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Continuous-Time Limit of the Binomial Model

Some of our twelve derivations lead to the Black-Scholes partial differential equation, and some to the idea of the option value as the present value of the option payoff under a risk-neutral random walk. The following simple model (Figure 7.1) does both.

The model.

Figure 7.1: The model.

In the binomial model the asset starts at S and over a time step St either rises to a value u x S or falls to a value v x S, with 0 < v < 1 < u. The probability of a rise is p and so the probability of a fall is 1 — p.

We choose the three constants u, v and p to give the binomial walk the same drift, x, and volatility, o, as the asset we are modelling. This choice is far from unique and here we use the choices that result in the simplest formulae:

Having defined the behaviour of the asset we are ready to price options.

Suppose that we know the value of the option at the time t + St. For example, this time may be the expiration of the option. Now construct a portfolio at time t consisting of one option and a short position in a quantity A of the underlying. At time t this portfolio has value

where the option value V is for the moment unknown. At time t + St the option takes one of two values, depending on whether the asset rises or falls

At the same time the portfolio of option and stock becomes either

Having the freedom to choose A, we can make the value of this portfolio the same whether the asset rises or falls. This is ensured if we make

This means that we should choose

for hedging. The portfolio value is then

Let's denote this portfolio value by

This just means the original portfolio value plus the change in value. But we must also have Sn = rnSt to avoid arbitrage opportunities. Bringing all of these expressions together to eliminate n, and after some rearranging, we get

This is an equation for V given V + and V , the option values at the next time step, and the parameters r and a.

The right-hand side of the equation for V can be interpreted, rather clearly, as the present value of the expected future option value using the probabilities p for an up move and 1 — p for a down.

Again this is the idea of the option value as the present value of the expected payoff under a risk-neutral random walk. The quantity p is the risk-neutral probability, and it is this that determines the value of the option not the real probability. By comparing the expressions for p and p we see that this is equivalent to replacing the real asset drift x with the risk-free rate of return r.

We can examine the equation for V in the limit as St — 0. We write

Expanding these expressions in Taylor series for small St we find that

and the binomial pricing equation for V becomes

This is the Black-Scholes equation.

CAPM

This derivation, originally due to Cox & Rubinstein (1985) starts from the Capital Asset Pricing Model in continuous time. In particular it uses the result that there is a linear relationship between the expected return on a financial instrument and the covariance of the asset with the market. The latter term can be thought of as compensation for taking risk. But the asset and its option are perfectly correlated, so the compensation in excess of the risk-free rate for taking unit amount of risk must be the same for each.

For the stock, the expected return (dividing by dt)is /x.Its risk is o.

From Ito we have

Therefore the expected return on the option in excess of the risk-free rate is

and the risk is

Since both the underlying and the option must have the same compensation, in excess of the risk-free rate, for unit risk

Now rearrange this. The x drops out and we are left with the Black-Scholes equation.

 
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