View
199
Download
0
Category
Tags:
Preview:
DESCRIPTION
1) Volatility modelling 2) Local stochastic volatility models: stochastic volatility, jumps, local volatility 3) Calibration of parametric local volatility models using partial differential equation (PDE) methods 4) Calibration of non-parametric local volatility volatility models with jumps and stochastic volatility using PDE methods 5) Numerical methods for PDEs 6) Illustrations using SPX and VIX data
Citation preview
Efficient Numerical PDE Methods to Solve Calibration andPricing Problems in Local Stochastic Volatility Models
Artur SeppBank of America Merrill Lynch, London
artur.sepp@baml.com
Global Derivatives Trading & Risk Management 2011Paris
April 12-15, 2011
1
Plan of the presentation
1) Volatility modelling
2) Local stochastic volatility models: stochastic volatility, jumps,local volatility
3) Calibration of parametric local volatility models using partial dif-ferential equation (PDE) methods
4) Calibration of non-parametric local volatility volatility models withjumps and stochastic volatility using PDE methods
5) Numerical methods for PDEs
6) Illustrations using SPX and VIX data
2
References
Some theoretical and practical details for my presentation can befound in:
1) Sepp, A. (2012) An Approximate Distribution of Delta-HedgingErrors in a Jump-Diffusion Model with Discrete Trading and Trans-action Costs, Quantitative Finance, 12(7), 1119-1141http://ssrn.com/abstract=1360472
2) Sepp, A. (2008) VIX Option Pricing in a Jump-Diffusion Model,Risk Magazine April, 84-89http://ssrn.com/abstract=1412339
3) Sepp, A. (2008) Pricing Options on Realized Variance in the He-ston Model with Jumps in Returns and Volatility, Journal of Compu-tational Finance, Vol. 11, No. 4, pp. 33-70http://ssrn.com/abstract=1408005
3
Literature
Local volatility: Dupire (1994)
Local volatility with jumps: Andersen-Andreasen (2000)
Local stochastic volatility with jumps: Lipton (2002)
4
Volatility modelling I
What is important for a competitive pricing and hedging volatilitymodel?
1) Consistency with the observed market dynamics implies stablemodel parameters and hedges
2) Consistency with vanilla option prices ensures that the modelfits the risk-neutral distribution implied by these prices
3) Consistency with market prices of liquid exotic options
5
Volatility modelling II. Volatility models1) Non-parametric local volatility models (Dupire (1994), Derman-Kani (1994), Rubinstein (1994))”+” LV models are consistent with vanilla prices by construction”-” but they tend to be poor in replicating the market dynamics ofspot and volatility (implied volatility tends to move too much givena change in the spot, no mean-reversion effect)”-” especially, it is impossible to tune-up the volatility of the impliedvolatility as there is simply no parameter for that!
2) Parametric stochastic volatility models (Heston (1993))”+” SV models tend to be more aligned with the market dynamics”+” introduce the term-structure (by mean-reversion parameters)and the volatility of the variance (by vol-of-vol parameters)”-” need a least-squares calibration to today’s option prices”-” unfortunately, any change in any of the parameters of the mean-reversion or the vol-of-vol requires re-calibration of other parameters
3) Stochastic local volatility models (JP Morgan (1999), Blacher(2001), Lipton (2002), Ren-Madan-Qian (2007))LSV models aim to include ”+”’s and cross ”-”’s of first two models
6
Volatility modelling III. LSV model specification
1) Global factors - specify the dynamics for the instantaneous volatil-ity, include jumps or default risk if this is relevant for product risk
(Guess) Estimate or calibrate model parameters for the dynamics ofthe instantaneous volatility and jumps using either historical or marketdata
Parameters are updated infrequently
2) Local factors - specify local factors for either parametric or non-parametric local volatility
Parameters of local factors are updated frequently (on the run) tofit the risk-neutral distributions implied by market prices of vanillaoptions
3) Mixing weight - specify mixing weight between stochastic andlocal volatilities, adjust vol-of-vol and correlation accordingly
7
LSV model dynamics. We consider a specific version of the LSVdynamics with jumps under pricing measure:
dS(t) = µ(t)S(t−)dt+ σ(loc,svj)(t−, S(t−))ϑ(t−, Y (t−))S(t−)dW (1)(t)
+((e−ν − 1)dN(t) + λνdt
)S(t−), S(0) = S
