17.3 Risk Neutral

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The previous spreadsheet model, Binomial Option Pricing Multi-Period, determined the price of an

option by constructing a replicating portfolio, which combines a stock and a bond to replicate the payoffs

of the option. An alternative way to price an option is the Risk Neutral method. Both techniques give you

the same answer. The main advantage of the Risk Neutral method is that it is faster and easier to

implement. The Replicating Portfolio method required the construction of four trees (stock prices, shares

of stock bought (sold), money lent (borrowed), and option prices). The Risk Neutral method will only

require two trees (stock prices and option prices).

FIGURE 17.7 Spreadsheet Model of Binomial Option Pricing - Risk Neutral - Call Option.

How To Build This Spreadsheet Model.

1. Start with the Multi-Period Spreadsheet. Open the spreadsheet that you created for Binomial

Option Pricing – Multi-Period and immediately save the spreadsheet under a new name using the

File Save As command.

2. Risk Neutral Probability. Calculate the Risk Neutral Probability = (Riskfree Rate / Period -

Down Movement / Period) / (Up Movement / Period - Down Movement / Period). Enter =(B8-

B7)/(B6-B7) in cell F4.

3. The Option Price Tree. At each point in the 8-by-9 range, you need to determine if you are on

the tree or off the tree. There are two possibilities:

o When the corresponding cell in the Stock Price area is blank, then show a blank.

o When the corresponding cell in the Stock Price area has a number, then (in the absence of

arbitrage) the Option Price At Each Node = Expected Value of the Option Price Next

Period (using the Risk Neutral Probability) Discounted At The Riskfree Rate = [(Risk

Neutral Probability) * (Stock Up Price) + (1 - Risk Neutral Probability) * ( Stock Down

Price)] / (1+ Riskfree Rate / Period).

Enter =IF(C28="","",($F$4*C27+(1-$F$4)*C28)/(1+$B$8)) in cell B27 (yielding a blank

output at first) and then copy this cell to the 8-by-8 range B27:I34. Be sure not to copy over

column J containing option payoffs at maturity. A binomial tree will form in the triangular

area from B27 to J27 to J35. Again the same procedure could create a binominal tree for any

number of periods. For appearances, delete rows 37 through 57 by selecting the range A37:A57,

clicking on Edit, Delete, selecting the Entire Row radio button on the Delete dialog box, and

clicking on OK.

We see that the Risk Neutral method predicts an eight-period European call price of $3.93. This is

identical to Replicating Portfolio Price. Now let's check the put.

FIGURE 17.8 Spreadsheet Model of Binomial Option Pricing - Risk Neutral - Put Option.

4. Put Option. Enter 0 in cell B4.

We see that the Risk Neutral method predicts an eight-period European put price of $6.39. This is

identical to Replicating Portfolio Price. Again, we get the same answer either way. The advantage of the

Risk Neutral method is that we only have to construct two trees, rather than four trees.