an empirical characterization of the market process · appendix: derivation of the black-scholes...
TRANSCRIPT
![Page 1: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/1.jpg)
An empirical An empirical characterization of the characterization of the
market processmarket process
(T(The fractional volatility model)he fractional volatility model)
![Page 2: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/2.jpg)
Contents1. Market prices are non-differentiable2. Geometric Brownian motion ?3. Volatility as a process. Volatility models4. The induced volatility process5. Time scales and pdf’sAppendix: Derivation of the Black-Scholes formula6. Option pricing. Risk-neutral approachMathExc1 – Stochastic integration with respect to fBm7. An option pricing equation using fBm stochastic integrationMathExc2 – An introduction to fractional calculus8. Leverage, fBm representation and fractional calculus interpretation
![Page 3: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/3.jpg)
1. Market prices are non-differentiable
![Page 4: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/4.jpg)
Detrending is needed for stationarity
![Page 5: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/5.jpg)
![Page 6: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/6.jpg)
The returns process
r(t)=log(S(t+1))-log(S(t))
Automatically detrended
)())(log( )(1 tStS dt
dtSdt
d =
![Page 7: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/7.jpg)
2. Geometric Brownian motion ?
?(a basis for most mathematical finance studies –Black-Scholes, etc.)Consequences:
(i)
Price changes would be lognormal
(ii)
Self-similar (Law(Xat)=Law(aH Xt)) with Hurst coefficient = 1/2
E S t S tS t
H( ) ( )( )
+ − − ≈Δ Δ Δμ
))(2
))(2
(lnexp(
)(21)(ln 2
22
2 tT
tTSS
tTSSp t
T
t
T
−
⎥⎦
⎤⎢⎣
⎡−−−
−−
=σ
σμ
πσ
dWdtSdS
t
t σμ +=
![Page 8: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/8.jpg)
2. Geometric Brownian motion ?Empirical tests :
P(r1) is not lognormal
r1=log(S(t+1)/S(t))
Deviations from scaling
Larger deviations for high-frequency data
![Page 9: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/9.jpg)
σ is not constantReturn
Conclusion : Nor do returns follow geometric Brownian motionnor is σ constant (not even a smooth function of S and t)
Δ≈Δ−=Δ Δ− dtdS
SrSSr t
ttttt
1)()log()log()(
![Page 10: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/10.jpg)
“Stilized” experimental facts(i) Returns have nearly no autocorrelation(ii) The autocorrelations of |rt|d decline slowly withincreasing lag. Long memory effect(iii) Leptokurtosis : asset returns have distributionswith fat tails and excess peakedness at the mean(iv) Autocorrelations of sign rt are insignificant(v) Volatility clustering : tendency of large changes to follow large changes and small changes to follow smallchanges. Volatility occurs in bursts.(vi) Volatility is mean-reversing and the distribution isclose to lognormal or inverse gamma(vii) Leverage effect : volatility tends to rise more following a large price fall than following a price rise(viii) Why volatility is important : Uncertainty and riskare the driving factors for investors’ behavior
![Page 11: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/11.jpg)
3. Volatility as a processWhen the future is uncertain investors are less likely to invest. Therefore uncertainty (volatility) would have to be changing over time.“... build a forecasting model for variance and make it a well-definedprocess ...” (Robert Engle – 1982)Structural model
Conditional variance
Homoscedasticity = variance of errors is constantHeteroscedasticity = variance of errors is not constant
uxxy ++++= ...33221 βββ
termerrorufactors...,, 321 βββ
[ ],..., 2122
−−= tttt uuuEσ
![Page 12: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/12.jpg)
