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Archive for November, 2015

Deterministic volatility models were championed by Dupire, Derman-Kani, and Rubinstein in 1994 — see dupire-PricingSmileLocalVolatilityPricing

I implemented the numerical PDE solution for the Dupire-Derman-Kani approach with parametric volatility following

V(t) = \sigma_0^2 \int_0^t S_\alpha(s) ds

where S_{\alpha}(s) = s^{\alpha-1} E_{\alpha,\alpha}(-\kappa s^{\alpha}) and also the fractional (stochastic) Heston model with a fractional volatility but with the stochastic parameters \sigma=\rho=0.  The latter fit the smile much better than the former in the exercise but there may be problems with the numeric solution of PDE that I have not checked yet.

 

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There is a history for deterministic volatility models to explain the smile with work by Bruno Dupire, Mark Rubinstein, Derman and Kani from 1994.  My approch is to seek the form of volatility functions of the form \int_0^t S_{\alpha}(-\kappa s^\alpha) ds since this is the reduction of a fractional Heston model removing the stochasticity.  A simple method to check models of deterministic volatility is to compare the candidate deterministic volatility function with the empirical volatility function using Dupire's formula  latex \sigma(T,K)^2 = \sqrt{2\frac{\frac{\partial C}{\partial T} + rK\frac{\partial C}{\partial K}}{K^2 \frac{\partial^2 C}{\partial K^2}}$

For local volatility \sigma(t,x) the Black-Scholes PDE is

v_t(t,x) + rxv_x(t,x) +\frac{1}{2}\sigma^2(t,x) x^2 v_{xx}(t,x) - rv=0

with boundary conditions v(t,L) = 0 and v(T,x) = (x-K)^+ which we can solve numerically with python.

 

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The following shows approximate fit of the fractional Heston model with deterministic volatility by setting volvol to 0.00001 and rho=0.0.  This is a surprising result since it is often assumed that stochastic volatility is responsible for the implied volatility curvature.

 

C-jan-15-2016-implied-vol-versus-deterministic-fractional-volatility

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Let \gamma_1(t),\gamma_2(t) be the market price of risk associated to the correlated Brownian motions w^1(t),w^2(t) in the Heston model.  The condition for the existence of an equivalent martingale measure is

\mu - r = \rho \gamma_1(t) + \sqrt{1-\rho^2}\gamma_2(t)

and Heston additionally imposed \gamma_1(t) = \lambda\sqrt{V(t)}.  (See MeasureChange-HestonModel) for details.  For a fractional Heston model the problem of determining the appropriate conditions is still open pending more numerical work.  The risk-neutral martingale measure is given in (1.2) in the above reference.

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The test is very simple.  Black (1976) leverage effect is a negative correlation that is said to exist for equities.  It’s probably something special about S&P 500 index and other popular academic datasets but it’s very simple to check that if we define volatility as log(r_t^2) then the empirical data does not show that the correlation is negative for a large sample of 1900 stocks:

import pandas as pd
import numpy as np

p=pd.DataFrame.from_csv(‘volatility.csv’,header=0,sep=’,’).as_matrix()

K = 1899
for k in range(1899):
r=np.diff(np.log(p[:,k]))
r[np.isnan(r)]=0
r[abs(r)>0.6]=0
v=np.log(r**2+1e-6)
rho = np.corrcoef(r,v)[0,1]
print(‘k=’,k,’ rho=’,rho)

