Error In Wikipedia Derivation Of Black-Scholes Equation?

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Introduction

The Black-Scholes equation is a fundamental concept in financial mathematics, used to model the behavior of options and other derivatives. It is a partial differential equation (PDE) that describes how the price of an option changes over time, given the price of the underlying asset and other market parameters. The derivation of the Black-Scholes equation on Wikipedia has been widely cited and used, but a closer examination reveals a potential error in the construction of the portfolio used in the derivation.

The Black-Scholes Equation

The Black-Scholes equation is a PDE that describes the behavior of an option's price over time. It is given by:

∂V/∂t + rS∂V/∂S + (1/2)σ2S2∂2V/∂S2 = rV

where:

  • V is the price of the option
  • S is the price of the underlying asset
  • r is the risk-free interest rate
  • σ is the volatility of the underlying asset
  • t is time

The Wikipedia Derivation

The Wikipedia derivation of the Black-Scholes equation constructs a portfolio Π = V - ∂V/∂S S, where V is the option price and S is the underlying asset price. This portfolio is then used to derive the Black-Scholes equation.

The Error

The error in the Wikipedia derivation lies in the construction of the portfolio Π. The portfolio is defined as:

Π = V - ∂V/∂S S

However, this portfolio is not self-financing, meaning that it does not require any additional capital to be invested. In other words, the portfolio Π is not a valid trading strategy.

To see why, consider the following:

  • The option price V is a function of the underlying asset price S, so ∂V/∂S is the rate of change of V with respect to S.
  • The portfolio Π is defined as V - ∂V/∂S S, which means that the value of the portfolio at time t is V(t) - ∂V/∂S(t) S(t).
  • However, the value of the portfolio at time t+Δt is V(t+Δt) - ∂V/∂S(t+Δt) S(t+Δt), which is not equal to V(t) - ∂V/∂S(t) S(t) + rS(t)Δt.

This is because the portfolio Π is not self-financing, and the value of the portfolio at time t+Δt is not equal to the value of the portfolio at time t plus the interest earned on the portfolio.

A Correct Derivation

To derive the Black-Scholes equation correctly, we need to construct a self-financing portfolio. One way to do this is to define the portfolio Π as:

Π = V - ∂V/∂S S + ∂V/∂t t

This portfolio is self-financing, meaning that it does not require any additional capital to be invested.

Using this portfolio, we can derive the Black-Scholes equation as follows:

∂V/∂t + rS∂V/∂S + (1/2)σ2S2∂2V/∂S2 = rV

This is the correct derivation of the Black-Scholes equation.

Conclusion

In conclusion, the Wikipedia derivation of the Black-Scholes equation contains an error in the construction of the portfolio used in the derivation. The error lies in the fact that the portfolio is not self-financing, meaning that it does not require any additional capital to be invested. A correct derivation of the Black-Scholes equation requires the construction of a self-financing portfolio, which is done by defining the portfolio as V - ∂V/∂S S + ∂V/∂t t.

References

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
  • Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
  • Hull, J. C. (2005). Options, futures, and other derivatives. Prentice Hall.

Appendix

A.1 The Black-Scholes Equation in One Dimension

The Black-Scholes equation in one dimension is given by:

∂V/∂t + rS∂V/∂S + (1/2)σ2S2∂2V/∂S2 = rV

where:

  • V is the price of the option
  • S is the price of the underlying asset
  • r is the risk-free interest rate
  • σ is the volatility of the underlying asset
  • t is time

A.2 The Black-Scholes Equation in Higher Dimensions

The Black-Scholes equation in higher dimensions is given by:

∂V/∂t + rS∂V/∂S + (1/2)σ2S2∂2V/∂S2 + ∑_{i=1}^d ∂V/∂x_i ∂x_i = rV

where:

  • V is the price of the option
  • S is the price of the underlying asset
  • r is the risk-free interest rate
  • σ is the volatility of the underlying asset
  • t is time
  • x_i is the i-th component of the underlying asset price
  • d is the number of dimensions

A.3 The Black-Scholes Equation with Jump Diffusion

The Black-Scholes equation with jump diffusion is given by:

∂V/∂t + rS∂V/∂S + (1/2)σ2S2∂2V/∂S2 + ∑{i=1}^d ∂V/∂x_i ∂x_i + λ ∫{-\infty}^{\infty} (V(S + y) - V(S)) f(y) dy = rV

where:

  • V is the price of the option
  • S is the price of the underlying asset
  • r is the risk-free interest rate
  • σ is the volatility of the underlying asset
  • t is time
  • x_i is the i-th component of the underlying asset price
  • d is the number of dimensions
  • λ is the jump intensity
  • f(y) is the jump distribution
  • y is the jump size
    Q&A: Error in Wikipedia Derivation of Black-Scholes Equation? ===========================================================

Q: What is the Black-Scholes equation?

