the pricing kernel and option pricing

7/27/2019 The Pricing Kernel and Option Pricing

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IV. The Pricing Kernel and Option Pricing

Xiaoquan Liu

Department of Statistics and FinanceUniversity of Science and Technology of China

Spring 2011

Pricing Kernel (USTC) 1 / 13


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Outline

What is a pricing kernel?

Liu, Shackleton, Taylor, and Xu (2009)

Kuo, Liu, and Coakley (2010)



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Option pricing

in general we have

c(K) = erTEQ[(ST K)+]

= erT

0

(x K)+fQ(x)dx

= erT

0(x K)+

fQ(

x)fP(x)fP(x)dx

=

0

(x K)+m(x)fP(x)dx

= EP[m(ST)(ST K)+]

the pricing kernel, also called the stochastic discount factor, is the randomvariable m(ST)

m(x) = erTfQ(x)

fP(x)



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A representative agent

In a general equilibrium, assume a representative agent faces the problem ofmaximizing expected utility over consumption, given budget constraint andmarket clearing conditions,

maxEt[j=0

ju(ct+j)]

where ct+j is consumption at time t+ j and is a time discount factor



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Representative agents problem

If we simplify the problem to a single period from t to t+ 1, the problem is

max

u(ct) + Et[u(ct+1)]

subject to

ct = et pt

ct+1 = et+1 + xt+1

where

: the holding of the asset by the agent

e: the original consumption level if nothing is tradedp: price of the asset

x: payoff of the asset



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Solving the problem

Forming a Lagrangian

L = u(ct) + Et[u(ct+1)] + t(et ct pt) + t+1(et+1 ct+1 + xt+1)

taking the first order conditions,

L

ct= 0 u(ct) t = 0

L

ct+1= 0 Et[u

(ct+1)] t+1 = 0

L

= 0 tpt + t+1xt+1 = 0



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The pricing kernel

simplify to get

ptu(ct) = Et[u

(ct+1)xt+1]

pt = Et

u(ct+1)

u(ct)xt+1

so the pricing kernel can also be written as

mt+1 = u(ct+1)

u(ct)

in discrete we have the relationship

s = s(1 + rF)u(c1s)

u(c0)



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The pricing kernel

the basic pricing formula for all assets is

pt = Et[mt+1xt+1]

or simplyE[mR] = 1

the pricing kernel must be positive

it must be downward sloping



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Kuo, Liu, and Coakley (2010)

specify functional form for the pricing kernel for LIBOR futures options

follows the intertemporal CAPM of Merton (1973) and makes use of statevariables, which are economic variables whose innovations can forecast futurereturns

sample period is from 2000.01 to 2008.2, monthly frequency



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Kuo, Liu, and Coakley (2010): Methodology

Test the following two functional forms with GMM

A power function of returns on wealth and an exponential function of the sv

m = c0(1 + RW)ec1r+c2+c3

with moment condition

E[c0(1 + RW)ec1r+c2+c3(1 + Ri,t+1)|It] = 1



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Kuo, Liu, and Coakley (2010): Methodology

A sum of orthogonal Chebyshev polynomials in wealth and an exponentialfunction of the sv (Chapman (1997) and Rosenberg and Engle (2002))

m = (RW)ec1r+c2+c3

(RW) = 0C0(1 + RW) +

nk=1

kCk(1 + RW)



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To Recap

the pricing kernel contains information essential for asset pricing E(mR) = 1

it can be derived as the ratio between RND and physical distribution, henceintimate relation between the pricing kernel, RND, and physical distribution

alternatively, we can assume specific functional forms that contains ameasure for investor risk aversion

using econometric method, we can estimate a pricing kernel that conforms toeconomic theory by being everywhere positive and monotonically downwardslopingh



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Empirical Pricing Kernels for FTSE 100 Index Options from 1993.07 to 2003.12

0

1

2

3

0.9 0.95 1 1.05 1.1

X/F

MLN/Historical GB2/Historical


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FIGURE 1. Market-related component of the pricing kernels estimated by the GMM

The pricing kernels are either with real interest rate (1 sv), with real interest rate and maximum Sharpe ratio (2 sv), or with real interest rate,

maximum Sharpe ratio, and volatility (3 sv), where sv stands for state variables.

0.9 0.95 1 1.05 1.1

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

Market return

Isoelastic form

1 sv

2 sv

3 sv

0.9 0.95 1 1.05 1.120

15

10

5

0

5

10

15

Market return

3term polynomial form

1 sv

2 sv

3 sv

0.9 0.95 1 1.05 1.120

15

10

5

0

5

10

15

20

Market return


1 sv

2 sv

3 sv

41


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FIGURE 2. Market-related component of the pricing kernels estimated by minimizing the second HJ distance

The pricing kernels are either with real interest rate (1 sv), with real interest rate and maximum Sharpe ratio (2 sv), or with real interest rate,

maximum Sharpe ratio, and volatility (3 sv), where sv stands for state variables.

0.95 1 1.050.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Market return

Isoelastic form

1 sv

2 sv

3 sv

0.95 1 1.050

0.2

0.4

0.6

0.8

1

Market return


1 sv

2 sv

3 sv

0.95 1 1.050

0.2

0.4

0.6

0.8

1

Market return


1 sv

2 sv

3 sv42

the pricing kernel and option pricing

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