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- 1. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Lattice Approximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en March 16, 2014 Bachelor thesis in mathematics
- 2. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Agenda 1 About Our Thesis 2 Introduction 3 Lattice Binomial Tree Trinomial Tree 4 Convergence of Binomial Models to GBM Part i Part ii Part iii 5 Lattice Approaches in Discrete Time Binomial Models Trinomial Models 6 Case of Equivalence 7 Conclusion
- 3. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion About Our Thesis • Why did we choose our topic?
- 4. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion About Our Thesis • Why did we choose our topic? • Knowledge and understanding
- 5. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion About Our Thesis • Why did we choose our topic? • Knowledge and understanding • Ability to search, collect, evaluate and interpret
- 6. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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
- 7. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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
- 8. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Introduction • Derivatives, Securities and Options
- 9. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Introduction • Derivatives, Securities and Options • Option Pricing Via Discrete and Continuous Time
- 10. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Introduction • Derivatives, Securities and Options • Option Pricing Via Discrete and Continuous Time • Lattice Approach in Discrete Time
- 11. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Introduction • Derivatives, Securities and Options • Option Pricing Via Discrete and Continuous Time • Lattice Approach in Discrete Time • Geometric Brownian Motion in Continuous Time
- 12. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Binomial Tree S0 S0u S0d S0u2 S0ud S0d2 S0u3 S0u2d S0ud2 S0d3 p 1 − p ∆T ∆t ∆t ∆t t0 t1 t2 T Figure : Three-Step Binomial Tree
- 13. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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
- 14. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Convergence of Binomial Models to Geometric Brownian Motion • The sequence of Binomial Models and its Convergence to Geometric Brownian Motion (Part i)
- 15. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Convergence of Binomial Models to Geometric Brownian Motion • The sequence of Binomial Models and its Convergence to Geometric Brownian Motion (Part i) • The sequence of Binomial Models and its Convergence to Black-Scholes Formulae Under Risk-Neutral Probability (Part ii)
- 16. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Convergence of Binomial Models to Geometric Brownian Motion • The sequence of Binomial Models and its Convergence to Geometric Brownian Motion (Part i) • The sequence of Binomial Models and its Convergence to Black-Scholes Formulae Under Risk-Neutral Probability (Part ii) • Mean and Variance of a Random Variable Which is Log-normally Distributed (Part iii)
- 17. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Central Limit Theorem Let Y1, Y2,. . . , Yn be independent and identically distributed random variables with E[Yi ] = µ and V [Yi ] = σ2 < ∞. Define Un = n i=1 Yi − nµ σ √ n = Y − µ σ/ √ n whereY = 1 n n i=1 Yi Then the distribution function of Un converges to the standard normal distribution function as n → ∞. That is lim n→∞ P(Un ≤ u) = u −∞ 1 √ 2π e−t2/2 dt for allu
- 18. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Geometric Brownian Motion • E[Y ] = E t k=1 Yn,k = E ln Sn,t Sn,0 = E [Yn,1 + Yn,2 + ... + Yn,t] , 1 ≤ t ≤ n
- 19. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Geometric Brownian Motion • E[Y ] = E t k=1 Yn,k = E ln Sn,t Sn,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 ] = σ2 T
- 20. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Geometric Brownian Motion • E[Y ] = E t k=1 Yn,k = E ln Sn,t Sn,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 ] = σ2 T • Denoting xn = ln un and yn = ln dn. E[Y ] = n [pxn + (1 − p)yn] = µT V [Y ] = np(1 − p)(xn − yn)2 = σ2 T
- 21. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Geometric Brownian Motion • xn = µT n + σ 1−p p T n yn = µT n − σ p 1−p T n ⇒
- 22. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Geometric Brownian Motion • xn = µT n + σ 1−p p T n yn = µT n − σ p 1−p T n ⇒ • lim n→∞ P Yn,1 + Yn,2 + ... + Yn,n − nE[Yn,1] nV [Yn,1] ≤ x =p ln(ST /S0) − µT σ √ T ≤ x = Φ(x)
- 23. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Geometric Brownian Motion • xn = µT n + σ 1−p p T n yn = µT n − σ p 1−p T n ⇒ • lim n→∞ 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 the normal distribution with mean µT and σ2T.
- 24. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Black-Scholes Model Under Risk-Neutral Probability • lim n→∞ E∗ [Y ] = lim n→∞ n[p∗ xn + (1 − p∗ )yn] = r − σ2 2 T
- 25. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Black-Scholes Model Under Risk-Neutral Probability • lim n→∞ E∗ [Y ] = lim n→∞ n[p∗ xn + (1 − p∗ )yn] = r − σ2 2 T • lim n→∞ V ∗ [Y ] = lim n→∞ np ∗ (1 − p∗ )(xn − yn)2 = σ2 T
- 26. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Black-Scholes Model Under Risk-Neutral Probability • lim n→∞ P∗ Y − nµn] σn √ n ≤ x =p∗ ln(ST /S0) − (r − σ2 2 )T σ √ T ≤ x = Φ(x)
- 27. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The sequence of Binomial Models and its Convergence to Black-Scholes Model Under Risk-Neutral Probability • lim n→∞ 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, our stochastic process (binomial models) at time T converges to normal distribution with mean (r − σ2 2 )T and variance σ2T.
