Lecture 8. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 1


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1 Lecture 8 Sergei Fedotov Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) / 1
2 Lecture 8 1 OneStep Binomial Model for Option Price 2 RiskNeutral Valuation 3 Examples Sergei Fedotov (University of Manchester) / 1
3 OneStep Binomial Model Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d ( u > 1;d < 1). Sergei Fedotov (University of Manchester) / 1
4 OneStep Binomial Model Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d ( u > 1;d < 1). At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. Sergei Fedotov (University of Manchester) / 1
5 OneStep Binomial Model Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d ( u > 1;d < 1). At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. Sergei Fedotov (University of Manchester) / 1
6 OneStep Binomial Model Initial stock price is S 0. The stock price can either move up from S 0 to S 0 u or down from S 0 to S 0 d ( u > 1;d < 1). At time T, let the option price be C u if the stock price moves up, and C d if the stock price moves down. The purpose is to find the current price C 0 of a European call option. Sergei Fedotov (University of Manchester) / 1
7 Riskless Portfolio Now, we set up a portfolio consisting of a long position in shares and short position in one call Π = S C Sergei Fedotov (University of Manchester) / 1
8 Riskless Portfolio Now, we set up a portfolio consisting of a long position in shares and short position in one call Π = S C Let us find the number of shares that makes the portfolio Π riskless. Sergei Fedotov (University of Manchester) / 1
9 Riskless Portfolio Now, we set up a portfolio consisting of a long position in shares and short position in one call Π = S C Let us find the number of shares that makes the portfolio Π riskless. The value of portfolio when stock moves up is S 0 u C u The value of portfolio when stock moves down is S 0 d C d Sergei Fedotov (University of Manchester) / 1
10 Riskless Portfolio Now, we set up a portfolio consisting of a long position in shares and short position in one call Π = S C Let us find the number of shares that makes the portfolio Π riskless. The value of portfolio when stock moves up is S 0 u C u The value of portfolio when stock moves down is S 0 d C d If portfolio Π = S C is riskfree, then S 0 u C u = S 0 d C d Sergei Fedotov (University of Manchester) / 1
11 NoArbitrage Argument The number of shares is = Cu C d S 0 (u d). Sergei Fedotov (University of Manchester) / 1
12 NoArbitrage Argument The number of shares is = Cu C d S 0 (u d). Because portfolio is riskless for this, the current value Π 0 can be found by discounting: Π 0 = ( S 0 u C u )e rt, where r is the interest rate. Sergei Fedotov (University of Manchester) / 1
13 NoArbitrage Argument The number of shares is = Cu C d S 0 (u d). Because portfolio is riskless for this, the current value Π 0 can be found by discounting: Π 0 = ( S 0 u C u )e rt, where r is the interest rate. On the other hand, the cost of setting up the portfolio is Π 0 = S 0 C 0. Therefore S 0 C 0 = ( S 0 u C u )e rt. Sergei Fedotov (University of Manchester) / 1
14 NoArbitrage Argument The number of shares is = Cu C d S 0 (u d). Because portfolio is riskless for this, the current value Π 0 can be found by discounting: Π 0 = ( S 0 u C u )e rt, where r is the interest rate. On the other hand, the cost of setting up the portfolio is Π 0 = S 0 C 0. Therefore S 0 C 0 = ( S 0 u C u )e rt. Finally, the current call option price is where = Cu C d S 0 (u d) C 0 = S 0 ( S 0 u C u )e rt, (NoArbitrage Argument). Sergei Fedotov (University of Manchester) / 1
15 RiskNeutral Valuation Alternatively where (RiskNeutral Valuation) C 0 = e rt (pc u +(1 p)c d ), p = ert d u d. Sergei Fedotov (University of Manchester) / 1
16 RiskNeutral Valuation Alternatively where (RiskNeutral Valuation) C 0 = e rt (pc u +(1 p)c d ), p = ert d u d. It is natural to interpret the variable 0 p 1 as the probability of an up movement in the stock price, and the variable 1 p as the probability of a down movement. Sergei Fedotov (University of Manchester) / 1
17 RiskNeutral Valuation Alternatively where (RiskNeutral Valuation) C 0 = e rt (pc u +(1 p)c d ), p = ert d u d. It is natural to interpret the variable 0 p 1 as the probability of an up movement in the stock price, and the variable 1 p as the probability of a down movement. Fair price of a call option C 0 is equal to the expected value of its future payoff discounted at the riskfree interest rate. Sergei Fedotov (University of Manchester) / 1
18 RiskNeutral Valuation Alternatively where (RiskNeutral Valuation) C 0 = e rt (pc u +(1 p)c d ), p = ert d u d. It is natural to interpret the variable 0 p 1 as the probability of an up movement in the stock price, and the variable 1 p as the probability of a down movement. Fair price of a call option C 0 is equal to the expected value of its future payoff discounted at the riskfree interest rate. For a put option P 0 we have the same result P 0 = e rt (pp u +(1 p)p d ). Sergei Fedotov (University of Manchester) / 1
19 Example A stock price is currently $40. At the end of three months it will be either $44 or $36. The riskfree interest rate is 12%. What is the value of threemonth European call option with a strike price of $42? Use noarbitrage arguments and riskneutral valuation. Sergei Fedotov (University of Manchester) / 1
20 Example A stock price is currently $40. At the end of three months it will be either $44 or $36. The riskfree interest rate is 12%. What is the value of threemonth European call option with a strike price of $42? Use noarbitrage arguments and riskneutral valuation. In this case S 0 = 40, u = 1.1, d = 0.9, r = 0.12, T = 0.25, C u = 2, C d = 0. Sergei Fedotov (University of Manchester) / 1
21 Example A stock price is currently $40. At the end of three months it will be either $44 or $36. The riskfree interest rate is 12%. What is the value of threemonth European call option with a strike price of $42? Use noarbitrage arguments and riskneutral valuation. In this case S 0 = 40, u = 1.1, d = 0.9, r = 0.12, T = 0.25, C u = 2, C d = 0. Noarbitrage arguments: the number of shares = C u C d S 0 u S 0 d = ( ) = 0.25 Sergei Fedotov (University of Manchester) / 1
22 Example A stock price is currently $40. At the end of three months it will be either $44 or $36. The riskfree interest rate is 12%. What is the value of threemonth European call option with a strike price of $42? Use noarbitrage arguments and riskneutral valuation. In this case S 0 = 40, u = 1.1, d = 0.9, r = 0.12, T = 0.25, C u = 2, C d = 0. Noarbitrage arguments: the number of shares = C u C d S 0 u S 0 d = and the value of call option ( ) = 0.25 C 0 = S 0 (S 0 u C u )e rt = ( ) e = Sergei Fedotov (University of Manchester) / 1
23 Example Riskneutral valuation: one can find the probability p p = ert d u d = e = Sergei Fedotov (University of Manchester) / 1
24 Example Riskneutral valuation: one can find the probability p and the value of call option p = ert d u d = e = C 0 = e rt [pc u +(1 p)c d ] = e [ ] = Sergei Fedotov (University of Manchester) / 1
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