local volatility dynamic models - columbia university · local volatility dynamic models ... 4...
TRANSCRIPT
![Page 1: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/1.jpg)
Local Volatility Dynamic Models
Rene Carmona∗
∗Bendheim Center for FinanceDepartment of Operations Research & Financial Engineering
Princeton University
Columbia November 9, 2007
14th CAP 2007 Local Volatility Dynamic Models
![Page 2: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/2.jpg)
Contents
Joint work with Sergey Nadtochyi
Motivation1 Understanding Market Models for Credit Portfolios
(Shoenbucher or SPA?)2 Choosing Time Evolution for a NonStationary Markov Process3 Let’s do it for Equity Markets4 Understanding Derman-Kani & Setting Dupire in Motion
R.C. review article in 4th Paris-Princeton Lecture Notes in MathematicalFinance. Lecture Notes in Math #1919
R.C. & S. Nadtochy, submitted
14th CAP 2007 Local Volatility Dynamic Models
![Page 3: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/3.jpg)
Contents
Joint work with Sergey Nadtochyi
Motivation1 Understanding Market Models for Credit Portfolios
(Shoenbucher or SPA?)2 Choosing Time Evolution for a NonStationary Markov Process3 Let’s do it for Equity Markets4 Understanding Derman-Kani & Setting Dupire in Motion
R.C. review article in 4th Paris-Princeton Lecture Notes in MathematicalFinance. Lecture Notes in Math #1919
R.C. & S. Nadtochy, submitted
14th CAP 2007 Local Volatility Dynamic Models
![Page 4: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/4.jpg)
Contents
Joint work with Sergey Nadtochyi
Motivation1 Understanding Market Models for Credit Portfolios
(Shoenbucher or SPA?)2 Choosing Time Evolution for a NonStationary Markov Process3 Let’s do it for Equity Markets4 Understanding Derman-Kani & Setting Dupire in Motion
R.C. review article in 4th Paris-Princeton Lecture Notes in MathematicalFinance. Lecture Notes in Math #1919
R.C. & S. Nadtochy, submitted
14th CAP 2007 Local Volatility Dynamic Models
![Page 5: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/5.jpg)
Contents
Joint work with Sergey Nadtochyi
Motivation1 Understanding Market Models for Credit Portfolios
(Shoenbucher or SPA?)2 Choosing Time Evolution for a NonStationary Markov Process3 Let’s do it for Equity Markets4 Understanding Derman-Kani & Setting Dupire in Motion
R.C. review article in 4th Paris-Princeton Lecture Notes in MathematicalFinance. Lecture Notes in Math #1919
R.C. & S. Nadtochy, submitted
14th CAP 2007 Local Volatility Dynamic Models
![Page 6: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/6.jpg)
Equity Market Model
{St}t≥0 price process0 interest rate (discount factor βt ≡ 1)No dividend
Classical ApproachSpecify dynamics for St , e.g. GBM in Black Scholes case
dSt = Stσt dWt
Compute prices of derivatives by expectation, e.g.
C0(T , K ) = E{(ST − K )+}
14th CAP 2007 Local Volatility Dynamic Models
![Page 7: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/7.jpg)
Actively/Liquidly Traded Instrument
Main AssumptionsAt each time t ≥ 0 we observe Ct(T , K ) the market price at timet of European call options of strike K and maturity T > t .Market prices by expectation
Ct(T , K ) = E{(ST − K )+|Ft}
for some measure (not necessarily unique) PEmpirical FactMany observed option price movements cannot be attributed tochanges in St
Fundamental market data: Surface {Ct(T , K )}T ,K instead of St
14th CAP 2007 Local Volatility Dynamic Models
![Page 8: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/8.jpg)
Remarks
No arbitrage impliesC0(T , K ) increasing in TC0(T , K ) non-increasing and convex in KlimK↗∞C0(T , K ) = 0limK↘0 C0(T , K ) = S0
Realistic Set-UpWe actually observe
C0(Ti , Kij) i = 1, · · · , m, j = 1, · · · , ni
Davis-Hobson & references therein.
14th CAP 2007 Local Volatility Dynamic Models
![Page 9: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/9.jpg)
More Remarks
Switch to notation τ = T − t for time to maturity (Musiela)Call surface {Ct(τ, K )} of prices Ct(T , K ) parameterized byτ ≥ 0 and K ≥ 0.
Ct(τ, K ) = E{(St+τ − K )+|Ft} = EPt{(St+τ − K )+}.
Ct(τ, K ) =
∫ ∞
0(x − K )+ dµt,t+τ (dx)
Crucial Fact (Breeden-Litzenberger)For each τ > 0, the knowledge of all the prices Ct(τ, K ) completelydetermines the marginal distribution µt,t+τ of St+τ w.r.t. Pt .
