Question

Consider a European put option on AAPL stock with a strike of 300 that expires on 9/18/20. As of close on 4/27, the spot price is 283.17 and the option premium is 25.90.

a. What is the current intrinsic and extrinsic value of this option?

b. If spot interest rates for a bond expiring on 9/18/20 is 0.5% annually, what should be the price of a call option with the same strike and expiration?

c. What is the Black-Scholes-implied volatility of this option?

d. What is the Black-Scholes delta, gamma, and theta of this option?

Answer 1

As per question given that

S = Spot = $283.17

X = Strike Price = $300

t = Time to expiration = 144 days

P = Put option permium = $25.9

r = Interest rate = 0.5% annually

Answer for (1)

intrinsic value of put option = Strike price - spot price ; = $300 - $283.17 ; = $16.83

Extrinsic value of put option = Option permium - intrinsic value ; = $25.9 - $16.83 ; = $9.07

Answer for (2)

since we have vlaue of put option we can calculate the price of call option using put call parity theory.

P + S = C + PV(X)

where,

C = call option value

PV(X) = present value of strike price

Calculation of discounting factor

= 1 / (1 + (0.5% * 144 / 365)) = 1 / 1.001973 = 0.998031

$25.9 + $283.17 = C + 0.99031 * $300

C = $9.660616 or $9.66

Answer for (3)

black scholes formula for call option

C = S * ​N(d1​) −X * e−rt * N(d2​)

Where,

d1​ = ​ln(S / X) ​​+ (r +​ (σ^2 / 2)) * t

​d2 ​= d1 ​− σ * (t)^0.5

N = normal distribution​

σ = Voltility

since we have all the value except Volitility we can calculate it by reverse calculation or by trial end error.

by trial and error we can arrive at Implied volatility of 22.66% at which our call option value will be $9.66 and put option value will be $25.9.

Answer for (4)

Calculation of delta

delta for call option = N(d1) ; = 0.3742

delta for put option = N(d1) - 0.3742 = -0.6258

Calculation of gamma

gamma for call and put option will be the same by using derivation we can arrive arrive at 0.0094

gamma for long option contract is positive and for short option constract is negative

Calculation of theta

we can calculate gamma for call option by using derivation. We can arrive arrive at -19.8383

for put option theta is -18.3412

theta for both call and put option is negtive because it represent time factor which will decline as we reach near to expiry date.