dY (t) = −κY (t)dt+ εdW (2)(t) + ηdN(t), Y (0) = 0(1)
S(t) is the underlying and Y (t) is the factor for instantaneous stochas-tic volatility with dW (1)(t)dW (2)(t) = ρdt
σ(loc,svj)(t, S(t)) is the local volatility (to be considered later)
ϑ(t, Y (t)) is volatility mapping with V[Y (t)] being variance of Y (t):
ϑ(t, Y (t)) = eY (t)−V[Y (t)]
For ρ = 0, E[ϑ2(t, Y (t)) |Y (0) = 0
]= 1, so that ϑ(t, Y (t)) introduces
”volatility-of-volatility” effect without affecting the local volatilityclose to the spot
8
Stochastic volatility factor IQ: Can we observe the dynamics of stochastic volatility factor Y (t)?A: Consider the model implied ATM volatility squared:
V (t) = σ20e
2Y (t)−2V[Y (t)] ≈ σ20(1 + 2Y (t))
Thus, given daily time series of implied ATM volatility σATM(tn):
Y (tn) =1
2
σ2ATM(tn)
σ2ATM
− 1
where σ2
ATM is the average ATM volatility during the period
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
Jan
-07
Ap
r-07
Jul-
07
Oct
-07
Jan
-08
Ap
r-08
Jul-
08
Oct
-08
Jan
-09
Ap
r-09
Jul-
09
Oct
-09
Jan
-10
Ap
r-10
Jul-
10
Oct
-10
Jan
-11
ATM vol
-1.00
0.00
1.00
2.00
3.00
4.00
Jan
-07
Ap
r-07
Jul-
07
Oct
-07
Jan
-08
Ap
r-08
Jul-
08
Oct
-08
Jan
-09
Ap
r-09
Jul-
09
Oct
-09
Jan
-10
Ap
r-10
Jul-
10
Oct
-10
Jan
-11
Y(t)
Left: One-month ATM implied volatility for SPX; Right: Y(t)9
Stochastic volatility factor II. Estimation
To obtain historical estimates for the mean-reversion and vol-of-vol,apply regression model:
Y (tn) = βY (tn−1) + θςn
where ςn is iid normals
Thus:κ = −252 ln(β)/2
ε =√
252θ(2κ)/(1− e−2κ/252)
For SPX (using data for last 4 years):R2 = 74%κ = 4.49ε = 2.30
10
Stochastic volatility factor III. Fit to implied SPX volatilities (23-March-1011) using flat local volatility (log-normal SV model) andhistorical parameters with ρ = −0.8Conclusion: pure stochastic volatility model is adequate to describelonger terms skew but need local factors to ”correct” mis-pricings
1M
0%
10%
20%
30%
40%
50%
60%
70%
50% 70% 90% 110% 130% 150%
K/S
Market, 1mModel, 1m
3M
0%
10%
20%
30%
40%
50%
60%
50% 70% 90% 110% 130% 150%
K/S
Market, 3mModel, 3m
6M
0%
10%
20%
30%
40%
50%
50% 70% 90% 110% 130% 150%
K/S
Market, 6mModel, 6m
1Y
0%
10%
20%
30%
40%
50%
50% 70% 90% 110% 130% 150%
K/S
Market, 1yModel, 1y
2Y
0%
10%
20%
30%
40%
50% 70% 90% 110% 130% 150%
K/S
Market, 2yModel, 2y
3Y
0%
10%
20%
30%
40%
50% 70% 90% 110% 130% 150%
K/S
Market, 3yModel, 3y
11
Jumps I
We assume that jumps in S(t) and Y (t) are simultaneous and dis-crete with magnitudes −ν < 0 and η > 0
Q: Why do we need jumps in addition to local stochastic volatility?A: Market prices of exotics (eg cliques) imply very steep forward skewsso the value of vol-of-vol parameter must be very large (I will showan example in a bit)
However, local stochastic volatility with a large vol-of-vol parametercannot be calibrated consistently to given local volatility
Calibration to both exotics and vanilla can be achieved by addinginfrequent but large negative jumps
Q: Why simultaneous jumps?A: Realistic, do not decrease the implied spot-volatility correlation
12
Jumps IIQ: Why discrete jumps?A: Check instantaneous correlation between X(t) = lnS(t) and Y (t):
ς ≡< dX(t), dY (t) >√
< dX(t), dX(t) >√< dY (t), dY (t) >
=ρεσ(loc,svj)(t, S(t))ϑ(t, Y (t)) + νηλ√
(σ(loc,svj)(t, S(t))ϑ(t, Y (t)))2 + ν2λ√ε2 + η2λ
In equity markets, ς∗ ∈ [−0.5,−0.9]If Y (t)→∞ or λ→ 0 then ς → ρIf Y (t)→ 0 or λ→∞ then ς → sign(ν) = −1 if ν < 0
For discrete jumps, the correlation is the smallest possible: ς ≈ −1For exponential jumps: ς → 0.5sign(ν) = sign(ν) = −0.5 if ν < 0
Jump variance leads to de-correlation (not good for stability of ρ)
Experience: even for 1-d jump-diffusion Merton model, adding jumpvariance to model calibration leads to less stable calibration
13
Jumps III. Q: How to estimate jump parameters?Consider a jump-diffusion model under the historic measure:
dS(t)/S(t) = µdt+ σdW (t) +(eJ − 1
)dN(t), S(t0) = S,
N(t) is Poisson process with intensity λ
Jumps PDF is normal mixture: w(J) =∑Ll=1 pln(J; νl, υl),
∑Ll=1 pl =
1L, L = 1,2, .., is the number of mixturespl, 0 ≤ pl ≤ 1, is the probability of the l-th mixtureνl and υl are the mean and variance of the l-th mixture, l = 1, .., L
Cumulative probability function can be represented as weighted sumover normal probabilities (see Sepp (2011))
Can be applied for estimation of jumps
14
Jumps IV. Empirical estimation for the S&P500 using last 10 yearsFit empirical quantiles of sampled daily log-returns to model quantilesby minimizing the sum of absolute differences
