Lattice Option Pricing Beyond Black Scholes Model

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1 Lattice Option Pricing Beyond Black Scholes Model Carolyne Ogutu 2 School of Mathematics, University of Nairobi, Box , Nairobi, Kenya ( cogutu@uonbi.ac.ke) April 26, 2017 ISPMAM workshop 2017, Mälardalen University April 26, / 18

2 Outline Introduction What we study Trinomial Case ISPMAM workshop 2017, Mälardalen University April 26, / 18

3 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

4 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

5 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

6 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

7 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

8 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

9 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. ISPMAM workshop 2017, Mälardalen University April 26, / 18

10 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. Principle of no arbitrage: ISPMAM workshop 2017, Mälardalen University April 26, / 18

11 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. Principle of no arbitrage: a mathematical model of a financial market should not allow for arbitrage possibility ISPMAM workshop 2017, Mälardalen University April 26, / 18

12 Introduction Definitions An option is a financial security that gives the holder the right to buy or sell a specified quantity of a specified asset at a specified price on or before a specified date. Principle of no arbitrage: The possibility of making profit in financial market without risk and without net investment of capital. ISPMAM workshop 2017, Mälardalen University April 26, / 18

13 Introduction Definitions ISPMAM workshop 2017, Mälardalen University April 26, / 18

14 Introduction Lattice models A lattice/tree model is used to discretize the life of a derivative (e.g. an option) into a number of time steps and model the underlying asset dynamics at each time step. ISPMAM workshop 2017, Mälardalen University April 26, / 18

15 Introduction Lattice models A lattice/tree model is used to discretize the life of a derivative (e.g. an option) into a number of time steps and model the underlying asset dynamics at each time step. The financial derivative can then be evaluated using backward induction. ISPMAM workshop 2017, Mälardalen University April 26, / 18

16 Introduction Lattice models A lattice/tree model is used to discretize the life of a derivative (e.g. an option) into a number of time steps and model the underlying asset dynamics at each time step. The financial derivative can then be evaluated using backward induction. Useful for dealing with nonstandard payoff functions and checking early exercising possibilities for American-style derivatives. ISPMAM workshop 2017, Mälardalen University April 26, / 18

17 Introduction CRR (1979) - pioneered lattice approach Rendleman and Barter (1979) Tian (1993) Jarrow and Rudd (1993) Boyle (1986) - first trinomial model Boyle (1988) Tian (1993) ISPMAM workshop 2017, Mälardalen University April 26, / 18

18 Introduction asymmetric heavy tails volatility clustering ISPMAM workshop 2017, Mälardalen University April 26, / 18

19 Introduction Jump Diffusion Models Merton (1976) Amin (1993) Hilliard and Schwartz (2005) Primbs et al (2007) Ssebugenyi et al (2012) ISPMAM workshop 2017, Mälardalen University April 26, / 18

20 What we study Adopt the parameterization of Primbs et al ISPMAM workshop 2017, Mälardalen University April 26, / 18

21 What we study Adopt the parameterization of Primbs et al St = S 0 e Xt ISPMAM workshop 2017, Mälardalen University April 26, / 18

22 What we study Adopt the parameterization of Primbs et al St = S 0 e Xt Z = m1 + (2l L 1) α with probability p l ISPMAM workshop 2017, Mälardalen University April 26, / 18

23 What we study Adopt the parameterization of Primbs et al St = S 0 e Xt Z = m1 + (2l L 1) α with probability p l Require the moments of Z match those of X L ((2l L 1) α) j p l = µ j (1) l=1 p 1 x p L x L = µ 1 p 1 x p L xl 2 = µ 2. p 1 x1 L p L xl L = µ L ISPMAM workshop 2017, Mälardalen University April 26, / 18

24 What we study Matrix formulation Matching the moment and ensuring that the probabilities sum up to one gives the equation Ap = µ Where: (1 L)α (2n L 1)α (L 1)α ((1 L)α) 2 ((2n L 1)α) 2 ((L 1)α) 2 A = ((1 L)α) 3 ((2n L 1)α) 3 ((L 1)α) ((1 L)α) L ((2n L 1)α) L ((L 1)α) L ISPMAM workshop 2017, Mälardalen University April 26, / 18

25 What we study Let x i = (2i L 1)α, 1 n L and the Vandermonde matrix will be A with the last row missing. ISPMAM workshop 2017, Mälardalen University April 26, / 18

