A stock trades at $30. Over the first six months it could go up to $35 or down t
ID: 2750059 • Letter: A
Question
A stock trades at $30. Over the first six months it could go up to $35 or down to $25. In the subsequent six-month period, if the stock starts at $35 it could go to $40 or it could fall to $30. If the stock starts at $25, it could go up to $30 or it might fall to $20. Value a call option on this stock with X = $28 using binomial method. Assume an annual risk-free rate of 1%. Suppose now that the actual market value of the call with X = $28 is $5. Demonstrate in detail how you could engage in arbitrage to take advantage of this mispricing.
Explanation / Answer
This can be represented as a binomial tree as follows
Current
n = 6 months or 0.5 year
Probability
p
N = 6 months or 0.5 year
Probability p1
Associated probability p*p1
Sup = $ 40
0.50
0.25
Sup = $ 35
0.50
S = $ 30
S down = $ 30
0.50
0.25
S up = $ 30
0.50
0.25
Sdown = $ 25
0.50
S down = $ 20
0.50
0.25
Let p be the probability that stock could go up during the first six months. Then (1-p) will be the probability of the stock going down in first six months
Current Price = p * expected Sup price + (1-p) * expected Sdown price
$ 30 = p * $ 35 + (1-p) * $ 25
$ 30 = p* ($ 35 - $ 25) + $ 25
P * $ 10 = $ 30 - $ 25 = $5
p = $ 5 / $10 = 0.50
1-p = 1-0.50 = 0.50
Probability of Sup in first six months changing in second 6 months
$ 35 = p1 * $ 40 + (1-p1) * $ 30
$ 35 = p1 * ($40-$ 30) + $ 30
$ 35-$30 = p1* $ 10
p1 = $ 5 / $ 10 = 0.50
(1-p1) = 0.50
Probability of Sdow in first six months changing in second 6 months
$ 25 = p1 * $ 30 + (1-p1) * $ 20
$ 25 = p1 * ($30-$ 20) + $ 20
$ 25-$20 = p1* $ 10
p1 = $ 5 / $ 10 = 0.50
(1-p1) = 0.50
From the associated probabilities, the stock has a 0.25 probability to go to $ 45, (0.25+0.25 =) 0.50 probability to go to $ 30 or 0.25 probability to go down to $ 20
Binomial Value of Option = (0.25 * $ 45 + 0.5 * $ 30 + 0.25 * $ 20)* e^-r*change in time
Binomial Value of option = (11.25 + 15 + 5)* (2.71828)^-0.01* 1 year
= $ 31.25 * 2.71828^-0.01
= $ 31.25 * 0.99004984
= $ 30.9390575 or $ 30.94
Value of the call Option = Binomial Value - Strike price = $ 30.94 - $ 28 = $ 2.94
If the call option is currently priced at $5, then
One can sell the call option at a strike price of $ 28 and received $ 5 . As this is a sale of call option, the call option seller is obliged to sell the stock to option buyer at the strike price.
Simultaneously the option seller can buy the stock at current price of $ 30.
At the call option expiry, if the expected price turns to be $ 30.94, the call option buyer demands delivery at $ 28. If it is a covered call, then the option seller can deliver the stock from his holdings
Loss on delivery under call option sold = Strike Price - purchase price = $ 28 - $ 30 = -$2
Premium received on sale of call option = $ 5
Net gain on covered call = $5 -$ 2 = $ 3
In case of sale of a naked call, ie., without simultaneous buying the underlying stock, the option seller needs to buy the stock in the market and deliver at the strike price to option buyer. If the stock price turns out to be $ 30.94 after 1 year, then
Loss on delivery to stock to option buyer = $28 - $30.94 = -$2.94
Premium received on sale of call option = $5
Net gain = $ 5 - $ 2.94 = $ 2.06
Current
n = 6 months or 0.5 year
Probability
p
N = 6 months or 0.5 year
Probability p1
Associated probability p*p1
Sup = $ 40
0.50
0.25
Sup = $ 35
0.50
S = $ 30
S down = $ 30
0.50
0.25
S up = $ 30
0.50
0.25
Sdown = $ 25
0.50
S down = $ 20
0.50
0.25
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