Question

Historical data suggest the standard deviation of an all-equity strategy is about 5.5% per month. Suppose the risk-free rate is now 2.9% per month and market volatility is at its historical level. What would be a fair monthly fee to a perfect market timer, according to the Black-Scholes formula?

Answer 1

S0 = underlying price	$1
E = strike price = (1 + rf) = (1 + 2.90%)	$1.03
? = volatility (% p.a.)	0.055
?^2 = variance (% p.a.)	0.00
r = continuously compounded risk-free interest rate (% p.a.)	2.90%
d = continuously compounded dividend yield (% p.a.)	0
t = time to expiration (years in %) =	1.0000
C = S × e–dt × N(d1) – E × e–Rt × N(d2)
d1 = [ln(S/E) + (R – d + ?2 / 2) × t] / (? × sqrt(t))
d2 = d1 – ? × sqrt(t)
d1 = [ln($1/$1.03+ (.029 - 0 + .00/2) × 1] / (.06× sqrt (1 )	3.50%
d2 =3.50% - .055 x sqrt(1)	-2.00%
Normal Distribution N( d1) using NormDIST	0.5140
Normal Distribution N(d2)	0.49
e-dt	1
e-rt	0.971416464
C = $1 x 1 x .5140 - $1.03 x .9714 x .49	2.21%