2. Volatility modelsARCH(q) (Autoregressive conditionally heteroscedastic)
GARCH (1,1) (Generalized ARCH)
IGARCH (Integrated GARCH)Leverage :GJR (Glosten, Jagannathan, Runkle)
EGARCH (exponential GARCH)
11 =+βα
2222
2110
2 ... qtqttt uuu −−− ++++= αααασ
21
2110
2−− ++= ttt u βσαασ
otherwiseuifIuI
tt
tttt
0;01)(
11
21
21110
2
=<=+++=
−−
−−− βσγαασ
⎥⎥⎦
⎤
⎢⎢⎣
⎡−+++=
−
−
−
−− πσ
ασ
γσβωσ 2)ln()ln(2
1
1
21
121
2
t
t
t
ttt
uu
![Page 13: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/13.jpg)
Stochastic volatility modelsIn GARCH models, the conditional volatility is a deterministicfunction of past quantities. In Stochastic Volatility models it isitself a random process.Heston model
Two-time scales model (Perello, Masoliver)
Comte, Renault
W’ is fractional Brownian motion
')()(
0')(20
22 dWdtd
dWdWdWdtSdS
ttt
ttt
γσσσσ
ρσμ
+−Ω−=
<=+=
'')(0''')(
0')(
000000
0
dWdtddWdWdWdtd
dWdWdWedtSdS
tt
ttt
ttt
γξξξγξξξ
ρμ ξ
+−Ω−==+−Ω−=
<=+=
')ln()(ln)(
dWdtkddWdtSdS
tt
ttt
γσθσσμ
+−=+=
![Page 14: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/14.jpg)
4. The induced volatility processLet logSt be a stochastic process defined on a tensor product probability space Ω⊗Ω’logSt(ω,ω’) with Ω being Wiener space(M1) Then, if logSt(ω,ω’) is square integrable in Ω
for fixed ω’
σt(ω, ω’) is called the “Induced volatility”
(E1) Obtained from the dataσt
2 (·, ω’)≈ var(log St)/(T0 - T1)
(μ=0)
tt
t dBdtS
dS )',()',( ωσμω •+=•
![Page 15: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/15.jpg)
What does the data suggest for σt ?σt is not self similar
However Rσ(t) is Σ log σ(nδ) = β t + Rσ(t)H ≈ 0.8 - 0.9
E t tt
Hσ σσ
( ) ( )( )
+ − ≠Δ Δ
E R t R tR t
Hσ σ
σ
( ) ( )( )
+ − =Δ Δ
![Page 16: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/16.jpg)
What does the data suggest for σt ?Recall:If a process Xt has finite variance, stationary increments and is self-similar, thenCov(Xs,, Xt)=(|s|2H+|t| 2H-|s-t| 2H)E(X1
2)(M2) The simplest such process is a zero-mean Gaussian process, Fractional Brownian motion BH
twith long-range dependence for H>1/2Conclusion :log σt = β + (k/δ) ( BH
t - BHt - δ )
σt modeled by a stochastic exponential of fractional noise
![Page 17: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/17.jpg)
The fractional volatility model (FVM)Two coupled processes :
dSt = μ St dt + σt St dBt
log σt = β + (k/δ) ( BHt - BH
t - δ )
log σt driven by fractional noise, not by fractional Brownian motion
![Page 18: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/18.jpg)
5. Time scales and pdf’sFrom
log σt is a Gaussian process with mean β and covariance
Then
( ))()(ln δδ
βσ −−+= tBtBkHHt
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠⎞
⎜⎝⎛−−−= H
tktBtBk 2
2
21)()(exp δ
δδ
δθσ
{ }HHH ussuuskus 2222
2
22
),( −−+−++−= δδδ
ψ
( )⎭⎬⎫
⎩⎨⎧ −−= −− 222
2
1 2logexp
21)( HH kk
pδ
βσδσπ
σδ
![Page 19: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/19.jpg)
5. Time scales and pdf’sand for the returns
with
The probability distribution of the returns depends on the observation time scale δ
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
Δ
⎟⎟⎠
⎞⎜⎜⎝
⎛Δ⎟⎟⎠
⎞⎜⎜⎝
⎛−−
Δ=
Δ+
Δ+2
22
2 22
logexp
21)(log
σ
σμ
πσσ
t
t
t
tS
S
SSp
∫∞
Δ+Δ+ ≅0
)(log)()(logt
t
t
t
SSppd
SSP σδδ σσ
![Page 20: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/20.jpg)
5. Time scales and pdf’s (NYSE 1973-2000)H=0.83 k=0.59 β= - 5 δ=1 Δ=1
![Page 21: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/21.jpg)
6. Time scales and pdf’s (NYSE 1973-2000)
![Page 22: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/22.jpg)
6. Time scales and pdf’s (USD-Euro 05-06 2001)
![Page 23: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/23.jpg)
6. Time scales and pdf’s (Scaling ??)
![Page 24: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/24.jpg)
