Question

A stock has a current price of $60. Each month the price rises by 10% or falls by 5%, with equal probability. The monthly risk-free rate is 1%.

a) Find the current price of a European put option on the stock with two months until expiration and exercise price of $65. (5 points)

b) Consider an option trader who has sold a call option on the stock with exercise price of $58 and two months until expiration. How many shares of stock must he hold at time zero to hedge his option position? (5 points)

c) Describe the parameters of the Black-Scholes option pricing model and whether marginal changes in each parameter affect the price of a call option positively or negatively. (5 points)

Answer 1

Ans:

Stock Price, S = $60

U =1.1

D=0.95

Risk Free Rate, Rf=0.01

1. Probability of up move =( e^(0.01*2/12) – 0.95)/(1.1-0.95) =0.3445

Probability of down move = 1- 0.3445 = 0.6555

  

Up Move = 60*1.1 = 66

Down Move = 60*0.95 = 57

Exercise Price, X = 65

Value of the Put option = [ (65-57)* 0.6555 + 0* 0.3445)* e^(-0.01*2/12) = 5.235

2. Value of the call option = [ (66 -58)*0.3445 + 0* 0.6555) * e^(-0.01*2/12) = 2.7514

               Total Number of shares he must hold to hedge his position =100*58/2.7514 = 2108 shares

3. The parameters of Black Scholes

• the strike price of an option,
• the current stock price,
• the time to expiration,
• the risk-free rate,
• and the volatility

Since Volatility is not present, the call and put option value cannot be calculated using Black-Scholes option pricing model

The marginal changes in each parameter affect the price of a call option and put option as follows:

		European call option	European Put option	American call option	American Put option
Strike Price, X	Increases	Increases	Decreases	Increases	Decreases
	Decreases	Decreases	Increases	Decreases	Increases
Stock Price, S	Increases	Decreases	Increases	Decreases	Increases
	Decreases	Increases	Decreases	Increases	Decreases
Time to Expiration	Increases	No Impact	No Impact	Increases	Increases
	Decreases	No Impact	No Impact	Decreases	Decreases
Risk Free Rate	Increases	Increases	Decreases	Increases	Decreases
	Decreases	Decreases	Increases	Decreases	Increases
Volatility	Increases	Increases	Increases	Increases	Increases
	Decreases	Decreases	Decreases	Decreases	Decreases