"A Deep Dive into the Basics of Real Options Analysis -Part 2 - Valuation of Real Options through Black-Scholes Option Pricing Model (excel /python(

Refer Earlier article : "A Deep Dive into the Basics of Real Options Analysis - Key Concepts for Strategic Decision-Making" - Part1 https://www.dhirubhai.net/feed/update/urn:li:activity:7278957287656857600/

Valuation of Real Options through Adjusted Black-Scholes Option Pricing Model

The Black-Scholes model, originally developed for financial options, can be adapted to value real options by mapping its parameters to real-world business variables:

1. Stock Price (S) → Present Value of Expected Cash Flows
2. Strike Price (K) → Investment Cost
3. Time to Expiration (t) → Time Until Opportunity Expires
4. Risk-free Rate (r) → Risk-free Interest Rate (Continuous Compounding)
5. Volatility (σ) → Uncertainty of Project Value

The fundamental Black-Scholes formula for a call option becomes:

C = S N(d1) - K × e^(-rt) * N(d2)

Where: d1 = [ln(S/K) + (r + σ2/2)*t] / (σ*√t) ; d2 = d1 - σ*√t

Advantages of Using Black-Scholes for Real Options

1. Quantitative Precision The model provides a systematic framework for valuing flexibility in business decisions, transforming qualitative strategic thinking into quantitative analysis.
2. Time Value Recognition It explicitly accounts for the time value of managerial flexibility, recognizing that longer decision horizons typically increase option value.
3. Risk Integration The model incorporates both market risk and project-specific risk through its volatility parameter.

Challenges and Limitations

Despite its utility, applying Black-Scholes to real options presents several challenges:

1. Parameter Estimation Difficulty in estimating volatility for non-traded assets Uncertainty in determining the appropriate time horizon Challenges in calculating present values of expected cash flows
2. Model Assumptions The assumption of geometric Brownian motion may not hold for real assets The model assumes European-style options, while real options often have American-style exercise features Perfect markets assumption may not apply to real business situations
3. Complexity of Real-World Applications Multiple interacting options Path-dependent decisions Competitive reactions

Implementation Framework

To effectively implement real options valuation using Black-Scholes:

1. Identify and Classify Options Map out all available strategic options and their characteristics.
2. Gather Input Data Collect necessary information for parameter estimation, including market data and project-specific information.
3. Adjust for Real-World Conditions Modify standard Black-Scholes assumptions to better reflect business realities.
4. Calculate and Interpret Apply the model and analyze results in the context of broader strategic decision-making.

Applications Across Industries

Real options valuation using Black-Scholes has proven particularly valuable in:

• Natural Resource Exploration: Valuing mineral rights and timing of extraction
• Real Estate Development: Analyzing land development timing and phasing
• Research & Development: Evaluating staged investment in new technologies
• Infrastructure Projects: Assessing expansion and modification options
• Manufacturing: Valuing production flexibility and capacity decisions

Best Practices for Implementation

1. Rigorous Data Collection Ensure robust processes for gathering and validating input data.
2. Sensitivity Analysis Conduct thorough sensitivity analyses to understand how parameter changes affect option values.
3. Regular Review and Update Periodically reassess assumptions and update valuations as new information becomes available.
4. Integration with Traditional Methods Use real options valuation as a complement to, not replacement for, traditional DCF analysis.

Evaluating Expansion Opportunities with Black-Scholes Option Pricing Model (through Excel)

Background: A manufacturing firm is exploring the construction of a new factory to enhance its production capabilities. The factory comes with significant upfront costs but offers the potential for future expansion based on market success. The management team is tasked with determining whether the factory project is financially viable, factoring in both the initial investment and the optionality of future expansion.

Scenario Details: The proposed factory requires an upfront investment of $300,000. Once operational, the factory is expected to generate $35,000 in annual cash flows for the next 20 years, with the first cash flow starting one year from today.

If the product proves successful within the first three years, the firm can opt to expand the facility at an additional cost of $150,000. The expansion is projected to produce $15,000 in annual cash flows for 17 years, with the first cash flow materializing four years from today (i.e., one year after the expansion decision is made).