The best fit, keeping L as small as possible, is obtained with L = 4Model estimates: σ = 0.1348, λ = 46.4444, υl is uniforml pl νl
√υl plλ expected frequency
1 0.0208 -0.0733 0.0127 0.9641 every year2 0.5800 -0.0122 0.0127 26.9399 every two weeks3 0.3954 0.0203 0.0127 18.3661 every six weeks4 0.0038 0.1001 0.0127 0.1743 every five years
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10X
Fre
qu
ency
Empirical PDFNormal PDFJDM PDF
-0.12
-0.08
-0.04
0.00
0.04
0.08
0.12
-0.12 -0.08 -0.04 0.00 0.04 0.08 0.12
JDM quantiles
Dat
a q
uan
tile
s
15
Jumps V. Implied Calibration of Jumps
Remove intermediate jumps (l = 2,3) and large positive jumps (l = 4)- the stochastic local volatility takes care of ”moderate” skew
Jump magnitude from historical estimation ν = −0.0733 is too smallto fit one-month SPX skew
An estimate implied from one-month SPX options is (approximately)ν = −0.25
Assume only large negative jumps with λ = 0.2
The magnitude of jump in volatility η is calibrated to match VIXskews
Typically η = 1.0 implying that ATM volatility will double following ajump
16
Local VolatilityUsing local volatility we fit vanilla options while having flexibility tospecify the dynamics for stochastic volatility and jumps
Local volatility parametrization:
1) Parametric local volatility
Specify a functional form for local volatility (CEV, shifted-lognormal,quadratic, etc)
Calibrate by boot-strap and least-squares
Good for single stocks where implication of Dupire non-parametriclocal volatility is problematic and unstable due to lack of quotes
2) Non-parametric local volatility
Fit the model local volatility to Dupire local volatility
Good for indices with liquid option market
17
Parametric local volatility IConsider the following parametric form:
σloc(t, S) = σatm(t)σskew(t, S)σsmile(t, S) (2)
where σatm(t) is at-the-money forward volatility
Skew function is specified as ratio of two CEVs:
σskew(t, S) =(S/S0)β(t)−1 + a
(S/S0)β(t)−1 + b
=1
2
((1 + q) + (1− q) tanh
(1 + q
1− q(β(t)− 1) ln(S/S0)
))where β(t) is the skew parameter, q is weight parameter, 0 < q < 1
Smile function is quadratic:
σsmile(t, S) =√
1 + (α(t) ln(S/S0))2
where α(t) is smile parameter
18
Parametric local volatility IIQ: What is intuition about the model parameters and initial guessesfor model calibration?A: From the given market implied volatilities, compute ATM volatility,skew and smile as follows:ϑatm(T ) = σ(T,K = S(T ))ϑskew(T ) ∝ σ(T,K = S(T )+)− σ(T,K = S(T )−)ϑsmile(T ) ∝ σ(T,K = S(T )+)− 2σ(T,K = S(T )) + σ(T,K = S(T )−)
Consider local volatility at S = S0 to obtain:
σatm(t) =2ϑatm(t)
1 + q
β(t) =2ϑskew(t)
ϑatm(t)+ 1
α2(t) =2ϑsmile(t)
ϑatm(t)
(3)
σatm(t) is proportional to the ATM volatilityβ(t) is the skew relative to ATM volatilityα(t) is the smile relative to ATM volatility
19
Calibration of parametric models I. Backward equationConsider general pricing equation for options on S(t) under stochasticlocal volatility modelPV function U(t, S, Y ;T,K), 0 ≤ t ≤ T , solves the backward equationas function of (t, S, Y ):
Ut + LU(t, S, Y ) = 0
U(T, S, Y ) = u(S)(4)
where u(S) is pay-off functionL is infinitesimal backward operator:
LU(t, S, Y ) =1
2(σ(loc,svj)(t, S)ϑ(t, Y )S)2USS + µ(t)SUS
+1
2ε2(Y )UY Y + θ(Y )UY
+ ρσ(loc,svj)(t, S)ϑ(t, Y )ε(Y )SUSY + λI(S)
I(S) =∫ ∞−∞
∫ ∞−∞
U(SeJ , Y + Υ)$(J)ς(Υ)dJdΥ− νSUS − U
For generality, jump sizes in S and Y have PDFs $(J) and ς(Υ),respectively; ν is jump compensator
20
Calibration of parametric models II. Forward equationTransition probability density function, G(0, S0, Y0; t, S′, Y ′) solves theforward equation as function of (t, S′, Y ′):
Gt(t, S′, Y ′)− L†G(t, S′, Y ′) = 0,
G(0, S′, Y ′) = δ(S′ − S0)δ(Y ′ − Y0)(5)
where δ() is Dirac delta function
L† is forward operator adjoint to L:
L†G(t, S′, Y ′) = −1
2
((σ(loc,svj)(T, S′)ϑ(T, Y ′)S′)2G
)S′S′
+(µ(T )S′G
)S′
−1
2
(ε2(Y ′)G
)Y ′Y ′
+(θ(Y ′)G
)Y ′
−(ρσ(loc,svj)(T, S′)ϑ(T, Y ′)ε(Y ′)S′G
)S′Y ′− λI(S′)
I(S′) =∫ ∞−∞
∫ ∞−∞
G(S′e−J , Y ′ −Υ)e−J$(J)ς(Υ)dJdΥ + (νS′G)S′ −G
21
Calibration of parametric models III Option PV can be computedby convolution (Duhamel’s or Feynman-Kac formula):
U(0, S0, Y0;T,K) =∫ ∞−∞
∫ ∞−∞
G(0, S0, Y0;T, S′, Y ′)u(S′)dS′dY ′ (6)
Calibration by bootstrap: For parametric local volatility, model pa-rameters are assumed to be piece-wise constant in time with jumpsat times {Tm}, {Tm} is set of maturity times of listed options
