Chapter 5
Black-Scholes Differential Equation
Abstract After deriving the Black-Scholes equation for a call option from the
requirement to make a portfolio risk-free, the equation is solved using a number
of variable substitutions, which transforms it into a diffusion equation. Using the latter’s Green’s function is then used to value European call options. The resemblance
of the solution found in this chapter to that in Chapter 4 stimulates the discussion of
martingale processes. In order to better understand the mechanics of using options
for hedging, a MATLAB simulation for the temporal evolution of stocks, options
and bank deposits is presented.
The Black-Scholes differential equation contributed significantly to the rapid expansion of the derivative market in the 1970s, because it enabled traders to set a fair price
on a large number of options and other financial products. Consequently, the trust
in these products increased and boosted their attractivity. Later, the Black-Scholes
equation was also blamed to have caused market crashes [1], because traders relied
on the existence of risk-free portfolios, which is a corner stone of the theory. But the
traders missed that some of the prerequisites were not valid any more. For example,
the unlimited liquidity, the availability of funds at the risk-free rate, is no longer true.
Moreover, the underlying stochastic dynamics is not necessarily Gaussian in times
of financial distress. In Chap. 9 we will address some of the criticism, but first let
us discuss the standard Black-Scholes theory and calculate a fair price for standard,
often called “vanilla” call and put options.
5.1 Derivation
Let us therefore consider the temporal variation of option prices and assume that the
option c(S, t) depends on the stock value S and on the time t. Thus c is a derivative
of the underlying asset S, which is a stochastic variable itself and obeys the Langevin
equation from (4.7). Thus, also c is a stochastic variable and we need to derive a
stochastic differential equation for it in order to analyze how it evolves in time. To
first order in the temporal increment dt, we write the Taylor expansion of c
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_5
47
Black-Scholes Differential Equation
Abstract After deriving the Black-Scholes equation for a call option from the
requirement to make a portfolio risk-free, the equation is solved using a number
of variable substitutions, which transforms it into a diffusion equation. Using the latter’s Green’s function is then used to value European call options. The resemblance
of the solution found in this chapter to that in Chapter 4 stimulates the discussion of
martingale processes. In order to better understand the mechanics of using options
for hedging, a MATLAB simulation for the temporal evolution of stocks, options
and bank deposits is presented.
The Black-Scholes differential equation contributed significantly to the rapid expansion of the derivative market in the 1970s, because it enabled traders to set a fair price
on a large number of options and other financial products. Consequently, the trust
in these products increased and boosted their attractivity. Later, the Black-Scholes
equation was also blamed to have caused market crashes [1], because traders relied
on the existence of risk-free portfolios, which is a corner stone of the theory. But the
traders missed that some of the prerequisites were not valid any more. For example,
the unlimited liquidity, the availability of funds at the risk-free rate, is no longer true.
Moreover, the underlying stochastic dynamics is not necessarily Gaussian in times
of financial distress. In Chap. 9 we will address some of the criticism, but first let
us discuss the standard Black-Scholes theory and calculate a fair price for standard,
often called “vanilla” call and put options.
5.1 Derivation
Let us therefore consider the temporal variation of option prices and assume that the
option c(S, t) depends on the stock value S and on the time t. Thus c is a derivative
of the underlying asset S, which is a stochastic variable itself and obeys the Langevin
equation from (4.7). Thus, also c is a stochastic variable and we need to derive a
stochastic differential equation for it in order to analyze how it evolves in time. To
first order in the temporal increment dt, we write the Taylor expansion of c
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Ziemann, Physics and Finance, Undergraduate Lecture Notes in Physics,
https://doi.org/10.1007/978-3-030-63643-2_5
47