Financial Inputs:(Assumptions)

• The standard deviation of returns for the factory expansion is 30%, which is higher than the 25% standard deviation for the initial factory.
• Treasury strip returns by maturity:1-year: 4.0%,3-year: 4.2%,4-year: 4.4%,20-year: 5.5%.
• The required rate of return for both the factory and expansion is 10.0% per year.

Key Question: To make an informed decision, the firm seeks to quantify the value of the expansion option embedded in the project using the Black-Scholes Option Pricing Model (BSOPM). The team must evaluate whether the factory should be built, factoring in the potential value of expansion.

Considerations for Analysis:

1. The expansion represents a real option with uncertain outcomes based on future market conditions.
2. The valuation will involve modelling the option to expand as a call option, where the option to invest in the expansion depends on future success.
3. The inputs to the Black-Scholes Model include the expansion costs, expected cash flows, volatility of returns, and discount rates derived from treasury strip yields.

Discussion Point:By valuing the optionality inherent in the project, the firm can determine whether the additional flexibility of expansion justifies the initial investment in the factory. Should the calculated value of the expansion option significantly enhance the overall project value, the factory project may present a compelling investment opportunity.

Evaluate Flexibility of Valuation of Expansion through Adjusted Black Schole Option Pricing Model

Evaluating Expansion Opportunities with Black-Scholes Option Pricing Model (through Python)

import numpy as np
from scipy.stats import norm

def black_scholes_call(S: float, K: float, T: float, r: float, sigma: float) -> float:
    """
    Calculate call option value using Black-Scholes formula.

    Parameters:
        S (float): Present value of additional cash flows from expansion
        K (float): Exercise price (future cost of expansion, not pre-discounted)
        T (float): Time to expiration in years
        r (float): Risk-free interest rate (as a decimal)
        sigma (float): Volatility of underlying asset (as a decimal)

    Returns:
        float: Value of the call option.
    """
    # Calculate d1 and d2
    d1 = (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)

    # Calculate call option value
    call_value = S * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)
    return call_value

def calculate_present_value(cash_flows: float, years: int, rate: float) -> float:
    """
    Calculate the present value of an annuity.

    Parameters:
        cash_flows (float): Annual cash flow amount
        years (int): Number of years for the annuity
        rate (float): Discount rate (as a decimal)

    Returns:
        float: Present value of the annuity.
    """
    pv = cash_flows * (1 - (1 + rate) ** -years) / rate
    return pv

def evaluate_project_with_option():
    # Inputs
    cost_of_factory = 300_000  # Upfront cost of building the factory
    annual_cash_flows = 35_000  # Annual cash inflows
    project_years = 20  # Number of years of project life
    rate_of_return = 0.10  # Expected rate of return (10%)

    # Expansion inputs
    expansion_cost = 150_000  # Additional investment for expansion (at T=3)
    expansion_cash_flows = 15_000  # Cash inflows from expansion
    expansion_years = 17  # Number of years of expansion cash flows
    time_to_expansion = 3  # Time to expansion in years
    volatility = 0.25  # Volatility (25%)
    risk_free_rate = 0.042  # Risk-free rate for 3 years (4.2%)

    # Static NPV calculation (without option)
    pv_cash_flows = calculate_present_value(annual_cash_flows, project_years, rate_of_return)
    static_npv = pv_cash_flows - cost_of_factory

    # Present value of expansion cash flows at T=0
    pv_expansion_cash_flows_t3 = calculate_present_value(expansion_cash_flows, expansion_years, rate_of_return)
    pv_expansion_cash_flows_t0 = pv_expansion_cash_flows_t3 / (1 + rate_of_return) ** time_to_expansion

    # Option value calculation using BSOPM
    option_value = black_scholes_call(
        S=pv_expansion_cash_flows_t0,
        K=expansion_cost,  # Pass K as it is; discounting is handled inside BSOPM
        T=time_to_expansion,
        r=risk_free_rate,
        sigma=volatility
    )

