Lecture Notes on Option Pricing

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• Usual rules apply for quizzes:
  • Quizzes can be picked up once graded.
  • Quiz 2 and Quiz 1 are still available for pickup.
  • Quiz 3 will be available soon.

Overview of Option Pricing

• Option pricing is often seen as mysterious.
• Two main concepts:
  • Replication: Creating something identical to the option using the underlying asset.
  • Arbitrage: Ensuring identical items have the same price.

Models of Option Pricing

• Binomial Pricing Model:

  • Stock prices can jump to one of two points over time.
  • Example: Stock price starting at $50 can go to $70 or $35.
  • Value a call option with a strike price of $40.

• Arbitrage Pricing:

  • Create a replicating portfolio with the same cash flows as the option by borrowing and buying shares.
  • Solve simultaneous equations (e.g., Delta and borrowing amount) to replicate cash flows.
  • Calculate the cost of the replicating portfolio.

Binomial Model to Black-Scholes Model

• Binomial model becomes impractical with many possible price changes.
• Black-Scholes Model:
  • Assumes continuous price changes, leading to a normal distribution.
  • Uses five inputs: S (current stock price), K (strike price), R (risk-free rate), T (time to expiration), and sigma (volatility).
  • Lacks dividend consideration because it was designed to value dividend-protected options.

Understanding the Black-Scholes Model

• Key Terms:

  • N(d1), N(d2): Areas under the normal distribution.
  • Delta: Number of shares to replicate the option.
  • Exercise price adjusted for present value.

• Limitations:

  • Assumes European options (exercised only at expiration).
  • Continuous price changes are unrealistic.

Adjustments for Dividends

• Dividends reduce the value of the underlying asset.
• Adjust Black-Scholes to include dividend yield.

Use of Decision Trees in Option Pricing

• Decision trees offer an alternative to complex option pricing models.
• Useful for intuitive understanding and presenting to non-finance professionals.

Real Options Analysis

Key Tests for Real Options

1. Existence of an Option: Confirm the presence of an option.
2. Economic Value: Check if the option has exclusivity and thus value.
3. Applicability of Pricing Models: Determine if an option pricing model can be used.

Types of Real Options

1. Option to Delay: Rights to a project that may become viable.
2. Option to Expand: Initial project leads to further opportunities.
3. Option to Abandon: Ability to stop a project if it underperforms.

Example: Valuing a Patent

• Underlying Asset: Product from the patent.

• Inputs for Valuation:

  • Present value of cash flows as the underlying asset's value.
  • Cost of developing as the strike price.
  • Uncertainty as sigma (volatility).
  • Remaining life of the patent as the option life.

• Case Study: Biogen

  • Valued the drug Avonex using the Black-Scholes model.
  • Considered existing products and R&D as separate entities.
  • Developed the drug despite option value, likely due to competitive pressures.

Conclusion

• Real options are complex but essential for assessing potential future projects.
• The Black-Scholes model provides a framework but requires understanding of assumptions and limitations.