Question

Cassiopeia inc. is currently trading at $100 per share. After examining the stock of Cassiopeia, you have determined that in each 3 month period its price will either increase to 25% or decrease by 20%. The interest rate is 3% every 3 months.

a) A six month European call option on Cassiopeia has an exercise price of $90. What is the value of this call option?

b) What is the value of a six month European put option on Cassiopeia with an exercise price of $90?

c) Verify that put-call parity holds.

d) Now suppose that Cassiopeia pays a dividend equal to $25 in three months. What is the value of a six month American call option on Cassiopeia with an exercise price of $90? Would you ever want to exercise this option early?

Answer 1

a.

Pu or Probability of price going up =

Pd or probabilitu of price going Down = 1-Pu

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Pu at Node 1 = = 0.51

Pd at Node 1 = 1-Pu = 1-0.51 =0.49

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Pu at Node 2A = = 0.51

Pd at Node 2A = 1-Pu = 1-0.51 =0.49

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Pu at Node 2B = = 0.51

Pd at Node 2B = 1-Pu = 1-0.51 =0.49

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Calculation Value of the call option at the end Nodes

Price of share at end Nodes	Call option exercise price	Valu of call option
156.25	90	66.25
100	90	10
64	90	nil

Value of the option =

Value of call option at Note 2A =

= = $37.56

Value of call option at Note 2B=

= = $4.95

Value of call option at Note 1=

= = $20.95

Hence value of the call option today means at Node 1 is $20.95

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b.

Pu or Probability of price going up =

Pd or probabilitu of price going Down = 1-Pu

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Pu at Node 1 = = 0.51

Pd at Node 1 = 1-Pu = 1-0.51 =0.49

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Pu at Node 2A = = 0.51

Pd at Node 2A = 1-Pu = 1-0.51 =0.49

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Pu at Node 2B = = 0.51

Pd at Node 2B = 1-Pu = 1-0.51 =0.49

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Calculation Value of the call option at the end Nodes

Price of share at end Nodes	Put option exercise price	Valu of Put option
156.25	90	0
100	90	0
64	90	26

Value of the option =

Value of call option at Note 2A =

= = $0

Value of call option at Note 2B=

= = $12.37

Value of call option at Note 1=

= = $5.88

Hence value of the Put option today means at Node 1 is $5.88

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c.

As per the put call parity theorem the call and put option price on a same share, same exercise price and same maturity period shall have the following relationship

=>($100+$5.88-$20.95) * (1+0.03*2)

=>$84.93 * 1.06

=> $ 90 which is equal to the exercise price of $ 90

Hence Put call parity theory holds Good.

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d.

In case dividend amount is given then while calculating the option value add the dividend receivable in the upper and lower prices

in this case the company is paying dividend of $ 25 in each 3 months .

Note- Americal options can be exercised at any point of time.

Pu or Probability of price going up =

Pd or probabilitu of price going Down = 1-Pu

-------------------------------------------------------------

Pu at Node 1 = = -0.044

Pd at Node 1 = 1-Pu = 1- (-0.044) =1.044

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Pu at Node 2A = = 0.524

Pd at Node 2A = 1-Pu = 1-0.524 =0.476

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Pu at Node 2B = = 0.532

Pd at Node 2B = 1-Pu = 1-0.532 =0.468

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Calculation Value of the call option at the end Nodes

Price of share at end Nodes	Call option exercise price	Valu of call option
181.25	90	91.25
125	90	35
29	90	nil

Value of the option =

Value of call option at Note 2A =

= = $62.60

Value of call option at Note 2B=

= = $18.08

Value of call option at Note 1=

= = $15.66

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Americal call option Exercise analysis

Node	Discounted value of option	Intrinsic value of option	Exercise Early or not
2A	62.60	($150-$90) = $60	Do not exercise early as the discounted value is more than the Interisic value
2B	18.08	($105-$90)= $15	Do not exercise early as the discounted value is more than the Interisic value
Node 1	15.66	($100-$90)=$10	Do not exercise Early

Note- Intrinsic value means value of option if Exercised today.