local volatility models- ankirchner

7/31/2019 Local Volatility Models- Ankirchner

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Option PricingChapter 12 - Local volatility models -

Stefan Ankirchner

University of Bonn

Stefan Ankirchner Option Pricing 1


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Agenda

The volatility surface

Local volatility

Dupires formulaPractical 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.



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Implied vola

Implied vola varies with

different strikes

different maturities

Recall the volatility smile:



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Volatility surface

Volatility surface = graph of the the implied volatility as a

function of t2m and moneyness

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


L l l ili d i i d i i


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Local volatility and transition densitiesProof of Dupires formula

Definition of local volatility

One way to make model prices consistent with market prices for

Plain Vanilla Options is to assume that the volatility is a functionof time and the underlying price:

dSt = rStdt + St(t,St)dWQt . (1)

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

We will see that

European call / put prices uniquely determine the local

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


L l l tilit d t iti d iti


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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

(T,K) =

2 CT + rKCKK2

2CK2

(2)

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


Local volatility and transition densities


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Transition probabilities

Let (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) = B

(0, x; T, y)dy.

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

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




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Kolmogorov forward equation

Theorem (Kolmogorov forward equation)

Suppose that St satisfies

dSt = rStdt + St(t,St)dWQt , S0 = x.

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

T(T, y) =

y(ry(T, y)) +

1

2

2

y2

y22(T, y)(T, y)

.

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




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Option prices

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

C(T, K) = erTEQ

(ST K)+

= erTR

(y K)+(T, y)dy

= erT

K

(y K)(T, y)dy.




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Partial derivatives

C(T,K) = erT

K

(y K)(T, y)dy

Observe that

TC(T,K) = rC(T, K)+erT

K

(y K) T

(T, y)dy(3)

KC(T,K) = erT

K

(T, y)dy (4)

2

K2C(T,K) = erT(T, K) (5)

Proof.




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yProof of Dupires formula

Proof of Dupires formula

With the Kolmogorov forward equation we get

TC(T,K)

= rC(T,K) + erT

K (y K)

T(T, y)dy

= rC(T,K) erT

K

(y K)

y(ry(T, y))

1

2

2

y2

y

2

2

(T, y)(T, y)

dy (6)




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yProof of Dupires formula

Proof of Dupires formula contd

Suppose that

A1) limy (y K) y

y22(T, y)(T, y)

= 0,

A2) limy y22(T, y)(T, y)

= 0,

A3) limy (y K)ry(T, y) = 0.Then we can show

Dupires formula: (T,K) =

2

CT

+rKCK

K2 2C

K2

Proof.



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Local volatility for pricing

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

Derive the local volatility function from standard European

options,

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

and solve the PDE numerically.



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

Assume that

dSt = rStdt + St(t,St)dWQt , S0 = x.

and that (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 assumption

that 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

22(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 functionL l l f BS i li d l


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Local vola from BS implied vola

How to find the local vola function?

Dupires formula requires market prices for all strikes andmaturities!In reality we have market prices for only a finite number of Plain

Vanilla 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.


time-space parameterization of the local vola functionL l l f BS i li d l


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A simple time-space parameterization

A simple parameterization of the local vola function is given by

(t, y) = a0 + a1y + a2y2 + a3t + a4t

2 + 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 expiration

Ti and strike Ki,j.


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


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A simple time-space parameterization contd

Assume that the local vola function satisfies

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

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

Ci,j = erTiKi,j

(y Ki,j)(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) =mi=1

n(i)j=1

(C(Ti,Ki,j) Ci,j)2




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A simple time-space parameterization contd

With a search algorithm one can find (a0, . . . , a

5) such that

error(a0, . . . , a

5) min(a0,...,a5)

error(a0, . . . , a5).

The local volatility function is then approximately given by

(t, y) = a0 + a

1y + a

2y2 + a3t + a

4t2 + a5ty.




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BS implied vola and local volatility

Notation: BS(T,K) = BS implied vola for a call with strike K

and exp. date TNote that the market call price C(T,K) satisfies

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

We will show that the local volatility function (T,K) is uniquelydetermined by the BS implied volatility function BS(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.




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Linking local vola and BS implied vola

Some new variables:log moneyness y = log

K

erTS0

implied total variance w(T, y) = T2BS(T, e

rTS0ey)

LemmaThe local volatility function satisfies

2(T, erTS0ey) (7)

=

w

T

(T, y)

1 yw

wy

(T, y) + 122wy2

(T, y) + 14 (14 1w + y2

w)(w

y)2(T, y)




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p

Steps in the proof of Equation (7)

a) First write the BS call price as a function of log moneyness and

implied total variance

CBS(y,w) := BS call(S0, erTS0e

y, T,

w

T, r).

Observe that

CBS(y, w) = S0

( yw

+ w2

) ey( yw w

2)

The partial derivatives of CBS satisfy

2CBS

w2(y, w) =

2CBS

yw(y, w) =

2CBS

y2(y, w)

CBS

y(y, w) =




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

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

Dupires formula implies

1

22(T, erTS0e

y)(2C

y2 C

y)(T, y) = CT(T, y). (8)

Proof of (8).




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c) Observe that by definition we have

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

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



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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


local volatility models- ankirchner

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