For all questions below assume time discount rate of zero. Q1. You believe stock
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For all questions below assume time discount rate of zero. Q1. You believe stock price by yearend will have the following multinomial distribution: Price Probability 80 10% 90 20% 100 40% 110 20% 120 10% Qla. What should be the stock price TODAY? Q1b. what is the prob that a 95 strike put will expire ITM? Qlc. what is the conditional average price of underlying stock when 95 strike put expire ITM? Qld. based on Qlb and Qlc, how much should the 95 put be priced at? Qle. redo Qlb-d for 95 call. Q2. Underlying at S100 and MAD at $IO0 PUT option has strike of s95 Q2a. What is the probability for PUT to be in the money? Q2b. What is the average price of the underlying conditional on PUT expiring ITM? Q2c. Based on Q2a and Q2b, how much should the 95 PUT be priced at? Q2d. For Q2c PUT price, how much of it is time value and how much is intrinsic value (aka exercise value)? Q3. Underlying at SI00 and MAD at $10 CALL option has strike of S95 Q3a. What is the probability for CALL to be in the money at expiration? Q3b. What is the average price of the underlying conditional on CALL expiring ITM? Q3c. Based on Q3a and Q3b, how much should the 95 CALL be priced at? Q3d. For Q3c CALL price, how much of it is time value and how much is intrinsic value (aka exercise value)?Explanation / Answer
Given the probability of expected prices of the stock at the end of the year
- > Stock price at the end of the year = E(S) = Expected Value of stock price based on the given probabilities
-> E(S) = Sum of the product of each price and its probability of occurrence
-> E(S) = (80*10%)+(90*20%)+(100*40%)+(110*20%)+(120*10%)
-> E(S) = 100
So, stock price at the end of the year = 100
1a) Since given that time discount rate is zero; the stock price at the beginning of the year i.e today = 100/(1+r) = 100/(1+0) = 100
Therefore, stock price today = 100
1b) A Put option is said to be (ITM) in-the-money when its strike price is greater than the stock price so that the option is in profit even if the stock price remains constant (Put option is profitable only when the stock price goes below the strike price else the payoff from put option will be zero)
Therefore, the 95 strike price PUT option will be ITM when the stock price is < 95
Stock price < 95 has two possibilities i.e stock price of 80 & 90 according to the given table of values.
The probability of 95 strike put will expire ITM = Probability that stock price is < 95 = probability of stock price is 80 + Probability of stock price is 90
The probability of 95 strike put will expire ITM = 10% + 20%
-> The probability of 95 strike put will expire ITM = 30%
1c) As we saw above that the condition for the underlying stock when 95 strike put expires ITM is that the stock price should be either 80 or 90.
Therefore, the conditional average price of the stock = Average of the stock price of either 80 or 90
-> the conditional average price of stock = (80 + 90)/2 = 85
1d) Based ob 1b & 1c, the 95 strike put price = Ke-r(T-t) -St
where K = Strike price = 95, St = Stock Price today = 85, r = 0 (time discount rate) & (T-t) = 1
Price = (95*e-0*1) - 85= (95*1) - 85 = 95 - 85 = 10
Therefore Price of the 95 strike put = 10
1e) redoing 1b,1c & 1d for 95 Call
A Call option is said to be (ITM) in-the-money when its strike price is less than the stock price so that the option is in profit even if the stock price remains constant (Call option is profitable only when the stock price goes above the strike price else the payoff from call option will be zero)
Therefore, the 95 strike price CALL option will be ITM when the stock price is > 95
Stock price > 95 has three possibilities i.e stock price of 100, 110 & 120 according to the given table of values.
The probability of 95 strike call will expire ITM = Probability that stock price is > 95 = probability of stock price is 100 + Probability of stock price is 110 + Probability of stock price is 120
The probability of 95 strike call will expire ITM = 40% + 20% + 10% = 70%
-> The probability of 95 strike call will expire ITM = 70%
As we saw above that the condition for the underlying stock when 95 strike call expires ITM is that the stock price should be either 100, 110 or 120.
Therefore, the conditional average price of the stock = Average of the stock price of either 100 or 110 or 120
-> the conditional average price of stock = (100+110+120)/3 = 110
the 95 strike call price = St- Ke-r(T-t)
where K = Strike price = 95, St = Stock Price today = 110, r = 0 (time discount rate) & (T-t) = 1
Price = 110 - (95*e-0*1) = 110 - (95*1) = 110 - 95 = 15
Therefore Price of the 95 strike call = 15
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