Question

There is an American put option on a stock that expires in two months. The stock price is $55, and the standard deviation of the stock returns is 64 percent. The option has a strike price of $62, and the risk-free interest rate is an annual percentage rate of 4.4 percent. What is the price of the option? Use a two-state model with one-month steps. (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

What is Put value?

Answer 1

Given:

Parameters
Stock Price S0	55
Exercise Price X	62
Interest Rate r	0.044
Volatility	0.64
Maturity (Time T)	0.08333(2/12*2)
Number of Steps	2
Dividend Yield	0

Note: Here in the time, I’ve divided by 2 because its a two step tree. For a three step tree this needs to be divided by 3 etc.

Tree:

Calculations:

Initial Calculation:

u (called as uptick factor) = e ^*T = e^{(0.64) * (2/12*2)} = 1.20292

d (called as downtick factor) = 1/u = 1/1.20292 = 0.83131

Using u & d start constructing the tree like below (first with Spot prices for each node)

Spot price calculation:

Spot price (First node):

• 55 * 1.20292 = 66.16
• 55 * 0.83131 = 45.72

Spot price (second node):

• 66.16 * 1.20292 = 79.59
• 66.16 * 0.83131 = 55
• 45.72 * 1.20292 = 55
• 45.72 * 0.83131 = 38.01

What are we doing in the above steps?

We are trying to identify the spot prices for each time period (1M & 2M) using uptick and down tick factors. Basically it derives the two possibility of the spot price for each time period

Option value calculation:

Once you have filled up the spot prices in tree, now comes the option value calculation. Option value calculation should start from the last node (i.e. reverse from right to left).

Option value (last node):

This node calculation is pretty easy, as we just valuate the option assuming its exercised at maturity or in other words we just need to calculate the “exercise value” alone for this node.

Exercise value (or called payoff) = Max (K-S,0) for a put option, similarly for a call option its Max (S-K,0)

• 79.59 Max (62– 79.59,0) = 0, because the option is Out of money
• 55 Max (62– 55,0) = 7
• 38.01 Max (62– 38.01,0) = 23.99, because it’s In the money

Option value (First node):

This is a bit different unlike the straight forward calculation like above, because the investor has a choice to exercise it or leave it until maturity, this is where we need to calculate the “binomial value”

Binomial value formula = [(V_u * p + V_d * (1-p) ] * e^{-R *T} , where e^{-R *T} is the discount factor

In order to calculate the binomial value, we first need to calculate the risk neutral probability (p).

Probability (up), p = e^{R *T} – d / (u-d) = e^{(0.044) * 0.0833} - 0.83131 / (1.20292 - 0.83131) = 0.46383 or 46.38%

Probability (down) , 1-p = 1- 0.46383 = 0.53617 or 53.62%

• 66.16 (0 * 46.38% + 7 * 53.62%) * e^{-0.044* 0.0833} = 3.74
• 45.72 (7 * 46.38% + 23.99 * 53.62%) * e^{-0.044* 0.08335}= 16.05 but I put in the tree 16.28 why?

The tree has a rule Max (binomial value, exercise value), whichever is greater between these two values will be the option value in the tree. Binomial value = 16.05, exercise value = 16.28 (62– 45.72) and in this case exercise value is greater and the same has been used in the tree.

Why not use the binomial value itself in the tree instead of following the tree rules? How does it impact?

Binomial tree valuation for American options follows the “no arbitrage rule”, meaning the option should be valuated in such a way so that no one can benefit out of an arbitrage situation. Let us assume we set the option value in the tree as 16.05, an arbitrager will benefit out of this situation by buying underlying @ 45.72 + buying the put option for 16.05 & exercising it immediately to sell the underlying for 62 to attain a riskless profit of 0.23 ( 62-45.72 - 16.05 )

American options enjoy the choice of being exercised at any point during the maturity month (early exercise), whereas European option can be exercised only on the maturity

Option value (initial):

Same process as above node.

• 55 (3.74 * 46.38% + 16.28 * 53.62%) * e^{-0.044* 0.0833} = 10.42

Price of the put option = 10.42 (rounded off)

Hit thumbs up if you find this useful, any questions feel free to put it in the comment section and ill reply asap.