6. Time scales and pdf’s (Scaling ??)
![Page 25: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/25.jpg)
Closed form and return asymptoticsFrom
one obtains
Asymptotic behavior :
( )2
20222
2)(8θ
λδθ β
Δ−Δ=== − rrkCe H
∫∞
Δ=Δ0
))(()())(( rppdrP σδδ σσ
21
log1
1)(1
41))((
2
=
⎟⎠⎞
⎜⎝⎛ −−
−Γ
Δ=Δ
z
dzd
CH
zek
rPλ
δ λδθπ
( )2log111))((λ
δ λCerP
−
Δ≈Δ
![Page 26: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/26.jpg)
Appendix:Derivation of the Black-Scholes formula
Assumptions: 1) The price of the underlying instrument St follows a geometric Brownian motion defined by
where Wt is a Wiener process with constant drift μ and volatility σ. 2) It is possible to short sell the underlying stock. 3) There are no arbitrage opportunities. 4) Trading in the stock is continuous. 5) There are no transaction costs or taxes. 6) All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share). 7)It is possible to borrow and lend cash at a constant risk-free interest rate. 8) The stock does not pay a dividend
![Page 27: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/27.jpg)
Let V(S,σ) be the value of a call option. By Itō's lemma we have
Now consider a trading strategy under which one holds one optionand continuously trades in the stock in order to hold
shares. At time t, the value of these holdings will be
The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Let R denote the accumulated profit or loss from following this strategy. Then over the time period [t, t + dt], the instantaneous profit or loss is
Substituting dV and dS from the equations above we are left with
![Page 28: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/28.jpg)
This last equation contains no dW term. That is, it is entirely riskless (delta neutral). Thus, given that there is no arbitrage, the rate of return on this portfolio must be equal to the rate of return on any other riskless instrument. Assuming the risk-free rate of return to be r we must have over the time period [t, t + dt]
If we now insert the expression for П and divide through by dt we obtain the Black–Scholes PDE:
This is the law of evolution of the value of the option. With the assumptions of the Black–Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to Sand once with respect to t.
![Page 29: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/29.jpg)
Solution of the Black-Scholes equationFor a call option the PDE above has the boundary condition
Introduce the change-of-variables
Then the Black–Scholes PDE becomes a diffusion equation
![Page 30: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/30.jpg)
After some algebra we obtain
where
and Φ is the standard normal cumulative distribution function.The formula for the price of a put option follows from this via put-call parity
)()(),()( tStPTtBKtV +=⋅+
![Page 31: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/31.jpg)
The Greeks under Black–Scholes:
)()( tVtC = )()( tPtC =
![Page 32: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/32.jpg)
Option pricing in FVM. “Risk-neutral approach”
![Page 33: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/33.jpg)
Option pricing in FVM. “Risk-neutral approach”
![Page 34: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/34.jpg)
Option pricing in FVM. “Risk-neutral approach”
![Page 35: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/35.jpg)
Option pricing in FVM. “Risk-neutral approach”
![Page 36: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/36.jpg)
![Page 37: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/37.jpg)
The option pricing equation in FVM
![Page 38: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/38.jpg)
The option pricing equation in FVM
![Page 39: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/39.jpg)
The option pricing equation in FVM
![Page 40: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/40.jpg)
Option pricing in FVM. Numerical solutions
![Page 41: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/41.jpg)
Option pricing in FVM. Numerical solutions
![Page 42: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/42.jpg)
The option pricing in FVM. Analytical solution
![Page 43: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/43.jpg)
10/23/2008
8. Leverage and the fractional calculus interpretation
![Page 44: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/44.jpg)
10/23/2008
![Page 45: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/45.jpg)
10/23/2008
![Page 46: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/46.jpg)
10/23/2008
Fractional Brownian motion and fractional calculus
![Page 47: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/47.jpg)
10/23/2008
![Page 48: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/48.jpg)
10/23/2008
![Page 49: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/49.jpg)
10/23/2008
The fractional calculus interpretation of the FVM
![Page 50: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/50.jpg)
10/23/2008
![Page 51: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/51.jpg)
![Page 52: An empirical characterization of the market process · Appendix: Derivation of the Black-Scholes formula Assumptions: 1) The price of the underlying instrument St follows a geometric](https://reader033.vdocuments.us/reader033/viewer/2022052017/60303a4898cc70600e2aa9c2/html5/thumbnails/52.jpg)
ReferencesR V M and M. J. Oliveira; A data-reconstructedfractional volatility model, Economics, discussionpaper 2008-22R V M and M. J. Oliveira; Fractional volatility andoption pricing, arxiv:cond-mat/0404684R. V. M.; The fractional volatility model: An agent-based interpretation, Physica A: Stat. Mech. and Applic. 387 (2008) 3987-3994R. V. M.; A fractional calculus interpretation of the fractional volatility model, Nonlinear Dynamics, doi:10.1007/s11071-008-9372-0