1) Given calibrated set of piece-wise constant model parameters attime Tm−1 and known values of G(Tm−1, S, Y ), make a guess for pa-rameters at time Tm and compute G(Tm, S, Y )
2) Compute PV-s of vanilla options using discrete version of (6)
3) By changing parameters for time Tm, minimize the sum of squareddifferences between model and market prices
4)After convergence, store G(Tm, S, Y ) and go to the next time slice
22
Calibration of parametric models IV. Numerical schemesConsistency condition for adjoint operators L and L† (can be checkedby integration by parts) is based on theoretical arguments:∫ T
0
∫ ∞−∞
∫ ∞−∞
[{L†G(0, S0, Y0; t′, S′, Y ′)
}U(t′, S′, Y ′)
−G(0, S0, Y0; t′, S′, Y ′){LU(t′, S′, Y ′)
}]dS′dY ′dt′ = 0
(7)
It is not necessarily true for discrete numerical schemes (Lipton (2007))
Implications for numerical schemes:1) For adjoint operators, if M is spacial discretisation of L then MT
should be spacial discretisation of L†2) If MT is diagonally dominant with positive diagonal and non-positive off-diagonal elements (for 1-d problems with zero correla-tion), then (MT )−1 > 0 and computed probabilities are non-negativeand sum up to one (Andreasen (2010), Nabben (1999))3) Both forward and backward equations should be solved using thesame schemes4) L and L† have the same values of operator norm, so the conver-gence of the forward scheme implies convergence of backward scheme
23
Calibration of parametric models V. Non-linear least square fitModel parameters: Θ = (θ1, ..., θq), Model prices: U(Θ) = (u1, ..., un)
Jacobian: Jn×q = (Jn′q′): Jn′q′ =∂un′∂θq′
(computed numerically)
Market prices: M = (m1, ...,mn), Weight (diagonal) matrix: Wn×n
Gauss-Newton normalized equations:
(JTWJ)(∆Θ) = JTW(∆U), ∆U = M−U(Θ(k)), Θ(k+1) = Θ(k)+∆Θ
Use SVD for stability, dimensionality is small so calibration is fast
Levenberg-Marquardt method:
(JTWJ + λI)(∆Θ) = JTW(∆U)
λ is Marquardt parameter to make the matrix diagonally-dominantLarge value of λ - the steepest descent (tendency to be slow)How to set λ? (NR in C (Press (1988)) use deterministic value)One possibility: λ = 1/trace(JTWJ)
Experience: for carefully chosen functional form of the local volatilitywith robust initial estimates, Gauss-Newton works fast
24
Calibration of parametric models VI. IllustrationImplied volatilities for parametric volatility with vol-of-vol reduced by50%: the fit is acceptable
1M
0%
5%
10%
15%
20%
25%
30%
35%
40%
75% 85% 95% 105% 115% 125%
K/S
Market, 1mModel, 1m
3M
0%
5%
10%
15%
20%
25%
30%
35%
75% 85% 95% 105% 115% 125%
K/S
Market, 3mModel, 3m
6M
0%
10%
20%
30%
40%
50%
50% 70% 90% 110% 130% 150%
K/S
Market, 6mModel, 6m
1Y
0%
10%
20%
30%
40%
50%
50% 70% 90% 110% 130% 150%
K/S
Market, 1yModel, 1y
2Y
0%
10%
20%
30%
40%
50% 70% 90% 110% 130% 150%
K/S
Market, 2yModel, 2y
3Y
0%
10%
20%
30%
40%
50% 70% 90% 110% 130% 150%
K/S
Market, 3yModel, 3y
25
Local volatility calibration I. Diffusion model
Consider a one-dimensional diffusion:
dS(t) = µ(t)S(t)dt+ σ(loc,dif)(t, S(t))S(t)dW (t), S(0) = S (8)
where σ(loc,dif) is the local volatility of the diffusion
Calibration objective: how we should specify σ(loc,dif)(T, S′)?
In terms of call option prices, we have Dupire equation (1994):
σ2(loc,dif)(T,K) =
CT (T,K) + µ(T )KCK(T,K)− µ(T )C(T,K)12K
2CKK(T,K)(9)
26
Local volatility calibration II. Jump-diffusion model. Andersen-Andreasen (2000) consider Merton jump-diffusion with local volatility:
dS(t) =µ(t)S(t−)dt+ σ(loc,jd)(t, S(t−))S(t−)dW (t)
+((eJ − 1)dN(t)− λνdt
)S(t−), S(0) = S
(10)
Here σ2(loc,jd) is specified by Andersen-Andreasen equation (2000):
σ2(loc,jd)(T,K) =
CT (T,K) + µ(T )KCK(T,K)− µ(T )C(T,K)− λI†(K)12K
2CKK(T,K)
= σ2(loc)(T,K)−
λI†(K)12K
2CKK(T,K)(11)
I†(K) =∫ ∞−∞
C(Ke−J′)eJ
′$(J ′)dJ ′+ νKCK − (ν + 1)C (12)
Calibration: i) Given Dupire local volatility σ(loc,dif)(T,K) solve for-ward equation for call prices C(T,K) using FD for 1-d problemii) Compute I†(K) and imply σ(loc,jd)(T,K)
27
Local volatility calibration III. Stochastic local volatility:
dS(t) = µ(t)S(t)dt+ σ(loc,sv)(t, S(t))ϑ(t, Y (t))S(t)dW (1)(t), S(0) = S,
dY (t) = θ(Y (t))dt+ ε(Y (t))dW (2)(t), Y (0) = Y
The model is consistent with the Dupire local volatility if
σ2(loc,sv)(T,K) =
σ2(loc,dif)(T,K)
V (T,K)(13)
where V (T, S′) is the conditional variance:
V (T, S′) =
∫∞−∞ ϑ
2(T, Y ′)G(t, S, Y ;T, S′, Y ′)dY ′∫∞−∞G(t, S, Y ;T, S′, Y ′)dY ′
(14)
