Please show all steps and explain

Problem 5. Derive the option pricing formula for a. standard buttery spread with K2 : % in an arbitrage-free binomial model. K2 and one short put option with strike K1 s: Kg [see Figure 4.11]. Both options have the same exercise date. The payoffs are described in the table below. Stock Price 3: Short Put Payo: Long Put Payoff: Total Payo': The payoff is positive for low future stock prices. The urn-arbitrage price of the bear spread is equal to PEI:ng PE{K1}. An investor who expects that the future stock price will change insignicantly and stay in an interval [H 1, K3] may choose a buttery spread that is constructed from three options of the same kind expiring on the same date with strike prices It}, i = 1, 2,3. so that K1 c: Kg c. K3. One way to construct a buttery spread is to combine one long call with strike K1. two short calls with strike Hg. and one long call with strike K3 (see Figure 4.12}. Assuming that K2 = iii-girl, the payoff function of the buttery spread is u HSsKh EurKl lle'ngE-s 1'1 = 331:5} KsS are c: as In. El ifS:'-'K3. The no-arbitrage price is CE{K1}+ UfKa} ZUE[K3'J. A combination is an option portfolio that involym taking a position in both calls and puts on the same underlying security. Some examples are as follows. Straddle combines one put and one call with the same strike and expiry date. Strip combinm one long call and two long puts with the same strike and expiry date. Strap combines two long calls and one long put with the same strike and expiry date. Strangle is a combination of a put and a call with the same expiry date but different strike prices.