Question

Suppose: 1) 1 year futures price=$1000, 2) interest rate = 0%, and 3)for K=$1000, the premium on a 1 year call (C) is $100 and on a 1year put (P) $150.

a. Does put-call parity hold? If not, relative to each other which option is overpriced and which underpriced?

b. Describe a profitable and risk free arbitrage, that is, what would you long today and what would you short?

c. For your arbitrage described in question b. what are your profits/losses if the price of the asset in one year takes the following values: $800, $900, $1000, $1100, $1200, or $1300?

d. As you and others undertake the trades described in question b. how will prices adjust?

Answer 1

Solution:

It is given that the future price = 1000

interest rate = 0%

Hence spot price will be eual to the future price S = 1000

P =150

C = 100

Part A)

Using put call parity formula

C + X /Exp(interest * Time) = P +S

100 + 1000 / exp(0%*1) = 150 +1000

100 + 1000 =1150

1100 = 1150

Since both the sides are not eual hence put call parity does not hold. Call option is undrpriced and put option is overpriced

Part B )

Strategy to get arbitrage profit : Sell hogher side and buy lower side

• Sell put option and earn a premium = 150
• Sell a stock at 1000
• Buy call option at 100
• and buy security that has 0% interest worth 1000

Arbitrage profit = 1150-1100 = 50

Part C )

Arbitrage profit will be same at all the price i.e. 50

When stock price = 800

1. We Sold the put option and earn a premium = 150.

Currently the premium of the option will be 1000-800 = 200

Hence we will make a loss of 50

2. We sold the stock at 1000 and current price = 800 hence profit = 1000-800 = 200

3. We bought a call option but this call option will expire worthless as Current price < Strike price . Hence we will make a loss of 100

4. Security price will remain at a value of 1000 as interest rate = 0

Total profit = -50 + 200 -100 +0 = 50

When stock price = 900

1. We Sold the put option and earn a premium = 150.

Currently the premium of the option will be 1000-900 = 100

Hence we will make a profit of 50

2. We sold the stock at 1000 and current price = 900 hence profit = 1000-900 = 100

3. We bought a call option but this call option will expire worthless as Current price < Strike price . Hence we will make a loss of 100

4. Security price will remain at a value of 1000 as interest rate = 0

Total profit = 50+ 100-100 +0 = 50

When stock price = 1000

1. We Sold the put option and earn a premium = 150.

Currently the premium of the option will be 1000-1000 = 0

Hence we will make a profit of 150

2. We sold the stock at 1000 and current price = 100 hence profit = 1000-1100 = -100

3.We bought a call option but the value of the call option will be zero as Current price = Strike price . Hence we will make a loss of 100

4. Security price will remain at a value of 1000 as interest rate = 0

Total profit = 150+ 0-100 +0 = 50

When stock price = 1100

1. We Sold the put option and earn a premium = 150.

Since Spot price > Strike price

Hence we will make a profit of 150

2. We sold the stock at 1000 and current price = 100 hence profit = 1000-1000 = 0

3.

We bought a call option and Current price > Strike price . Hence the value of call option = 1100-1000 = 100. Initially we purchased this at 100 hence overall profit = 0

4. Security price will remain at a value of 1000 as interest rate = 0

Total profit = 150+ 0-100 +0 = 50

We can similarly calculate for all the prices and profit will be same always

Part D )

When people start to take arbitrage trade then the arbitrage will reduce and prices of the option and asset comes to the stage where arbitrage profit would be minimum or zero.