108156.pdf

150 Mathematics for Finance 7.2 Put-Call Parity In this section we shall make an important link between the prices of European c ...

Options: General Properties 151 buy one put option forPE; write and sell one call option forCE; invest the sumCE−PE−S(0) (or bo ...

152 Mathematics for Finance lying stock is $20.37 and the interest rate is 7.48%. Find an arbitrage opportunity. Remark 7.1 We c ...

Options: General Properties 153 Exercise 7.7 For the data in Exercise 6.5, find the strike price for European calls and puts t ...

154 Mathematics for Finance If the put is not exercised at all, then we can buy a share forXby exercising the call at timeTand c ...

Options: General Properties 155 Figure 7.3 Scenario in which an American call can bring a positive payoff, but a European call ...

156 Mathematics for Finance other hand, we have the lower bound S(0)−Xe−rT≤CE, which follows immediately by put-call parity, sin ...

Options: General Properties 157 Exercise 7.12 For dividend-paying stock sketch the regions of call and put prices de- termined ...

158 Mathematics for Finance to gainS(t)−Xby exercising an American call option ifS(t)>Xat time t<T, this is not so with a ...

Options: General Properties 159 all, the final balance will also be positive,PAerT>0, at expiry. These results canbesummari ...

160 Mathematics for Finance 7.4.1 European Options Dependence on the Strike Price. We shall consider options on the same underly ...

Options: General Properties 161 Subtracting, we get ( CE(X′)−CE(X′′) ) + ( PE(X′′)−PE(X′) ) =(X′′−X′)e−rT. Since, by Propositi ...

162 Mathematics for Finance We can write and sell an option with strike priceX, and purchaseαoptions with strike priceX′and 1−αo ...

Options: General Properties 163 Remark 7.4 Even though options on a portfolio of stocks are of little practical significance, ...

164 Mathematics for Finance Proof We employ put-call parity (7.1): CE(S′′)−PE(S′′)=S′′−Xe−rT, CE(S′)−PE(S′)=S′−Xe−rT. Subtractin ...

Options: General Properties 165 Proof We putS=αS′+(1−α)S′′for brevity. LetS′=x′S(0),S′′=x′′S(0) and S=xS(0), sox=αx′+(1−α)x′′. ...

166 Mathematics for Finance Proposition 7.13 Suppose thatX′<X′′.Then CA(X′)−CA(X′′)<X′′−X′, PA(X′′)−PA(X′)<X′′−X′. Proo ...

Options: General Properties 167 Dependence on the Underlying Asset Price.Once again, we shall con- sider options on a portfoli ...

168 Mathematics for Finance On subtracting, we obtain ( CA(S′′)−CA(S′) ) + ( PA(S′)−PA(S′′) ) ≤S′′−S′+X(1−e−rT) ≤S′′−S′. Each of ...

Options: General Properties 169 Proof Suppose thatCA(T′)>CA(T′′). We write and sell one option expiring at timeT′and buy on ...