Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by...

Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Model-Free Static Replication

Christopher Ting

Christopher Ting

http://www.mysmu.edu/faculty/christophert/

k: christopherting@smu.edu.sgT: 6828 0364ÿ: LKCSB 5036

April 8, 2017

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Table of Contents

1 Introduction

2 Dirac’s Delta “Function”

3 Static Replication

4 Applications

5 Conclusions

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Introduction

M If a corporate client wants to buy a product that produces adesired payoff at maturity, is there a way to replicate thespecified payoff?

M As a quant, it is your job to figure out a way that satisfiesthe client’s requirements by designing a bespoke solution.

M What is the cost of replication?

M What is the price of the product?

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What is a Unit Step Function?

N Definition

1x>0 :=

1 if x > 0

0 if x ≤ 0.0

1

N Mathematical identity

1x>0 + 1x≤0 = 1.

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Integration of Step Function

N There is nothing sacrosanct about (0, 0). You can shift theorigin to a non-zero number.

N Recall that x+ := max(x, 0).

N Let λ and a be a positive real number. Then∫ λ

−∞1x>a dx = (x− a)+

∣∣∣∣λ−∞

= (λ− a)+.

N Likewise ∫ ∞λ

1x≤a dx = (a− x)+∣∣∣∣∞λ

= (a− λ)+.

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Dirac’s Delta “Function”

N Definition 1 ∫ ∞−∞

δ(x)dx = 1

δ(x) = 0 for x 6= 0.

(1)

N The most important property of δ(x) is exemplified by thefollowing equation,∫ ∞

−∞f(x)δ(x)dx = f(0), (2)

where f(x) is any continuous function of x.

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Dirac’s Delta “Function” (Cont’d)

N By making a shift of origin to a for Dirac’s δ function, wecan deduce the formula∫ ∞

−∞f(x)δ(x− a)dx = f(a). (3)

N The process of multiplying a function of x by δ(x− a) andintegrating over all x is equivalent to the process ofsubstituting a for x.

N The range of integration need not be from −∞ to∞. Anydomain, say the interval (−g2, g1) containing the criticalpoint at which δ(x) does not vanish, will do.

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Dirac’s Alternative Definition of δ(x)

N Consider the differential coefficient ε′(x) of the stepfunction ε(x) given by

1x>0 :=

ε(x) = 1 if x > 0

ε(x) = 0 if x ≤ 0.

(4)

N Substitute ε′(x) for δ(x) in the left side of (3). For positiveg1 and g2, integration by parts leads to∫ g1

−g2f(x)ε′(x) dx = f(x)ε(x)

∣∣∣∣g1−g2−∫ g1

−g2f ′(x)ε(x) dx

= f(g1)−∫ g1

0f ′(x) dx

= f(0)

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Who is Dirac?

Theoretical physicist who predicted the existence of anti-matter.

Picture source: Bubble Chamber

Dirac’s equation

i}γµ∂µψ = mcψ

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Which One is Paul Adrien Maurice Dirac?

Picture source: Paul Dirac and the religion of mathematical beauty

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Mathematics and Beauty

We must admit that religion is ajumble of false assertions, with nobasis in reality. The very idea ofGod is a product of the humanimagination.

— Dirac (1927), atheist

Source: Physics and Beyond : Encounters and Conversations

(1971) by Werner Heisenberg, pp. 85-86

God used beautiful mathematics increating the world.

— Dirac (1963), ex-atheist

Source: The Cosmic Code: Quantum Physics As The Language

Of Nature (2012) by Heinz Pagels, pp. 295

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Conversion by Mathematics

“God is a mathematician of a veryhigh order, and He used veryadvanced mathematics inconstructing the universe.”

