Taskin: This Assignment Will Solve Problems On No Arbitrage

Taskin This Assignment You Will Solve Problems On No Arbitrage Restri

Taskin This Assignment You Will Solve Problems On No Arbitrage Restri

Task In this assignment, you will solve problems on No-arbitrage Restrictions, Early Exercise and Put-Call Parity. Instructions Use your textbook to answer the following questions from Chapter 10: Exercise 21 and 22. Please, upload xls, xlsx file. Please, use the full computing power of Excel.

21. stock is trading at S = 50. There are one-month European calls and puts on the stock with a strike of 50. The call is trading at a price of CE = 3. Assume that the one-month rate of interest (annualized) is 2% and that no dividends are expected on the stock over the next month. (a) What should be the arbitrage-free price of the put? (b) Suppose the put is trading at a price of PE = 2.70. Are there any arbitrage opportunities?

22. A stock is trading at S = 60. There are one-month American calls and puts on the stock with a strike of 60. The call costs 2.50 while the put costs 1.90. No dividends are expected on the stock during the options’ lives. If the one-month rate of interest (annualized) is 3%, show that there is an arbitrage opportunity available and explain how to take advantage of it.

Paper For Above instruction

The concepts of no-arbitrage restrictions, put-call parity, and early exercise features are fundamental in options pricing and trading strategies. This paper examines two specific scenarios involving European and American options, utilizing the principles of arbitrage-free valuation to identify potential mispricings and trading opportunities within the frameworks described by the no-arbitrage principle.

Part 1: European Options and Put-Call Parity

Given a stock trading at \(S = 50\), with European call and put options both with a one-month maturity and a strike price of \(K=50\), the European call price \(C_E\) is \$3. The risk-free interest rate is 2% per annum, with no dividends expected over the next month. The put-call parity for European options, expressed as:

P = C + Ke^{-rT} - S

where \(P\) is the price of the put, \(C\) the price of the call, \(K\) the strike price, \(r\) the risk-free rate, and \(T\) the time to maturity in years, allows us to determine the arbitrage-free price of the put.

Substituting the known values:

P = 3 + 50  e^{-0.02  (1/12)} - 50

Calculating \(e^{-0.02/12}\), which approximates to 0.99833, the computation becomes:

P ≈ 3 + 50 * 0.99833 - 50 ≈ 3 + 49.917 - 50 ≈ 2.917

Thus, the arbitrage-free price of the put is approximately \$2.92. If the put market price deviates from this, arbitrage opportunities may exist.

In scenario (b), if the put is traded at \$2.70, which is lower than the arbitrage-free valuation (\$2.92), an arbitrageur could buy the undervalued put and simultaneously execute a synthetic short position in the stock combined with a short forward contract to lock in riskless profit.

Part 2: American Options and Arbitrage Opportunities

Considering a stock priced at \(S=60\), with American call and put options both with a one-month maturity and strike \(K=60\). The American call costs \$2.50, and the American put costs \$1.90. The risk-free rate is 3% per annum, with no dividends expected during the option's life.

According to the no-arbitrage bounds, the American call should reflect at least the European call's value, and the put's value should at least reflect the intrinsic value. Because American options can be exercised early, their prices should not violate put-call parity adjusted for early exercise premiums.

In particular, the sum of the American call price \(C_A\) and the present value of the strike \(K e^{-rT}\) should not exceed the sum of the American put \(P_A\) and the current stock price \(S\), i.e.,

C_A + K e^{-rT} ≥ P_A + S

Calculations yield:

K e^{-rT} = 60  e^{-0.03/12} ≈ 60  0.9975 ≈ 59.85

Substituting known values:

2.50 + 59.85 ≈ 62.35


1.90 + 60 = 61.90

Since 62.35 > 61.90, there exists a potential arbitrage opportunity. Specifically, an arbitrageur can exploit the mispricing by simultaneously selling the overpriced combination or constructing a synthetic position with the standard options and underlying asset, taking advantage of the violation.

Conclusion

The analysis illustrates that adherence to no-arbitrage principles and put-call parity constrains option pricing. Deviations signal potential arbitrage opportunities, which can be exploited through carefully structured trades. Practical implementation requires precise calculations especially involving interest rates and option maturities, emphasizing the importance of tools like Excel for comprehensive analysis.

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