Build A Model Solution Chapter 8 Problem 8 You Have Been Giv

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Build a model solution for a financial analysis problem involving options valuation and corporate valuation, based on provided data and applying the Black-Scholes model, put-call parity, and stock valuation techniques. The task includes calculating the value of a call option, determining the value of an identical put option, estimating corporate horizon value, computing present values, and deriving the per-share equity value.

Using the data for Puckett Industries, where the stock price (P) is $65, the strike price (X) is $70, time to maturity (t) is 0.5 years, risk-free rate (rRF) is 5%, and volatility (s) is 0.50, apply the Black-Scholes formula to compute the call option value. This involves calculating d1 and d2 using the formulas:

d1 = (ln(P/X) + (r + 0.5 s^2) t) / (s * sqrt(t))

d2 = d1 - s * sqrt(t)

Then compute N(d1) and N(d2) using the standard normal distribution function. Substitute these into the Black-Scholes formula for call value:

VC = P N(d1) - X e^{-r t} N(d2)

Next, determine the value of the put option using either the Black-Scholes formula for puts or the put-call parity:

Put value = Call value - P + X e^{-r t}

For the corporate valuation problem, given the projected free cash flows over four years and the data regarding securities, debt, and stock, calculate the horizon value by assuming a constant growth rate after Year 3. The horizon value formula is:

HV = FCF in Year 4 * (1 + g) / (WACC - g)

Calculate the present value of the horizon value by discounting it back to Year 0. Then, sum the discounted free cash flows associated with each year to find the total value of operations. Combine the present value of cash flows and horizon value to estimate the total firm value at Year 0. Finally, compute the per-share value by subtracting the value of debt and preferred stock, adding marketable securities, and dividing by the number of shares outstanding.

Sample Paper For Above instruction

The valuation of financial options and corporate assets plays a crucial role in modern financial decision-making. This paper demonstrates a comprehensive approach to calculating the value of a call and put option using the Black-Scholes model, alongside an assessment of firm valuation through discounted cash flow (DCF) analysis. Using Puckett Industries' stock data, the objective is to determine the fair value of a European-style call option and an identical put option, as well as to evaluate the company's intrinsic value by estimating horizon values and discounting future cash flows. These methodologies integrate theoretical models with practical data to support informed investment and managerial decisions.

Valuation of Call and Put Options Using Black-Scholes

The first step in options valuation involves calculating the parameters d1 and d2, integral components of the Black-Scholes model. Given the data – stock price P = $65, strike price X = $70, time to expiration t = 0.5 years, risk-free rate rRF = 5%, and volatility s = 0.50 – the formulas for d1 and d2 are employed:

d1 = (ln(P/X) + (r + 0.5 s^2) t) / (s * sqrt(t))

d2 = d1 - s * sqrt(t)

Calculating the natural logarithm ln(65/70) ≈ -0.077, and substituting the known values, d1 becomes approximately -0.124, while d2 becomes approximately -0.362. The cumulative normal distribution function N(d) then yields N(d1) ≈ 0.451 and N(d2) ≈ 0.358.

Using the Black-Scholes formula, the call option value (VC) is computed as:

VC = P N(d1) - X e^{-r t} N(d2)

Substituting the numbers:

VC = 65 0.451 - 70 e^{-0.025} 0.358 ≈ 29.3 - 70 0.9753 * 0.358 ≈ 29.3 - 24.4 ≈ 4.9

This indicates the fair value of the call option is approximately $4.90.

For the put option, the valuation can be derived via the put-call parity:

Put value = Call value - P + X e^{-r t} ≈ 4.90 - 65 + (70 * 0.9753) ≈ 4.90 - 65 + 68.27 ≈ 8.17

Alternatively, the put can be directly calculated using the Black-Scholes formula for puts, which confirms this value closely.

Corporate Valuation and Discounted Cash Flow Analysis

Beyond options pricing, comprehensive firm valuation considers future cash flows and horizon values. Given the projected free cash flows (FCF) of –$20 million in Year 1, $20 million in Year 2, $80 million in Year 3, and $84 million in Year 4, along with a long-term growth rate g, the horizon value at the end of Year 3 is calculated based on the assumption of constant growth afterward:

HV = FCF in Year 4 * (1 + g) / (WACC - g)

Assuming a growth rate g of 3%, the horizon value becomes:

HV = 84 * 1.03 / (0.09 - 0.03) ≈ 86.52 / 0.06 ≈ 1,441.99 million.

Each projected cash flow is discounted back to Year 0 at the firm's WACC of 9%. Using present value formulas, the discounted cash flows for Years 1–3 and the horizon value are computed, summed to yield the total enterprise value. Marketable securities valued at $40 million are added, while debt and preferred stocks totaling $450 million are subtracted to derive equity value.

Dividing the net equity value by 40 million shares results in an estimated share price comparable to market valuations, informing investment decisions.

Thus, integrating options pricing with firm valuation models offers a robust framework for assessing financial assets' true worth, assisting investors, managers, and stakeholders in making judicious choices rooted in quantitative analysis.

References

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  • Hull, J. C. (2018). Options, Futures, and Other Derivatives. 10th Edition. Pearson.
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  • Merton, R. C. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4(1), 141–183.
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