Calibration:(More details to follow)i) Stepping forward in time, solve for density G(t, S, Y ;T, S′, Y ′) usingFD for 2-d problemii) Compute V (T,K) and imply σ2
(loc,sv)(T,K) using (13)
28
Local volatility calibration IV. Stochastic local volatility withjumps specified by dynamics (1)Local stochastic volatility for jump-diffusion, σ2
(loc,svj)(T,K), is spec-
ified by Lipton equation (2002):
σ2(loc,svj)(T,K) =
CT (T,K) + µ(T )KCK(T,K)− µ(T )C(T,K)− λI†(K)12V (T,K)K2CKK(T,K)
=1
V (T,K)
σ2(loc,dif)(T,K)−
λI†(K)12K
2CKK(T,K)
=σ2
(loc,jd)(T,K)
V (T,K)(15)
Calibration: i) Given Dupire local volatility σ(loc,dif)(T,K) solve for-ward equation for call prices C(T,K) using FD for 1-d problem andimply σ(loc,jd)(T,K) using (11)
ii) Stepping forward in time, solve for density G(t, S, Y ;T, S′, Y ′) usingFD for 2-d problem with jumpsiii) compute V (T,K) using (14) and imply σ2
(loc,sv)(T,K) using (15)
29
PDE based methods
Non-parametric local stochastic volatility can only be implementedusing numerical methods for partial integro-differential equations
These methods should be flexible to handle jumps in one and twodimensions and the correlation term for two dimensions
I will review some of the available methods
30
1-d Problem. For the backward problem:
L = D+ λJHere D is the diffusion-convection operator:
DU(t, S) ≡1
2σ2(t, S)S2USS + µ(t)SUS + λU
J is the integral operator:
JU(t, S) ≡∫ ∞−∞
U(t, SeJ′)$(J ′)dJ ′
For the forward problem, we consider the operator L† adjoint to L:
L† = D†+ λJ †
For both problems, denote the discretized diffusion and integral op-erators by Dn and J, respectively
i) The diffusion operator is space and time dependentii) Care must be taken for discretization of the operator for the mean-reverting process
31
1-d Problem for diffusion problem. N - number of spacial stepsDn - discrete diffusion matrix at time tn with dimension N ×NUn - the solution vector at time tn with dimension N × 1Time-stepping with θ-method:(
I− θDn+1)Un+1 =
(I + θDn+1
)Un
i) θ = 0 - explicit method requires very fined grids and is highlyunstable - it should be avoidedii) θ = 1/2 - Crank-Nicolson scheme is unconditionally stable and isof order (δS)2 and (δt)2, but might lead to negative densitiesiii) θ = 1 - implicit method is unconditionally stable and does not leadto negative densities, but it is of order (δt) and becomes less accuratefor longer maturities
My favourite is the predictor-corrector scheme:(I−Dn+1
)U = Un(
I−Dn+1)Un+1 = Un +
1
2Dn+1(Un − U)
with order of (δS)2 and (δt)2; does not lead to negative densities
32
Numerics for jump-diffusions IConsider the forward equation with additive jumps (multiplicativejumps can be handled in the log-space):
JG(x) =∫ ∞−∞
G(x− j)$(J ′)dJ ′
Consider discrete negative jumps with size −η, η > 0:
JG(x) = G(x+ η)
Discretization is a simple linear interpolation:
JGi = wGk + (1− w)Gk+1
where k = min{k : xk+1 ≥ xi + ν} and w =xk+1−(xi+ν)xk+1−xk
33
Numerics for jump-diffusions IIIn the matrix form, for uniform spacial grid:
J =
0 0 ... w 1− w 0 ... 0 00 0 ... 0 w 1− w ... 0 0...0 0 ... 0 0 0 ... w 1− w0 0 ... 0 0 0 ... 0 1...0 0 ... 0 0 0 ... 0 0
J(p,N) = 1, where p = min{p : xp + ν ≥ xN}
In general, for two-sided jumps, J is a full matrix:The implicit method for the integral term leads to O(N3) complexityand is not practicalThe explicit method leads to O(N2) complexity but needs extracare for stability
For exponential jumps, Lipton (2003) develops recursive scheme withO(N) complexity
34
Numerics for jump-diffusions III
In case of discrete jumps, we have two-banded matrix with p rowsthat have non-zero elements
Consider solving a linear system:
(I− βJ)X = R
where β = λδt, 0 < β < 1
We can solve this system by back-substitution with cost of O(N)operations
35
Numerics for jump-diffusions IVConsider θ and θJ schemes for the diffusion and the jump parts:(
I− θDn+1 − θJλn+1J)Un+1 =
(I + θDn+1 + θJλn+1J
)Un
where λn+1 = (tn+1 − tn)λ(tn+1)
The accuracy with θ = θJ = 12 is supposed to be O(dx2) +O(dt2)
However, the matrix in the lhs will be full: the cost to invert is O(N3)
Implicit-explicit method:(I−Dn+1
)Un+1 =
(I + λn+1J
)Un
with accuracy of O(dx2) +O(dt)
θ-explicit method:(I− θDn+1
)Un+1 =
(I + θDn+1 + λn+1J
)Un
with accuracy of O(dx2) +O(dt)
36
Numerics for jump-diffusions V. Andersen-Andreasen (2000)scheme
Make the first half step with θ = 1 and θJ = 0 and the second halfstep with θ = 1 and θJ = 1:(
I−1
2Dn+1
)U =
(I +
1
2λn+1J
)Un(
I−1
2λn+1J
)Un+1 =
(I +
1
2Dn+1
)U
with accuracy of O(dx2) +O(dt2)