The Evolution of the Physicist’s Picture ofNature

Scientific American, May 1963

Source: http://ysfine.com/dirac/dirac44.jpg

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Dirac’s Delta Function in QF

Ý For any payoff function f(S), since δ(x) = δ(−x), we have,for any non-negative S and λ,

f(S) =

∫ ∞0

f(K)δ(K − S)dK

=

∫ λ

0f(K)δ(K − S)dK +

∫ ∞λ

f(K)δ(S −K)dK

Ý Integrating each integral by parts results in

f(S) = f(K)1S<K

∣∣∣∣λ0

−∫ λ

0f ′(K)1S<KdK

− f(K)1S≥K

∣∣∣∣∞λ

+

∫ ∞λ

f ′(K)1S≥KdK

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Replication by Bonds and Options

Ý Integrating each integral by parts once more!

f(S) = f(λ)1S<λ − f ′(K)(K − S)+∣∣∣∣λ0

+

∫ λ

0f ′′(K)(K − S)+dK

+ f(λ)1S≥λ − f ′(K)(S −K)+∣∣∣∣∞λ

+

∫ ∞λf ′′(K)(S −K)+dK

= f(λ) + f ′(λ)[(S − λ)+ − (λ− S)+

]+

∫ λ

0f ′′(K)(K − S)+dK +

∫ ∞λf ′′(K)(S −K)+dK.

= f(λ) + f ′(λ)(S − λ)

+

∫ λ

0f ′′(K)(K − S)+dK +

∫ ∞λf ′′(K)(S −K)+dK.

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Static Replication

Proposition 1

f(ST)= f(λ) + f ′(λ)(ST − λ)

+

∫ λ

0f ′′(K)(K − ST )+dK +

∫ ∞λf ′′(K)(ST −K)+dK.

(5)Ý The payoff f(S) contingent on the outcome S at maturity T

can be replicated by• f(λ): number of risk-free discount bonds, each paying $1 atT

• f ′(λ): number of forward contracts with delivery price λ

• (K − ST )+: European put option’s payoff at T of strike K

• (ST −K)+: European call option’s payoff at T of strike K

• f ′′(λ)dK is the number of put options of all strikes K < λ,and call options of all strikes K > λ

Ý The payoff replication is static, and model-free.Christopher Ting QF 604 Week 3 April 8, 2017 15/20

Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Risk Neutral Pricing

Ý The pricing formulas for options is

c0(K) = e−r0TEQ0

[(ST −K

)+]; (6)

p0(K) = e−r0TEQ0

[(K − ST

)+]. (7)

Ý Proposition 2 The price of replication is

EQ0

(f(ST )

)= f(λ)e−r0T

+

∫ λ

0f ′′(K) p0(K)dK +

∫ ∞λf ′′(K) c0(K)dK.

(8)

Note that at time 0, the price of a forward contract is zero.

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Important Example: Natural Log Contract

Ý Let f(x) = ln(x). Then f ′(x) =1

x, and f ′′(x) = − 1

x2. We

set S = ST , and λ = F0. It follows that

ln(ST ) = ln(F0) +1

F0(ST − F0)

−∫ ST

0

(K − F0)+

K2dK −

∫ ∞ST

(F0 −K)+

K2dK

Ý Accordingly,

ln

(STF0

)=

1

F0(ST − F0)−

∫ ST

0

(K − F0)+

K2dK

−∫ ∞ST

(F0 −K)+

K2dK. (9)

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Risk Neutral Expectation for Log Contract

Ý Under the risk neutral measure,

EQ0 (ST ) = F0

Ý Consequently,

EQ0

(1

F0

(ST − F0

))=

1

F0

[EQ0 (ST )− F0

]= 0.

Ý Therefore, under the risk-neutral measure Q, (9) becomes

EQ0

[ln

(STF0

)]= −er0T

∫ ∞F0

c0(K)

K2dK − er0T

∫ F0

0

p0(K)

K2dK.

(10)

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Example: Square Payoff

Ý Suppose the payoff function is f(x) = x2. Then f ′(x) = 2x,and f ′′ = 2.

Ý Moreover, we assume that λ = F0.

Ý By Proposition 1, the payoff function is

f(ST ) = F 20 + 2F0

(ST − F0

)+ 2

∫ F0

0(K − ST )+dK + 2

∫ ∞F0

(ST −K)+dK.

Ý The total cost of static replication is

F 20 e−r0T + 2

∑Ki≤F0

c0(Ki)∆Ki + 2∑

Kj≥F0

p0(Kj)∆Kj

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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions

Example: Takeaways

9 Static replication is one-off effort and easy to do.

9 Model-free approach frees you from model risks.

9 The biggest caveat obviously is the fact that option chainsare not continuous.

9 Most of the options are not liquidly traded and liquidity riskhas to be dealt with carefully.

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