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Stochastic volatility model of Hestonand the smile
Rafa l Weron
Hugo Steinhaus CenterWroc law University of TechnologyPoland
In collaboration with:Piotr Uniejewski (LUKAS Bank)Uwe Wystup (Commerzbank Securities)
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.im.pwr.wroc.pl/~rweron/
2
Agenda
1. FX markets and the smile
2. Heston’s model
3. Calibration
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 3
FX markets
EUR/USD and USD/JPY are two of the most
liquid underlying markets with trading in:
• Spot/forward (ca. 90% of activity, very smallmargins)
• Vanilla options (9%, small margins)
• Exotic options (1%, potentially high margins)
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 4
Global markets
USD/JPY market activity
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 5
Black-Scholes type formula
• Assumes that asset prices follow GBM:
dSt = St(µdt + σdBt) (1)
• European FX call option price(Garman and Kohlhagen, 1983):
Ct = Ste−rfτΦ(d1)−Ke−rτΦ(d2),
where d1 =log(St/K)+(r−rf+ 12σ
2)τσ√
τ, d2 = d1 − σ
√τ
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 6
BS formula is flawed
• Implied volatility σi is the volatility thatequates the BS price:
BS(St, K, r, σi, τ) = Option market price
• Model implied volatilities for different strikesand maturities are not constant
• Volatility smile or smirk/grin is observed
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 7
The smile1W, 1M, 3M, 6M, 1Y, 2Y EUR/USD smiles
20 40 60 80
Delta [%]
1010
.511
11.5
Impl
ied
vola
tility
[%
]
STFhes07.xpl
1W (black), 1M (red), 3M (green), 6M (blue), 1Y (cyan), and 2Y(yellow) EUR/USD implied volatility smiles on July 1, 2004
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.quantlet.de/codes//STFhes07.html
FX markets and the smile 8
The smirk/grin
0.6 0.8 1 1.2 1.4 1.60.3
0.4
0.5
0.6
0.7
0.8
0.9
1
moneyness=K/St
σi
ODAX options implied σ’s on Oct. 12, 1998
τ=10 days
τ=38 days
τ=66 days τ=157 days
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 9
Correcting the BS formula (1/3)
• Allow the volatility to be a deterministicfunction of time (Merton, 1973): σ = σ(t)
• Explains the different σi levels for differentτ ’s, but cannot explain the smile shape for
different strikes
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 10
Correcting the BS formula (2/3)
• Allow not only time, but also statedependence of σ (Dupire, 1994; Derman and
Kani, 1994; Rubinstein, 1994): σ = σ(t, St)
• Lets the local volatility surface to be fitted,but cannot explain the persistent smile shape
which does not vanish as time passes
Rafa l Weron: The Third Nikkei Econophysics Symposium
FX markets and the smile 11
Correcting the BS formula (3/3)
• Allow the volatility coefficient in the BSdiffusion equation (1) to be random: σ = σt
• Pioneering work of Hull and White (1987),Stein and Stein (1991), and Heston (1993)
led to the development of stochastic volatility
models
Rafa l Weron: The Third Nikkei Econophysics Symposium
Heston’s model 12
Heston’s model
dSt = St
(µ dt +
√vtdW
(1)t
), (2)
dvt = κ(θ − vt) dt + σ√
vtdW(2)t , (3)
dW(1)t dW
(2)t = ρ dt (4)
• Variance process (3) is non-negative andmean-reverting (as observed in the markets)
• It has CIR dynamics
Rafa l Weron: The Third Nikkei Econophysics Symposium
Heston’s model 13
GBM vs. Heston dynamics
0 0.5 1
Time [years]
0.7
0.75
0.8
0.85
0.9
Exc
hang
e ra
te
GBM vs. Heston volatility
0 0.5 1
Time [years]
1520
25
Vol
atili
ty [
%]
STFhes01.xpl
GBM vs. Heston: ρ = −0.05, initial (=GBM) variance v0 = 4%, longterm var. θ = 4%, speed of mean reversion κ = 2, vol of vol σ = 30%
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.quantlet.de/codes//STFhes01.html
Heston’s model 14
Gaussian vs. Heston densities
-1 -0.5 0 0.5 1
x
00.
