It’s quite challenging to agree on the accurate pricing of any tradable asset, even on present day. That’s why the stock prices keep constantly changing. In reality the company hardly changes its valuation on a day-to-day basis, but the stock price and its valuation change every second. This shows the difficultly in reaching a consensus about present day price for any tradable asset, which leads to arbitrage opportunities. However, these arbitrage opportunities are really short lived.

It all boils down to present day valuation – what is the right current price today for an expected future payoff?

In a competitive market, to avoid arbitrage opportunities, assets with identical payoff structures must have the same price. Valuation of options has been a challenging task and high variations in pricing are observed leading to arbitrage opportunities. Black-Scholes remains one of the most popular models used for pricing options. but has its own limitations. (For further information, see: Options Pricing ). Binomial option pricing model is another popular method used for pricing options. This article discusses a few comprehensive step-by-step examples and explains the underlying risk neutral concept in applying this model. (For related reading, see: Breaking Down The Binomial Model To Value An Option ).

This article assumes familiarity of the user with options and related concepts and terms.

Assume there exists a call option on a particular stock whose current market price is $100. The ATM option has strike price of $100 with time to expiry of one year. There are two traders, Peter and Paul, who both agree that the stock price will either rise to $110 or fall to $90 in one year’s time. They both agree on expected price levels in a given time frame of one year, but disagree on the probability of the up move (and down move). Peter believes that probability of stock price going to $110 is 60%, while Paul believes it is 40%.

Based on the above, who would be willing to pay more price for the call option?

Possibly Peter, as he expects high probability of the up move.

Let’s see the calculations to verify and understand this. The two assets on which the valuation depends are the call option and the underlying stock. There is an agreement among participants that the underlying stock price can move from current $100 to either $110 or $90 in one year’s time, and there are no other price moves possible.

In an arbitrage-free world, if we have to create a portfolio comprising of these two assets (call option and underlying stock) such that irrespective of where the underlying price goes ($110 or $90), the net return on portfolio always remains the same. Suppose we buy ‘d’ shares of underlying and short one call option to create this portfolio.

If the price goes to $110, our shares will be worth $110*d and we’ll lose $10 on short call payoff. The net value of our portfolio will be (110d – 10).

If the price goes down to $90, our shares will be worth $90*d, and option will expire worthless. The net value of our portfolio will be (90d).

If we want the value of our portfolio to remain the same, irrespective of wherever the underlying stock price goes, then our portfolio value should remain the same in either cases, i.e.:

=> (110d – 10) = 90d

=> d = ½

i.e. if we buy half a share (assuming fractional purchases are possible), we will manage to create a portfolio such that its value remains same in both possible states within the given timeframe of one year. (point 1)

This portfolio value, indicated by (90d) or (110d -10) = 45, is one year down the line. To calculate its present value, it can be discounted by risk free rate of return (assuming 5%).

=> 90d * exp(-5%*1 year) = 45* 0.9523 = 42.85 => Present value of the portfolio

Since at present, the portfolio comprises of ½ share of underlying stock (with market price $100) and 1 short call, it should be equal to the present value calculated above i.e.

=> 1/2*100 – 1*call price = 42.85

=> Call price = $7.14 i.e. the call price as of today.

Since this is based on the above assumption that portfolio value remains the same irrespective of which way the underlying price goes (point 1 above), the probability of up move or down move does not play any role here. The portfolio remains risk-free, irrespective of the underlying price moves.

In both cases (assumed to be up move to $110 and down move to $90), our portfolio is neutral to the risk and earns the risk free rate of return.

Hence both the traders, Peter and Paul, will be willing to pay the same $7.14 for this call option, irrespective of their own different perceptions of the probabilities of up moves (60% and 40%). Their individually perceived probabilities don’t play any role in option valuation, as seen from the above example.

If suppose that the individual probabilities matter, then there would have existed arbitrage opportunities. In real world, such arbitrage opportunities exist with minor price differentials and vanish in a short term.

But where is the much hyped volatility in all these calculations, which is an important (and most sensitive) factor affecting option pricing?

The volatility is already included by the nature of problem definition. Remember we are assuming two (and only two — and hence the name “binomial”) states of price levels ($110 and $90). Volatility is implicit in this assumption and hence automatically included – 10% either way (in this example).

Now let’s do a sanity check to see whether our approach is correct and coherent with the commonly used Black-Scholes pricing. (See: The Black-Scholes Option Valuation Model ).

Here are the screenshots of options calculator results (courtesy of OIC), which closely matches with our computed value.

Unfortunately, the real world is not as simple as “only two-states”. There are several price levels which can be achieved by the stock till the time to expiry.

Is it possible to include all these multiple levels in our binomial pricing model which is restricted to only two levels? Yes, it is very much possible, and to understand it, let’s get into some simple mathematics.

A few intermediate calculation steps are skipped to keep it summarized and focused on results.

To proceed further, let’s generalize this problem and solution:

‘X’ is the current market price of stock and ‘X*u’ and ‘X*d’ are the future prices for up and down moves ‘t’ years later. Factor ‘u’ will be greater than 1 as it indicates up move and ‘d’ will lie between 0 and 1. For above example, u=1.1 and d=0.9.

The call option payoffs are ‘P_up ’ and ‘P_dn ’ for up and down moves, at the time of expiry.

