Question

1. Using the Black-Scholes option pricing model, find the premium for a put on Uber. The stock currently trades for $26.70. The expiration is in 23 days. The strike price is $30. The risk free rate is 2.25% and the volatility (standard deviation) of the stock is .44.

Answer 1

put options give the option holder the right to sell the underlying stock for an agreed-upon price anytime between today and option expiration. Traders that think a stock is going to go down can buy these put options in the hopes of making money if the stock goes does.

The actual Black-Sholes formula looks complicated but is actually simple when you break it down to the basics. The main factors in the equation are:

The Black–Scholes-Merton formula of value

P(S​,t)=S0​N(d1​)−Ke−r(T−t)N(d2​),

where

• S-: is the stock price =$ 26.70
• P(S0,t): is the price of the put option as a formulation of the stock price and time;
• K is the exercise price;= $30
• (T-t) is the time to maturity, i.e. the exercise date TT, less the amount of time between now tt and then.
• N(d1​) and N(d2​) are cumulative distribution functions for standard normal distribution with the following formulation
• risk-free rate r = 2.25 %an
• volatility (s) standard deviation = .44

d1= [Ln (S / E) + (r + s2 / 2) X t] / s√t

d2= d1-s√t

hence in given question

d1= [Ln (S / E) + (r + s2 / 2) X t] / s√t

= [ ln 26.70/30 + ( .0225+ .442/2 ) 23/365] / .44√ 23/365

= [ ln 0.89 + ( .0225 + 0.0968 )* 0.0630 ] / 0.1104

=   -0.1165 + 0.2027 / 0.1104

= - 0.871

d2= d1-s√t

= - 0.8716- 0.1104

= - 0.982

N(d1​) and N(d2​) can be found by looking at a z-score table:

N(d1​) =-0.8078

N(d2​)= 0.8365

Value of Put option (S​,t)=S0​N(d1​)−Ke−r(T−t)N(d2​),

= 26.70 ( - 0.8078) - (30 *e(-0.0225*23/365) * (-0.8365)

=- 21.56 +30 * 1.0014 *(-0.8365)

=3.57