CFA 59: Risk Management Applications of Options Strategies

Consider a call option selling for $4 in which the exercise price is $50. Determine the value at expiration and the profit for a buyer under the following outcomes: The price of the underlying at expiration is $55. The price of the underlying at expiration is $51. The price of the underlying at expiration is $48. Determine the value at expiration and the profit for a seller under the following outcomes: The price of the underlying at expiration is $49. The price of the underlying at expiration is $52. The price of the underlying at expiration is $55. Determine the following: The maximum profit to the buyer (maximum loss to the seller). The maximum loss to the buyer (maximum profit to the seller). Determine the breakeven price of the underlying at expiration. Answer- Call buyer cT=max(0,ST−X)=max(0,55−50)=5 ∏=cT−c0=5−4=1 cT=max(0,ST−X)=max(0,51−50)=1 ∏=cT−c0=1−4=−3 cT=max(0,ST−X)=max(0,48−50)=0 ∏=cT−c0=0−4=−4 Call seller Value=−cT=−max(0,ST−X)=−max(0,49−50)=0 ∏=−cT+c0=−0+4=4 Value=−cT=−max(0,ST−X)=−max(0,52−50)=−2 ∏=−cT+c0=−2+4=2 Value=−cT=−max(0,ST−X)=−max(0,55−50)=−5 ∏=−cT+c0=−5+4=−1 Maximum and minimum Maximum profit to buyer (loss to seller) = ∞ Maximum loss to buyer (profit to seller) = c0 = 4 ST* = X + c0 = 50 + 4 = 54 Suppose you believe that the price of a particular underlying, currently selling at $99, is going to increase substantially in the next six months. You decide to purchase a call option expiring in six months on this underlying. The call option has an exercise price of $105 and sells for $7. Determine the profit under the following outcomes for the price of the underlying six months from now: $99. $104. $105. $109. $112. $115. Determine the breakeven price of the underlying at expiration. Check that your answer is consistent with the solution to Part A of this problem. Answer- cT=max(0,ST−X)=max(0,99−105)=0 ∏=cT−c0=0−7=−7 cT=max(0,ST−X)=max(0,104−105)=0 ∏=cT−c0=0−7=−7 cT=max(0,ST−X)=max(0,105−105)=0 ∏=cT−c0=0−7=−7 cT=max(0,ST−X)=max(0,109−105)=4 ∏=cT−c0=4−7=−3 cT=max(0,ST−X)=max(0,112−105)=7 ∏=cT−c0=7−7=0 cT=max(0,ST−X)=max(0,115−105)=10 ∏=cT−c0=10−7=3 ST* = X + c0 = 105 + 7 = 112 Clearly, this result is consistent with our solution above, where the profit is exactly zero in Part A(v), in which the price at expiration is 112. Consider a put option on the NASDAQ 100 selling for $106.25 in which the exercise price is 2100. Determine the value at expiration and the profit for a buyer under the following outcomes: The price of the underlying at expiration is 2125. The price of the underlying at expiration is 2050. The price of the underlying at expiration is 1950. Determine the value at expiration and the profit for a seller under the following outcomes: The price of the underlying at expiration is 1975. The price of the underlying at expiration is 2150. Determine the following: The maximum profit to the buyer (maximum loss to the seller). The maximum loss to the buyer (maximum profit to the seller). Determine the breakeven price of the underlying at expiration. Answer- Put buyer pT=max(0,X−ST)=max(0,2100−2125)=0 ∏=pT−p0=0−106.25=−106.25 pT=max(0,X−ST)=max(0,2100−2050)=50 ∏=pT−p0=50−106.25=−56.25 pT=max(0,X−ST)=max(0,2100−1950)=150 ∏=pT−p0=150−106.25=43.75 Put seller Value=−pT=−max(0,X−ST)=−max(0,2100−1975)=−125 ∏=−pT+p0=−125+106.25=−18.75 Value=−pT=−max(0,X−ST)=−max(0,2100−2150)=0 ∏=−pT+p0=−0+106.25=106.25 Maximum and minimum Maximumprofittobuyer(losstoseller)=X−p0=2100−106.25=1993.75 