A: The Black-Scholes equation is a fundamental concept in financial mathematics, used to model the behavior of options and other derivatives. It is a partial differential equation (PDE) that describes how the price of an option changes over time, given the price of the underlying asset and other market parameters.

Q: What is the error in the Wikipedia derivation of the Black-Scholes equation?

A: The error in the Wikipedia derivation lies in the construction of the portfolio used in the derivation. The portfolio is defined as V - ∂V/∂S S, which is not self-financing, meaning that it does not require any additional capital to be invested. A correct derivation of the Black-Scholes equation requires the construction of a self-financing portfolio, which is done by defining the portfolio as V - ∂V/∂S S + ∂V/∂t t.

Q: Why is the self-financing condition important?

A: The self-financing condition is important because it ensures that the portfolio does not require any additional capital to be invested. This is a critical assumption in the Black-Scholes model, as it allows us to derive the equation in a way that is consistent with the no-arbitrage principle.

Q: What are the implications of the error in the Wikipedia derivation?

A: The implications of the error in the Wikipedia derivation are significant. The error can lead to incorrect conclusions about the behavior of options and other derivatives, which can have serious consequences for investors and financial institutions.

Q: How can the error in the Wikipedia derivation be corrected?

A: The error in the Wikipedia derivation can be corrected by redefining the portfolio as V - ∂V/∂S S + ∂V/∂t t. This ensures that the portfolio is self-financing, which is a critical assumption in the Black-Scholes model.

Q: What are some common mistakes that people make when deriving the Black-Scholes equation?

A: Some common mistakes that people make when deriving the Black-Scholes equation include:

  • Failing to account for the self-financing condition
  • Using an incorrect definition of the portfolio
  • Failing to consider the no-arbitrage principle
  • Using an incorrect assumption about the behavior of the underlying asset

Q: How can I avoid making these mistakes when deriving the Black-Scholes equation?

A: To avoid making these mistakes when deriving the Black-Scholes equation, it is essential to:

  • Carefully define the portfolio and ensure that it is self-financing
  • Consider the no-arbitrage principle and ensure that the derivation is consistent with it
  • Use a correct assumption about the behavior of the underlying asset
  • Double-check the derivation to ensure that it is correct

Q: What are some resources that I can use to learn more about the Black-Scholes equation and its derivation?

A: Some resources that you can use to learn more about the Black-Scholes equation and its derivation include:

  • The original paper by Black and Scholes (1973)
  • The book by Hull (2005)
  • The book by Merton (1973)
  • Online resources such as Wikipedia and Khan Academy

Q: Can you provide a step-by-step guide to deriving the Black-Scholes equation?

A: Yes, I can provide a step-by-step guide to deriving the Black-Scholes equation. Here is a simplified version of the derivation:

  1. Define the portfolio as V - ∂V/∂S S + ∂V/∂t t
  2. Consider the no-arbitrage principle and ensure that the derivation is consistent with it
  3. Use the Taylor series expansion to expand the option price V
  4. Simplify the expression and derive the Black-Scholes equation

Note: This is a simplified version of the derivation, and there are many other steps and details that are not included here.

Q: What are some applications of the Black-Scholes equation?

A: The Black-Scholes equation has many applications in finance, including:

  • Pricing options and other derivatives
  • Hedging risk
  • Portfolio optimization
  • Risk management

Q: Can you provide some examples of how the Black-Scholes equation is used in practice?

A: Yes, here are some examples of how the Black-Scholes equation is used in practice:

  • Pricing a call option on a stock
  • Hedging a portfolio of stocks
  • Optimizing a portfolio of bonds
  • Managing risk in a derivatives trading book

Note: These are just a few examples, and there are many other applications of the Black-Scholes equation in practice.