Post on 11-Apr-2017
IntroductionFinite Different Method
Compare and ConclusionReference
Finite Difference method for Black-Scholesmodel
Zhihan Wei
FI 520University of Alabama
Email: zwei7@crimson.ua.edu
April 28th, 2016
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 1/20
IntroductionFinite Different Method
Compare and ConclusionReference
Outline
1 Introduction
2 Finite Different MethodMain Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
3 Compare and Conclusion
4 Reference
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 2/20
IntroductionFinite Different Method
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Black-Scholes-Merton equation
Black and Scholes revolutionized the pricing theory of optionsby showing how to hedge continuously the exposure on theshort position of an option.When a stock pays no dividend, the equation is given by:
∂f
∂t+ rS
∂f
∂S+
1
2σ2S2 ∂
2f
∂S2= rf (1)
When a stock pays known dividend q, the equation is given by:
∂f
∂t+ (r − q)S ∂f
∂S+
1
2σ2S2 ∂
2f
∂S2= rf (2)
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 3/20
IntroductionFinite Different Method
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Closed form of B-S-M equation
Because of the no early exercise feature for European options,the prices of European call and put options can be calculatedanalytically by a closed form
Call = S0e−qTN(d1)−Ke−rTN(d2) (3)
Put = Ke−rTN(−d2)− S0e−qTN(−d1) (4)
where
d1 =ln(S0/K)+(r−q+σ2
2)T
σ√T
and d2 = d1 − σ√T
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 4/20
IntroductionFinite Different Method
Compare and ConclusionReference
Main Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
Main idea of Finite Difference
To implement the explicit finite difference method, a small timeinterval, ∆t, and a small change in price ∆s are chosen. A gridis then constructed for considering values of f when S is equalto 0,∆s, ......, Smax and time is equal to T, T −∆t, ......, 0.Denoting the value of the deriative security at the (i, j) pointon the grid by fi,j , based on the fundamental of FiniteDifference method,
∂f
∂s=fi,j+1 − fi,j−1
2∆s(5)
∂2f
∂s2=fi,j+1 + fi,j−1 − 2fi,j
∆s2(6)
∂f
∂t=fi,j − fi−1,j
∆t(7)
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 5/20
IntroductionFinite Different Method
Compare and ConclusionReference
Main Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
Initial conditions:
Put: Ui,m = max(K − xi, 0) i = 0, 1, ......, nCall: Ui,m = max(xi −K, 0) i = 0, 1, ......, n
Boundary conditions:
Put: U0,j = max(K − xmin, 0), Un,j = max(K − xmax, 0)Call: U0,j = max(xmin −K, 0), Un,j = max(xmax −K, 0)
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 6/20
IntroductionFinite Different Method
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Main Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
Explicit Finite Different method
fi,j = a∗jfi+1,j−1 + b∗jfi+1,j + c∗jfi+1,j+1 (8)
where
a∗j = 11+r∆t(−
12(r − q)j∆t+ 1
2σ2j2∆t)
b∗j = 11+r∆t(1− σ
2j2∆t)
c∗j = 11+r∆t(
12(r − q)j∆t+ 1
2σ2j2∆t)
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 7/20
IntroductionFinite Different Method
Compare and ConclusionReference
Main Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
Explicit Finite Different method
The explicit finite difference method is equivalent to thetrinomial tree approach, three coefficients a∗j , b
∗j , c∗j can be seen
as probabilities of stock price decreasing, remaining unchangedand increasing in time ∆t.
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 8/20
IntroductionFinite Different Method
Compare and ConclusionReference
Main Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
Implicit Finite Different method
ajfi,j−1 + bjfi,j + cjfi,j+1 = fi+1,j (9)
where
aj = 12(r − q)j∆t− 1
2σ2j2∆t
bj = 1 + σ2j2∆t+ r∆tcj = −1
2(r − q)j∆t− 12σ
2j2∆t
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 9/20
IntroductionFinite Different Method
Compare and ConclusionReference
Main Idea of Finite DifferenceExplicit Finite Different MethodImplicit Finite Different MethodCrank-Nicolson Method
Crank-Nicolson method
ajfi,j−1+bjfi,j+cjfi,j+1 = a∗jfi+1,j−1+b∗jfi+1,j+c∗jfi+1,j+1 (10)
where
aj = ∆t(14(r− q)j− 1
4σ2j2), a∗j = ∆t(−1
4(r− q)j+ 14σ
2j2)
bj = 1 + 12σ
2j2∆t+ 12r∆t, b∗j = 1− 1
2σ2j2∆t− 1
2r∆t
cj = −∆t(14(r − q)j + 1
4σ2j2), c∗j = ∆t(1
4(r − q)j + 14σ
2j2)
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 10/20
IntroductionFinite Different Method
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S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5
Ns = 41, Nt = 41
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 11/20
IntroductionFinite Different Method
Compare and ConclusionReference
S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5
Ns = 41, Nt = 41
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 12/20
IntroductionFinite Different Method
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S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5
Ns = 81, Nt = 41
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 13/20
IntroductionFinite Different Method
Compare and ConclusionReference
S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5
Ns = 81, Nt = 41
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 14/20
IntroductionFinite Different Method
Compare and ConclusionReference
S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5
Ns = 81, Nt = 41
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 15/20
IntroductionFinite Different Method
Compare and ConclusionReference
S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5
Ns = 81, Nt = 41
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 16/20
IntroductionFinite Different Method
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S=60,K=55,r=0.2,q=0.05,σ = 0.3,T=1
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 17/20
IntroductionFinite Different Method
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Conclusion
Closed Form Explicit Implicit C-N
European√ √ √ √
American ×√ √ √
Stability stable conditionally stable stable stableSpeed fast relatively fast slow slow
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 18/20
IntroductionFinite Different Method
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Conclusion
Least-Square Monte Carlo methodAdvantages
(a) Applicable to any stochastical process
(b) No boundary conditions are needed
Disadvantages
(a) Slow convergence
Finite Difference methodAdvantages
(a) Fast convergence
(b) Easy computation of the Greek
Disadvantages
(a) Cannot be applied to any stochastical process
(b) Boundary conditions are needed
Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 19/20
IntroductionFinite Different Method
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Reference
1 J. Hull, A. White. Valuing derivative securities using theexplicit finite different method. Journal of Financial andQuantitative Analysis, 25, 1990
2 Daniel J. Duffy. A critique of the crank nicolson schemestrengths and weaknesses for financial instrument pricing.Wilmott magazine, 12, 2004
3 J.C. Cox, S.A. Ross, M. Rubinstein. Option pricing: asimplified approach. Journal of Financial Economics, 7,1979
4 G. Courtadon. A more accurate Finite Differenceapproximation for the valuation of options. Journal ofFinancial and Quantitative Analysis, 17, 1982
5 Ali Hirsa. Computational Methods in Finance. Chapman &Hall, 1, 2013
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