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Mathematics and Scientific Representation$

Christopher Pincock

Print publication date: 2012

Print ISBN-13: 9780199757107

Published to Oxford Scholarship Online: May 2012

DOI: 10.1093/acprof:oso/9780199757107.001.0001

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(p.303) Appendix B Black-Scholes Model

(p.303) Appendix B Black-Scholes Model

Source:
Mathematics and Scientific Representation
Publisher:
Oxford University Press

(p.303) Appendix B

Black-Scholes Model

Given that V is a function of both S and t, we can approximate a change in V for a small time-step δt using a series expansion known as a Taylor series

δ V = V t δ t + V s δ S + 1 2 V S S δ S 2
(B.1)

where additional higher-order terms are dropped. Given an interest rate of r for the assets held as cash, the corresponding change in the value of the replicating portfolio Π = DS + C of D stocks and C in cash is

δ = D δ S + r C δ t
(B.2)

The last two equations allow us to represent the change in the value of a difference portfolio, which buys the option and offers the replicating portfolio for sale. The change in value is

δ ( V ) = ( V t r C ) δ t + ( V S D ) δ S + 1 2 V S S δ S 2
(B.3)

The δS term reflects the random fluctuations of the stock price and if it could not be dealt with we could not derive a useful equation for V. But fortunately the δS term can be eliminated if we assume that at each time-step the investor can adjust the number of shares held so that

D = V S
(B.4)

Then we get

δ ( V ) = ( V t r C ) δ t + 1 2 V S S δ S 2
(B.5)

The δS 2 remains problematic for a given time-step, but we can find it for the sum of all the time-steps using our lognormal model. This permits us to simplify B.5 so that over the whole time interval ∆t,

(p.304)

( V ) = ( V t r C + 1 2 σ 2 S 2 V S S ) t
(B.6)

Strictly speaking, we are applying a result known as Itô’s lemma. This is glossed over somewhat in Almgren (2002), but discussed more fully in Wilmott (2007, ch. 5 and 6). What is somewhat surprising is that we have found the net change in the value of the difference portfolio in a way that has dropped any reference to the random fluctuations of the stock price S. This allows us to deploy the efficient market hypothesis and assume that ∆(V − Π) is identical to the result of investing V − Πin a risk-free bank account with interest rate r. That is,

( V ) = r ( V ) t
(B.7)

But given that V − Π = VDSC and D = V s, we can simplify the right-hand side of this equation to

r ( V r V S S r C ) t
(B.8)

Given our previous equation for the left-hand side, we get 7.2 after all terms are brought to the left-hand side.