Question

The spot price of BOA is $150. The risk free rate is 3%. Assume that BOA follows a log-normal model with volatility 20% and expected rate of return 2%. Consider a 1-year 150-180 bull spread on BOA made of calls. What is the probability that the payoff this bull-spread is $20 or more?

a. none of the above
b. between 0.42 and 0.46
c. below 0.42
d. above 0.46

Answer 1

The probability of the payoff is greater than equal to $ 20 that will be fulfilled when the price of the stock crosses 170. But at price 180, the payoff will get stabilize.

Now the price of the BOA follows lognormal distribution, the continuous returns would follow the Normal distribution.

∴ So for the payoff of $ 20 or more than that, return has to be
`$\ln(170/150)=12.5163 \%$` Pa or more than that. So, x = 12.5163 %

`$Z=\frac{x-\text{mean}}{s_d}$`

`$=(12.5163-10)/(20)$`

= 0.12582

∴ Using Z table, the cumulative probability to the left side =
`$0.550061$`

But the probability of the return which is more than 12.5163 % is given by :

= 1 - 0.55061

= 0.449939

= 0.45