26 What we study Let x i = (2i L 1)α, 1 n L and the Vandermonde matrix will be A with the last row missing. The formula for the inverse can be simplified to ( V 1 L )ij = ( 1)j i 2 L 2 α j 1 σ L j,i (i 1)!(L i)! where σ j,i = σ j,i α j ISPMAM workshop 2017, Mälardalen University April 26, / 18

27 What we study Solving the equation V L p = µ gives p i = L ( V 1 ) ij µ j 1 = j=1 L j=1 ( 1) j i σ L j,i 2 L 1 α j i (i 1)!(n i)! µ j 1 ISPMAM workshop 2017, Mälardalen University April 26, / 18

28 What we study Consider the missing row from A. µ L = = = L (2i L 1) L p i = i=1 L L (2i L 1) L α L i=1 L L i=1 j=1 j=1 ( 1) j i 2 L 1 α j i (2i L 1) L ( 1) j i 2 L 2 (i 1)!(n i)! σ L j,iµ j 1 α L j+i σ L j,i (i 1)!(n i)! µ j 1 = α is a real positive root of this polynomial. ISPMAM workshop 2017, Mälardalen University April 26, / 18

29 Trinomial Case Let L = 3 ISPMAM workshop 2017, Mälardalen University April 26, / 18

30 Trinomial Case Let L = 3 A = α 0 2α ( 2α) 2 0 (2α) 2 ( 2α) 3 0 (2α) 3 ISPMAM workshop 2017, Mälardalen University April 26, / 18

31 Trinomial Case Let L = 3 A = V 1 3 = α 0 2α ( 2α) 2 0 (2α) 2 ( 2α) 3 0 (2α) α 8α α 0 1 4α 1 8α 2 ISPMAM workshop 2017, Mälardalen University April 26, / 18

32 Trinomial Case Let u n = exp ( m 1 L + α t) and d n = exp ( m 1 L α t) then u = exp{2/3µ µ3 µ 1 } ISPMAM workshop 2017, Mälardalen University April 26, / 18

33 Trinomial Case Let u n = exp ( m 1 L + α t) and d n = exp ( m 1 L α t) then u = exp{2/3µ µ3 µ 1 } m = exp (2/3µ 1 ) ISPMAM workshop 2017, Mälardalen University April 26, / 18

34 Trinomial Case Let u n = exp ( m 1 L + α t) and d n = exp ( m 1 L α t) then u = exp{2/3µ µ3 µ 1 } m = exp (2/3µ 1 ) d = exp{2/3µ µ3 µ 1 } ISPMAM workshop 2017, Mälardalen University April 26, / 18

35 Trinomial Case Let u n = exp ( m 1 L + α t) and d n = exp ( m 1 L α t) then u = exp{2/3µ µ3 µ 1 } m = exp (2/3µ 1 ) d = exp{2/3µ µ3 µ 1 } The probabilities µ3 α = 1, (2) 2 µ 1 p 1 = 1 2 µ 1µ 2 µ 3 µ 3 1 µ 3, (3) p 2 = 1 µ 1µ 2, (4) µ 3 p 3 = 1 2 µ 1µ 2 µ 3 + µ 3 1 µ 3. (5) ISPMAM workshop 2017, Mälardalen University April 26, / 18

36 Trinomial Case ISPMAM workshop 2017, Mälardalen University April 26, / 18

37 Trinomial Case References P. Boyle Option valuation using a three-jump process. International Option Journal 3, 7 12, J. C. Cox, S.A. Ross and M. Rubinstein Option pricing: A simplifed approach. Journal of Financial Economics 7, , October J.A. Primbs, M.Rathinam and Y.Yamada. Option Pricing with a Pentanomial Lattice Model that Incorporates Skewness and Kurtosis. Applied Mathematical Finance, 14:1, 1-17, N. Macon and A. Spitzbart. Inverses of Vandermonde matrices. The American Mathematical Monthly, 65(2), , ISPMAM workshop 2017, Mälardalen University April 26, / 18

38 Trinomial Case Thank you for your attention! ISPMAM workshop 2017, Mälardalen University April 26, / 18

Pricing Options with Binomial Trees

Pricing Options with Binomial Trees MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will learn: a simple discrete framework for pricing options, how to calculate risk-neutral