• London School of Economics

• FM 322

 7.1 ⋆ Barrier option in a binomial tree Consider an underlying asset whose price evolves according to the binomial tree in Figure 1. The riskfree interest rate is 10% for each period (with no compounding within a period). Price an up-and-in European call option on this underlying with barrier 55 and strike price 40. . ....... ....... ........ ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... .... ....... ....... ........ ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... 50 ... . ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... .... ....... ....... ........ ........ ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... .... 60 . ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... ..... ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... ........ ....... ....... ....... .... 40 70 50 30 Figure 1: Binomial tree Since this is a path-dependent option, we use the corresponding non-recombining tree. The terminal date payoffs are as follows: Vuu = 30, Vud = 10, Vdu = 0, Vdd = 0. Note that the option is knocked-in if and only if the underlying goes up in the first period. The risk-neutral probabilities are given by 50 = [q0 × 60 + (1 − q0) × 40]/1.1 ⇒ q0 = 3/4, 60 = [qu × 70 + (1 − qu) × 50]/1.1 ⇒ qu = 4/5. Note that the risk-neutral probabilities are time-varying and state-dependent (we don’t need the risk-neutral probability qd because the option is deactivated in that part of the tree). We work backwards through the tree to get: Vu = [quVuu + (1 − qu)Vud]/1.1 = 23.64, Vd = 0, V0 = [q0Vu + (1 − q0)Vd]/1.1 = 16.12.