(‘k=’, 0, ‘ rho=’, -0.027987450523234373)
(‘k=’, 1, ‘ rho=’, 0.030049260925158625)
(‘k=’, 2, ‘ rho=’, 0.01039618926295386)
(‘k=’, 3, ‘ rho=’, 0.0025948401889503113)
(‘k=’, 4, ‘ rho=’, 0.018330227574373512)
(‘k=’, 5, ‘ rho=’, 0.035302021200324843)
(‘k=’, 6, ‘ rho=’, 0.025111606783465721)
(‘k=’, 7, ‘ rho=’, -0.0045806166056756225)
(‘k=’, 8, ‘ rho=’, 0.0054141120369151459)
(‘k=’, 9, ‘ rho=’, 0.027338946041420571)
(‘k=’, 10, ‘ rho=’, 0.014071826784366108)
(‘k=’, 11, ‘ rho=’, 0.0049781232029495843)
(‘k=’, 12, ‘ rho=’, 0.026629241315977041)
(‘k=’, 13, ‘ rho=’, 0.013074100488701303)
(‘k=’, 14, ‘ rho=’, -0.0012052758929136619)
(‘k=’, 15, ‘ rho=’, -0.013505751355903841)
(‘k=’, 16, ‘ rho=’, 0.024571944796759542)
(‘k=’, 17, ‘ rho=’, -0.017538820015073914)
(‘k=’, 18, ‘ rho=’, -0.0014578100389434868)
(‘k=’, 19, ‘ rho=’, 0.0049229640476859508)
(‘k=’, 20, ‘ rho=’, -0.0053475795313882753)
(‘k=’, 21, ‘ rho=’, 0.0070619061614987499)
(‘k=’, 22, ‘ rho=’, -0.00083464226514642665)
(‘k=’, 23, ‘ rho=’, -0.11747703279630202)
(‘k=’, 24, ‘ rho=’, 0.026062008742990344)
(‘k=’, 25, ‘ rho=’, 0.039202071483586745)
(‘k=’, 26, ‘ rho=’, 0.033469117037219387)
(‘k=’, 27, ‘ rho=’, 0.0062450212086564193)
(‘k=’, 28, ‘ rho=’, 0.030960939210050702)
(‘k=’, 29, ‘ rho=’, 0.011125761246834242)
(‘k=’, 30, ‘ rho=’, 0.05487401072981643)
(‘k=’, 31, ‘ rho=’, 0.00015254945112039731)
(‘k=’, 32, ‘ rho=’, -0.0024729254732480141)
(‘k=’, 33, ‘ rho=’, 0.016618766169179591)
(‘k=’, 34, ‘ rho=’, 0.0096053584396495629)
(‘k=’, 35, ‘ rho=’, 0.028196000756367703)
(‘k=’, 36, ‘ rho=’, 0.00088576633668703206)
(‘k=’, 37, ‘ rho=’, 0.018102999265135478)
(‘k=’, 38, ‘ rho=’, 0.010528849790133951)
(‘k=’, 39, ‘ rho=’, 0.0036843579498243829)
(‘k=’, 40, ‘ rho=’, -0.0073301803615446748)
(‘k=’, 41, ‘ rho=’, -0.024441111990310196)
(‘k=’, 42, ‘ rho=’, 0.028342864379294411)
(‘k=’, 43, ‘ rho=’, 0.022432950218482104)
(‘k=’, 44, ‘ rho=’, 0.00097968803130992908)
(‘k=’, 45, ‘ rho=’, -0.0056298022481729498)
(‘k=’, 46, ‘ rho=’, 0.0082631009830472675)
(‘k=’, 47, ‘ rho=’, 0.021902450049008579)
(‘k=’, 48, ‘ rho=’, -0.018436480989089215)
(‘k=’, 49, ‘ rho=’, 0.017365655736967595)
(‘k=’, 50, ‘ rho=’, 0.0035435328501678418)
(‘k=’, 51, ‘ rho=’, 0.021634189257362661)
(‘k=’, 52, ‘ rho=’, 0.00024351913977668758)
(‘k=’, 53, ‘ rho=’, 0.011855737551799776)
(‘k=’, 54, ‘ rho=’, -0.022208358439892582)
(‘k=’, 55, ‘ rho=’, 0.013596290487406005)
(‘k=’, 56, ‘ rho=’, 0.11305791772126209)
(‘k=’, 57, ‘ rho=’, 0.0093284677436124876)
(‘k=’, 58, ‘ rho=’, 0.0051976315180729945)
(‘k=’, 59, ‘ rho=’, 0.018910127890756412)
(‘k=’, 60, ‘ rho=’, 0.003070565812542603)
(‘k=’, 61, ‘ rho=’, -0.0045344255149375708)
(‘k=’, 62, ‘ rho=’, 0.009524618308610391)
(‘k=’, 63, ‘ rho=’, 0.0074155261055746154)
(‘k=’, 64, ‘ rho=’, 0.012479459736983743)