Maximumlosstobuyer(profittoseller)=p0=106.25 ST* = X - p0 = 2100 - 106.25 = 1993.75 Suppose you believe that the price of a particular underlying, currently selling at $99, will decrease considerably in the next six months. You decide to purchase a put option expiring in six months on this underlying. The put option has an exercise price of $95 and sells for $5. Determine the profit for you under the following outcomes for the price of the underlying six months from now: $100. $95. $93. $90. $85. Determine the breakeven price of the underlying at expiration. Check that your answer is consistent with the solution to Part A of this problem. What is the maximum profit that you can have? At what expiration price of the underlying would this profit be realized? Answer- pT=max(0,X−ST)=max(0,95−100)=0 ∏=pT−p0=0−5=−5 pT=max(0,X−ST)=max(0,95−95)=0 ∏=pT−p0=0−5=−5 pT=max(0,X−ST)=max(0,95−93)=2 ∏=pT−p0=2−5=−3 pT=max(0,X−ST)=max(0,95−90)=5 ∏=pT−p0=5−5=0 pT=max(0,X−ST)=max(0,95−85)=10 ∏=pT−p0=10−5=5 ST* = X - p0 = 95 - 5 = 90 Clearly, this result is consistent with our solution above, where the profit is exactly zero in Part A(iv), in which the price at expiration is 90. Maximum profit (to put buyer) = X - p0 = 95 - 5 = 90. This profit would be realized in the unlikely scenario of the price of the underlying falling all the way down to zero. You simultaneously purchase an underlying priced at $77 and write a call option on it with an exercise price of $80 and selling at $6. What is the term commonly used for the position that you have taken? Determine the value at expiration and the profit for your strategy under the following outcomes: The price of the underlying at expiration is $70. The price of the underlying at expiration is $75. The price of the underlying at expiration is $80. The price of the underlying at expiration is $85. Determine the following: The maximum profit. The maximum loss. The expiration price of the underlying at which you would realize the maximum profit. The expiration price of the underlying at which you would incur the maximum loss. Determine the breakeven price at expiration. Answer- This position is commonly called a covered call. VT=ST−max(0,ST−X)=70−max(0,70−80)=70−0=70 ∏=VT−V0=70−(S0−c0)=70−(77−6)=70−71=−1 VT=ST−max(0,ST−X)=75−max(0,75−80)=75−0=75 ∏=VT−V0=75−(S0−c0)=75−(77−6)=4 VT=ST−max(0,ST−X)=80−max(0,80−80)=80−0=80 ∏=VT−V0=80−(S0−c0)=80−(77−6)=9 VT=ST−max(0,ST−X)=85−max(0,85−80)=85−5=80 ∏=VT−V0=80−(S0−c0)=80−(77−6)=9 Maximum profit = X - S0 + c0 = 80 - 77 + 6 = 9 Maximum loss = S0 - c0 = 77 - 6 = 71 The maximum profit would be realized if the expiration price of the underlying is at or above the exercise price of $80. The maximum loss would be incurred if the underlying price drops to zero. ST* = S0 - c0 = 77 - 6 = 71 Suppose you simultaneously purchase an underlying priced at $77 and a put option on it, with an exercise price of $75 and selling at $3. What is the term commonly used for the position that you have taken? Determine the value at expiration and the profit for your strategy under the following outcomes: The price of the underlying at expiration is $70. The price of the underlying at expiration is $75. The price of the underlying at expiration is $80. The price of the underlying at expiration is $85. The price of the underlying at expiration is $90. Determine the following: The maximum profit. The maximum loss. The expiration price of the underlying at which you would incur the maximum loss. Determine the breakeven price at expiration. Answer- This position is commonly