14th CAP 2007 Local Volatility Dynamic Models
![Page 10: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/10.jpg)
Black-Scholes FormulaDynamics of the underlying asset
dSt = StσdWt , S0 = s0
Wiener process {Wt}t , σ > 0.
Price of a call option
Ct(τ, K ) = StΦ(d1)− KΦ(d2)
with
d1 =− log Mt + τσ2/2
σ√
τ, d1 =
− log Mt − τσ2/2σ√
τ
Mt = K/St moneyness of the option
Φ error function
Φ(x) =1√2π
∫ x
−∞e−y2/2 dy , x ∈ R.
14th CAP 2007 Local Volatility Dynamic Models
![Page 11: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/11.jpg)
Implied Volatility
Classical Black-Scholes frameworkOn any given day t fix
maturity T (or time to maturity τ )strike K
price is an increasing function of the parameter σ
σ � C(BS)t (τ, K ) one-to-one
In general case, given an option price C quoted on the market, itsimplied volatiltiy is the unique number σ = Σt(τ, K ) for which
Ct(τ, K ) = C.
Used by ALL market participants as a currency for options
the wrong number to put in the wrong formula to get the right price.(Black)
14th CAP 2007 Local Volatility Dynamic Models
![Page 12: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/12.jpg)
Implied Volatility Code-Book
{Ct(τ, K ); τ > 0, K > 0} � {Σt(τ, K ); τ > 0, K > 0}
Static (t = 0) ”No arbitrage” conditions difficult to formulate(B. Dupire, Derman-Kani, P.Carr, ....)
Dynammic ”No arbitrage conditions” difficult to check in adynamic framework
(Derman-Kani for tree models)
14th CAP 2007 Local Volatility Dynamic Models
![Page 13: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/13.jpg)
Search for another Option Code-Book
dSt = Stσt dWt , S0 = s0
If t > 0 is fixed, for any τ1 and τ2 such that 0 < τ1 < τ2, then for anyconvex function φ on [0,∞) we have (Jensen)∫ ∞
0φ(x)µt,t+τ1(dx) ≤
∫ ∞
0φ(x)µt,t+τ2(dx)
Orµt,t+τ1 � µt,t+τ2
{µt,t+τ}τ>0 non-decreasing in the balayage order(Kellerer) Existence of a Markov martingale {Yτ}τ≥0 withmarginal distributions {µt,t+τ}τ>0.NB{Yτ}τ≥0 contains more information than the mere marginaldistributions {µt,t+τ}τ>0
14th CAP 2007 Local Volatility Dynamic Models
![Page 14: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/14.jpg)
Local Volatility Code-Book
On Wiener space (in Brownian filtration)Martingale Property implies
Yτ = Y0 +
∫ τ
0Ysa(s) dBs
Markov Property implies
a(s, ω) = at(s, Ys(ω))
At each time t , I choose surface {at(τ, K )}τ>0,K>0 as an alternativecode-book for {C(τ, K )}τ>0,K>0.{at (τ, K )}τ>0,K>0 was introduced in a static framework (i.e. for t = 0) simultaneouslyby Dupire and Derman and Kani – called local volatility surface
14th CAP 2007 Local Volatility Dynamic Models
![Page 15: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/15.jpg)
PDE Code, I
AssumedYτ = Yτ at(τ, Yτ )dBτ , τ > 0
with initial conditionY0 = St
and µt,t+τ has density gt(τ, x).
Breeden-Litzenberger argument (specific to the hockey-stick pay-offfunction)
Ct(τ, K ) =
∫ ∞
0(x − K )+gt(τ, x)dx
Differentiate both sides twice with respect to K
∂2
∂K 2 Ct(τ, K ) = gt(τ, K ). (1)
14th CAP 2007 Local Volatility Dynamic Models
![Page 16: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/16.jpg)
PDE Code, II
Tanaka’s formula:
(Yτ − K )+ = (Y0 − K )+ +
∫ τ
01[K ,∞)(Ys)dYs +
12
∫ τ
0δK (Ys) d [Y , Y ]s
and taking Et - expectations on both sides using the fact that Y is amartingale satisfying d [Y , Y ]s = Y 2
s at(s, Ys)2ds, we get:
Ct(τ, K ) = (St − K )+ +12
∫ τ
0Et{δK (Ys)Y 2
s at(s, Ys)2}ds
= (St − K )+ +12
∫ τ
0K 2at(s, K )2gt(s, K ) ds.
Take derivatives with respect to τ on both sides
∂C(τ, K )
∂τ=
12
K 2at(τ, K )2gt(τ, K ).
