Question

A stock is currently priced at $37.00. The risk free rate is 5% per annum with continuous compounding. In 7 months, its price will be either $42.18 or $31.82. Using the binomial tree model, compute the price of a 7 month bear spread made of European puts with strike prices $41.00 and $45.00.

Answer 1

For strike 41:

probbaility of upmove=(e^(rt)-Sd/S0)/(Su/S0-Sd/S0)=(e^(0.05*7/12)-31.82/37)/(42.18/37-31.82/37)
Call payoff in case of upmove=MAX(Su-Strike price,0)=MAX(42.18-41,0)
Call payoff in case of downmove=MAX(Sd-Strike price,0)=MAX(31.82-41,0)

Price of call=(probability of upmove*payoff in case of upmove+(1-probability of upmove)*payoff in case of downmove)*e^(-rt)=((e^(0.05*7/12)-31.82/37)/(42.18/37-31.82/37)*MAX(42.18-41,0)+(1-(e^(0.05*7/12)-31.82/37)/(42.18/37-31.82/37))*MAX(31.82-41,0))*e^(-0.05*7/12)=0.694181632

Price of put=C+Strike price*e^(-rt)-S0=0.694181632+41*e^(-5%*7/12)-37=3.515619216

For strike 45:

probbaility of upmove=(e^(rt)-Sd/S0)/(Su/S0-Sd/S0)=(e^(0.05*7/12)-31.82/37)/(42.18/37-31.82/37)
Call payoff in case of upmove=MAX(Su-Strike price,0)=MAX(42.18-45,0)
Call payoff in case of downmove=MAX(Sd-Strike price,0)=MAX(31.82-45,0)

Price of call=(probability of upmove*payoff in case of upmove+(1-probability of upmove)*payoff in case of downmove)*e^(-rt)=((e^(0.05*7/12)-31.82/37)/(42.18/37-31.82/37)*MAX(42.18-45,0)+(1-(e^(0.05*7/12)-31.82/37)/(42.18/37-31.82/37))*MAX(31.82-45,0))*e^(-0.05*7/12)=0

Price of put=C+Strike price*e^(-rt)-S0=0+45*e^(-5%*7/12)-37=6.706455885

Price of bear spread=Buy put of strike 45 and sell put of strike 41=Price of put of strike 45-Price of put of strike 41=6.706455885-3.515619216=3.190836669