# Lecture 15. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 6

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1 Lecture 15 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 6

2 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static and Dynamic Risk-Free Portfolio Sergei Fedotov (University of Manchester) / 6

3 Replicating Portfolio The aim is to show that the option price V (S,t) satisfies the Black-Scholes equation V t σ2 S 2 2 V V + rs S2 S rv = 0. Sergei Fedotov (University of Manchester) / 6

4 Replicating Portfolio The aim is to show that the option price V (S,t) satisfies the Black-Scholes equation V t σ2 S 2 2 V V + rs rv = 0. S2 S Consider replicating portfolio of shares held long and N bonds held short. Sergei Fedotov (University of Manchester) / 6

5 Replicating Portfolio The aim is to show that the option price V (S,t) satisfies the Black-Scholes equation V t σ2 S 2 2 V V + rs rv = 0. S2 S Consider replicating portfolio of shares held long and N bonds held short. The value of portfolio: Π = S NB. Recall that a pair (,N) is called a trading strategy. How to find (,N) such that Π t = V t? Sergei Fedotov (University of Manchester) / 6

6 Replicating Portfolio The aim is to show that the option price V (S,t) satisfies the Black-Scholes equation V t σ2 S 2 2 V V + rs rv = 0. S2 S Consider replicating portfolio of shares held long and N bonds held short. The value of portfolio: Π = S NB. Recall that a pair (,N) is called a trading strategy. How to find (,N) such that Π t = V t? SDE for a stock price S(t) : ds = µsdt + σsdw. Equation for a bond price B (t) : db = rbdt. Sergei Fedotov (University of Manchester) / 6

7 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. Sergei Fedotov (University of Manchester) / 6

8 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt Sergei Fedotov (University of Manchester) / 6

9 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt = ( µs rnb)dt+ σsdw Sergei Fedotov (University of Manchester) / 6

10 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt = ( µs rnb)dt+ σsdw Equating the last two equations dπ = dv, we obtain Sergei Fedotov (University of Manchester) / 6

11 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt = ( µs rnb)dt+ σsdw Equating the last two equations dπ = dv, we obtain = V S, Sergei Fedotov (University of Manchester) / 6

12 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt = ( µs rnb)dt+ σsdw Equating the last two equations dπ = dv, we obtain = V V S, rnb = t σ2 S 2 2 V. S 2 Sergei Fedotov (University of Manchester) / 6

13 Derivation of the Black-Scholes Equation By using the Ito s lemma, we find the change in the option value ( V V dv = + µs t S + 1 ) 2 σ2 S 2 2 V S 2 dt + σs V S dw. By using self-financing requirement dπ = ds NdB, we find the change in portfolio value dπ = ds NdB = (µsdt+σsdw) rnbdt = ( µs rnb)dt+ σsdw Equating the last two equations dπ = dv, we obtain = V V S, rnb = t σ2 S 2 2 V. S 2 Since NB = S Π = ( S V S V ), we get the classical Black-Scholes equation V t σ2 S 2 2 V V + rs rv = 0. S2 S Sergei Fedotov (University of Manchester) / 6

14 Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. Sergei Fedotov (University of Manchester) / 6

15 Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. The value of this portfolio is Π = S + P C. Sergei Fedotov (University of Manchester) / 6

16 Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. The value of this portfolio is Π = S + P C. The payoff for this portfolio is Π T = S + max(e S,0) max(s E,0) Sergei Fedotov (University of Manchester) / 6

17 Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. The value of this portfolio is Π = S + P C. The payoff for this portfolio is Π T = S + max(e S,0) max(s E,0) = E The payoff is always the same whatever the stock price is at t = T. Sergei Fedotov (University of Manchester) / 6

18 Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. The value of this portfolio is Π = S + P C. The payoff for this portfolio is Π T = S + max(e S,0) max(s E,0) = E The payoff is always the same whatever the stock price is at t = T. Using No Arbitrage Principle, we obtain S t + P t C t = Ee r(t t), where C t = C(S t,t) and P t = P (S t,t). Sergei Fedotov (University of Manchester) / 6

19 Static Risk-free Portfolio Let me remind you a Put-Call Parity. We set up the portfolio consisting of long position in one stock, long position in one put and short position in one call with the same T and E. The value of this portfolio is Π = S + P C. The payoff for this portfolio is Π T = S + max(e S,0) max(s E,0) = E The payoff is always the same whatever the stock price is at t = T. Using No Arbitrage Principle, we obtain S t + P t C t = Ee r(t t), where C t = C(S t,t) and P t = P (S t,t). This is an example of complete risk elimination. Definition: The risk of a portfolio is the variance of the return. Sergei Fedotov (University of Manchester) / 6

20 Dynamic Risk-Free Portfolio Put-Call Parity is an example of complete risk elimination when we carry out only one transaction in call/put options and underlying security. Sergei Fedotov (University of Manchester) / 6

21 Dynamic Risk-Free Portfolio Put-Call Parity is an example of complete risk elimination when we carry out only one transaction in call/put options and underlying security. Let us consider the dynamic risk elimination procedure. We could set up a portfolio consisting of a long position in one call option and a short position in shares. The value is Π = C S. Sergei Fedotov (University of Manchester) / 6

22 Dynamic Risk-Free Portfolio Put-Call Parity is an example of complete risk elimination when we carry out only one transaction in call/put options and underlying security. Let us consider the dynamic risk elimination procedure. We could set up a portfolio consisting of a long position in one call option and a short position in shares. The value is Π = C S. We can eliminate the random component in Π by choosing = C S. This is a -hedging! It requires a continuous rebalancing of a number of shares in the portfolio Π. Sergei Fedotov (University of Manchester) / 6

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