Question

This question is designed to make sure you know how to derive the Black-Scholes partial differential equation (pde). a) Write down the key assumptions of the Black-Scholes model. b) Why is Ito's lemma needed? c) Derive the Black-Scholes pde, d) What is the role of hedging? Why is this hedge portfolio also called a replicating portfolio? e) Why does the Black-Scholes pde not have a stochastic component?

Answer 1

a) Assumptions of the BSM:

• The underlying should follow a lognormal random walk.
• The risk free interest rate is a known function of time.
• There are no dividends on the underlying.
• Continuous delta hedging is carried out.
• No Arbitrage opportunities.
• No transaction costs are there.

b) When we have work out the continuous change in any value we use Taylor's extension in order to work out the continuous change we use ITo ' s lemma as the rules of deterministic world does not apply in here .

c)

Please find the derivation in the below attached images.

Solution: Desiring Bm E 97 We write option value as vest; o, My & T; 2) S and t variables o and M d Asset parameters E and T Contract details & associated with currency. We use it to denote value of the folio of one long option position and a short position in some quantity o, detta of the underlying.) 11 - vis,+) - AS. long option Short oposition in Asset. - We know that a togrosmal sandom wealk = ds-M.s.dt to.s.dx Anding the change in tolio value. d = dv - Ads. from to we have dr- dr.de & ar. ds + 1 8252 2. dit Ot ds 2 252 This the portfolio changes by : do- dr, at & dv ds t1 8252 de dt - Aids DE DS 2 OS? To eliminate sisk (randomness we use delta Medging (1 - 1). ds , Now if we choose Dad , sandomnes becomes 200. lutting Dadr in e l S

dll & dr. dt + 125² der dt et. - 6 dsz arbitrage le s We have an assumption of No do -.18.dt Putting values of da, it from & and respectively (b + c ) (y-s...)- dr & l se 52 de ers. dr -rv-ol Our required BSM egn. 1. Voy

d) Hedging is carried in order to decrease the degree of randomness by exploiting the correlations between two instruments.

Hedge portfolio is called a replicating portfolio because the hedge portfolio approximately replicates the payoff of the original portfolio but with the reduced randomness or risk.