14th CAP 2007 Local Volatility Dynamic Models
![Page 17: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/17.jpg)
PDE Code, III
Equate both expressions of gt(τ, K )
at(τ, K )2 =2∂τ C(τ, K )
K 2∂2KK C(τ, K )
Smooth Call Prices ↪→ Local Volatilities
14th CAP 2007 Local Volatility Dynamic Models
![Page 18: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/18.jpg)
PDE Code IV
From local volatility surface {at(τ, K )}τ,K to call option prices
{Ct(τ, K )}τ,K solve PDE (Dupire’s PDE)
∂τ C(τ, K ) = 12 K 2 a2(τ, K )∂2
KK C(τ, K ), τ > 0, K > 0
C(0, K ) = (St − K )+
{Ct(τ, K ); τ > 0, K > 0} ↔ {at(τ, K ); τ > 0, K > 0}
Why would this approach be better?
NEED ONLY POSITIVITY for no arbitrage
14th CAP 2007 Local Volatility Dynamic Models
![Page 19: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/19.jpg)
Dupire Formula
IfdSt = Stσt dWt
for some Wiener process {Wt}t and some adapted non-negaitveprocess {σt}t , then
at(τ, K )2 = Et{σ2t+τ |St+τ = K}.
14th CAP 2007 Local Volatility Dynamic Models
![Page 20: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/20.jpg)
HJM Prescription
Proposed by Derman-Kani in 1998, but NEVER developed!
Compute a0(τ, K ) from market call prices (Initial condition)Define a dynamic model by defining the dynamics of the localvolatility surface
dat(τ, K ) = αt(τ, K )dt + βt(τ, K )dWt
14th CAP 2007 Local Volatility Dynamic Models
![Page 21: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/21.jpg)
Consistency
Question Under what conditions do the Call Prices computedfrom the dynamics of at(τ, K ) come from a model of the form ofthe form
dSt = StσtdB1t
with initial condition S0 = s the underlying instrument?Answer
σt = at(0, St)
14th CAP 2007 Local Volatility Dynamic Models
![Page 22: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/22.jpg)
No-Arbitrage Condition
Question Under what conditions on the dynamics of at(τ, K ) arethe call prices (local) martingales?Answer
(α +‖β‖2
2) · ∂2
∂K 2 C +∂
∂t〈a,
∂2
∂K 2 C〉t =∂
∂Ta · ∂2
∂K 2 C
Recall classical HJM drift condition
α(t , T ) = β(t , T ) ·∫ T
tβ(t , s)ds =
d∑j=1
β(j)(t , T )
∫ T
tβ(j)(t , s)ds.
14th CAP 2007 Local Volatility Dynamic Models
![Page 23: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/23.jpg)
Main Result Statement
The dynamic model of the local volatility surface given by the system ofequations
dat(τ, K ) = αt(τ, K )dt + βt(τ, K )dWt , t ≥ 0, (2)
is consistent with a spot price model of the form
dSt = StσtdBt
for some Wiener process {Bt}t , and does not allow for arbitrage if and onlyif a.s. for all t > 0:
•at(0, St) = σt (3)
•∂τ at(τ, K )∂2KK Ct(τ, K ) = (4)(
at(τ, K )αt(τ, K ) +‖βt(τ, K )‖2
2
)∂2
KK Ct(τ, K ) +ddt〈a·(τ, K )2, ∂2
KK C·(τ, K )〉t
〈 · · 〉t quadratic covariation of two semi-martingales.
14th CAP 2007 Local Volatility Dynamic Models
![Page 24: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/24.jpg)
Practical Monte Carlo Implementation
Start from a model for βt(τ, K ) (say a stochastic differentialequation);Get S0 and C0(τ, K ) from the market and compute ∂2
KK C0, a0 andβ0 from its model;Loop: for t = 0,∆t , 2∆t , · · ·
1 Get αt(τ, K ) from the drift condition;2 Use Euler to get
at+∆t (τ, K ) from the dynamics of the local volatility;St+∆t from St Dynamics;βt+∆t from its own model;
14th CAP 2007 Local Volatility Dynamic Models
![Page 25: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/25.jpg)
Markovian Spot Models (β ≡ 0)
αt(τ, K ) =ddt
at(τ, K ).
Drift condition reads∂τ at(τ, K ) = αt(τ, K )
Hence∂τ at(τ, K ) =
ddt
at(τ, K )
which shows that for fixed K , at(τ, K ) is the solution of a transport equationwhose solution is given by:
at(τ, K ) = a0(τ + t , K )
and the consistency condition forces the special form
σt = a0(t , St)
of the spot volatility. Hence we proved:
The local volatility is a process of bounded variation for eachτ and K fixed if and only if it is the deterministic shift of aconstant shape and the underlying spot is a Markov process.