When J is full, the second equation can be solved by the applicationof the discrete Fourier transform (DFT) with the cost of O(N logN)
Beware of deficiencies of the DFT
For discrete jumps, the second equation solved with cost of O(N)operations
37
Numerics for jump-diffusions VI. d’Halluin et al (2005) scheme
Consider fixed point iterations:
Set V 1 = Un
Iterate for p = 1, .., p (p = 2 is good enough):(I− θDn+1
)V p+1 =
(I + θDn+1
)Un + λn+1JV
p,
Set Un+1 = V p
The accuracy is O(dx2) +O(dt)
38
Numerics for jump-diffusions VII
My favourite is the implicit scheme with predictor-corrector appliedtwice:
U(0) =(I + Dn+1 + λn+1J
)Un(
I−Dn+1)U = U(0)(
I−Dn+1)Un+1 = U(0) +
1
2(Dn+1 + λn+1J)(U − Un)
The accuracy is O(dx2) +O(dt2)
39
Numerical Methods for Two Dimensional Problem I
We consider the forward problem for U(t, x1, x2):
UT −MU = 0
U(0, x1, x2) = δ(x1 − x1(0))δ(x2 − x2(0))(16)
where
M = D1 +D2 + C+ λJD1 and D2 are 1-d diffusion-convection operators in x1 and x2 direc-tions, respectivelyC is the correlation operatorJ is the integral operator for joint jumps in x1 and x2
Let D1 and D2 denote the discretized 1-d diffusion-convection oper-ators in x1 and x2 directions, respectively
C and J are the discretized correlation and integral operator, respec-tively
40
Douglas-Rachford (1956) scheme
Make a predictor and apply two orthogonal corrector steps:
U(0) = (I + C + λJ + D1 + D2)Un
(I− θD1)U = U(0) − θD1Un
(I− θD2)Un+1 = U − θD2Un
In the second line, for each fixed index j we apply the diffusion oper-ator in x1 direction; and solve the tridiagonal system of equations toget the auxiliary solution U(·, x2(j))
In the third line, keeping i fixed, we apply the diffusion step in x2direction and solve the system of tridiagonal equations to get thesolution Un+1(x1(i), ·) at time tn+1
Complexity is O(N1N2) per time step
41
Craig-Sneyd (1988) scheme
Start as with Douglas-Rachford scheme make a second predictor andagain apply two orthogonal corrector steps:
U(0) = (I + C + λJ + D1 + D2)Un
(I− θD1)U(1) = U(0) − θD1Un
(I− θD2)U(2) = U(1) − θD2Un
U(3) = U(0) +1
2(C + λJ)(U(2) − Un)
(I− θD1)U(4) = U(3) − θD1Un
(I− θD2)Un+1 = U(4) − θD2Un
42
Hundsdorfer-Verwer (2003) scheme (in’t Hout-Foulon (2008))
Similar to Craig-Sneyd scheme with predictor including D1 and D2and the second corrector applied on U(2):
U(0) = (I + C + λJ + D1 + D2)Un
(I− θD1)U(1) = U(0) − θD1Un
(I− θD2)U(2) = U(1) − θD2Un
U(3) = U(0) +1
2(C + λJ + D1 + D2)
(U(2) − Un
)(I− θD1)U(4) = U(3) − θD1U
(2)
(I− θD2)Un+1 = U(4) − θD2U(2)
43
Discretisation of integral term
Direct methods are infeasible because of O(N21N
22 ) complexity
DFT method (Clift-Forsyth (2008)) has O(N1N2 logN1N2) complex-ity but suffers from problems associated with the DFT
Explicit methods with O(N1N2) complexity are available for discreteand exponential jumps (Lipton-Sepp (2011))
The simplest case is if jumps are discrete with sizes η1 and η2:
JG = G(x1 − η1, x2 − η2)
This term is approximated by bi-linear interpolation with the secondorder accuracy leading to the O(N1N2) complexity
44
Summary I. LSV model calibration
1) Specify time and space grids
2) Compute the local volatility using Dupire equation (9) on the spec-ified grid (interpolating implied volatilities in strikes and maturities)
3) Calibrate local stochastic volatility on the specified grid
4) Use backward PDE methods or Monte-Carlo simulations of thelocal stochastic volatility model to compute values of exotic options(see Andreasen-Huge (2010) for consistent schemes)
45
Summary II. Local volatility calibration at time tngiven Gn−1(x1(i), x2(j)) and V (tn−1, x1(i))i) Compute the local variance by:
σ2(loc,sv)(tn, x1(i)) =
σ2(loc,dif)(tn, x1(i))
V (tn−1, x1(i))
ii) Compute Gn(x1(i), x2(j)) by solving the forward PDE
iii) Compute V (tn, x1(i)) by
V (tn, x1(i)) =
∑i∑j ϑ
2(tn, x2(j))Gn(x1(i), x2(j))∑j G
n(x1(i), x2(j))
and set
σ2(loc,sv)(tn, x1(i)) =
σ2(loc,dif)(tn, x1(i))
V (tn, x1(i))
D) Repeat B) and C) and go to the next time step
For stochastic volatility with jumps we use σ2(loc,jd)
46
Summary III. Predictor-corrector schemes
When we use predictor-corrector schemes:
i) Compute the predictor step using local stochastic volatility fromprevious time step
ii) Update local stochastic volatility and compute the corrector stepwith new volatility
iii) After the corrector step, compute new local stochastic volatilityand go to the next step
47
Summary IV. Discrete dividends