- 28. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Mean and Variance of a Random Variable Which is Log-normally Distributed • Random variable Y is normally distributed E[Y ] = (r − σ2 2 )T V [Y ] = σ2 T
- 29. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Mean and Variance of a Random Variable Which is Log-normally Distributed • Random variable Y is normally distributed E[Y ] = (r − σ2 2 )T V [Y ] = σ2 T • Random variable X = eY or Y = ln X is log-normally distributed E[X] = E[eY ] = e(µ+1 2 σ2)T V [X] = e(2µ+σ2)T eσ2T − 1
- 30. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Binomial Models • Cox-Ross-Rubinstein Model
- 31. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Binomial Models • Cox-Ross-Rubinstein Model • Jarrow-Rudd Model
- 32. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Binomial Models • Cox-Ross-Rubinstein Model • Jarrow-Rudd Model • Tian Model
- 33. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Binomial Models • Cox-Ross-Rubinstein Model • Jarrow-Rudd Model • Tian Model • Trigeorgis Model
- 34. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Binomial Models • Cox-Ross-Rubinstein Model • Jarrow-Rudd Model • Tian Model • Trigeorgis Model • Leisen-Reimer Model
- 35. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Binomial Models p CRR u d eσ √ ∆t e−σ √ ∆t er∆t −d u−d JR u d e(r−σ2 2 )∆t+σ √ ∆t e(r−σ2 2 )∆t−σ √ ∆t e σ2 2 ∆t −e−σ √ ∆t eσ √ ∆t −e−σ √ ∆t Ti u d MV 2 [V + 1 + √ V 2 + 2V + 3] MV 2 [V + 1 − √ V 2 + 2V + 3] M−d u−d Tri ∆X σ2∆t + r − σ2 2 2 (∆t)2 1 2 1 + r − σ2 2 ∆t ∆X LR u d un = rn pn pn dn = rn−pnun 1−pn pn = h−1(d1) pn = h−1(d2) Where in Tian’s Model, M = er∆t and V = eσ2∆t.
- 36. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Trinomial Models • Boyle’s Approach
- 37. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Trinomial Models • Boyle’s Approach • The Replicating Portfolio
- 38. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Trinomial Models • Boyle’s Approach • The Replicating Portfolio • Log-normal Transformation (Kamrad-Ritchken Model)
- 39. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Trinomial Models • Boyle’s Approach • The Replicating Portfolio • Log-normal Transformation (Kamrad-Ritchken Model) • The Explicit Finite Difference Approach (Brennan-Schwartz Approach)
- 40. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Different Trinomial Models pi B u λeσ √ ∆t pu pd (V + M2 − M)u − (M − 1) (u − 1)(u2 − 1) (V + M2 − M)u2 − u3(M − 1) (u − 1)(u2 − 1) KR v λσ √ ∆t pu pd 1 2λ2 + µ √ ∆t 2λσ 1 2λ2 − µ √ ∆t 2λσ BS pu pd − 1 2 rj∆t + 1 2 σ2j2∆t 1 2 rj∆t + 1 2 σ2j2∆t Where in Boyle’s Model M = er∆t and V = M2 eσ2∆t − 1 . Further, pm = 1 − pu − pd .
- 41. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion The Case of Equivalence Between Binomial and Trinomial Models Static binomial and trinomial trees with equal ∆t and T coincide, if we choose: • u = e √ σ2h−µ2h • p = 1 2 1 2(σ2h−µ2h2) + 1√ σ2h−µ2h2 µ √ 2h σ 1 2 (Other models exist, e.g. Derman)
- 42. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Conclusion • Knowledge and understanding
- 43. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Conclusion • Knowledge and understanding • Ability to search, collect, evaluate and interpret
- 44. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Conclusion • Knowledge and understanding • Ability to search, collect, evaluate and interpret • Identify, formulate and solve problems
- 45. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence Conclusion Conclusion • Knowledge and understanding • Ability to search, collect, evaluate and interpret • Identify, formulate and solve problems • Communication of our project to different groups
- 46. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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 its value within it
- 47. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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 its value within it • Plans to continue and develop this research
- 48. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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 its value within it • Plans to continue and develop this research • Questions?
- 49. Lattice Ap- proximations for Black-Scholes type models in Option Pricing Hossein Nohrouzian Anne Karl´en About Our Thesis Introduction Lattice Binomial Tree Trinomial Tree Convergence of Binomial Models to GBM Part i Part ii Part iii Lattice Approaches in Discrete Time Binomial Models Trinomial Models Case of Equivalence 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 its value within it • Plans to continue and develop this research • Questions? • Thanks!