Post on 18-Aug-2015
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
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
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
2/19
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
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
3/19
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
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
3/19
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
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
3/19
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
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
3/19
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
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
3/19
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
Introduction
• Derivatives, Securities and Options
• Option Pricing Via Discrete and Continuous Time
• Lattice Approach in Discrete Time
• Geometric Brownian Motion in Continuous Time
4/19
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
Introduction
• Derivatives, Securities and Options
• Option Pricing Via Discrete and Continuous Time
• Lattice Approach in Discrete Time
• Geometric Brownian Motion in Continuous Time
4/19
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
Introduction
• Derivatives, Securities and Options
• Option Pricing Via Discrete and Continuous Time
• Lattice Approach in Discrete Time
• Geometric Brownian Motion in Continuous Time
4/19
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
Introduction
• Derivatives, Securities and Options
• Option Pricing Via Discrete and Continuous Time
• Lattice Approach in Discrete Time
• Geometric Brownian Motion in Continuous Time
4/19
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
Binomial Tree
S0
S0u
S0d
S0u2
S0ud
S0d2
S0u3
S0u2d
S0ud2
S0d3
p
1− p
∆T
∆t ∆t ∆tt0 t1 t2 T
Figure : Three-Step Binomial Tree
5/19
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
Trinomial Tree
S0
S0u
S0pm
S0d
S0u2
S0u
S0
S0d
S0d2
S0u3
S0u2
S0u
S0
S0d
S0d2
S0d3
pu
pd
Figure : Three-Step Trinomial Tree
6/19
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
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)
7/19
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
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)
7/19
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
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)
7/19
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
Central Limit Theorem
Let Y1, Y2,. . . , Yn be independent and identically distributedrandom variables with E [Yi ] = µ and V [Yi ] = σ2 <∞. Define
Un =
∑ni=1 Yi − nµ
σ√n
=Y − µσ/√n
whereY =1
n
n∑i=1
Yi
Then the distribution function of Un converges to the standardnormal distribution function as n→∞. That is
limn→∞
P(Un ≤ u) =
∫ u
−∞
1√2π
e−t2/2dt for allu
8/19
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
The sequence of Binomial Modelsand its Convergence to Geometric
Brownian Motion•
E [Y ] = E
[t∑
k=1
Yn,k
]= E
[ln
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
9/19
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
The sequence of Binomial Modelsand its Convergence to Geometric
Brownian Motion•
E [Y ] = E
[t∑
k=1
Yn,k
]= E
[ln
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
9/19
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
The sequence of Binomial Modelsand its Convergence to Geometric
Brownian Motion•
E [Y ] = E
[t∑
k=1
Yn,k
]= E
[ln
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 = σ2T9/19
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
The sequence of Binomial Modelsand its Convergence to Geometric
Brownian Motion
• xn = µT
n + σ√
1−pp
√Tn
yn = µTn − σ
√p
1−p
√Tn
⇒
•
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 .
10/19
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
The sequence of Binomial Modelsand its Convergence to Geometric
Brownian Motion
• xn = µT
n + σ√
1−pp
√Tn
yn = µTn − σ
√p
1−p
√Tn
⇒
•
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 .
10/19
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
The sequence of Binomial Modelsand its Convergence to Geometric
Brownian Motion
• xn = µT
n + σ√
1−pp
√Tn
yn = µTn − σ
√p
1−p
√Tn
⇒
•
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 .
10/19
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
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
2
)T
•
limn→∞
V ∗[Y ] = limn→∞
np ∗ (1− p∗)(xn − yn)2 = σ2T
11/19
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
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
2
)T
•
limn→∞
V ∗[Y ] = limn→∞
np ∗ (1− p∗)(xn − yn)2 = σ2T
11/19
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
The sequence of Binomial Modelsand its Convergence to
Black-Scholes Model UnderRisk-Neutral Probability
•
limn→∞
P∗{Y − nµn]
σn√n≤ x
}=p∗
{ ln(ST/S0)− (r − σ2
2 )T
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 .