At ex-dividend time tn:
S(tn+) = S(tn−)−Dnwhere Dn is the cash dividend at time tn
Note that this corresponds to the negative jump in S(t) with a con-stant magnitude Dn at deterministic time tn
Therefore the developed method for discrete negative jumps are read-ily applied for discrete dividends
It is important that σ2(loc,dif) is consistent with the discrete dividends
48
Illustration I. Local volatilities for options on the S&P 500
0%
10%
20%
30%
40%
50%
60%
70%
0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.8 1.0 1.2S', spot = S'*forward(T)
Loca
l Vol
atili
ty
Dupire Local Volatility, T=1month
SV Local Volatility, T=1month
0%
10%
20%
30%
40%
50%
60%
70%
80%
0.2 0.2 0.3 0.3 0.4 0.5 0.6 0.8 1.0 1.2S', spot = S'*forward(T)
Loca
l Vol
atili
ty
Dupire Local Volatility, T=1year
SV Local Volatility, T=1year
Left: Dupire local volatility and LSV local volatilities at T=1 monthRight: ... at T=1 year
LSV local volatility is an adjustment to the skew implied by thestochastic volatility part and is mostly flat
49
Illustration II. Forward volatilities for options on the S&P 500
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5K, Strike=K*Forward(T)
Imp
lied
BS
M v
ola
tilit
y
6m Spot skew, Local vol
6m Spot skew, LSV
6m-6m Skew if S(6m)>1.1S(0), Local vol
6m-6m Skew if S(6m)>1.1S(0), LSV
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5K, Strike=K*Forward(T)
Imp
lied
BS
M v
ola
tilit
y
6m Spot skew, Local vol
6m Spot skew, LSV
6m-6m Skew if S(6m)<0.9S(0), Local vol
6m-6m Skew if S(6m)<0.9S(0), LSV
6m-6m skew: the implied volatility for a vanilla call option startingin 6 month and maturing in one year from now with strike fixed in 6months: K = S(T = 6m)Left: 6m-6m skew if S(T = 6m) > 1.1S(0): the implied volatility forthe forward-start option conditional that S(T = 6m) > 1.1S(0)Right: 6m-6m skew if S(T = 6m) < 0.9S(0): the implied volatilityfor the forward-start option conditional that S(T = 6m) < 0.9S(0)
50
Illustration III. Implications
Conditional that spot moves up, the LSV model preserves the shapeof the volatility skew while the pure local volatility does not
Conditional that spot moves down, the pure local volatility impliesthat the ATM volatility goes too high and the skew flattens unlikethe LSV model
The LSV is more closer to the actual dynamics!
51
Illustration IV. Quarterly 95−105% spread clique on SPX, T = 2y:
u =∑n
max
{min
{S(tn)
S(tn−1)− 1, c
}, −f
}c = f = 5%
Marking model1 5.42% considered as ”market price”1-d LocalVol 0.90% tiny, no fit to steep forward skew!
1-d LocalVol+jump3 2.40% still smallHeston 4.33% under-price
Heston+jump 2 6.08% over-priceLSV, 70% 3 3.78% too small
LSV, 100% 3 4.48% under-price, in line with HestonLSV, 130% 3 4.77% still under-price !
LSV+jump2 70% 3 5.37% closest to the marketLSV+jump2 100% 3 6.37% over-price
1internal ”local beta” model for cliques developed by Alex Langnau21 jump per 5 years: λ = 0.2, ν = −25%3vol-of-vol is set by x% ∗ ε, ε = 2.3
52
Illustration V. Implied volatilities of options on the daily realizedvariance of the S&P 500
0%
40%
80%
120%
160%
200%
23% 58% 92% 127% 162% 196% 231%
K, Strike=K*ExpectedVariance(T)
Imp
lied
log
-no
rmal
vo
ltili
ty
LSV, T=3m
Local Vol, T=3m
0%
40%
80%
120%
15% 37% 59% 81% 103% 125% 147%
K, Strike=K*ExpectedVariance(T)
Imp
lied
log
-no
rmal
vo
ltili
ty
LSV, T=1year
Local Vol, T=1year
Left: Implied log-normal volatilities on options on the daily realizedvariance of the S&P 500 with maturity T=3 monthRight: ... at T=1 year
LSV local volatility implies higher volatility of the realized variance adthe vol-of-vol parameter can be ”tuned-up” to match the market
53
Options on the VIX IAugment the model dynamics (1) with the third variable for the re-alized quadratic variance of S(t):
dI(t) =(σ(loc,svj)(t−, S(t−))ϑ(t−, Y (t−))
)2dt+ ν2dN(t), I(0) = 0
Consider the expected variance realized over period [T, T + Tvix]:
I(T, T + Tvix) =1
TvixE[∫ T+Tvix
TdI(t′)
]where Tvix = 30/365.25 is the annualized tenor of the VIX
A call option on the VIX maturing at T is computed by:
C(T,K) = E[(I(T, T + Tvix)−K
)+| I(T, T ) = 0
]
54
Options on the VIX II. ValuationConsider
I(T, T+Tvix) =1
TvixE[∫ T+Tvix
TdI(t′)
]≈
1
Tvix
∑tn∈[T,T+Tvix]
(E [V (tn)] + ν2λ
)dtn
where V (t) is instantaneous variance:
V (t) =(σ(loc,svj)(t, S(t))ϑ(t, Y (t))
)2
U(t, S, Y ) ≡ I(T, T+Tvix) solves backward problem (4), t ∈ [T, T+Tvix]:
Ut + LU(t, S, Y ) = −1
Tvix
(V (t) + ν2λ
)U(T + Tvix, S, Y ) = 0
