bachelor's pp presentation

Lattice Ap-proximations

forBlack-Scholestype models inOption Pricing

HosseinNohrouzianAnne Karlen

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Convergenceof BinomialModels toGBM

Part i

Part ii

Part iii

LatticeApproaches inDiscrete Time

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Lattice Approximations for Black-Scholes typemodels in Option Pricing

Hossein NohrouzianAnne Karlen

March 16, 2014

Bachelor thesis in mathematics

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Agenda

1 About Our Thesis

2 Introduction

3 LatticeBinomial TreeTrinomial Tree

4 Convergence of Binomial Models to GBMPart iPart iiPart iii

5 Lattice Approaches in Discrete TimeBinomial ModelsTrinomial Models

6 Case of Equivalence

7 Conclusion

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About Our Thesis

• Why did we choose our topic?

• Knowledge and understanding

• Ability to search, collect, evaluate and interpret

• Identify, formulate and solve problems

• Communication of our project to different groups

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About Our Thesis

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About Our Thesis

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About Our Thesis

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About Our Thesis

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Introduction

• Derivatives, Securities and Options

• Option Pricing Via Discrete and Continuous Time

• Lattice Approach in Discrete Time

• Geometric Brownian Motion in Continuous Time

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Introduction

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Introduction

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Introduction

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Binomial Tree

1− p

∆t ∆t ∆tt0 t1 t2 T

Figure : Three-Step Binomial Tree

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Trinomial Tree

Figure : Three-Step Trinomial Tree

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Convergence of Binomial Modelsto Geometric Brownian Motion

• The sequence of Binomial Models and its Convergence toGeometric Brownian Motion (Part i)

• The sequence of Binomial Models and its Convergence toBlack-Scholes Formulae Under Risk-Neutral Probability(Part ii)

• Mean and Variance of a Random Variable Which isLog-normally Distributed (Part iii)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Central Limit Theorem

Let Y1, Y2,. . . , Yn be independent and identically distributedrandom variables with E [Yi ] = µ and V [Yi ] = σ2 <∞. Define

∑ni=1 Yi − nµ

σ√n

=Y − µσ/√n

whereY =1

n∑i=1

Then the distribution function of Un converges to the standardnormal distribution function as n→∞. That is

limn→∞

P(Un ≤ u) =

−∞

1√2π

e−t2/2dt for allu

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

The sequence of Binomial Modelsand its Convergence to Geometric

Brownian Motion•

E [Y ] = E

Sn,tSn,0

]= E [Yn,1 + Yn,2 + ...+ Yn,t ] , 1 ≤ t ≤ n

E [Yn,t ] = p ln un + (1− p) ln dn

Y = µt + σW (t) 0 ≤ t ≤ T

E [Y ] = µT V [Y ] = σ2T

• Denoting xn = ln un and yn = ln dn.

E [Y ] = n [pxn + (1− p)yn] = µT

V [Y ] = np(1− p)(xn − yn)2 = σ2T

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Brownian Motion•

E [Y ] = E

Sn,tSn,0

]= E [Yn,1 + Yn,2 + ...+ Yn,t ] , 1 ≤ t ≤ n

Y = µt + σW (t) 0 ≤ t ≤ T

E [Y ] = µT V [Y ] = σ2T

E [Y ] = n [pxn + (1− p)yn] = µT

V [Y ] = np(1− p)(xn − yn)2 = σ2T

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Brownian Motion•

E [Y ] = E

Sn,tSn,0

]= E [Yn,1 + Yn,2 + ...+ Yn,t ] , 1 ≤ t ≤ n

Y = µt + σW (t) 0 ≤ t ≤ T

E [Y ] = µT V [Y ] = σ2T

E [Y ] = n [pxn + (1− p)yn] = µT

V [Y ] = np(1− p)(xn − yn)2 = σ2T9/19

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Brownian Motion

• xn = µT

n + σ√

1−pp

yn = µTn − σ

limn→∞

P{Yn,1 + Yn,2 + ...+ Yn,n − nE [Yn,1]√

nV [Yn,1]≤ x

}=p{ ln(ST/S0)− µT

σ√T

≤ x}

= Φ(x)

• This proves that binomial models at time T , follow thenormal distribution with mean µT and σ2T .