51
1.5
2
PDF(
x)
Gaussian vs. Heston log-densities
-1 -0.5 0 0.5 1
x
-10
-50
Log
(PD
F(x)
)STFhes02.xpl
Unlike Gaussian tails, tails of Heston’s marginals are exponential:log-densities resemble hyperbolas (Dragulescu and Yakovenko, 2002)
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.quantlet.de/codes//STFhes02.html
Heston’s model 15
Option pricing in Heston’s model
• PDE for the option price can be solvedanalytically using the method of characteristic
functions (Heston, 1993)
• Closed-form solution for vanilla options:
h(t) = HestonVanilla(κ, θ, σ, ρ, λ, rd, rf , v0, S0, K, τ)
= e−rfτStP+(φ)−Ke−rdτP−(φ) (5)
Rafa l Weron: The Third Nikkei Econophysics Symposium
Calibration 16
Calibration
1. Look at a time series of historical data:
• Use GMM, SMM, EMM, or BayesianMCMC to fit the price process
• Fit empirical distributions of returns to themarginal distributions
• Cannot estimate the market price of risk λ
2. Calibrate the model to derivative prices or
better to the volatility smile
Rafa l Weron: The Third Nikkei Econophysics Symposium
Calibration 17
Calibration to the smile
• Take the smile of the current vanilla optionsmarket as a given starting point
• Find the optimal set of model parameters fora fixed τ and a given vector of market BS
implied volatilities {σ̂i}ni=1 for a given set ofdelta pillars {∆i}ni=1
• No need to worry about estimating λ as it isalready embedded in the market smile
Rafa l Weron: The Third Nikkei Econophysics Symposium
Calibration 18
3M market and Heston volatilities
20 40 60 80
Delta [%]
10.2
10.4
10.6
10.8
11
Impl
ied
vola
tility
[%
]
6M market and Heston volatilities
20 40 60 80
Delta [%]
10.5
11
Impl
ied
vola
tility
[%
]STFhes06.xpl
EUR/USD volatility surface on July 1, 2004: the fit is very good formaturities between three and eighteen months
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.quantlet.de/codes//STFhes06.html
Calibration 19
1W market and Heston volatilities
20 40 60 80
Delta [%]
9.5
1010
.511
11.5
Impl
ied
vola
tility
[%
]
2Y market and Heston volatilities
20 40 60 80Delta [%]
10.5
1111
.5Im
plie
d vo
latil
ity [
%]
STFhes06.xpl
Unfortunately, Heston’s model does not perform satisfactorily forshort maturities and extremely long maturities
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.quantlet.de/codes//STFhes06.html
Calibration 20
Only 3 parameters to fit
Long-run variance and the smile
20 40 60 80
Delta [%]
9.5
1010
.511
11.5
Impl
ied
vola
tility
[%
]
θ(≈ v0) – ATM level of the smileCorrelation and the smile
20 40 60 80
Delta [%]
1010
.511
11.5
Impl
ied
vola
tility
[%
]
ρ – skew (quoted as risk reversals)Vol of vol and the smile
20 40 60 80
Delta [%]
910
1112
Impl
ied
vola
tility
[%
]
σ(≈ κ) – convexity (butterflies)
Rafa l Weron: The Third Nikkei Econophysics Symposium
Calibration 21
Application
1. Calibrate the model to vanilla options
2. Employ it for pricing exotics, like one-touch or
barrier options (finite difference, Monte Carlo)
Rafa l Weron: The Third Nikkei Econophysics Symposium
References 22
References
1. Hakala, J. and Wystup, U. (2002) Heston’s Stochastic
Volatility Model Applied to Foreign Exchange Options,
in J. Hakala, U. Wystup (eds.) Foreign Exchange Risk,
Risk Books.
2. Weron, R. and Wystup, U. (2005)
Heston’s model and the smile, in
P. Cizek, W. Härdle, R. Weron (eds.)
Statistical Tools for Finance
and Insurance, Springer.
http://www.xplore-stat.de/ebooks/ebooks.html
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Statistical Tools for Finance and Insurance
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1 237 83540 2218909
ISBN 3-540-22189-1
Statistical Tools for Finance and Insurance presents ready-to-use solutions, theoretical developments andmethod construction for many practical problems in quantitative finance and insurance. Written by practi-tioners and leading academics in the field this book offers a unique combination of topics from whichevery market analyst and risk manager will benefit.
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Pavel Čížek is an Assistant Professor of Econometrics at Tilburg University, Tilburg,The Netherlands. He is teaching general econometric courses as well as courses focusedon the analysis of (financial) time series for the Master’s Program in Economics and Econo-metrics. His research interests include, besides theoretical econometrics, nonparametricand computationally intensive methods, primarily with applications to finance.
Wolfgang Härdle is Professor of Statistics at Humboldt-Universität zu Berlin and Directorof CASE – Center for Applied Statistics and Economics. He teaches Quantitative Financeand Semiparametric Statistical methods. His research focuses on dynamic factor models,multivariate statistics in finance and computational statistics. He is an elected ISI memberand advisor to the Guanghua School of Management, Peking University.
Rafał Weron is Assistant Professor of Mathematics at Wrocław University of Technology,Poland. His research focuses on risk management in the power markets and computa-tional statistics as applied to finance and insurance. His professional experience includesdevelopment of insurance strategies for the Polish Power Grid Co. (PSE S.A.) and Hydro-storage Power Plants Co. (ESP S.A.) as well as implementation of option pricing softwarefor LUKAS Bank S.A. (Crédit Agricole Group). He has also been a consultant and executiveteacher to a large number of banks and corporations.
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eron
22189-1 Cizek U. 14.10.2004 9:28 Uhr Seite 1
Rafa l Weron: The Third Nikkei Econophysics Symposium
http://www.xplore-stat.de/ebooks/ebooks.html
Stochastic volatility model of Heston and the smileAgendaFX marketsGlobal marketsBlack-Scholes type formulaBS formula is flawedThe smileThe smirk/grinCorrecting the BS formula (1/3)Correcting the BS formula (2/3)Correcting the BS formula (3/3)Heston's modelOption pricing in Heston's modelCalibrationCalibration to the smileOnly 3 parameters to fitApplicationReferences