Valuation: i) Solve 2-d backward problem for I(T, T + Tvix) as func-tion of (S, Y )ii) Compute G(t, S, Y ;T, S′, Y ′) and evaluate C(T,K) by convolutioniii) Can price call with different strikes at a time
55
Options on the VIX III. VIX Implied volatilities
20%
21%
22%
23%
24%
1m 2m 3m 4m
VIX
Fu
ture
s
LSVLSV-JumpMarket
1M
80%
100%
120%
140%
160%
90% 100% 110% 120% 130% 140% 150%
LSVLSV-JumpMarket
2M
80%
100%
120%
140%
160%
90% 100% 110% 120% 130% 140% 150%
LSVLSV-JumpMarket
3M
60%
80%
100%
120%
90% 100% 110% 120% 130% 140% 150%
LSVLSV-JumpMarket
4M
60%
80%
100%
120%
90% 100% 110% 120% 130% 140% 150%
LSVLSV-JumpMarket
Conclusion: consistency with options on the SPX does not lead toconsistency with options on the VIX; Need extra flexibility for jumpsin volatility (time and/or space dependency)
56
Conclusions
I have presented theoretical and practical grounds for stochastic localvolatility models and highlighted details of model implementation
The presentation is based on my experience in implementation of theLSV model with jumps for BAML equity derivatives library
During this project I benefited from interactions with Alex Lipton,Hassan El Hady and other members of BAML quantitative analytics
Thank you for your attention!
57
ReferencesAndersen, L. and Andreasen J. (2000), “Jump-diffusion processes:Volatility smile fitting and numerical methods for option pricing”,Review of Derivatives Research 4, 231-262
Andreasen, J. (2010), “Volatility Interpolation and Finite DifferenceBased Simulation”, Presentation at ICBI Global Derivatives
Andreasen, J. and Huge N. (2010), “Finite Difference Based Calibra-tion and Simulation ”, Working paper, ssrn.com/abstract= 1697545
Blacher, G. (2001), “A new approach for designing and calibratingstochastic volatility models”, Presentation at ICBI Global Derivatives
Craig I. and Sneyd A. (1988), “An alternating-direction implicit schemefor parabolic equations with mixed derivatives ”, Comput. Math.Appl. 16(4) 341-350
Clift S. and Forsyth P. (2008), “Numerical solution of two asset jumpdiffusion models for option valuation”, Applied Numerical Mathemat-ics 58, 743-782
58
Derman, E., and Kani, I. (1994), “The volatility smile and its impliedtree”, Goldman Sachs Quantitative Strategies Research Notes.
d’Halluin Y., Forsyth P. and Vetzal K. (2005), “Robust numericalmethods for contingent claims under jump diffusion processes”, IMAJournal of Numerical Analysis 25, 87-112
Dupire, B. (1994), “Pricing with a smile”, Risk, 7, 18-20.
Douglas J. and Rachford H.H. (1956), “On the numerical solution ofheat conduction problems in two and three space variables”, Trans.Amer. Math. Soc. 82 421-439
Heston, S. (1993), “A closed-form solution for options with stochasticvolatility with applications to bond and currency options”, Review ofFinancial Studies 6, 327-343
in ’t Hout, K.J., Foulon S. (2008), “ADI finite difference schemes foroption pricing in the Heston model with correlation,” Working paper
Hundsdorfer W and Verwer J. G. (2003), “Numerical Solution ofTime-Dependent Advection-Diffusion- Reaction Equations”, Springer
JP Morgan (1999), “Pricing exotics under smile”, Risk, 11 72-75
Lipton, A. (2002). “The vol smile problem”, Risk, February, 81-85.
Lipton, A. (2003), “Evaluating the latest structural and hybrid modelsfor credit risk”, Presentation at ICBI Global Derivatives
Lipton, A. (2007), “Pricing of credit-linked notes and related prod-ucts”, Merrill Lynch research papers
Lipton A. and Sepp A. (2011), “Credit Value Adjustment in the Ex-tended Structural Default Model,” in The Oxford Handbook of CreditDerivatives, ed. Lipton A. and Rennie A., 406-463
Merton, R. (1973), “Theory of Rational Option Pricing”, The BellJournal of Economics and Management Science 4(1), 141-183.
Nabben, R. (1999), “On Decay Rates of Tridiagonal and Band Ma-trices”, SIAM J. Matrix Anal. Appl 20, 820-837
Press, W., Vetterling W., Teukolsky S., and Flannery B. (1988), “Nu-merical Recipes in C: The Art of Scientific Computing”, CambridgeUniversity Press
Ren Y., Madan D., Qian Q. (2007), “Calibrating and pricing withembedded local volatility models”, Risk, 9, 138-143
Rubinstein, M. (1994), “Implied binomial trees”, Journal of Finance,49, 771-818
Sepp, A. (2011) “An Approximate Distribution of Delta-Hedging Er-rors in a Jump-Diffusion Model with Discrete Trading and TransactionCosts,” Quantitative Finance, forthcoming, (ssrn.com/abstract=1360472)
Recommended