(‘k=’, 65, ‘ rho=’, -0.0040700544259990304)
(‘k=’, 66, ‘ rho=’, 0.015219317190378091)
(‘k=’, 67, ‘ rho=’, -0.0092322255656288894)
(‘k=’, 68, ‘ rho=’, -0.0002022526097535967)
(‘k=’, 69, ‘ rho=’, 0.010461201820221673)
(‘k=’, 70, ‘ rho=’, 0.021401451060686624)
(‘k=’, 71, ‘ rho=’, -0.0028211687040903279)
(‘k=’, 72, ‘ rho=’, 0.0021498161906205681)
(‘k=’, 73, ‘ rho=’, 0.0049852551865029087)
(‘k=’, 74, ‘ rho=’, 0.026398771770477631)
(‘k=’, 75, ‘ rho=’, 0.016933734141062889)
(‘k=’, 76, ‘ rho=’, -0.0018161161016975822)
(‘k=’, 77, ‘ rho=’, -0.012491358575926837)
(‘k=’, 78, ‘ rho=’, 0.0084004032069166044)
(‘k=’, 79, ‘ rho=’, 0.028952045775891548)
(‘k=’, 80, ‘ rho=’, 0.015943556493537785)
(‘k=’, 81, ‘ rho=’, 0.012458596238099175)
(‘k=’, 82, ‘ rho=’, 0.031347644055602662)
(‘k=’, 83, ‘ rho=’, 0.017363509207228105)
(‘k=’, 84, ‘ rho=’, 0.0045221367092907576)
(‘k=’, 85, ‘ rho=’, 0.015049725710054329)
(‘k=’, 86, ‘ rho=’, -0.0043665991012094294)
(‘k=’, 87, ‘ rho=’, -0.039332536262202324)
(‘k=’, 88, ‘ rho=’, 0.010960287569483508)
(‘k=’, 89, ‘ rho=’, 0.0031372633519927162)
(‘k=’, 90, ‘ rho=’, 0.025274002178746546)
(‘k=’, 91, ‘ rho=’, -0.021819617551753059)
(‘k=’, 92, ‘ rho=’, 0.012937346516384349)
(‘k=’, 93, ‘ rho=’, 0.019249750003852636)
(‘k=’, 94, ‘ rho=’, 0.018556585750246213)
(‘k=’, 95, ‘ rho=’, 0.011868780595876661)
(‘k=’, 96, ‘ rho=’, -0.0046381229388078888)
(‘k=’, 97, ‘ rho=’, 0.069359261455554919)
(‘k=’, 98, ‘ rho=’, 5.2814964088003243e-05)
(‘k=’, 99, ‘ rho=’, 0.0041085608216866768)
(‘k=’, 100, ‘ rho=’, 0.00094170209882686151)
(‘k=’, 101, ‘ rho=’, 0.017801105469348658)
(‘k=’, 102, ‘ rho=’, -0.0010676446191825223)
(‘k=’, 103, ‘ rho=’, 0.0069622746682286895)
(‘k=’, 104, ‘ rho=’, 0.023119818372009802)
(‘k=’, 105, ‘ rho=’, 0.032352553975480675)
(‘k=’, 106, ‘ rho=’, 0.023493277585020157)
(‘k=’, 107, ‘ rho=’, 0.011945244718858262)
(‘k=’, 108, ‘ rho=’, -0.045262670663240967)
(‘k=’, 109, ‘ rho=’, 0.018511455305311932)
(‘k=’, 110, ‘ rho=’, 0.035919072643766675)
(‘k=’, 111, ‘ rho=’, 0.026775723720277748)
(‘k=’, 112, ‘ rho=’, 0.0022014607573888792)
(‘k=’, 113, ‘ rho=’, -0.0066568183511904137)
(‘k=’, 114, ‘ rho=’, 0.00033985821858436384)
(‘k=’, 115, ‘ rho=’, -0.019996960831735567)
(‘k=’, 116, ‘ rho=’, 0.024447496220669412)
(‘k=’, 117, ‘ rho=’, 0.0017114201054675329)
(‘k=’, 118, ‘ rho=’, 0.012842996054216349)
(‘k=’, 119, ‘ rho=’, 0.016046386266568957)
(‘k=’, 120, ‘ rho=’, 0.0097334003467022395)
(‘k=’, 121, ‘ rho=’, 0.01909219393845284)
(‘k=’, 122, ‘ rho=’, 0.0074279005569835542)
(‘k=’, 123, ‘ rho=’, 0.0025669603475214509)
(‘k=’, 124, ‘ rho=’, 0.015461214426239839)
(‘k=’, 125, ‘ rho=’, -0.024335156987780219)
(‘k=’, 126, ‘ rho=’, -0.017062760038822386)
(‘k=’, 127, ‘ rho=’, -0.054802082042580733)
(‘k=’, 128, ‘ rho=’, 0.0065006106652015548)
