Taskin: This Assignment You Will Solve Problems On No Arbitr

Taskin This Assignment You Will Solve Problems On No Arbitrage Restri

In this assignment, you will analyze issues related to no-arbitrage restrictions, early exercise, and put-call parity based on specific problems from Chapter 9 of your textbook. You are asked to use Excel to perform comprehensive calculations and to ensure your solutions are precise and fully substantiated. Please upload your Excel file with all numerical computations.

Question 12: The current price of a stock is $60. The one-year call option on the stock at a strike price of $60 is trading at $10. Given a 10% interest rate, determine if the call price is free from arbitrage assuming the stock pays no dividends. Repeat the analysis assuming the stock pays a dividend of $5 one day before the option's maturity.

Question 13: The current price of ABC stock is $50. The interest rate curve is flat at 10% (continuously compounded). Calculate the six-month forward price (F). The six-month call with strike F is priced at $8, and the six-month put at strike F is priced at $7. Explain why arbitrage opportunities may exist with these prices.

Paper For Above instruction

Understanding no-arbitrage conditions in options pricing is fundamental in financial theory and practice. Arbitrage opportunities, when present, allow traders to make riskless profits, indicating inconsistencies in market prices and violating efficiency principles. This paper explores two important problems related to no-arbitrage restrictions, examining how theoretical models and real-world data interact, with a focus on options valuation, dividend considerations, and forward prices.

Question 12: Arbitrage-Free Call Pricing and Dividends

The valuation of European-style call options hinges on the no-arbitrage principle, which ensures that options are priced consistently with the underlying asset. The fundamental relationship guarantees that no riskless profit can be obtained by exploiting price discrepancies between the call, underlying stock, and bond positions.

Assuming the stock does not pay dividends, the no-arbitrage call price is connected to the underlying stock price (S), the risk-free interest rate (r), the time to maturity (T), and the strike price (K) through the classic put-call parity formula:

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

where C and P denote the call and put prices, respectively. Since only the call price is given, the arbitrage bounds can be derived from these relationships. The no-arbitrage upper bound for the call is the current stock price (S = $60). The lower bound is the maximum between zero and (S - K e^{-rT}); in this case, since e^{-0.10} ≈ 0.9048, the lower bound is ($60 - $60 * 0.9048) ≈ $5. This means that if the call price exceeds $60, there's an arbitrage opportunity, and if it falls below $5, negative arbitrage could exist.

Given the actual call price is $10, it lies within this arbitrage bounds, suggesting that, absent dividends, the market is consistent and free of arbitrage. If additional dividends are paid, particularly a dividend of $5 one day before maturity, the precise replicating strategies shift, as dividends decrease the stock price pre-maturity, affecting the put-call parity conditions. Adjusting for dividends involves subtracting the present value of the dividend from the stock price when evaluating arbitrage bounds. Since the dividend occurs just one day prior to maturity, its present value (PV) at the close of the period is approximately $5 / e^{0.10*(364/365)} ≈ $4.995, nearly the dividend amount itself, discounted at the risk-free rate.

Thus, with dividends, the adjusted stock price right before maturity becomes roughly ($60 - $5) = $55 at the dividend date, and the boundaries for arbitrage considerations shift accordingly, reinforcing the importance of including dividends in options valuation models for accuracy and consistency.

Question 13: Forward Prices, Option Prices, and Arbitrage

The current stock price of $50 and the flat interest rate of 10% (continuously compounded) permit the calculation of the fair forward price for six months. The forward price without arbitrage is given by:

F = S e^{rT} = 50 e^{0.10 0.5} ≈ 50 e^{0.05} ≈ 50 1.0513 ≈ $52.57

However, the observed prices of the options provide insight into potential arbitrage opportunities. The call option with a strike equal to the forward price (F ≈ $52.57) is priced at $8, and the put at the same strike is $7. According to put-call parity for European options:

C - P = S - K e^{-rT}

where K = F and S = $50. Calculating the right side:

$50 - $52.57 e^{-0.10 0.5} ≈ $50 - $52.57 * 0.9512 ≈ $50 - $50 ≈ $0

thus, the difference (C - P) should be approximately zero if no arbitrage exists. Yet, the actual difference is $8 - $7 = $1, indicating an inconsistency. This discrepancy suggests an arbitrage opportunity because the observed prices violate the put-call parity relationship.

Specifically, since the parity demands that C - P ≈ $0, but the market prices show C - P = $1, traders could exploit this inconsistency by constructing arbitrage strategies such as buying the undervalued option and shorting the overvalued one, or by simultaneously taking positions in underlying assets and options to lock in riskless profits.

Conclusion

These analyses underline the delicate balance maintained by no-arbitrage conditions in options and futures markets. When prices deviate from theoretically derived bounds and relationships—factoring in dividends, interest rates, and other market parameters—opportunities for riskless profit arise, prompting arbitrageurs to act and restore market efficiency. Accurate modeling of dividends and interest rates is crucial in ensuring consistent pricing, and observed market prices serve as indicators of potential arbitrage opportunities, motivating continuous market adjustments and further refinements in pricing approaches.

References

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