# Ch 4. Binomial Tree Model

## One step binomial tree put option

As the standard deviation increases, so does the divide (dispersion) between stock prices in up and down states ($${ S }_{ U }$$ and $${ S }_{ D }$$, respectively). Suppose there was no deviation at all. Would we have a binomial tree in the first place? The answer is no.

## 1A One-Step Binomial Model

Payoff at expiration is different for calls and puts. If our CallPut cell contains the value 6, the option is a call, otherwise a put. Call payoff is underlying price at expiration (cell L9) minus strike put payoff is strike minus underlying price. If these differences are negative (option expires out of the money), payoff is zero, which is done with the MAX function. More explanation here.

### Lecture 6: Option Pricing Using a One-step Binomial Tree

It is a nested IF function. The outer IF uses our AmEur input as condition. If the option is American (AmEur is 6), the second item in the MAX is the option’s intrinsic value (the inner IF). Otherwise (AmEur is not 6), it is zero.

#### Binomial-tree Option Calculator - Jan Röman

The price of an exchange-quoted zero-dividend share is $85. Over the past year, the stock has exhibited a standard deviation of 67%. The continuously compounded risk-free rate is 5% per annum. Compute the value of a 6-year European call option with a strike price of$85 using a one-period binomial model:

##### Chapter 9: Two-step binomial trees - UCD

Therefore, we will first use dummy values for our move sizes and probabilities, create the trees first with these dummy values, and in the next parts of the tutorial we will replace these values with correct formulas for individual models (Cox-Ross-Rubinstein, Jarrow-Rudd, and Leisen-Reimer).

###### Understanding the Binomial Option Pricing Model

When dealing with options on currencies, a plausible assumption is that the return earned on a foreign currency asset is equal to the foreign risk-free rate of interest. As such, the probability of an up move is given by:

Probability of an up move=$${ \pi }_{ u }=\frac { { e }^{ \left( r-q \right) t }-d }{ u-d } =\frac { { e }^{ \left( - \right) 6 }- }{ - } =$$

Choosing #7 from the three layouts introduced above, our tree will have up moves horizontal (next cell to the right) and down moves diagonal (down and right). Therefore, if price moves up in the first step (from cell E9), it will end up in cell F9. The formula is:

Because a * b = b * a , or in our case up * down = down * up , it doesn’t matter whether a node is calculated as an up move from the node to the left, or as down move from the node to the left and up – both give the same result. It is best to be consistent though – I copy the up move formulas in the top row only, and use the down move formulas elsewhere.

If the option is a call (CallPut is 6), intrinsic value is underlying price minus strike. Otherwise (CallPut is not 6), it is strike minus underlying price. Two notes:

The formulas for up and down move sizes and probabilities are different in different binomial models, but everything else – structure and all formulas inside the binomial trees – are the same in all the models.

As the number of time steps is increased, the binomial tree model makes the same assumptions about stock price behavior as the Black– Scholes–Merton model. When the binomial tree is used to price a European option, the price converges to the Black–Scholes–Merton price as the number of time steps is increased.

The binomial option pricing model is a simple approximation of returns which, upon refining, converges to the analytic pricing formula for vanilla options. The model is also useful for valuing American options that can be exercised before expiry.

The last column in the underlying price tree contains different underlying prices at expiration. We will use them to calculate option payoffs at expiration for these different scenarios, which will be the last column in the option price tree.