12/19
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
The sequence of Binomial Modelsand its Convergence to
Black-Scholes Model UnderRisk-Neutral Probability
•
limn→∞
P∗{Y − nµn]
σn√n≤ x
}=p∗
{ ln(ST/S0)− (r − σ2
2 )T
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 .
12/19
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
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)
13/19
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
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)
13/19
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
Different Binomial Models
• Cox-Ross-Rubinstein Model
• Jarrow-Rudd Model
• Tian Model
• Trigeorgis Model
• Leisen-Reimer Model
14/19
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
Different Binomial Models
• Cox-Ross-Rubinstein Model
• Jarrow-Rudd Model
• Tian Model
• Trigeorgis Model
• Leisen-Reimer Model
14/19
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
Different Binomial Models
• Cox-Ross-Rubinstein Model
• Jarrow-Rudd Model
• Tian Model
• Trigeorgis Model
• Leisen-Reimer Model
14/19
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
Different Binomial Models
• Cox-Ross-Rubinstein Model
• Jarrow-Rudd Model
• Tian Model
• Trigeorgis Model
• Leisen-Reimer Model
14/19
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
Different Binomial Models
• Cox-Ross-Rubinstein Model
• Jarrow-Rudd Model
• Tian Model
• Trigeorgis Model
• Leisen-Reimer Model
14/19
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
Different Binomial Models
p
CRRud
eσ√
∆t
e−σ√
∆ter∆t−du−d
JRud
e(r−σ2
2)∆t+σ
√∆t
e(r−σ2
2)∆t−σ
√∆t
eσ2
2 ∆t−e−σ√
∆t
eσ√
∆t−e−σ√
∆t
Tiud
MV2 [V + 1 +
√V 2 + 2V + 3]
MV2 [V + 1−
√V 2 + 2V + 3]
M−du−d
Tri ∆X
√σ2∆t +
(r − σ2
2
)2(∆t)2 1
2
[1 +
(r − σ2
2
)∆t∆X
]LR
ud
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 .
15/19
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
Different Trinomial Models
• Boyle’s Approach
• The Replicating Portfolio
• Log-normal Transformation (Kamrad-Ritchken Model)
• The Explicit Finite Difference Approach(Brennan-Schwartz Approach)
16/19
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
Different Trinomial Models
• Boyle’s Approach
• The Replicating Portfolio
• Log-normal Transformation (Kamrad-Ritchken Model)
• The Explicit Finite Difference Approach(Brennan-Schwartz Approach)
16/19
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
Different Trinomial Models
• Boyle’s Approach
• The Replicating Portfolio
• Log-normal Transformation (Kamrad-Ritchken Model)
• The Explicit Finite Difference Approach(Brennan-Schwartz Approach)
16/19
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
Different Trinomial Models
• Boyle’s Approach
• The Replicating Portfolio
• Log-normal Transformation (Kamrad-Ritchken Model)
• The Explicit Finite Difference Approach(Brennan-Schwartz Approach)
16/19
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
Different Trinomial Models
pi
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
1
2λ2+µ√
∆t
2λσ1
2λ2− µ√
∆t
2λσ
BSpupd
−1
2rj∆t +
1
2σ2j2∆t
1
2rj∆t +
1
2σ2j2∆t
Where in Boyle’s Model M = er∆t and V = M2(eσ
2∆t − 1)
.
Further, pm = 1− pu − pd .
17/19
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
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
[1
2(σ2h−µ2h2)+ 1√
σ2h−µ2h2
µ√
2hσ
] 12
(Other models exist, e.g. Derman)
18/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19
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
Conclusion
• Knowledge and understanding
• Ability to search, collect, evaluate and interpret
• Identify, formulate and solve problems
• Communication of our project to different groups
• Ability to put our work into a societal context and itsvalue within it
• Plans to continue and develop this research
• Questions?
• Thanks!
19/19