- The Assumptions Behind the Black-Scholes Equation
- Self-financing, Replicating Hedging Strategies
- Risk-Neutral Valuation
- Ito's Lemma
- The Black-Scholes Equation
- Solution of the Black-Scholes Equation

The pioneering work of Black and Scholes[7] started the serious study of the theory of option pricing. All further advances in this field have been extensions and refinements of the original idea expressed in that paper. Considering its immense importance in the field of option pricing, we will study the theory behind the Black-Scholes equation in some detail. We will also introduce the quantum mechanical formulation of the Black-Scholes theory which will be of great use in the following chapters as we consider stochastic volatility.

- The efficient market hypothesis is assumed to be satisfied. In other words, the markets are assumed to be liquid, have price-continuity, be fair and provide all players with equal access to available information. This implies that zero transaction costs are assumed in the Black-Scholes analysis.
- It is assumed that the underlying security is perfectly divisible and that short selling with full use of proceeds is possible.
- Constant risk-free interest rates are assumed. In other words, we assume that there exists a risk-free security which returns $1 at time when $ is invested at time .
- As we will see later, the Black-Scholes analysis requires continuous trading. This is, of course, not possible in practice as the more frequently one trades, the larger the transaction costs.
- The prinicple of no arbitrage is assumed to be satisfied.
- The price of the underlying security is assumed to follow a geometric Brownian process of the form where is white noise.

Considering the above, it may seem strange that we pay so much importance to the Black-Scholes theory. There are several reasons for this. The most important one is that the concepts behind the Black-Scholes analysis provide the framework for thinking about option pricing. All the research in option pricing since the Black-Scholes analysis has been done either to extend it or to generalize it. Another important reason for studying the Black-Scholes theory is that the financial world uses it as a standard. In fact, traders quote Black-Scholes volatility to each other, not the actual price of the options! Further, Black-Scholes prices still give very good approximations to the prices of options.

For example, suppose we have a 2-year coupon bond with payoffs of $1 every half a year and $11 at the maturity. We can replicate this bond using the following portfolio : one zero-coupon bond (we assume that all zero-coupon bonds have a payoff of $1 at maturity) each of maturity 0.5 year, 1 year, 1.5 years and 11 zero-coupon bonds of maturity 2 years. As each of the zero-coupon bonds mature, they replicate the payoff of the coupon-bearing bond. The portfolio in this example is perfectly hedged since the underlying price of the bonds is known at maturity. The present value of the bonds is not important for our purposes since our aim is only to replicate the payoff of the coupon-bearing bond.

By the principle of no arbitrage, any portfolio that can replicate the payoff of an option (or, for that matter, any derivative) must have the same value as that of the option. In the above example, the price of the coupon bond must be the same as the combination of zero-coupon bonds. The Black-Scholes analysis makes a portfolio which replicates the payoff of an option, hence solving the valuation problem.

We derive the put-call parity relation (1.4) using this principle. To do so, use the fact that the sum of the payoffs of a long call and a short put option with the same strike price and maturity (and, of course, on the same underlying security)is given by . Hence, the value of the call and put option is given by , the last equality coming from risk-neutral valuation. This gives us or which is the same as (1.4).

Ito's lemma states that if a variable follows a stochastic
process of the form

(43) |

We now present a non-rigorous definition of Ito's lemma using the
Taylor series formula. For a smooth function , the normal
Taylor series expansion goes as

(45) |

(46) |

(47) |

where (the average growth rate of the underlying security) and (the volatility) are constants. Using Ito's lemma, we see that

(49) |

(50) |

The initial (or, in finance, usually final) conditions determine the kind of derivative that we are pricing. For a call option, the final condition we have to use is . We see that the principle of risk-neutral valuation is clearly satisfied in this case since the Black-Scholes equation is independent of , the expected rate of growth of the underlying security price.

It is important to note that the portfolio represents a self-financing, replicating, hedging strategy. It replicates a risk-free investment and it is hedged since it has no stochastic component.

(52) |

(53) |

which shows that follows a lognormal distribution.

We are now almost done. The principle of risk-neutral valuation
implies that the present value of the option is the expected final
value
of the option discounted at the risk-free
interest rate. So, we have

where has been replaced by in accordance with the principle of risk-neutral valuation. We can easily verify that this solution satisfies the principle of risk-neutral valuation by evaluating .

The value of the integral (3.13) can be found with a bit of
algebraic manipulation and is

(58) |

While the result looks very complicated, it has an intuitive
interpretation. Equation (3.15) can be written as

(59) |

We can also form a self-financing, replicating hedging strategy for the call option. The portfolio in this case is made up of units of the underlying security and zero-coupon bonds with the same maturity as the option. This approach is discussed in great detail in Chriss[1]. We do not consider this approach here as it is unintuitive since the result must be known in advance.

We can recast the Black-Scholes equation (3.9) into a
Schrödinger-like equation. We identify , the value of the option
as a wave function dependent on time and the price of the underlying
security. Then, the Schrödinger equation becomes

(61) |

(62) |

The price of the European call option is then given by

We now change the basis to the momentum basis where is
diagonal. The transformation from the ``position'' basis to the
``momentum'' basis is defined by

(64) |

The Hamiltonian in the momentum basis is given by

(65) |

where . After changing variables from to and noting that time runs backward here, we see that this distribution is the same as (3.14) as it should be. Hence, the solution is the same as the one using the risk-neutral valuation method as it should be.

While the advantages of the quantum mechanical method may not be very obvious here, they will become obvious when we consider stochastic volatility (i.e., when we allow to follow a stochastic process).