Pricing Options Using the Binomial Tree (Risk Neutral Valuation Approach)

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In finance, the binomial options pricing model provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein in 1979, six years after the Black Scholes model. Essentially, the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying financial instrument.

The Binomial options pricing model approach has been widely used since it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the model is based on a description of an underlying instrument over a period of time rather than a single point in time. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software or even in spreadsheets like Excel.

Although computationally slower than the Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets.

The Risk-Neutral Binomial Tree Approach
The concept of a portfolio made up of some portion of stock and some portion of a derivative on that stock gives rise to the ability to generate equivalent cash flows at end nodes of binomial trees, and this certainty of cash flows allows us to discount the cash flows at the risk-free rate. The ability to discount cash flows in the future at a known risk-free rate gives us the concept of risk-neutral valuation.

Risk neutral valuation is a powerful concept in derivatives pricing which enables valuation of assets based on their expected payoffs at different points in time and with different scenarios of underlying asset price movements.

Risk-neutral valuation is applicable whenever you can create a portfolio including the underlying plus a derivative on the same underlying. It cannot be extrapolated to find the value of derivatives on other underlyings. It relies on the portfolio instruments having a level of dependency on one another.
This model is very flexible and powerful because we don’t need to know the real probability of the upside scenario, or the real likelihood of the downside economic scenario—it is not required to maintain our certainty of cash flows at the end point.

An easy mistake to make is to confuse this constructed probability distribution with real-world probability. They will be different, but the method of risk-neutral pricing is, like many other useful computational tools, convenient and powerful. The approach of risk-neutral valuation makes sense for valuation purposes but only works when all of the instruments included in the valuation model depend on the same underlying and thus are exposed to the same risks, though held in different proportions. We are pricing the option in terms of the underlying stock, thus risk preferences are taken into account in the pricing of the underlying.

The risk-neutral binomial tree approach is mathematically equivalent to the portfolio approach previously covered, and gives us the exact same valuation. The risk neutral binomial tree valuation approach to value the derivative, f, is as follows.

With some algebra, you can show that the risk-neutral formula is mathematically equivalent to the portfolio approach where  of shares is calculated to generate riskless outcomes at maturity, T.

Although we are not making any assumptions about the probabilities of the returns on the underlying, the element p in this formula can be interpreted as the “risk-neutral probability of an up move” and 1–p as the “risk-neutral probability of a down move.”
The value today of the derivative, f, can thus be read as the present value of the derivative at the up node times the risk-neutral probability of an up move plus the value of the derivative at the down node times the risk-neutral probability of a down move.

In our prior example of an underlying that has a price of $50, where we know at the end of three months that the underlying will be at one of two prices (either $70 or $30), we will price a European Call with a strike of $50, and one month to expiration and a risk-free interest rate of 5%.

Here p = 0.5052 based on the above formula where p is a function of the risk free rate and time to maturity T, as well as u and d. Then the call value at time zero is f = e(-rT)x[pfu] = $10.0624, the same as in the portfolio valuation approach above.

#Risk Neutral Valuation Approach