![Using a Nonrectangular tree
• Rubinstein has suggested nonrectangular method to build
three dimensional tree for two correlated stock prices.
• Let, S1 and S2 be the stock prices with a chance of .25 of
moving to each of the following:
• (S1u1,S2A), (S2u1,S2B), (S1d1,S2C), (S2d1, S2D)
• Where u1= exp[(r-q1- σ2
1/2) Δt + σ1√Δt]
• d1= exp[(r-q1- σ2
1/2) Δt - σ1√Δt]
• A = exp[(r-q2- σ2
2/2) Δt + σ2√Δt(ρ + √1- ρ2)]
• B = exp[(r-q2- σ2
2/2) Δt + σ2√Δt(ρ - √1- ρ2)]
• C =exp[(r-q2- σ2
2/2) Δt - σ2√Δt(ρ - √1- ρ2)]
• D =exp[(r-q2- σ2
2/2) Δt - σ2√Δt(ρ + √1- ρ2)]](https://image.slidesharecdn.com/optionontwocorrelatedassets-150124105736-conversion-gate02/85/Option-on-two-correlated-assets-5-320.jpg)
Option on two correlated assets
- 1. Option On Two Correlated Assets Presented By- Yasha Singh 4113007007
- 2. Option On Two Correlated Assets • This technique is used for valuing American options dependent on two assets whose price are correlated. • Under this technique there are three methods- • Transformation Variables • Using a Nonrectangular Tree • Adjusting the probabilities
- 3. Transforming Variables • In this technique three dimensional tree is created. Which represents the movements of two uncorrelated variables. • Firstly, two dimensional tree is created for each variable and then these trees are combined into single three dimensional tree. • Probabilities on the branches of the three dimensional tree are the product of the corresponding probabilities on the two dimensional tree.
- 4. Transforming Variables • For example variables are stock prices S1 and S2 . • For S1 p1 is the upward moving probability by proportional amount u1 and 1-p1 is the downward moving probability by proportional amount d1. • For S2 p2 is the upward moving probability by proportional amount u2 and 1-p2 is the downward moving probability by proportional amount d2. • Three dimensional probabilities- • p1p2: S1 increases ; S2 increases • P1(1-p2): S1 increases ; S2 decreases • (1- p1)p2: S1 decreases ; S2 increases • (1- p1)(1- p2): S1 decreases ; S2 decreases
- 5. Using a Nonrectangular tree • Rubinstein has suggested nonrectangular method to build three dimensional tree for two correlated stock prices. • Let, S1 and S2 be the stock prices with a chance of .25 of moving to each of the following: • (S1u1,S2A), (S2u1,S2B), (S1d1,S2C), (S2d1, S2D) • Where u1= exp[(r-q1- σ2 1/2) Δt + σ1√Δt] • d1= exp[(r-q1- σ2 1/2) Δt - σ1√Δt] • A = exp[(r-q2- σ2 2/2) Δt + σ2√Δt(ρ + √1- ρ2)] • B = exp[(r-q2- σ2 2/2) Δt + σ2√Δt(ρ - √1- ρ2)] • C =exp[(r-q2- σ2 2/2) Δt - σ2√Δt(ρ - √1- ρ2)] • D =exp[(r-q2- σ2 2/2) Δt - σ2√Δt(ρ + √1- ρ2)]
- 6. Adjusting the Probabilities • This approach first assumes no correlation and then adjust probabilities at each node to reflect correlation. • Table shows the combination of binomial assuming no correlation. S2- Moves S1- Moves Down Up Up 0.25 0.25 Down 0.25 0.25
- 7. Adjusting the Probabilities • Table shows the binomials assuming correlation of ρ. S2-Moves S1-Moves Down Up Up 0.25(1-ρ) 0.25(1+ρ) Down 0.25(1+ρ) 0.25(1-ρ)