Option pricing based on B-S-M model

IntroductionFinite Different Method

Compare and ConclusionReference

Finite Difference method for Black-Scholesmodel

Zhihan Wei

FI 520University of Alabama

Email: [email protected]

April 28th, 2016

Zhihan Wei — April 28th, 2016 Finite Difference method for Black-Scholes model 1/20



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




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)




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




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)





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)





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)





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.





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





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)




S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5

Ns = 41, Nt = 41




S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5

Ns = 41, Nt = 41




S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5

Ns = 81, Nt = 41




S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5

Ns = 81, Nt = 41




S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5

Ns = 81, Nt = 41




S=60,K=55,r=0.1,q=0,σ = 0.25,T=0.5

Ns = 81, Nt = 41




S=60,K=55,r=0.2,q=0.05,σ = 0.3,T=1




Conclusion

Closed Form Explicit Implicit C-N

European√ √ √ √

American ×√ √ √

Stability stable conditionally stable stable stableSpeed fast relatively fast slow slow




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




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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Option pricing based on B-S-M model

Economy & Finance