Question

The current price of a non-dividend-paying stock is $30. Over the next three months it is expected to rise to $36 or fall to $26. Assume that the risk-free rate is 10% per annum (continuously compounded). What is the risk-neutral probability of the stock price moving up to $36?

a) .40

b) .48

c) .50

d) .60

Answer 1

Solution :

The formula for calculating the risk – neutral probability of an upward movement in the stock price is

P = ( e ( r * t ) - d ) / ( u – d)

Where

P = Risk neutral probability of an upward movement in the stock price

r = risk free rate per annum ; t = time period in years   ; e = 2.71828

u = Probability of a rise in price = ( Probable upward price / Current price of stock )

d = Probability of a fall in price = ( Probable downward price / Current price of stock )

As per the information given in the question we have

r = 10 % = 0.10 ; t = 3 months = 3/12 years = 0.25 years ; u = $ 36 / $ 30 = 1.2 ; d = $ 26 / $ 30 = 0.866667

Applying the above information in the formula we have

= ( 2.71828 ( 0.10 * 0.25 ) - 0.866667 ) / ( 1.2 – 0.866667 )

= ( 2.71828 ( 0.025 ) - 0.866667 ) / ( 1.2 – 0.866667 )

= ( 1.025315 – 0.866667 ) / ( 1.2 – 0.866667 )

= 0.158648 / 0.333333

= 0.475945

= 0.48 ( when rounded off to two decimal places )

Thus the Risk neutral probability of an upward movement in the stock price = 0.48

The solution is Option b) .48

Note : The value of ( 2.71828 ) 0.025 is calculated using the excel formula =POWER(Number, Power)

=POWER(2.71828,0.025) = 1.025315