    # Expanded NPV calculation
    expanded_npv = static_npv + option_value

    # Results
    results = {
        "PV of Cash Flows (t=0)": pv_cash_flows,
        "Static NPV (without option)": static_npv,
        "PV of Expansion Cash Flows (t=3)": pv_expansion_cash_flows_t3,
        "PV of Expansion Cash Flows (t=0)": pv_expansion_cash_flows_t0,
        "Real Option Value": option_value,
        "Expanded NPV": expanded_npv
    }

    return results

# Run the evaluation
results = evaluate_project_with_option()

# Print results
print("\nReal Options Analysis - Expansion Option")
print("=" * 50)
for key, value in results.items():
    print(f"{key}: ?{value:,.2f}")

# Decision recommendation
print("\nDecision Analysis:")
if results["Expanded NPV"] > 0:
    if results["Static NPV (without option)"] < 0:
        print("PROCEED WITH CAUTION: While the static NPV is negative, "
              "the expansion option adds sufficient value to make the project viable.")
    else:
        print("PROCEED: Both static NPV and expanded NPV are positive. "
              "The expansion option adds significant value.")
else:
    print("DO NOT PROCEED: Even with the expansion option, "
          "the project does not generate sufficient value to justify the investment.")
        

Output from the Above Code :

Real Options Analysis - Expansion Option ================================================== PV of Cash Flows (t=0): ?297,974.73 Static NPV (without option): ?-2,025.27 PV of Expansion Cash Flows (t=3): ?120,323.30 PV of Expansion Cash Flows (t=0): ?90,400.68 Real Option Value: ?4,875.61 Expanded NPV: ?2,850.34

Decision Analysis: PROCEED WITH CAUTION: While the static NPV is negative, the expansion option adds sufficient value to make the project viable.

Evaluating Abandonment Option with Black-Scholes Option Pricing Model (through Excel)

Case Study: Evaluating Real Options for a Strategic Project

Scenario

A company is evaluating a new investment project that requires an initial outlay of $120,000. The project is expected to generate annual cash flows of $15,000 for 10 years, starting one year from today. The firm requires a 12% annual return on this project, and the standard deviation of returns is 40% over the next 2 years and 30% over its 10-year life. If the project fails to meet expectations, the facility can be sold for $60,000 at any point within the next 2 years.

Treasury Strip Yields

The annual yields on Treasury strips are:

• 1-year: 3.5%
• 2-year: 4.0%
• 10-year: 5.2%

Objective

Using real options analysis, determine whether the firm should undertake the project, considering the potential to abandon the project within the first 2 years.

Analysis Steps

1. Calculate the Static Net Present Value (NPV): Use the expected cash flows, initial investment, and the firm's required return to compute the static NPV of the project.
2. Valuation of the Option to Abandon: The option to sell the facility provides downside risk protection. Using the 2-year standard deviation, risk-free rate (Treasury strip yield), and the sale value of $60,000, apply the Black-Scholes Option Pricing Model to value this abandonment option.
3. Extended NPV Analysis: Combine the static NPV and the value of the abandonment option to arrive at the Extended NPV. If the Extended NPV is positive, the project becomes a viable investment.
4. Compare Outcomes: Assess the results under various scenarios:

Evaluating Abandonment Option with Black-Scholes Option Pricing Model (through Python)

import math
from scipy.stats import norm

def calculate_npv(initial_cost, annual_cash_flow, years, required_return):
    """Calculate the basic NPV of the project"""
    npv = -initial_cost
    for t in range(1, years + 1):
        npv += annual_cash_flow / (1 + required_return) ** t
    return npv

def black_scholes_put(S, K, T, r, sigma):
    """Calculate put option value using Black-Scholes model"""
    d1 = (math.log(S/K) + (r + sigma**2/2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)

    put_value = K * math.exp(-r * T) * norm.cdf(-d2) - S * norm.cdf(-d1)
    return put_value

# Project parameters
initial_cost = 120000
annual_cash_flow = 15000
years = 10
required_return = 0.12
salvage_value = 60000
early_sigma = 0.40  # Volatility for first 3 years
option_years = 2
rf_3year = 0.04  # 2-year risk-free rate

# Calculate basic NPV
basic_npv = calculate_npv(initial_cost, annual_cash_flow, years, required_return)

# Calculate present value of expected cash flows
pv_cash_flows = sum([annual_cash_flow / (1 + required_return) ** t for t in range(1, years + 1)])

