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Option PricingChapter 12 - Local volatility models -

Stefan Ankirchner

University of Bonn

last update: 13th January 2014

Stefan Ankirchner Option Pricing 1

Agenda

The volatility surface

Local volatility

Dupire’s formula

Practical note: how to calibrate the loc. volatility function

Further reading:Chapter 1 in J. Gatheral: The volatility surface, 2006.

B. Dupire: Pricing with a smile. Risk 7, 1994.


Implied vola

Implied vola varies with

different strikes

different maturities

Recall the volatility smile:


Volatility surface

Volatility surface = graph of the the implied volatility as afunction of t2m and moneyness

Data: Exxon Mobile call prices, CBO on 5/12/2010.


Local volatility and transition densitiesProof of Dupire’s formula

Definition of local volatility

One way to make model prices consistent with market prices forPlain Vanilla Options is to assume that the volatility is a functionof time and the underlying price:

dSt = rStdt + Stσ(t, St)dW Qt . (1)

Definition: the mapping (t, x) 7→ σ(t, x) is called local volatilityfunction.

We will see that

European call / put prices uniquely determine the localvolatility function,

our pricing PDEs for American and exotic options remainalmost the same: only the constant vola has to be replacedwith the local volatility function.



Local volatility is determined by European Call Prices

Suppose that the market call prices C (T ,K ) are known for allpossible expiration dates T > 0 and strike prices K ≥ 0. Then thelocal volatility function is given by

σ(T ,K ) =

√√√√2∂C∂T + rK ∂C

∂K

K 2 ∂2C∂K2

(2)

Equation (2) is called Dupire’s formula. For the proof we use thetransition density of the price process St .



Transition probabilities

Let φ(0, x ; T , y) be the probability density of ST at y conditionalto S0 = x . This means that for any nice B ⊂ R

P0,x(ST ∈ B) =

∫Bφ(0, x ; T , y)dy .

Definition: φ(0, x ; T , y) is called transition density of the processSt .

Notation: In the following we fix the initial condition (0, x) andsimply write φ(T , y) = φ(0, x ; T , y).



Kolmogorov forward equation

Theorem (Kolmogorov forward equation)

Suppose that St satisfies

dSt = rStdt + Stσ(t,St)dW Qt , S0 = x .

Then the transistion density φ(T , y) of St satisfies the PDE

∂

∂Tφ(T , y) = − ∂

∂y(ryϕ(T , y)) +

1

2

∂2

∂y2

(y2σ2(T , y)φ(T , y)

).

Remark: The Kolmogorov forward equation is sometimes alsocalled Fokker-Planck equation.



Option prices

The price of a call option expiring at T and struck at K satisfies

C (T ,K ) = e−rTEQ[(ST − K )+

]= e−rT

∫R

(y − K )+φ(T , y)dy

= e−rT∫ ∞K

(y − K )φ(T , y)dy .



Partial derivatives

C (T ,K ) = e−rT∫ ∞K

(y − K )φ(T , y)dy

Observe that

∂

∂TC (T ,K ) = −rC (T ,K )+e−rT

∫ ∞K

(y − K )∂

∂Tφ(T , y)dy (3)

∂

∂KC (T ,K ) = −e−rT

∫ ∞K

φ(T , y)dy (4)

∂2

∂K 2C (T ,K ) = e−rTφ(T ,K ) (5)

Proof.



Proof of Dupire’s formula

With the Kolmogorov forward equation we get

∂

∂TC (T ,K )

= −rC (T ,K ) + e−rT∫ ∞K

(y − K )∂

∂Tφ(T , y)dy

= −rC (T ,K )− e−rT∫ ∞K

(y − K )

[∂

∂y(ryϕ(T , y))

−1

2

∂2

∂y2

(y2σ2(T , y)φ(T , y)

)]dy (6)



Proof of Dupire’s formula cont’d

Suppose that

A1) limy→∞ (y − K ) ∂∂y

(y2σ2(T , y)φ(T , y)

)= 0,

A2) limy→∞(y2σ2(T , y)φ(T , y)

)= 0,

A3) limy→∞ (y − K )ryφ(T , y) = 0.

Then we can show

Dupire’s formula: σ(T ,K ) =

√2

∂C∂T

+rK ∂C∂K

K2 ∂2C∂K2

Proof.


Local volatility for pricing

Question: How to price American or exotic options that are notactively traded?

Derive the local volatility function from standard Europeanoptions,

use the local vola function in the pricing PDE for theAmerican or exotic option considered,

and solve the PDE numerically.


