Forward and backward equations usually refer to the differential equations governing the transition probability density function for a stochastic process. They are diffusion equations and must therefore be solved in the appropriate direction in time, hence the names.

Example

An exchange rate is currently 1.88. What is the probability that it will be over 2 by this time next year? If you have a stochastic differential equation model for this exchange rate then this question can be answered using the equations for the transition probability density function.

Long answer

Let us suppose that we have a random variable y evolving according to a quite general, one-factor stochastic differential equation

Here A and B are both arbitrary functions of y and t.

Many common models can be written in this form, including the lognormal asset random walk, and common spot interest rate models.

The transition probability density function p(y, t; y',t)is the function of four variables defined by

This simply means the probability that the random variable y lies between a and b at time t in the future, given that it started out with value y at time t. You can think of y and t as being current or starting values with y' and t being future values.

The transition probability density function p(y, t; y', ^satisfies two equations, one involving derivatives with respect to the future state and time (y' and t') and called the forward equation, and the other involving derivatives with respect to the current state and time (y and t) and called the backward equation. These two equations are parabolic partial differential equations not dissimilar to the Black-Scholes equation.

The forward equation

Also known as the Fokker-Planck or forward Kolmogorov equation this is

This forward parabolic partial differential equation requires initial conditions at time t and to be solved for t > t.

Example An important example is that of the distribution of equity prices in the future. If we have the random walk

then the forward equation becomes

A special solution of this representing a variable that begins with certainty with value S at time t is

This is plotted as a function of both S' and t in Figure 2.7.

Figure 2.7: The probability density function for the lognormal random walk evolving through time.

The backward equation

Also known as the backward Kolmogorov equation this is

This must be solved backwards in t with specified final data. For example, if we wish to calculate the expected value of some function F(S)at time T we must solve this equation for the function p(S, t)with

Option prices

If we have the lognormal random walk for S, as above, and we transform the dependent variable using a discount factor according to

then the backward equation for p becomes an equation for V which is identical to the Black-Scholes partial differential equation. Identical but for one subtlety, the equation contains a x where Black-Scholes contains r. We can conclude that the fair value of an option is the present value of the expected payoff at expiration under a risk-neutral random walk for the underlying. Risk neutral here means replace // with r.

References and Further Reading

Feller, W 1950 Probability Theory and Its Applications. John Wiley & Sons Inc.

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

What is the Black—Scholes Equation?

Short answer

The Black-Scholes equation is a differential equation for the value of an option as a function of the underlying asset and time.

Example

The basic equation is

where V (S, t) is the option value as a function of asset price S and time t.

There have been many extensions to this model, some people call them 'improvements.' But these extensions are all trivial compared with the breakthrough in modelling that was the original equation.

Long answer

Facts about the Black-Scholes equation:

• The equation follows from certain assumptions and from a mathematical and financial argument that involves hedging.

• The equation is linear and homogeneous (we say 'there is no right-hand side,' i.e. no non-V terms) so that you can value a portfolio of derivatives by summing the values of the individual contracts.

• It is a partial differential equation because it has more than one independent variable, here S and t.

• It is of parabolic type, meaning that one of the variables, t, only has a first-derivative term, and the other S has a second-derivative term.

• It is of backward type, meaning that you specify a final condition representing the option payoff at expiry and then solve backwards in time to get the option value now. You can tell it's backward by looking at the sign of the t-derivative term and the second S-derivative term, when on the same side of the equals sign they are both the same sign. If they were of opposite signs then it would be a forward equation.

• The equation is an example of a diffusion equation or heat equation. Such equations have been around for nearly two hundred years and have been used to model all sorts of physical phenomena.

• The equation requires specification of two parameters, the risk-free interest rate and the asset volatility. The interest rate is easy enough to measure, and the option value isn't so sensitive to it anyway. But the volatility is another matter, rather harder to forecast accurately.

• Because the main uncertainty in the equation is the volatility one sometimes thinks of the equation less as a valuation tool and more as a way of understanding the relationship between options and volatility.

• The equation is easy to solve numerically, by finite-difference or Monte Carlo methods, for example.

• The equation can be generalized to allow for dividends, other payoffs, stochastic volatility, jumping stock prices, etc.

And then there are the Black-Scholes formula which are solutions of the equation in special cases, such as for calls and puts.

The equation contains four terms:

d V

— = time decay, how much the option value changes by dt

if the stock price doesn't change

1 2 2d2v

2° 9S2" = convex 'ty term, how much a hedged position makes on average from stock moves

dV

rS—— = drift term allowing for the growth in the stock at the dS

risk-free rate

and

—rV = the discounting term, since the payoff is received at expiration but you are valuing the option now.

References and Further Reading

Black, F & Scholes, M 1973 The pricing of options and corporate liabilities. Journal of Political Economy 81 637-659

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