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The current price of a stock is $50. In 1 year, the price will be either $65 or $35. The annual risk-free rate is 10%. Find the price of a call option on the stock that has an exercise price of $55 and that expires in 1 year. (Hint: Use daily compounding)

The exercise price on one of Chrisardan Company’s call options is $20, its exercise value is $27, and its time value is $8. What are the option’s market value and the price of the stock? Submit your answers in a Word document. Text book:

Sample Paper For Above instruction

Introduction

Options are vital financial derivatives that grant investors the right, but not the obligation, to buy or sell an underlying asset at a specified strike price before or at expiration. Pricing these options accurately entails understanding various financial theories and models, especially when dealing with uncertain future prices and risk-free rates. This paper tackles two core problems: determining the fair value of a call option based on binomial assumptions with daily compounding and estimating the stock price from an option's market value components.

Part 1: Calculating the Call Option Price Using the Binomial Model with Daily Compounding

Problem Restatement

The stock currently trades at $50. After one year, its price will either rise to $65 or fall to $35, with a risk-free interest rate of 10% annually compounded daily. The goal is to determine the fair value of a call option with a strike price of $55 and expiry of one year.

Methodology

The binomial model is a discrete-time model used in options pricing that assumes the underlying asset can only move to two possible prices over each period. To incorporate daily compounding, the risk-free rate must be converted to a daily rate, recognizing that 365 days approximate a year (ignoring leap years for simplicity). Specifically, the daily risk-free rate (r_d) is computed as:

r_d = (1 + r_annual)^{1/365} - 1

where r_annual = 0.10. Plugging in the values:

r_d = (1 + 0.10)^{1/365} - 1 ≈ 0.000263

The risk-neutral probabilities are then calculated as:

p = ( (1 + r_d) - d ) / ( u - d )

where u = 1.3 (since 65/50) and d = 0.7 (since 35/50).

Calculations yield:

p ≈ (1.000263 - 0.7) / (1.3 - 0.7) ≈ 0.5004

Using these, the expected payoff of the call option at expiration and discounting it back to the present provides its fair value. The payoff at maturity if the stock rises to $65:

  • Payoff = max(0, stock price - strike price) = max(0, 65 - 55) = $10

If the stock falls to $35:

  • Payoff = max(0, 35 - 55) = $0

The expected payoff under the risk-neutral measure is:

Expected payoff = p  10 + (1 - p)  0 ≈ 0.5004  10 + 0.4996  0 ≈ $5.004

Finally, discounting this payoff at the daily rate over 365 days gives:

Price ≈ Expected payoff  (1 + r_d)^{-365} ≈ 5.004  (1.000263)^{-365} ≈ $5.004 / 1.10 ≈ $4.55

Conclusion

The approximated fair value of the call option, considering daily compounding risk-free rate, is roughly $4.55.

Part 2: Estimating Stock Price from Option Market Values

Given Data

  • Exercise price = $20
  • Exercise value (intrinsic value) = $27
  • Time value = $8

Analysis

The market value of the option equals the sum of intrinsic value and time value, which is:

Market value = Intrinsic value + Time value = $27 + $8 = $35

Given that the intrinsic value is the difference between the underlying stock price and the strike (when positive), the current stock price can be deduced as:

Stock price = Exercise value + (Market value - Exercise value) = $20 + ($35 - $27) = $20 + $8 = $28

This indicates that the current stock price, based on valuation components, is approximately $28.

Discussion

Determining the correct stock price from options involves understanding how market values are composed of intrinsic and time values. The intrinsic value reflects immediate profit potential if the option were exercised, while the time value accounts for future opportunities and volatility. The derived stock price aids investors in assessing whether the option is undervalued or overvalued relative to the current market conditions.

Implications and Limitations

The calculations presented rest on simplified assumptions—such as stable volatility and interest rates—and ignore transaction costs, dividends, and other market factors. In real-world settings, models like Black-Scholes or binomial trees incorporate more complex variables, leading to more refined estimates. Nonetheless, these fundamental calculations provide essential insights into option valuation processes.

Conclusion

By applying the binomial model with daily compounding interest and analyzing option components, we approximate the fair value of options and infer underlying stock prices. Investors and financial analysts must understand these models for effective decision-making and hedging strategies in dynamic markets.

References

  • Hull, J. C. (2018). Options, Futures, and Other Derivatives. Pearson.
  • McDonald, R. (2013). Derivatives Markets. Pearson.
  • Natenberg, S. (1994). Option Volatility & Pricing. McGraw-Hill.
  • Bakas, N. (2014). Financial Calculus: An Introduction to Derivative Pricing. Cambridge University Press.
  • Boyle, P. (2017). Derivative Pricing and Risk Management. Cambridge University Press.
  • Rogers, L. C. G., & Selden, T. (2012). Introduction to Derivatives and Risk Management. Pearson.
  • Choudhry, M. (2018). An Introduction to Financial Markets and Securities. Wiley.
  • Hull, J. C. (2017). Options, Futures, and Other Derivatives (10th ed.). Pearson.
  • Jarrow, R., & Turnbull, S. (2014). Derivative Securities. South-Western College Publishing.
  • Martin, J. (2020). Quantitative Finance: Theory, Implementation, and Applications. Wiley.

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