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Brownian Motion

• xn = µT

n + σ√

1−pp

yn = µTn − σ

limn→∞

P{Yn,1 + Yn,2 + ...+ Yn,n − nE [Yn,1]√

nV [Yn,1]≤ x

σ√T

≤ x}

= Φ(x)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Brownian Motion

• xn = µT

n + σ√

1−pp

yn = µTn − σ

limn→∞

P{Yn,1 + Yn,2 + ...+ Yn,n − nE [Yn,1]√

nV [Yn,1]≤ x

σ√T

≤ x}

= Φ(x)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

The sequence of Binomial Modelsand its Convergence to

Black-Scholes Model UnderRisk-Neutral Probability

limn→∞

E ∗[Y ] = limn→∞

n[p∗xn + (1− p∗)yn] =

(r − σ2

limn→∞

V ∗[Y ] = limn→∞

np ∗ (1− p∗)(xn − yn)2 = σ2T

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

limn→∞

E ∗[Y ] = limn→∞

n[p∗xn + (1− p∗)yn] =

(r − σ2

limn→∞

V ∗[Y ] = limn→∞

np ∗ (1− p∗)(xn − yn)2 = σ2T

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

limn→∞

P∗{Y − nµn]

σn√n≤ x

}=p∗

{ ln(ST/S0)− (r − σ2

σ√T

≤ x}

= Φ(x)

• which means, under risk-neutral probability measure, ourstochastic process (binomial models) at time T converges

to normal distribution with mean (r − σ2

2 )T and varianceσ2T .

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

limn→∞

P∗{Y − nµn]

σn√n≤ x

}=p∗

{ ln(ST/S0)− (r − σ2

σ√T

≤ x}

= Φ(x)

• which means, under risk-neutral probability measure, ourstochastic process (binomial models) at time T converges

to normal distribution with mean (r − σ2

2 )T and varianceσ2T .

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Mean and Variance of a RandomVariable Which is Log-normally

Distributed

• Random variable Y is normally distributed

E [Y ] = (r − σ2

2)T V [Y ] = σ2T

• Random variable X = eY or Y = lnX is log-normallydistributed

E [X ] = E [eY ] = e(µ+ 12σ2)T

V [X ] = e(2µ+σ2)T(eσ

2T − 1)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Mean and Variance of a RandomVariable Which is Log-normally

Distributed

• Random variable Y is normally distributed

E [Y ] = (r − σ2

2)T V [Y ] = σ2T

• Random variable X = eY or Y = lnX is log-normallydistributed

E [X ] = E [eY ] = e(µ+ 12σ2)T

V [X ] = e(2µ+σ2)T(eσ

2T − 1)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Different Binomial Models

• Cox-Ross-Rubinstein Model

• Jarrow-Rudd Model

• Tian Model

• Trigeorgis Model

• Leisen-Reimer Model

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Tian Model

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Tian Model

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Tian Model

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Tian Model

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

eσ√

e−σ√

∆ter∆t−du−d

e(r−σ2

2)∆t+σ

√∆t

e(r−σ2

2)∆t−σ

√∆t

2 ∆t−e−σ√

eσ√

∆t−e−σ√

MV2 [V + 1 +

√V 2 + 2V + 3]

MV2 [V + 1−

√V 2 + 2V + 3]

M−du−d

Tri ∆X

√σ2∆t +

(r − σ2

)2(∆t)2 1

(r − σ2

)∆t∆X

un = rnp′npn

dn = rn−pnun1−pn

p′n = h−1(d1)pn = h−1(d2)

Where in Tian’s Model, M = er∆t and V = eσ2∆t .

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

Different Trinomial Models

• Boyle’s Approach

• The Replicating Portfolio

• Log-normal Transformation (Kamrad-Ritchken Model)

• The Explicit Finite Difference Approach(Brennan-Schwartz Approach)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

B u λeσ√

∆t pupd

(V + M2 −M)u − (M − 1)

(u − 1)(u2 − 1)(V + M2 −M)u2 − u3(M − 1)

(u − 1)(u2 − 1)

KR v λσ√

∆tpupd

2λ2+µ√

2λσ1

2λ2− µ√

BSpupd

2rj∆t +

2σ2j2∆t

2rj∆t +

2σ2j2∆t

Where in Boyle’s Model M = er∆t and V = M2(eσ

2∆t − 1)

Further, pm = 1− pu − pd .

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

The Case of Equivalence BetweenBinomial and Trinomial Models

Static binomial and trinomial trees with equal ∆t and Tcoincide, if we choose:

• u = e√σ2h−µ2h

• p = 12

2(σ2h−µ2h2)+ 1√

σ2h−µ2h2

(Other models exist, e.g. Derman)

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Ability to put our work into a societal context and itsvalue within it

• Plans to continue and develop this research

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

About OurThesis

Introduction

Lattice

Binomial Tree

Trinomial Tree

Part i

Part ii

Part iii

Binomial Models

TrinomialModels

Case ofEquivalence

Conclusion

• Questions?

• Thanks!

bachelor's pp presentation

Documents

lattice approaches

blackscholes type models

lattice ap proximations

bachelor thesis