Conditionally on the past and the entire volatility pathlnST /St ∼ N
(r − (1/2)
∫ Ttσ2sds,
∫ Ttσ2sds).
So,
Ct = St
{EQ[Φ(xtVt,T
+ Vt,T2
)]− e−xtEQ
[Φ(xtVt,T
− Vt,T2
)]},
where xt = ln(e−rTSt/K
)and Vt,T =
(∫ Ttσ2sds)1/2
.
– Imperfect Delta-Sigma hedging strategy
∆t(xt, σt) = EQ[Φ(xtVt,T
+ Vt,T2
)]
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Hedging
Two notions of Implied Volatility1 Black-Scholes Implied Volatility:
The unique solution to
Ct(x, σ) = CBSt (x, σi(x, σ)),
i.e. the volatility parameter that equates the BS price to the HW.
2 Hedging Volatility:The unique solution to
∆t(x, σ) = ∆BSt (x, σh(x, σ))
i.e. the volatility parameter that equates the BS hedge ratio againstthe underlying asset variations to the HW one.
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Hedging
Two notions of Implied Volatility1 Black-Scholes Implied Volatility:
The unique solution to
Ct(x, σ) = CBSt (x, σi(x, σ)),
i.e. the volatility parameter that equates the BS price to the HW.
2 Hedging Volatility:The unique solution to
∆t(x, σ) = ∆BSt (x, σh(x, σ))
i.e. the volatility parameter that equates the BS hedge ratio againstthe underlying asset variations to the HW one.
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Hedging
Hedging Bias (Definition)The difference between the BS implied volatility-based hedging ratio andthe HW one:
Bias = ∆BSt (x, σi(x, σ))−∆t(x, σ)
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Hedging
Theorem(a) Sign of Hedging Bias:
σh(−x, σ) ≤ σi(−x, σ) = σi(x, σ) ≤ σh(x, σ)
(b) Accuracy of approximation of partial hedging ratio by the BSimplicit volatility-based hedging ratio:
∆BS(x, σ) ≤ ∆(x, σ)∆BS(−x, σ) ≥ ∆(−x, σ)∆BS(0, σ) = ∆(0, σ)
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Hedging
Theorem(a) Sign of Hedging Bias:
σh(−x, σ) ≤ σi(−x, σ) = σi(x, σ) ≤ σh(x, σ)
(b) Accuracy of approximation of partial hedging ratio by the BSimplicit volatility-based hedging ratio:
∆BS(x, σ) ≤ ∆(x, σ)∆BS(−x, σ) ≥ ∆(−x, σ)∆BS(0, σ) = ∆(0, σ)
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Thank you!