Question

You are interested to value a put option with an exercise price of $100 and one year to expiration. The underlying stock pays no dividends, its current price is $100, and you believe it either increases to $120 or decreases to $80. The risk-free rate of interest is 10%. Calculate the put option's value using the binomial pricing model, presenting your calculations and explanations as follows:

a. Draw tree-diagrams to show the possible paths of the share price and put payoffs over one year period. (Note: Show the numbers that are known and use letter(s) for what is unknown in your diagrams.)

b. Compute the hedge ratio.

c. Find the put option price. Explain your calculations clearly.

Answer 1

We use the binominal method ( Risk less model)

Step 1: We create portfolio by buying one share and selling one call option.

Step 2 : Calculate the value of portfolio as on expiry using both expiry price.

Step 3 : Find out the numer of shares buy using details of 'Step 2'

Step 4 : Calculate the value of portfolio on expiry.

Step 5 : Calculate the value of portfolio today. ( Mean present value of portfolio on expiry)

Step 6 : Using step 5 and step 1 find out value of call option.

Step 7 : Calculate the Put option value using put-call-parity.

Binominal tree

Not known is call option and put option value at strike price $100 after 1 Year.

Call (x) = $100 t = 1 Year r = 10%

We create portfolio by buying 'h' no. of shares from market and to protect the same we sell 1 call.

The value of portfolo today = $100h - c

Valaue of portfolio as on expiry as below

Expiry Price		$120	$80
		$120h - $20	$80h - $0

If expiry price $120 then value of share bought = $120h and

Value of call option sold = Expiry price - Strike price = $120 -$100 =$20

If expiry price $80 then value of share bought = $80 and

Value of call option sold = Expiry price - Strike price = $80 - $100 = 0 ( Beacause call option buyer not exercise the call option as the buyer of option can buy stock @ $80 then why he exercise call option and buy share @$100, so option buyer let the lapse the call option so value of that option is '0'.

risk less portfolio so ( value of both expiry price is equal because of riskless)

$120h - $20 = $80h - 0

h = 0.5

or we can calculate 'h' by using option

if h = 0.5 then value of portfolio on expiry = $80h - 0 = $80 x 0.5 - 0 = $40

or

if h = 0.5 then value of portfolio on expiry = $120h -$20 = $120 x 0.5 - $20 = $60 -$20 = $40

There fore value of portfolio today is present value of $40

= $40 x 1 / (1+r)

= $40 x 1 / (1+0.10)

= $40 x 1 / 1.10

= $36.3637

But we know that value of portfolio today is ($100h - c)

so

$100h - c = $36.3637

$100 x 0.5 - c = $36.3637

c = $13.6363 or say $13.64

Value of call option = $13.64

Now for find out put value we use put call parity

so P + S = C + PV of (x)

Where:

P = Put option value

S = Current market price

C = Call option value

PV of (x) = Present value of strike price

P + $100 = $13.64 + $100 / (1+r)

P = $13.64 + ($100 / 1.10) - $100

P= $13.64 + $90.91 - $100

P = $4.55

So value of put at strike price $100 = $4.55

Here we taken simple interest for calculation.

If any help require regarding this question please comment i will help you.