Question

6. Derivatives 6a. The price of a European put that expires in four months and has a strike price of $30 is $3. The underlying stock price is $28. The term structure is flat, with all risk-free interest rates being 3%. What is the price of a European call option that expires in four months and has a strike price of $30? 6b. What if the price of the call is $1.5, any arbitrage opportunity? Please show. 6c. What if for this put, a dividend of $0.50 is expected in two months and again in five months, please rework part a and b. Are there any arbitrage opportunities? Please show.

Answer 1

6a) calculation of value of the European call option:

With out dividend the put call parity equation is:

''value of call + present value of strike price = value of put option + value of share ''

Facts: Strike price = $30, Spot price = $28, Value of put = $3, term = 4 months, Risk free rate= 3%

present value of strike price = futurevalue of strike price * e ^-tr_f

⁼ $ 30 * e ^{-(4/12)(0.03)} = $ 30 * e ^-0.01 =  $ 30 * 1/e ^0.01 = $ 30 * 1/1.01005 = $ 29.7

^{value of e 0.01}= 1+ 0.01/1 + (0.01)² /2 ! + (0.01)³/3!      

= 1.01005   

Subsitute the values in the equation we get V_c + Pv of strike price = V_p + Value of share

V c + $ 29.7 = $3 + $ 28

= $ 1.3

6b) Arbitrage Opportunity exists or not:

Assume   Vc + PV of Srike price as Porfolio A and Vp + Value of share as Portfolio B.

If the value of call option is $ 1.5 then equation becomes $1.5+ $29.7 = $3 +$28

$31.2 = $ 31

Since the LHS is not equal to RHS there exists an arbitrage opportunity.. The Portfolio B was under priced and hence it should be bought so the strategy is "" Buy the put option and Buy the share"".

6c) In case there exists dividend then the situation differs as follows: When dividend occurs during the option period then the value of share is cumulative dividend share value and while Vc and Vp are created on Ex-dividend share ..we should revise the share price by reducing the present value of dividend computed as below:

step 1: Calculation of Present values of dividends: dividend 1 that expected in 2 months

=$ 0.5 * e -^(2/12*0.03) = $ 0.5 * e ^-0.005 = $ 0.5 * 1.005 = $ 0.5025.

(value of e 0.005 was calculated as in 6a )

Dividend 2 that is expected in 5 months

= $ 0.5 * e ^{-(5/12 * 0.03)} = $ 0.5 * e ^-0.0125 = $ 0.5 * 1.01257 = $ 0.5063 (value of e 0.0125 was calculated as in 6a )

Total Present value of Dividend = $ 0.5025 + $ 0.5063 = $ 1.0088.

Reworking 6a) Subsitute the values in the equation we get V_c + Pv of strike price = V_p + (Value of share- Present value of dividends)

Vc + $ 29.7 = $ 3 + ($ 28- $ 1.0088)

= $ 0.2912

Reworking 6b)

Arbitrage Opportunity exists or not:

Assume   Vc + PV of Srike price as Porfolio A and Vp + Value of share as Portfolio B.

If the value of call option is $ 1.5 then equation becomes $1.5+ $29.7 = $3 +$28 - $1.0088

$31.2 = $ 30.00

Since the LHS is not equal to RHS there exists an arbitrage opportunity.. The Portfolio B was under priced and hence it should be bought so the strategy is "" Buy the put option and Buy the share""