2. A binary option is an exotic option which pays $1 if some condition is statisfied, or $0 if the condition is not statisfied. For example, a European call binary option will pay $1 if the underlying asset price is greater than the strike at expiry, but $0 otherwise. We can similarly define European put binary options. American binary options are also possible, and like the vanilla American options can be exercised any time up to expiry. Usually, for binary options, the strike is equal to the initial value of the underlying asset K = S(0). (a) Construct a four step binomial model for an American binary call option with strike $30, where the underlying asset under Cox-Ross-Rubenstein notation fol- lows S = 30, u = 1.15 and d=1/u, and the return over each time step is R= 1.1. What is the premium of this American binary call? (b) Premiums of European binary call options can be calculated in a variety of ways, for example, using the binomial models, state prices, or a modified Black-Scholes equation. Consider a European binary call option otherwise identical to the American binary call discussed in part (a). Calculate all state prices at expiry in a four step model and then use these state prices to calculate the premium of the European binary call. (c) Now consider a variant on a binary option. This option has two strikes Ki and K2, with K2 > Ki, and pays $1 when the underlying is between Ki and K2, but zero otherwise. Again, there is a European version which only pays at expiry and an American version which may be exercised at any time up to expiry. However, we do not define a put or a call since this option combines aspects of both puts and calls. Use the same underlying asset as in part (a) and R= 1.1 to construct a four-step binomial pricing tree for an American option of this type where the two strikes are K = $20 and K2 = $30. What is the premium of this American option? 2. A binary option is an exotic option which pays $1 if some condition is statisfied, or $0 if the condition is not statisfied. For example, a European call binary option will pay $1 if the underlying asset price is greater than the strike at expiry, but $0 otherwise. We can similarly define European put binary options. American binary options are also possible, and like the vanilla American options can be exercised any time up to expiry. Usually, for binary options, the strike is equal to the initial value of the underlying asset K = S(0). (a) Construct a four step binomial model for an American binary call option with strike $30, where the underlying asset under Cox-Ross-Rubenstein notation fol- lows S = 30, u = 1.15 and d=1/u, and the return over each time step is R= 1.1. What is the premium of this American binary call? (b) Premiums of European binary call options can be calculated in a variety of ways, for example, using the binomial models, state prices, or a modified Black-Scholes equation. Consider a European binary call option otherwise identical to the American binary call discussed in part (a). Calculate all state prices at expiry in a four step model and then use these state prices to calculate the premium of the European binary call. (c) Now consider a variant on a binary option. This option has two strikes Ki and K2, with K2 > Ki, and pays $1 when the underlying is between Ki and K2, but zero otherwise. Again, there is a European version which only pays at expiry and an American version which may be exercised at any time up to expiry. However, we do not define a put or a call since this option combines aspects of both puts and calls. Use the same underlying asset as in part (a) and R= 1.1 to construct a four-step binomial pricing tree for an American option of this type where the two strikes are K = $20 and K2 = $30. What is the premium of this American option