Question

Problem 1: - Using the Black/Scholes formula and put/call parity, value a European put option on the equity in Amgen, which has the following characteristics. Expiration: Current stock price of Amgen: Strike Price: Volatility of Amgen Stock price: Risk-free rate (continuously compounded): Dividends: 3 months (i.e., 60 trade days) $53.00 $50.00 26% per year 2% None If the market price of the Amgen put is actually $2.00 per share, is the above estimate of volatility higher or lower than the market's estimate of volatility? In fact, what volatility value is consistent with the Black/Scholes formula and the market price of the Amgen puts being $2?

Answer 1

Put option value calculation:

INPUTS		Outputs	Value
Standard deviation (Annual) σ	26.00%	d1	0.5517
Expiration (in Years) T	0.25	d2	0.4217
Risk free rates (annual) r	2.00%	N(d1)	0.7094
Current stock price (S0)	$53.00	N(d2)	0.6634
Strike price (X)	$50.00	B/S call Price	4.5960
Dividend yield (annual)	0	B/S Put Price	1.3466

Using Black-Scholes formula, value of European put option is $1.3466

If the actual value of European put option is $2.00; then the volatility estimates will be higher than the market estimate of volatility

Now to calculate the volatility consistent with above put price, we will use trial and error method (use different volatilities and check the price of put option) –

INPUTS		Outputs	Value
Standard deviation (Annual) σ	33.02%	d1	0.4657
Expiration (in Years) T	0.25	d2	0.3006
Risk free rates (annual) r	2.00%	N(d1)	0.6793
Current stock price (S0)	$53.00	N(d2)	0.6182
Strike price (X)	$50.00	B/S call Price	5.2494
Dividend yield (annual)	0	B/S Put Price	2.0000

The new volatility estimate is 33.02%

Formulas used in excel calculation

A B C Outputs d1 Value 1 INPUTS (LN(B5/B6)+(B4-B7+0.5*B2^2)*B3)/(B2*SQRT(B3)) -D2-B2*SQRT(B3) 2 Standard deviation (Annual) o 0.26 3/12 3 Expiration (in Years) T 4 Risk free rates (annual) r d2 N(d1) N(d2) B/S call Price B/S Put Price ENORMSDIST(D2) 0.02 ENORMSDIST(D3) =B5*EXP(-B7*B3)*D4- B6*EXP(- B4 B3) *D5 B6*EXP(-B4*B3)*(1-D5) - B5 * EXP ( - B7*B3) *(1- D4) 5 Current stock price (S0) 53 6 Strike price (X) 7 Dividend yield (annual) 50