Question

A bushel of corn is currently valued at $3.87. Its volatility is 18.54%. The risk-free rate is 3.2%. You are considering synthetic positions with 105 days to expiration. What is the at-the-money forward (ATMF) price for corn? How would you use this information to set up synthetic positions?

Answer 1

Answer-

Value of bushel of corn = K = $ 3.87
Volaility = 18.54 %
Risk free rate = r = 3.2 % = 0.032
Days to expiration = 105 days

S = e - r ( T- t) K

(T-t) = 105 / 365 = 0.2877

S = e - 0.032 x 0.2877 x $ 3.87

S = e - 0.009206 x $ 3.87

S = 0.99084 x $ 3.87

S = $ 3.8345

At the money forward ( ATMF) in Black scholes

ATMF = S [ 2 N (d) - 1]

d = [ volatility x ( T -t )1/2 ] / 2

d = [ 18.54 % x (0.2877)0.5 ] / 2

d = [ 18.54 % x 0.5364 ] / 2

d = (0.1854 x 0.5364 ) / 2

d = 0.09945 / 2 = 0.049725

ATMF = $ 3.8345 x [ 2 N(d) - 1]

ATMF = $ 3.8345 x [ 2 x 0.049725 - 1]

ATMF = $ 3.8345 x [ 0.09945 - 1]

ATMF = $ 3.8345 x - 0.90055

ATMF = - $ 3.45316

This can be used in synthetic positions by taking Synthetic short position as the price is expected to fall to $ 3.45316 from $ 3.8345.