Question

Stock price = £60. In 2 months, two months the price will be either £66 or £54. The risk-free interest rate is 10% p.a on a continuous compounding basis.

What will be the value of a 2-month European put option with a strike price of £62?

Please provide a step by step explanation as I would like to fully understand and not just copy the answer. Thank you :)

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

Call (x) = £62 t = 2 Months r = 10% CCI

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 = £60h - c

Valaue of portfolio as on expiry as below

Expiry Price		£66	£54
		£66h - £4	£54h - £0

If expiry price £66 then value of share bought = £66h and

Value of call option sold = Expiry price - Strike price = £66 -£62 =£4

If expiry price £54 then value of share bought = £54h and

Value of call option sold = Expiry price - Strike price = £54 -£62 = 0 ( Beacause call option buyer not exercise the call option as the buyer of option can buy stock @ £54 then why he exercise call option and buy share @£62, 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)

£66h - £4 = £54h - 0

h = 0.3333

or we can calculate 'h' by using option

if h = 0.3333 then value of portfolio on expiry = £54h - 0 = £54 x 0.3333 - 0 = £17.9982

or

if h = 0.3333 then value of portfolio on expiry = £66h -£4 = £66 x 0.3333 - £4 = £21.9978 -£4 = £17.9978 or say £17.9982

There fore value of portfolio today is present value of £17.9982

= £17.9982 x 1 / er x t

= £17.9982 x 1 / e0.10 x 2/12

= £17.9982 x 1 / e0.016667

= £17.9982 x 1/ 1.01681

= £17.7007

But we know that value of portfolio today is (£60h - c)

so

£60h - c = £17.7007

£21.9978 - c = £17.7007

c = £4.2971

Value of call option = £4.2971

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 + £60 = £4.2971 + £62 / er x t

P = £4.2971 + (£62 / 1.01681) - £60

P= £4.2971 + £60.9750 - £60

P = £5.2721

So value of put at strike price £62 = £5.2721

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