Question

Today’s price of a non-dividend paying stock is $60. Use a two-step tree to value an American put option on the stock with a strike price of $50 that expires in 12 months. Each step is 6 months, the risk free rate is 10% per annum, and the volatility is 15%. Assume that the option is written on 100 shares of stock.

What is the option price today?

How would you hedge a position where you sell the put option today?

Answer 1

P = probability for price increase =( i -d) / (u - d ),

i = e^rt, r is the risk free rate = 10% per annum,

t = Each step is 6 months = 0.5,,

i = e ^(0.1X0.5),

= 1.05317176392796,

d = 1-0.15 = 0.85,

u = 1+0.15 = 1.15,

p = (1.05317176392796 - 0.85) / (1.15 - 0.85),

p=0.6772392130932,

1-p = probability for price decrease = 1- 0.6772392130932,

1-p=0.3227607869068,

The expected put value at each node = (Value of upper sub node X p) +( Value of lower sub node X (1-p)) X e ^-rt,

e ^-rt = 0.949512733108557,

The expected value at $ 69 is zero, becaue its all subnodes value is zero,

The expected value at $ 51 = (( 0 X 0.6772392130932) + ($6.65 X 0.3227607869068)) X 0.949512733108557,

=2.03799542149236. Since this option is american option, we have to check the early exercising of the option. here there is no skope of early exercise the option at both of $51 price node and $69 price node because these price are above the exercise price, so the option holder cannot exercise the option. so at $51 price node, the computed value is only considered.

The expected value at $ 60 = (( 0 X 0.6772392130932) + ($2.03799542149236 X 0.3227607869068)) X 0.949512733108557,

=0.624575238800575, this will be the value of put option per share and it is sprcified in the question that number of option is written on 100 shares of stock. So Value per option = 100 X 0.624575238800575 = $62.4575238800575 per option , rounded to 2 decimal places = $62.46,

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