Binomial Option Pricing Model Definition

What Is the Binomial Choice Pricing Mannequin?

The binomial choice pricing mannequin is an choices valuation methodology developed in 1979. The binomial choice pricing mannequin makes use of an iterative process, permitting for the specification of nodes, or closing dates, in the course of the time span between the valuation date and the choice’s expiration date.

The mannequin reduces potentialities of value adjustments and removes the likelihood for arbitrage. A simplified instance of a binomial tree may look one thing like this:

Fundamentals of the Binomial Choice Pricing Mannequin

With binomial choice value fashions, the assumptions are that there are two potential outcomes—therefore, the binomial a part of the mannequin. With a pricing mannequin, the 2 outcomes are a transfer up, or a transfer down. The key benefit of a binomial choice pricing mannequin is that they’re mathematically easy. But these fashions can turn into advanced in a multi-period mannequin.

In distinction to the Black-Scholes mannequin, which supplies a numerical outcome based mostly on inputs, the binomial mannequin permits for the calculation of the asset and the choice for a number of durations together with the vary of potential outcomes for every interval (see under).

The benefit of this multi-period view is that the person can visualize the change in asset value from interval to interval and consider the choice based mostly on choices made at completely different closing dates. For a U.S-based choice, which may be exercised at any time earlier than the expiration date, the binomial mannequin can present perception as to when exercising the choice could also be advisable and when it needs to be held for longer durations. 

By trying on the binomial tree of values, a dealer can decide upfront when a choice on an train might happen. If the choice has a optimistic worth, there’s the potential of train whereas, if the choice has a price lower than zero, it needs to be held for longer durations.

Calculating Value with the Binomial Mannequin

The fundamental methodology of calculating the binomial choice mannequin is to make use of the identical chance every interval for achievement and failure till the choice expires. Nonetheless, a dealer can incorporate completely different chances for every interval based mostly on new info obtained as time passes.

A binomial tree is a great tool when pricing American choices and embedded choices. Its simplicity is its benefit and drawback on the identical time. The tree is straightforward to mannequin out mechanically, however the issue lies within the potential values the underlying asset can absorb one time frame. In a binomial tree mannequin, the underlying asset can solely be price precisely one in every of two potential values, which isn’t lifelike, as belongings may be price any variety of values inside any given vary.

For instance, there could also be a 50/50 probability that the underlying asset value can improve or lower by 30 p.c in a single interval. For the second interval, nevertheless, the chance that the underlying asset value will improve might develop to 70/30.

For instance, if an investor is evaluating an oil nicely, that investor is just not positive what the worth of that oil nicely is, however there’s a 50/50 probability that the value will go up. If oil costs go up in Interval 1 making the oil nicely extra beneficial and the market fundamentals now level to continued will increase in oil costs, the chance of additional appreciation in value might now be 70 p.c. The binomial mannequin permits for this flexibility; the Black-Scholes mannequin doesn’t.

Actual-World Instance of Binomial Choice Pricing Mannequin

A simplified instance of a binomial tree has just one step. Assume there’s a inventory that’s priced at $100 per share. In a single month, the value of this inventory will go up by $10 or go down by $10, creating this example:

• Inventory value = $100
• Inventory value in a single month (up state) = $110
• Inventory value in a single month (down state) = $90

Subsequent, assume there’s a name choice out there on this inventory that expires in a single month and has a strike value of $100. Within the up state, this name choice is price $10, and within the down state, it’s price $0. The binomial mannequin can calculate what the value of the decision choice needs to be at present.

For simplification functions, assume that an investor purchases one-half share of inventory and writes or sells one name choice. The whole funding at present is the value of half a share much less the value of the choice, and the potential payoffs on the finish of the month are:

• Price at present = $50 – choice value
• Portfolio worth (up state) = $55 – max ($110 – $100, 0) = $45
• Portfolio worth (down state) = $45 – max($90 – $100, 0) = $45

The portfolio payoff is equal regardless of how the inventory value strikes. Given this consequence, assuming no arbitrage alternatives, an investor ought to earn the risk-free price over the course of the month. The associated fee at present should be equal to the payoff discounted on the risk-free price for one month. The equation to resolve is thus:

• Choice value = $50 – $45 x e ^ (-risk-free price x T), the place e is the mathematical fixed 2.7183.

Assuming the risk-free price is 3% per yr, and T equals 0.0833 (one divided by 12), then the value of the decision choice at present is $5.11.

The binomial choice pricing mannequin presents two benefits for choice sellers over the Black-Scholes mannequin. The primary is its simplicity, which permits for fewer errors within the industrial software. The second is its iterative operation, which adjusts costs in a well timed method in order to cut back the chance for consumers to execute arbitrage methods.

For instance, because it supplies a stream of valuations for a spinoff for every node in a span of time, it’s helpful for valuing derivatives akin to American choices—which may be executed anytime between the acquisition date and expiration date. It is usually a lot less complicated than different pricing fashions such because the Black-Scholes mannequin.