# Calculate abandonment option value using Black-Scholes put option formula
put_option_value = black_scholes_put(
    S=pv_cash_flows,  # Present value of expected cash flows
    K=salvage_value,  # Strike price (salvage value)
    T=option_years,   # Time to expiration (2 years)
    r=rf_3year,       # Risk-free rate
    sigma=early_sigma # Volatility
)

# Calculate strategic NPV (basic NPV + option value)
strategic_npv = basic_npv + put_option_value

# Print results
results = f"""
Real Options Analysis Results:
-----------------------------
Basic NPV: ${basic_npv:,.2f}
Abandonment Option Value: ${put_option_value:,.2f}
Strategic NPV: ${strategic_npv:,.2f}
"""

print(results)        

Real Options Analysis Results: -----------------------------

Basic NPV: $-35,246.65

Abandonment Option Value: $4,950.49

Strategic NPV: $-30,296.16

Limitations of Black-Scholes for Real Options

1. Project Volatility is Not Constant Over Time

The application of Black-Scholes to real options valuation represents a significant advancement in strategic investment analysis. While the approach presents certain challenges, its ability to quantify managerial flexibility and strategic options makes it an invaluable tool for modern business decision-making. Success in implementation requires careful attention to parameter estimation, recognition of model limitations, and integration with broader strategic analysis frameworks.

The continued evolution of real options valuation techniques, including adaptations of the Black-Scholes model, promises to further enhance our ability to make informed strategic investment decisions in an increasingly complex business environment.

2. No Definitive Expiration Date for the Option

• BSOPM Assumption: The option has a fixed expiration date (TTT).
• Reality in Real Options: Real options, such as investment or expansion decisions, often have flexible timelines. Decision-makers might delay the decision indefinitely if conditions are unfavorable.

Impact: A lack of definitive expiration invalidates the finite-time framework of BSOPM.

3. Both Asset Value and Strike Price Behave Stochastically

• BSOPM Assumption: The underlying asset value (SSS) is stochastic, but the strike price (KKK) is deterministic.
• Reality in Real Options: Both the project value (asset value) and costs (strike price) fluctuate due to market dynamics, input cost variations, or changes in technology.

Impact: BSOPM cannot handle stochastic strike prices, leading to inaccurate valuations.

4. Returns Are Not Normally Distributed

• BSOPM Assumption: Asset returns follow a normal distribution.
• Reality in Real Options: Returns in real-world scenarios often have fat tails (kurtosis) or skewness, reflecting extreme events or asymmetry in outcomes.

Impact: BSOPM's reliance on normal distribution fails to capture these non-standard behaviors.

5. The Random Walk of Real Assets is Not Symmetric; There Are Jumps

• BSOPM Assumption: Asset prices follow a continuous and symmetric random walk (geometric Brownian motion).
• Reality in Real Options: Real assets exhibit "jumps" due to sudden market shocks, regulatory changes, or innovations.

Impact: Jump diffusion or other advanced stochastic processes would better capture these dynamics than BSOPM.

Alternative Methods

Given these limitations, alternative models are often used for valuing real options:

1. Binomial Lattice Models: Handle changing volatilities and stochastic cash flows.
2. Monte Carlo Simulations: Capture complex stochastic processes, including jumps and non-normal distributions.
3. Stochastic Dynamic Programming: Useful for options with indefinite expiration.
4. Modified BSOPM: Adjusted to account for some real-world complexities like varying volatility or stochastic strike prices.

Conclusion

While the Black-Scholes model provides a foundational approach to valuing options, it falls short when applied to real options due to its rigid assumptions. Adopting more flexible and realistic models is essential for accurate valuation in real-world scenarios. Let me know if you'd like further insights or alternative model explanations

Part 1 : Refer

"A Deep Dive into the Basics of Real Options Analysis - Key Concepts for Strategic Decision-Making" - Part1 https://www.dhirubhai.net/feed/update/urn:li:activity:7278957287656857600/

Part 3:

Continued Part 3 : Valuation of Real Options - using Binomial Option Pricing Model approach