Example: Pricing D&O call with a local volatility model

Assume that

dSt = rStdt + Stσ(t,St)dW Qt , S0 = x .

and that σ(t, x) has been calibrated from market prices ofstandard calls or puts.Let v(t, x) be the time t D&O call value under the assumptionthat it has not been knocked out before t and that St = x . Thenv(t, x) satisfies the Black-Scholes PDE

vt(t, x) + rxvx(t, x) +1

2σ2(t, x)x2vxx(t, x)− rv(t, x) = 0,

with boundary conditions

v(t, L) = 0, 0 ≤ t ≤ T ,

v(T , x) = (x − K )+, x > L.


time-space parameterization of the local vola functionLocal vola from BS implied vola

How to find the local vola function?

Dupire’s formula requires market prices for all strikes andmaturities!In reality we have market prices for only a finite number of PlainVanilla Calls.

Idea: Parameterize the local vola function and choose theparameters such that the model prices for European calls & putsare as close as possible to the real market prices.



A simple time-space parameterization

A simple parameterization of the local vola function is given by

σ(t, y) = a0 + a1y + a2y2 + a3t + a4t2 + a5ty

Let

� T1 ≤ · · · ≤ Tm be the maturities of traded calls.

� Ki ,1, . . . ,Ki ,n(i) strikes of traded calls expiring at Ti ,1 ≤ i ≤ m.

� C (Ti ,Ki ,j) = market price of the traded call with expirationTi and strike Ki ,j .



A simple time-space parameterization cont’d

Assume that the local vola function satisfies

σ(t, y) = a0 + a1y + a2y2 + a3t + a4t

2 + a5ty .

For a given set of parameters (a0, . . . , a5) one can calculate themodel call prices

Ci ,j = e−rTi

∫ ∞Ki,j

(y − Ki ,j)φ(0, x ; Ti , y)dy .

for all 1 ≤ i ≤ m and 1 ≤ j ≤ n(i).The quadratic distance to the market prices is given by

error(a0, a1, a2, a3, a4, a5) =m∑i=1

n(i)∑j=1

(C (Ti ,Ki ,j)− Ci ,j)2



A simple time-space parameterization cont’d

With a search algorithm one can find (a∗0, . . . , a∗5) such that

error(a∗0, . . . , a∗5) ≈ min

(a0,...,a5)error(a0, . . . , a5).

The local volatility function is then approximately given by

σ(t, y) = a∗0 + a∗1y + a∗2y2 + a∗3t + a∗4t2 + a∗5ty .



BS implied vola and local volatility

Notation: σBS(T ,K ) = BS implied vola for a call with strike Kand exp. date TNote that the market call price C (T ,K ) satisfies

C (T ,K ) = BS call(S0,K ,T , σBS(T ,K ), r).

We will show that the local volatility function σ(T ,K ) is uniquelydetermined by the BS implied volatility function σBS(T ,K ).Therefore: instead of parameterizing and calibrating the local volafunction directly, one can proceed as follows:

Parameterize the BS implied vola function

Calibrate the BS implied vola function to market prices

Calculate the local vola function from the calibrated BSimplied vola function.



Linking local vola and BS implied vola

Some new variables:

log moneyness y = log(

KerTS0

)implied total variance w(T , y) = Tσ2BS(T , erTS0ey )

Lemma

The local volatility function satisfies

σ2(T , erTS0ey ) (7)

=∂w∂T (T , y)

1− yw

∂w∂y (T , y) + 1

2∂2w∂y2 (T , y) + 1

4(−14 −

1w + y2

w )(∂w∂y )2(T , y)



Steps in the proof of Equation (7)

a) First write the BS call price as a function of log moneyness andimplied total variance

CBS(y ,w) := BS call(S0, erTS0ey ,T ,

√w

T, r).

Observe that

CBS(y ,w) = S0

[Φ(− y√

w+

√w

2)− eyΦ(− y√

w−√

w

2)

]The partial derivatives of CBS satisfy

∂2CBS

∂w 2(y ,w) =

∂2CBS

∂y∂w(y ,w) =

∂2CBS

∂y 2(y ,w) − ∂CBS

∂y(y ,w) =




b) Write the market call price as a function of T and logmoneyness:

C (T , y) := C (T , erTS0ey ).

Dupire’s formula implies

1

2σ2(T , erTS0ey )(

∂2C

∂y2− ∂C

∂y)(T , y) = CT (T , y). (8)

Proof of (8).




c) Observe that by definition we have

C (T , y) = CBS(y ,w(y ,T )).

With this one can rewrite Dupire’s formula in terms of CBS andthen derive (7).Proof.


Objections to local volatility models

Local volatility models are criticized because:bad statistical properties:� the local volatility functions change considerably over time.

E.g. parameters usually change from one week to the next� bad prediction properties

See Dumas, Fleming, Whaley. Implied Volatility Functions: Empirical

Tests. Journal of Finance, 6. 1998.

The local volatility model predicts the smile/skew to move inthe opposite direction as the underlying; in reality, both movein the same direction.See Hagan, Kumar, Lesniewski, Woodward. Managing Smile Risk.

Wilmott Magazine, 2002.

hedging based on volatility models is inconsistent

ad hoc model; no economic explanation of the local volatilityfunction


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