(‘k=’, 129, ‘ rho=’, 0.0011568983583537895)
(‘k=’, 130, ‘ rho=’, 0.034842008331972409)
(‘k=’, 131, ‘ rho=’, -0.0012816088196605608)
(‘k=’, 132, ‘ rho=’, -0.0013412612306394484)
(‘k=’, 133, ‘ rho=’, 0.0069534628807618121)
(‘k=’, 134, ‘ rho=’, -0.0065170711926642085)
(‘k=’, 135, ‘ rho=’, -0.037208789556520946)
(‘k=’, 136, ‘ rho=’, 0.010381290531440978)
(‘k=’, 137, ‘ rho=’, 0.01077294419726753)
(‘k=’, 138, ‘ rho=’, 0.050843367476408712)
(‘k=’, 139, ‘ rho=’, -0.0038083355907019317)
(‘k=’, 140, ‘ rho=’, 0.0062781363712549878)
(‘k=’, 141, ‘ rho=’, 0.035608651886921437)
(‘k=’, 142, ‘ rho=’, 0.003253755732145406)
(‘k=’, 143, ‘ rho=’, 0.0088885044484776476)
(‘k=’, 144, ‘ rho=’, 0.0032848752758806816)
(‘k=’, 145, ‘ rho=’, 0.013042671696445437)
(‘k=’, 146, ‘ rho=’, 0.014511212979655947)
(‘k=’, 147, ‘ rho=’, 0.027254951977276752)
(‘k=’, 148, ‘ rho=’, 0.016854880767799087)
(‘k=’, 149, ‘ rho=’, -0.012744174132990779)
(‘k=’, 150, ‘ rho=’, 0.0020097295617016071)
(‘k=’, 151, ‘ rho=’, 0.0052533846507287008)
(‘k=’, 152, ‘ rho=’, 0.013207808769294902)
(‘k=’, 153, ‘ rho=’, 0.0047483730664030525)
(‘k=’, 154, ‘ rho=’, -0.012235247538663953)
(‘k=’, 155, ‘ rho=’, -0.02844767732034309)
(‘k=’, 156, ‘ rho=’, 0.0034711008226906343)
(‘k=’, 157, ‘ rho=’, 0.0098648441888563466)
(‘k=’, 158, ‘ rho=’, -0.0046004838891438018)
(‘k=’, 159, ‘ rho=’, 0.017503172042320693)
(‘k=’, 160, ‘ rho=’, -0.004099810745114338)
(‘k=’, 161, ‘ rho=’, 0.04116123074779348)
(‘k=’, 162, ‘ rho=’, 0.012070880036133696)
(‘k=’, 163, ‘ rho=’, 0.036679578291325739)
(‘k=’, 164, ‘ rho=’, 0.001142046736075342)
(‘k=’, 165, ‘ rho=’, 0.0006067453946188954)
(‘k=’, 166, ‘ rho=’, 0.015554750417883682)
(‘k=’, 167, ‘ rho=’, -0.045460854111633663)
(‘k=’, 168, ‘ rho=’, 0.02731012004106138)
(‘k=’, 169, ‘ rho=’, -0.00051200195948666297)
(‘k=’, 170, ‘ rho=’, 0.003354363661831171)
(‘k=’, 171, ‘ rho=’, -0.0035452194591270299)
(‘k=’, 172, ‘ rho=’, -0.0059931892852148104)
(‘k=’, 173, ‘ rho=’, -0.0055779074580996216)
(‘k=’, 174, ‘ rho=’, -0.015531910606195914)
(‘k=’, 175, ‘ rho=’, 0.01364332893516142)
(‘k=’, 176, ‘ rho=’, 0.0041910097457555301)
(‘k=’, 177, ‘ rho=’, 0.026749167934988823)
(‘k=’, 178, ‘ rho=’, 0.012863402486124055)
(‘k=’, 179, ‘ rho=’, 0.022826661291181511)
(‘k=’, 180, ‘ rho=’, 0.017601808693374195)
(‘k=’, 181, ‘ rho=’, 0.0098120299706947467)
(‘k=’, 182, ‘ rho=’, 0.022001791825991195)
(‘k=’, 183, ‘ rho=’, 0.0045668172312326397)
(‘k=’, 184, ‘ rho=’, 0.014047700472253353)
(‘k=’, 185, ‘ rho=’, 0.014010717595984476)
(‘k=’, 186, ‘ rho=’, 0.036558427307108544)
(‘k=’, 187, ‘ rho=’, -0.0040412726045759564)
(‘k=’, 188, ‘ rho=’, -0.01072269722025284)
(‘k=’, 189, ‘ rho=’, -0.011267824502122404)
(‘k=’, 190, ‘ rho=’, 0.046105862986669147)
(‘k=’, 191, ‘ rho=’, 0.015970832016939985)
(‘k=’, 192, ‘ rho=’, 0.020042508712023111)