If we build a portfolio of ‘s’ shares purchased today and short one call option, then after time ‘t’:

Value of portfolio in case of up move = s*X*u – P_up

Value of portfolio in case of down move = s*X*d – P_dn

For similar valuation in either case of price move,

=> s*X*u – P_up = s*X*d – P_dn

=> s = (P_up — P_dn )/(X*(u-d)) = the no. of shares to purchase for risk free portfolio

The future value of portfolio at the end of ‘t’ years will be

In case of up move = s*X*u – P_up = (P_up — P_dn )/(X(u-d))* X*u – P_up

The present day value of above can be obtained by discounting it with risk free rate of return:

This should match the portfolio holding of ‘s’ shares at X price, and short call value ‘c’ i.e. present day holding of (s* X — c) should equate to above. Solving for c finally gives c as:

IF WE SHORT THE CALL PREMIUM SHOULD BE ADDITION TO PORTFOLIO NOT SUBTRACTION.

Another way to write the above equation is by rearranging it as follows:

Taking q as

then above equation becomes

Rearranging the equation in terms of “q” has offered a new perspective.

“q” can now be interpreted as the probability of the up move of the underlying (as “q” is associated with P_up and “1-q” is associated with P_dn ). Overall, the above equation represents the present day option price i.e. the discounted value of its payoff at expiry.

How is this probability “q” different from the probability of up move or down move of the underlying?

The value of stock price at time t = q* X*u +(1-q)*X*d

Substituting the value of q and rearranging, the stock price at time t comes to

i.e. in this assumed world of two-states, the price of stock simply rises by risk free rate of return, i.e. exactly like a risk free asset and hence it remains independent of any risk. All investors are indifferent to risk under this model, and this constitutes the risk neutral model.

Probability “q” and (1-q) are known as risk neutral probabilities and the valuation method is known as risk neutral valuation model.

The above example has one important requirement — the future payoff structure is required with precision (level $110 and $90). In real life, such clarity about step based price levels is not possible; rather the price moves randomly and may settle at multiple levels.

Let’s expand the example further. Assume that two step price levels are possible. We know the second step final payoffs and we need to value the option today (i.e. at initial step)

Working backwards, the intermediate first step valuation (at t=1) can be made using final payoffs at step two (t=2), and then using these calculated first step valuation (t=1), the present day valuation (t=0) can be reached using the above calculations.

To get option pricing at no. 2, payoffs at 4 and 5 are used. To get pricing for no. 3, payoffs at 5 and 6 are used. Finally, calculated payoffs at 2 and 3 are used to get pricing at no. 1.

Please note that our example assumes same factor for up (and down) move at both steps — u (and d) are applied in compounded fashion.

Here is a working example with calculations:

Assume a put option with strike price $110 currently trading at $100 and expiring in one year. Annual risk free rate is at 5%. Price is expected to increase 20% and decrease 15% every six months.

Let’s structure the problem:

Here, u=1.2 and d = 0.85, X=100, t =0.5

using above derived formula of , we get q = 0.35802832

value of put option at point 2,

At P_upup condition, underlying will be = 100*1.2*1.2 = $144 leading to P_upup = zero

At P_updn condition, underlying will be = 100*1.2*0.85 = $102 leading to P_updn = $8

At P_dndn condition, underlying will be = 100*0.85*0.85 = $72.25 leading to P_dndn = $37.75

p₂ = 0.975309912*(0.35802832*0+(1-0.35802832)*8) = 5.008970741

Similarly, p₃ = 0.975309912*(0.35802832*8+(1-0.35802832)*37.75) = 26.42958924

And hence value of put option, p₁ = 0.975309912*(0.35802832*5.008970741+(1-0.35802832)* 26.42958924) = $18.29.

Similarly, binomial models allow one to break the entire option duration to further refined multiple steps/levels. Using computer programs or spreadsheets one can work backwards one step at a time, to get the present value of the desired option.

Let’s conclude with one more example involving three steps for binomial option valuation:

Assume a put option of European type, having 9 months to expiry with strike price of $12 and current underlying price at $10. Assume risk free rate of 5% for all periods. Assume each 3 months, the underlying price can move 20% up or down, giving us u=1.2, d=0.8, t=0.25 and 3 step binomial tree.

The figures in red indicate underlying prices, while the ones in blue indicate the payoff of put option.

Risk neutral probability q computes to 0.531446.

Using the above value of q and payoff values at t=9 months, the corresponding values at t=6 months are computed as:

Further, using these computed values at t=6, values at t=3 and then at t=0 are:

giving the present day value of put option as $2.18, which is pretty close to the one computed using Black-Scholes model ($2.3)

The Bottom Line

Although use of computer programs can make a lot of these intensive calculations easy, the prediction of future prices remains a major limitation of binomial models for option pricing. The finer the time intervals, the more difficult it gets to precisely predict the payoffs at the end of each period. However, the flexibility to incorporate changes as expected at different periods of time is one added plus, which makes it suitable for pricing the American options, including early exercise valuations. The values computed using the binomial model closely match the ones computed from other commonly used models like the Black-Scholes, which indicates the usefulness and accuracy of binomial models for option pricing. Binomial pricing models can be developed according to a trader’s preference and works as an alternative to Black-Scholes.