108156.pdf

170 Mathematics for Finance option at expiryTcoincides with the intrinsic value. The price of an American option prior to expiry ...

Options: General Properties 171 Figure 7.8 Time valueCE(S)−(S−X)+of a European call option For a European option we have to wa ...

172 Mathematics for Finance value of the option is zero. SinceCE(S) is an increasing function ofSby Proposition 7.9, this means ...

8. Option Pricing.............................................. By aEuropean derivative securityorcontingent claim with stockSas ...

174 Mathematics for Finance our arbitrage profit. IfD(0)<V(0),then we take the opposite positions, with V(0)−D(0) the resulti ...

Option Pricing 175 The initial value of the replicating portfolio isx(1)S(0) +y(1). By Theorem 8.1 D(0) =x(1)S(0) +y(1) = f(Su ...

176 Mathematics for Finance Proof This is an immediate consequence of (8.1): D(0) = f(Su)−f(Sd) u−d + (1 +u)f(Sd)−(1 +d)f(Su) (u ...

Option Pricing 177 Figure 8.1 Branchings in the two-step binomial tree For each of the three subtrees in Figure 8.1 we can use ...

178 Mathematics for Finance Theorem 8.3 The expectation of the discounted payoff computed with respect to the risk- neutral prob ...

Option Pricing 179 It follows that D(0) = 1 1+r [ p∗h(Su)+(1−p∗)h(Sd)) ] = 1 (1 +r)^2 [ p^2 ∗g(Suu)+2p∗(1−p∗)g(Sud)+(1−p∗)^2 g ...

180 Mathematics for Finance 8.1.4 Cox–Ross–Rubinstein Formula ....................... The payoff for a call option with strike p ...

Option Pricing 181 Theorem 8.5 (Cox–Ross–Rubinstein Formula) In the binomial model the price of a European call and put option ...

182 Mathematics for Finance To begin with, we shall analyse an American option expiring after 2 time steps. Unless the option ha ...

Option Pricing 183 The price of the American put will be denoted byPA(n)forn=0, 1 ,2. At expiry the payoff will be positive on ...

184 Mathematics for Finance induction: DA(N)=f(S(N)), DA(N−1) = max { f(S(N−1)), 1 1+r[p∗f(S(N−1)(1 +u)) +(1−p∗)f(S(N−1)(1 +d))] ...

Option Pricing 185 Exercise 8.11. The ex-dividend stock prices are n 012 158. 80 144. 00 < S(n) 120. 00 < 115. 60 ex-div ...

186 Mathematics for Finance treated in detail in more advanced texts. In place of this, we shall exploit an analogy with the dis ...

Option Pricing 187 a probabilityP∗such thatV(t)=W(t)+ ( m−r+^12 σ^2 ) t/σ (rather than W(t) itself) becomes a Wiener process u ...

188 Mathematics for Finance Above we have identified the risk-neutral probabilityP∗. Now we shall con- sider a European call opt ...

Option Pricing 189 with d 1 = lnSX(0)+ ( r+^12 σ^2 ) (T−t) σ √ T−t ,d 2 = lnSX(0)+ ( r−^12 σ^2 ) (T−t) σ √ T−t . (8.11) Exerci ...