(‘k=’, 193, ‘ rho=’, -0.0067351176718219727)
(‘k=’, 194, ‘ rho=’, 0.0091567037700150834)
(‘k=’, 195, ‘ rho=’, 0.010029380327033781)
(‘k=’, 196, ‘ rho=’, -0.02611749582443261)
(‘k=’, 197, ‘ rho=’, 0.017650653974853589)
(‘k=’, 198, ‘ rho=’, 0.019029661795288349)
(‘k=’, 199, ‘ rho=’, 0.0074098583087608477)
(‘k=’, 200, ‘ rho=’, -0.0096816081075251671)
(‘k=’, 201, ‘ rho=’, -0.0070847025638858686)
(‘k=’, 202, ‘ rho=’, -0.0026188290845147147)
(‘k=’, 203, ‘ rho=’, 0.015468985953913203)
(‘k=’, 204, ‘ rho=’, -0.0047437065587263511)
(‘k=’, 205, ‘ rho=’, 0.015417363634793655)
(‘k=’, 206, ‘ rho=’, -0.0084951618055395545)
(‘k=’, 207, ‘ rho=’, 0.0061331479409922869)
(‘k=’, 208, ‘ rho=’, 0.011948125025042481)
(‘k=’, 209, ‘ rho=’, 0.015992938911627619)
(‘k=’, 210, ‘ rho=’, 0.011226510096382864)
(‘k=’, 211, ‘ rho=’, 0.023059918357780807)
(‘k=’, 212, ‘ rho=’, -0.016007949135416616)
(‘k=’, 213, ‘ rho=’, 0.0076241421649379786)
(‘k=’, 214, ‘ rho=’, 0.012467598869426492)
(‘k=’, 215, ‘ rho=’, 0.029052372183837188)
(‘k=’, 216, ‘ rho=’, 0.0052925792032430755)
(‘k=’, 217, ‘ rho=’, 0.019736912328116633)
(‘k=’, 218, ‘ rho=’, 0.011933765388360591)
(‘k=’, 219, ‘ rho=’, 0.0023867492151204398)
(‘k=’, 220, ‘ rho=’, 0.011663443373919552)
(‘k=’, 221, ‘ rho=’, 0.0073783308163092726)
(‘k=’, 222, ‘ rho=’, 0.0029003221070863609)
(‘k=’, 223, ‘ rho=’, -0.00024437067981934409)
(‘k=’, 224, ‘ rho=’, -0.033733117722844139)
(‘k=’, 225, ‘ rho=’, -0.0092232497507164987)
(‘k=’, 226, ‘ rho=’, 0.0088576619987832655)
(‘k=’, 227, ‘ rho=’, 0.03617854170267349)
(‘k=’, 228, ‘ rho=’, -0.0062717630357007857)
(‘k=’, 229, ‘ rho=’, -8.0609642527184575e-05)
(‘k=’, 230, ‘ rho=’, 0.00036077355353058481)
(‘k=’, 231, ‘ rho=’, 0.015877709120859756)
(‘k=’, 232, ‘ rho=’, -0.00069291014577824389)
(‘k=’, 233, ‘ rho=’, 0.0054933861872744827)
(‘k=’, 234, ‘ rho=’, 0.0022172911391641439)
(‘k=’, 235, ‘ rho=’, 0.0096246562459923312)
(‘k=’, 236, ‘ rho=’, 0.011424819397695031)
(‘k=’, 237, ‘ rho=’, 0.016528246937788111)
(‘k=’, 238, ‘ rho=’, 0.033407366064212615)
(‘k=’, 239, ‘ rho=’, 0.0094670530467918963)
(‘k=’, 240, ‘ rho=’, 0.031563380731984669)
(‘k=’, 241, ‘ rho=’, -0.014627343230092774)
(‘k=’, 242, ‘ rho=’, -0.013354299182644745)
(‘k=’, 243, ‘ rho=’, -0.028217269193816903)
(‘k=’, 244, ‘ rho=’, -0.010630803428708995)
(‘k=’, 245, ‘ rho=’, 0.015393422647660979)
(‘k=’, 246, ‘ rho=’, -0.0089744682504334025)
(‘k=’, 247, ‘ rho=’, -0.0036485044375085657)
(‘k=’, 248, ‘ rho=’, 0.04490134891661024)
(‘k=’, 249, ‘ rho=’, 0.029110793724287397)
(‘k=’, 250, ‘ rho=’, -0.0047087895801443437)
(‘k=’, 251, ‘ rho=’, 0.020012272038265371)
(‘k=’, 252, ‘ rho=’, 0.014980356683300909)
(‘k=’, 253, ‘ rho=’, 0.026457206861300847)
(‘k=’, 254, ‘ rho=’, 0.010287733029434757)
(‘k=’, 255, ‘ rho=’, -0.020406384067415042)
