Question

Use the following Call Option Information for parts a-b: Strike (K) is $1.42, Maturity (T) is in 31 trading days (assume 260 trading days in the year), and a discount rate (r) of .0006.

a. What is the implied volatility (standard deviation) if the Spot price (S) is $1.3852 and the Call Premium is $0.01?

b. What is the Call Premium if the Spot price is $1.3891 and the standard deviation is 0.0057?

Answer 1

a.

Implied volatilty can be calculated by reverse calculation in Black-Scholes Model of option pricing.

Formula of Black Scholes Model:

Formula for calulating n i excel is NORM.S.DIST(value,True). In place of value we insert d1 and d2.

To get implied volatility we need to use trial and error method.

So to start with we will take volatility of 15%.

Spot (S)	1.3852		Step 1	Ln(S/X)	-0.0248123
Strike (X)	1.42		Step 2	(r + (SD^2)/2)*t	0.00141288
t	31	days	Step 3	Step 1 + Step 2	-0.0233995
	0.12	years	Step 4	SD * sqrt (t)	0.05179471
r	0.0006		Step 5	Step 3/Step 4	-0.451773	<<d1
SD	15.00%		Step 6	Step 5 - Step 4	-0.5035677	<<d2
			Step 7	N(d1)	0.32571626
				N(d2)	0.30728259
			Step 8	Call = S * N(d1) - X * exp (-rn) * N(d2)	0.0148721

So, now we got the price of call as 0.0148721. This means we are close to the exact implied volatility.

To get closer to the price of $0.01, we will now reduce the volatility to 12.5%.

Spot (S)	1.3852		Step 1	Ln(S/X)	-0.0248123
Strike (X)	1.42		Step 2	(r + (SD^2)/2)*t	0.00100303
t	31	days	Step 3	Step 1 + Step 2	-0.0238093
	0.12	years	Step 4	SD * sqrt (t)	0.04316226
r	0.0006		Step 5	Step 3/Step 4	-0.5516233	<<d1
SD	12.50%		Step 6	Step 5 - Step 4	-0.5947856	<<d2
			Step 7	N(d1)	0.29060323
				N(d2)	0.27599341
			Step 8	Call = S * N(d1) - X * exp (-rn) * N(d2)	0.01066099

Now the price is 0.01066, which means we are very close to our implied volatility.

Next trial will be of 12%.

Spot (S)	1.3852		Step 1	Ln(S/X)	-0.02481234
Strike (X)	1.42		Step 2	(r + (SD^2)/2)*t	0.00093
t	31	days	Step 3	Step 1 + Step 2	-0.02388234
	0.12	years	Step 4	SD * sqrt (t)	0.04143577
r	0.0006		Step 5	Step 3/Step 4	-0.57637007	<<d1
SD	12.00%		Step 6	Step 5 - Step 4	-0.61780584	<<d2
			Step 7	N(d1)	0.28218253
				N(d2)	0.26835167
			Step 8	Call = S * N(d1) - X * exp (-rn) * N(d2)	0.0098471

Now we are almost close enough, just to be closest we will perform the last iteration of 12.1%

Spot (S)	1.3852		Step 1	Ln(S/X)	-0.02481234
Strike (X)	1.42		Step 2	(r + (SD^2)/2)*t	0.00094437
t	31	days	Step 3	Step 1 + Step 2	-0.02386797
	0.12	years	Step 4	SD * sqrt (t)	0.04178107
r	0.0006		Step 5	Step 3/Step 4	-0.57126281	<<d1
SD	12.10%		Step 6	Step 5 - Step 4	-0.61304388	<<d2
			Step 7	N(d1)	0.28391075
				N(d2)	0.26992366
			Step 8	Call = S * N(d1) - X * exp (-rn) * N(d2)	0.0100090

So the implied volatility at spot price of $1.3852 is 12.10%=0.121.

b.

Now we need to calculate the call premium at Spot Price $1.3891 and standard deviation of 0.0057.

Spot (S)	1.3891		Step 1	Ln(S/X)	-0.02200082
Strike (X)	1.42		Step 2	(r + (SD^2)/2)*t	7.3475E-05
t	31	days	Step 3	Step 1 + Step 2	-0.02192734
	0.12	years	Step 4	SD * sqrt (t)	0.0019682
r	0.0006		Step 5	Step 3/Step 4	-11.1408144	<<d1
SD	0.0057		Step 6	Step 5 - Step 4	-11.1427826	<<d2
			Step 7	N(d1)	3.9695E-29
				N(d2)	3.8827E-29
			Step 8	Call = S * N(d1) - X * exp (-rn) * N(d2)	0.0000000

As the volatility is too low, the call premium is 0.