The Current Price Of A Stock Is $50. In 1 Year, The Price Wi ✓ Solved

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 current price of a stock is $50. In one year, the stock price will either be $65 or $35, representing a binomial model scenario. The risk-free interest rate is 10% per annum, compounded daily. The task is to determine the fair value of a call option with an exercise (strike) price of $55, expiring in one year. To accomplish this, we employ the binomial option pricing model, adjusted for daily compounding interest, and determine the risk-neutral probability of the stock's upward or downward movement.

Introduction

Option pricing fundamentally relies on valuing the expected payoff under risk-neutral probabilities, discounted at the risk-free rate. The binomial model provides a discrete-time framework suitable for this purpose, especially when the stock can only move to two possible prices in a given period. Here, the model's parameters—possible future stock prices, risk-free rate, and strike price—are critical for calculating the option's fair value.

Step 1: Determine the parameters of the binomial model

Up and Down Factors

The potential future stock prices are S_u = $65 and S_d = $35, with the current price S_0 = $50. The up (u) and down (d) factors are therefore:

  • u = S_u / S_0 = 65 / 50 = 1.3
  • d = S_d / S_0 = 35 / 50 = 0.7

Risk-Free Rate Adjustment with Daily Compounding

The annual risk-free rate is 10%. Over one year, with daily compounding (assuming 365 days), the accumulation factor is:

r_{\text{daily}} = (1 + 0.10)^{1/365} - 1

This translates to:

r_{\text{daily}} = e^{\frac{\ln(1 + 0.10)}{365}} - 1 ≈ e^{0.000263} - 1 ≈ 0.000263

Thus, the daily risk-free rate is approximately 0.0263%. The cumulative growth factor over one year is:

R = (1 + r_{\text{daily}})^{365} ≈ e^{\ln(1 + 0.10)} = 1.10

which confirms consistent with continuous compounding. For discounting purposes, the present-value factor is:

DF = 1 / (1 + r_{\text{daily}})^{365} ≈ 1 / 1.10 = 0.9091

Step 2: Calculate the risk-neutral probability

The risk-neutral probability (p) is the likelihood that under the risk-neutral measure, the stock will go up. It is given by:

p = (R - d) / (u - d) = (1.10 - 0.7) / (1.3 - 0.7) = 0.4 / 0.6 = 2/3 ≈ 0.6667

Step 3: Determine the possible payoff of the call option at expiration

The strike price (K) is $55.

  • If stock price rises to $65: Payoff = max(0, 65 - 55) = $10
  • If stock price falls to $35: Payoff = max(0, 35 - 55) = $0

Step 4: Calculate the expected value of the option under risk-neutral probability

The expected payoff (E) is:

E = p payoff_up + (1 - p) payoff_down = (2/3) $10 + (1/3) $0 = $20/3 ≈ $6.6667

Step 5: Discount to present value

The current fair value of the call option is:

Call value = Discounted expected payoff = E DF ≈ $6.6667 0.9091 ≈ $6.06

Conclusion

Using the binomial model adjusted for daily compounding, the fair price of the call option with a strike price of $55, expiring in one year, is approximately $6.06.

Additional Consideration: Validity of the Model

This valuation assumes no arbitrage opportunities, frictionless markets, and known probabilities under the risk-neutral measure. The binomial model's flexibility allows for such discrete-time approximations, and daily compounding provides a more precise estimate concerning real market conditions.

Additional Question: Calculating the Stock Price and Market Value of the Option

Given the information: exercise value of $27, exercise price of $20, and time value of $8, the total current market value of the option is the sum of its intrinsic value and time value: $27. The intrinsic value here corresponds to how much the stock is above the exercise price, indicating that the market value of the stock is at least the sum of the exercise value plus the difference in market and intrinsic value measures.

Specifically, if the exercise value is $27, and the exercise price is $20, then the market value of the option (call) is:

Market value = intrinsic value + time value = $27 + $8 = $35

The stock price, considering the intrinsic value, would be at least equal to the exercise value level, which is $27 in this context, but the precise stock price can be approximated based on the exercise price and market premium, in this case approximately $27 or above. Alternatively, if the current market value of the stock is related to the option’s value, then the stock’s market price is approximately $27.

Therefore, the market value of the option is approximately $35, and the stock price is approximately $27, according to the provided data.

References

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  • Hull, J. C. (2021). Risk Management and Financial Institutions. 6th Edition. Wiley.
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  • Jarrow, R. A., & Turnbull, S. M. (2000). Price of Default Risk and the Credit Spread Puzzle. The Journal of Financial Economics.
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