Question

Stock price = £30. In 2 months, two months the price will be either £33 or £27. 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 £31?

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) = £31 t = 2 Months r = 10% CCI

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

The value of portfolo today = £30h - c

Valaue of portfolio as on expiry as below

Expiry Price		£33	£27
		£33h - £2	£27h - £0

If expiry price £33 then value of share bought = £33h and

Value of call option sold = Expiry price - Strike price = £33 -£31 =£2

If expiry price £33 then value of share bought = £27h and

Value of call option sold = Expiry price - Strike price = £27 -£31 = 0 ( Beacause call option buyer not exercise the call option as the buyer of option can buy stock @ £27 then why he exercise call option and buy share @£31, 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)

£33h - £2 = £27h - 0

h = 0.3333

or we can calculate 'h' by using option

if h = 0.3333 then value of portfolio on expiry = £27h - 0 = £27 x 0.3333 - 0 = £8.9991

or

if h = 0.3333 then value of portfolio on expiry = £33h -£2 = £33 x 0.3333 - £2 = £10.9989 -£2 = £8.9989 or say £8.9991

There fore value of portfolio today is present value of £8.9991

= £8.9991 x 1 / er x t

= £8.9991 x 1 / e0.10 x 2/12

= £8.9991 x 1 / e0.016667

= £8.9991 x 1/ 1.01681

= £8.8503

But we know that value of portfolio today is (£33h - c)

so

£33h - c = £8.8503

£10.9989 - c = £8.8503

c = £2.1486

Value of call option = £2.1486

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 + £30 = £2.1486 + £31 / er x t

P = £2.1486 + (£31 / 1.01681) - £30

P= £2.1486 + £30.4875 - £30

P = £2.6361

So value of put at strike price £31 = £2.6361

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