(‘k=’, 256, ‘ rho=’, 0.012427657375528121)
(‘k=’, 257, ‘ rho=’, 0.030933855088680646)
(‘k=’, 258, ‘ rho=’, -0.023508952993352848)
(‘k=’, 259, ‘ rho=’, -0.0061787445954096503)
(‘k=’, 260, ‘ rho=’, 0.017809498083886575)
(‘k=’, 261, ‘ rho=’, 0.024526514517202426)
(‘k=’, 262, ‘ rho=’, 0.013005384029600171)
(‘k=’, 263, ‘ rho=’, 0.021973659139409712)
(‘k=’, 264, ‘ rho=’, 0.026898808367512177)
(‘k=’, 265, ‘ rho=’, 0.029515078750265728)
(‘k=’, 266, ‘ rho=’, -0.037720681517086713)
(‘k=’, 267, ‘ rho=’, 0.016053603632484308)
(‘k=’, 268, ‘ rho=’, 0.013171697975258892)
(‘k=’, 269, ‘ rho=’, -0.0026747689735233259)
(‘k=’, 270, ‘ rho=’, -0.00043727827075384782)
(‘k=’, 271, ‘ rho=’, 0.011600415563670521)
(‘k=’, 272, ‘ rho=’, 0.013949406278158851)
(‘k=’, 273, ‘ rho=’, 0.12038943714976875)
(‘k=’, 274, ‘ rho=’, 0.026158486455764879)
(‘k=’, 275, ‘ rho=’, 0.012559112901745823)
(‘k=’, 276, ‘ rho=’, -0.020281234576610664)
(‘k=’, 277, ‘ rho=’, 0.014984477542879711)
(‘k=’, 278, ‘ rho=’, 0.0040995968299132619)
(‘k=’, 279, ‘ rho=’, -0.00078364998467419784)
(‘k=’, 280, ‘ rho=’, -0.018239988272112789)
(‘k=’, 281, ‘ rho=’, 0.043349758014894026)
(‘k=’, 282, ‘ rho=’, 0.02275900792985101)
(‘k=’, 283, ‘ rho=’, 0.027555926185902524)
(‘k=’, 284, ‘ rho=’, 0.021883986418438906)
(‘k=’, 285, ‘ rho=’, 0.011773525758729073)
(‘k=’, 286, ‘ rho=’, 0.022645202042396874)
(‘k=’, 287, ‘ rho=’, 0.014164034857565358)
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(‘k=’, 1869, ‘ rho=’, -0.02394542353223128)
(‘k=’, 1870, ‘ rho=’, -0.0012215871633002875)
(‘k=’, 1871, ‘ rho=’, 0.0088327000475961707)
(‘k=’, 1872, ‘ rho=’, -0.010145207077391808)
(‘k=’, 1873, ‘ rho=’, 0.0097098744864301204)
(‘k=’, 1874, ‘ rho=’, 0.041726943249742507)
(‘k=’, 1875, ‘ rho=’, -0.0084841443116240121)
(‘k=’, 1876, ‘ rho=’, 0.030093495211848037)
(‘k=’, 1877, ‘ rho=’, 0.0084355858628549722)
(‘k=’, 1878, ‘ rho=’, -0.023354148015313301)
(‘k=’, 1879, ‘ rho=’, -0.0017002735771937841)
(‘k=’, 1880, ‘ rho=’, -0.0062071091647407269)
(‘k=’, 1881, ‘ rho=’, -0.0066541609187143304)
(‘k=’, 1882, ‘ rho=’, 0.053805483065180654)
(‘k=’, 1883, ‘ rho=’, -0.0070897877857596112)
(‘k=’, 1884, ‘ rho=’, 0.015610593475612965)
(‘k=’, 1885, ‘ rho=’, -0.0027750178446876326)
(‘k=’, 1886, ‘ rho=’, 0.012852669996528957)
(‘k=’, 1887, ‘ rho=’, 0.0095685043447754133)
(‘k=’, 1888, ‘ rho=’, 0.011414768683530609)
(‘k=’, 1889, ‘ rho=’, 0.025108256680230347)
(‘k=’, 1890, ‘ rho=’, 0.032387061544283273)
(‘k=’, 1891, ‘ rho=’, -0.03292216419144528)
(‘k=’, 1892, ‘ rho=’, 0.0076611596160343787)
(‘k=’, 1893, ‘ rho=’, -0.0034289842567330957)
(‘k=’, 1894, ‘ rho=’, -0.026464256288398581)
(‘k=’, 1895, ‘ rho=’, 0.015977222114513748)
(‘k=’, 1896, ‘ rho=’, 0.030712596535602082)
(‘k=’, 1897, ‘ rho=’, -0.012179341558460097)
(‘k=’, 1898, ‘ rho=’, -0.023350814238359708)

