# Option Pricing-Black Scholes, Put Call Parity

A. Use the Black-scholes formula to find the price of a three-month European call option on a non-dividend-paying stock with a current price of $50. Assume the exercise price is $51, the continuously compounded riskless interest rate is 8% per year, and standard deviation is .4.

b. What is the composition of the initial replicating portfolio for this call option?

c. Use the put-call parity relation to find the Black-Scholes formula for the price of the corresponding put option.

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a. use the Black-scholes formula to find the price of a three-month European call option on a non-dividend-paying stock with a current price of $50. Assume the exercise price is $51, the continuously compounded riskless interest rate is 8% per year, and s is .4.

Note: * refers to multiplication and ^ to raised to the power of

We will use Black Scholes Pricing Formula

Value of call= S N(d1) - X * e -r(T-t) * N(d2)

We therefore need to calculate the values of d1 and d2

d1= {ln (S/X) + ( r + ½ s2 ) x (T-t)}/ (s x square root of (T-t))

d2= {ln (S/X) + ( r - ½ s2 ) x (T-t)}/ (s x square root of (T-t)) =d1-s x square root of (T-t)

Inputs

Stock Price= ...

#### Solution Summary

Uses the Black-scholes formula to find the price of a three-month European call option on a non-dividend-paying stock. Uses the put-call parity relation to find the Black-Scholes formula for the price of the corresponding put option.

Option Pricing Models- Black-Scholes option-pricing model, Two-Period Binomial Model

1. A stock currently sells for $60 and pays no dividends. There is a call option (striking price = $65) on this security that expires in 41 days. At present U.S. Treasury bills are yielding 1 percent per year. You estimate the past volatility of the stock returns to be 39 percent (i.e., standard deviation). Using Black-Scholes option-pricing model, calculate the value of the call.

2. Using the arbitrage argument (i.e., put-call parity) for the value you derived for problem 1, calculate the implied value of the put on the same stock.

3. XYZ Corporation stock sells for $55 per share. The AUG option series has exactly three months until expiration. At the moment, the AUG 55 call sells for $4 and the AUG 55 put sells for $1 1/2. What is the annual interest rate implied in these prices?

4. Use the Two-Period Binomial Model to solve the following problem. Let the current stock price be $45 and the risk-free rate 5 percent. Each period the stock price can either go up or down by 10 percent. A call option expiring at the end of the second period has an exercise price of $ 40.

a. What is the initial hedge ratio?

b. What are the two possible hedge ratios at the end of the first period?

c. Illustrate by single example that the hedge portfolio works and earns the risk-free rate over the two periods.

Hint:

Given: S = $45, r = 5%, u = 10%., d = -10% and E = $40 where

'S' is stock price, 'r' is interest rate, 'u' is up factor, 'd' is down factor and 'E' is the exercise price.

5. On July 6 a July 160 call on a stock priced at $165.13 is selling for $6. The option will expire on July 17. Compute the intrinsic values, time values and lower bound of the call. (Assume American options for intrinsic and time value, and European option for lower bound).

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