14th CAP 2007 Local Volatility Dynamic Models
![Page 26: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/26.jpg)
A First Parametric Family
a2(τ, x ,Θ) =
∑2i=0 piσie−x2/(2τσ2
i )−τσ2i /8∑2
i=0(pi/σi)e−x2/(2τσ2i )−τσ2
i /8
forΘ = (σ0, σ1, σ2, p1, p2)
Mixture of Black-Scholes Call surfaces for 3 different volatilitiesSingularity when τ ↘ 0
14th CAP 2007 Local Volatility Dynamic Models
![Page 27: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/27.jpg)
Numerical Evidence of Singularity
14th CAP 2007 Local Volatility Dynamic Models
![Page 28: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/28.jpg)
A Second Parametric Family
As in Brigo-MercurioStill a mixture of Black-Scholes Call surfaces for 3 differentvolatilitiesEach volatility is time dependent t ↪→ σi(t)σ0(0) = σ1(0) = σ2(0)
a2(Θ, τ, x) =(1 − (p1 + p2)τ) σe−d2(σ)/2 + p1τσ1e−d2(σ1)/2 + p2τσ2e−d2(σ2)/2
(1 − (p1 + p2)τ) 1σ
e−d2(σ)/2 + p1τ1
σ1e−d2(σ1)/2 + p2τ
1σ2
e−d2(σ2)/2
where
d(σ) =s − x +
(r + 1
2σ2)τ
σ√
τ
Θ = (p1, p2, σ, σ1, σ2, s, r)
14th CAP 2007 Local Volatility Dynamic Models
![Page 29: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/29.jpg)
Fit to Real Data
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 30
1
2
30.02
0.03
0.04
0.05
0.06
0.07
log K
S&P500 April 3, 2006
T
LVol
14th CAP 2007 Local Volatility Dynamic Models
![Page 30: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/30.jpg)
Stochastic Volatility Models
dSt = σtStdWt
withdσ2
t = b(σ2t )dt + a(σ2
t )dWt
whered〈W , W 〉t = ρdt .
Usuallyb(σ2) = −κ(σ2 − σ2)
Special cases:
a(σ2) = γ, (Hull-White) a(σ2) = γ√
σ2 (Heston)
14th CAP 2007 Local Volatility Dynamic Models
![Page 31: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/31.jpg)
Local Volatility of SV Models
a2(τ, K ) =2∂τ C
K 2∂2KK C
= σ20
√1− ρ2 ·
E{
S σ2T
σTe−
d212
}E
{SσT
e−d2
12
}where σT = σT
σ0, and σT =
√1T
∫ T0 σ2
s ds
S = s0 exp(
ρσ0
σ(στ − 1)− 1
2σ2
0ρ2σ2
ττ
)and
d1 =log(s0)− log(K ) + ρσ0
σ(στ − 1) + ( 1
2 − ρ2)σ20 σ
2ττ√
1 − ρ2σ0στ√
τ
14th CAP 2007 Local Volatility Dynamic Models
![Page 32: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/32.jpg)
First Example: ρ = 0.5
0
1
2
3
−5
0
50.05
0.1
0.15
0.2
0.25
0.3
T
Rho=0.5
log K
LVol
14th CAP 2007 Local Volatility Dynamic Models
![Page 33: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/33.jpg)
Second Example: ρ = −0.1
0
1
2
3
−5
0
50
0.1
0.2
0.3
0.4
rho=−0.1
14th CAP 2007 Local Volatility Dynamic Models
![Page 34: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/34.jpg)
Third Example: ρ = −0.75
0
1
2
3
−5
0
50
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
T
Rho=−0.1, N=100,000
log K
LVol
14th CAP 2007 Local Volatility Dynamic Models
![Page 35: Local Volatility Dynamic Models - Columbia University · Local Volatility Dynamic Models ... 4 Understanding Derman-Kani & Setting Dupire in Motion R.C. review article in 4th Paris-Princeton](https://reader034.vdocument.in/reader034/viewer/2022052319/5bc02d7009d3f2e72d8cd379/html5/thumbnails/35.jpg)
Comparing SV Models
00.5
11.5
22.5
3
−5
0
50
0.05
0.1
0.15
0.2
T
Rho=−0.75, N=10,000
log K
LV
ol
0
1
2
3
−50
50
0.05
0.1
0.15
0.2
T
Rho=−0.75, N=10,000
log K/SL
Vo
l
14th CAP 2007 Local Volatility Dynamic Models