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The key to extending the Heston model to a fractional stochastic volatility model is to consider the equation for the characteristic function

f(x,v,t) = \exp( C(t) + G(t) v + i\phi x)

in Heston’s notation (Heston-original) but with

G(t) = \int_0^t S_{\alpha}(s) D'(s) ds

where D(t) solves the Heston’s Riccati equation.  The difference between this solution and the Comte-Renault approach is that they fractionally integrate the volatility of Heston’s model while this approach produces directly a closed form solution as follows: solve for D(t) using Heston’s Riccati equation and then integrate by S_{\alpha}(t) = t^{\alpha-1} E_{\alpha,\alpha}(-\kappa t^\alpha) to obtain the solution of an exact PDE for long memory fractional stochastic volatility.

Some more details are here (fracsv)

 

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ClosedFormHeston

For the model

dS(t)=S(t) \sqrt{V(t)} dw^1(t)

D^\alpha_t V(t) = -\kappa(V(t)-\theta) dt + \nu\sqrt{V(t)} dw^2(t)

Corr(dw^1(t),dw^2(t)) = \rho dt

The solution in the above screenshot from Heston’s paper remains valid with the change \rho \rightarrow \rho S_{\alpha}(T-t) and \nu \rightarrow \nu S_{\alpha}(T-t) where

S_{\alpha}(s) = s^{\alpha-1}E_{\alpha,\alpha}(-\kappa s^\alpha)

This follows from the computation of the Black-Scholes PDE computation in my last blog.

C(t,\phi) = r\phi i t + \frac{a}{\sigma S_{\alpha}(T-t)^2}( (b_j - \rho \nu S_{\alpha}(T-t)^2}\phi i + d) t - 2 \log(\frac{1-g e^{dt}}{1-g}))

D(t,\phi) = \frac{b_j - S_{\alpha}(t)^2\rho\nu \phi i + d}{S_{\alpha}(t)^2\nu^2}( \frac{1-ge^{dt}}{1-g})

g = \frac{b_j + d - \rho\nu S_{\alpha}(t)^2 \phi i}{b_j - \rho\nu S_{\alpha}(t)^2 \phi i}

d = \sqrt{ (S_{\alpha}(t)^2 \rho\nu \phi i - b_j)^2 - \nu^2(2 u_f \phi i - \phi^2)}

f_j(x,v,t,\phi) = \exp( C(t,\phi) + D(t,\phi) v + i\phi)

The probabilities that the option expires in-the-money are then exactly the same as the solution provided by Heston

P_j(x,v,t,\phi) = \frac{1]{2} +\frac{1}{\pi} \int_0^\infty \frac{\exp(-i \phi\log(K) f_j(x,v,T,\phi)}{i\phi} d\phi

The value of a call option is then:
C(S,v,t) = SP_1 + e^{-r t}KP_2

 

 

 

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