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Introduction Dirac’s Delta “Function” Static Replication Applications Conclusions Model-Free Static Replication Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ k: [email protected] T: 6828 0364 : LKCSB 5036 April 8, 2017 Christopher Ting QF 604 Week 3 April 8, 2017 1/20

Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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Page 1: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Model-Free Static Replication

Christopher Ting

Christopher Ting

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

k: [email protected]: 6828 0364ÿ: LKCSB 5036

April 8, 2017

Christopher Ting QF 604 Week 3 April 8, 2017 1/20

Page 2: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 2/20

Page 3: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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?

Christopher Ting QF 604 Week 3 April 8, 2017 3/20

Page 4: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

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.

Christopher Ting QF 604 Week 3 April 8, 2017 4/20

Page 5: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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)+.

Christopher Ting QF 604 Week 3 April 8, 2017 5/20

Page 6: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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.

Christopher Ting QF 604 Week 3 April 8, 2017 6/20

Page 7: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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.

Christopher Ting QF 604 Week 3 April 8, 2017 7/20

Page 8: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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)

Christopher Ting QF 604 Week 3 April 8, 2017 8/20

Page 9: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 9/20

Page 10: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 10/20

Page 11: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 11/20

Page 12: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 12/20

Page 13: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 13/20

Page 14: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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.

Christopher Ting QF 604 Week 3 April 8, 2017 14/20

Page 15: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Page 16: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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.

Christopher Ting QF 604 Week 3 April 8, 2017 16/20

Page 17: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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)

Christopher Ting QF 604 Week 3 April 8, 2017 17/20

Page 18: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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)

Christopher Ting QF 604 Week 3 April 8, 2017 18/20

Page 19: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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

Christopher Ting QF 604 Week 3 April 8, 2017 19/20

Page 20: Model-Free Static Replication · Source:Physics and Beyond : Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86 God used beautiful mathematics in creating the world

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.

Christopher Ting QF 604 Week 3 April 8, 2017 20/20