Trying to finish homework for a risk management class.

Trying to finish homework for a risk management class.
Risk Management Homework Short answers (2-points each) When E -S < 0 a put option is said to be The Put-Call Parity points to a mispricing. The Fiduciary Call value is greater than the value of the Protective Put. To take advantage of the arbitrage opportunity you should sell the and buy the When S > E the Call option is said to be Applying the Put-Call Parity relationship, create a synthetic Call Problems: Show all work for full credit. If I cannot read your writing, it will be considered wrong. Tabulate the value at expiration for each of the following portfolios. Graph each value and Payoff function. Be sure to label breakeven points. All options are European, written on the same stock, with the same expiration date. Long put E = 55, P55=$3.25, two Short Puts E = $60, P60 = $7, long put E=65 P65 = $10 (10-pts.) A straddle is a long Call E=$35 C35=$2.50 and a long Put E = $35 P35=$1.75 (10-pts) A stock is trading for $63 per share. A 3-month (90-day) call and put options are written on the stock with an exercise price of $60 are trading for $6.00 and $2 respectively. Create a 90- day Synthetic Investment. (8-points) Calculate your return on investment (the implied RF in the prevailing prices). ( 8-points) A stock is trading for $172. A July Put and Call expiring in 50 days with an exercise price E=$175 are trading on the stock. The 50-day Risk Free Rate is 2.0% , the Call Premium, C=$ 6.0 and the Put price, P= $5.50 Applying the Put Call parity, does an arbitrage opportunity exist? If so, how would you take advantage of the mispricing? (10-points) Calculate the arbitrage profit. (10-points) A stock is currently trading at $82. The stock volatility as expressed by its standard deviation is 18%. A Call with an exercise price of $80 is expiring in 90 days. The 3-month treasury is selling to yield 2%. All options are European, written on the same stock, with the same expiration date. (36- points) Create a 3- period binomial lattice. Provide the Call values at each node. For the first and second periods, calculate the time value of the option at each node. What should be the fair value of the Call option today? Given the fair value of the Call option, calculate the fair value of the Put option today. Page 2 of 2
Trying to finish homework for a risk management class.
Spreads A trade is referred to as a “spread” when it involves the purchase of one option and the sale of the other on the same underlying asset. Buying a spread entails a cash outflow i.e. COF > CIF called a debit spread Selling a spread entails a cash inflow i.e. CIF > COF called a credit spread Spreads limit risk while offering a potential for small profits. The reduced risk is a consequence of being both in a long and a short position simultaneously. Types of Spreads: Vertical Spread or Money Spread Options have different exercise prices. Horizontal or calendar spreads Options have different time to expiration Bull Money Spreads A bull money spread is a combination of options designed to profit if the price of the UA rises. 1. Bull money spread with calls Same underlying asset, same time to maturity. Long call E1 ( E1 < ST at t=0 ) Short call E2 ( E2 > ST at t=0 ) E1 < E2 Remember a trade is referred to as a “spread when it involves buying an option and selling an option on the same UA.   S E2 Long Call E1 S – E1 S – E1 Short Call E2 -(S – E2)   S – E1 E2 – E1 Note: Both options payoff when S > E2 Bull money spread the stock price > E1 +C1 – C2 Bull money spread limits the traders risk in comparison to a long position in the UA Bull money spreads also limit the profit to E2 – E1 Holding Period: Early exercise – not a problem Short holding period has the lowest range of profits When the stock price is low: Short holding period has the lowest loss. Long HP produces the higher loss The E1 option is always worth more than the E2 option however the relative time values differ Note that the option’s time value is greatest when the call is at the money. In other words when E = ST. When the stock price is high, E2 will have the greatest time value. When the stock price is low, E1 will have the greatest time value. If we close out the spread prior to expiration i.e. sell the call with E1 and buy the call with E2, The long call E1 will sell for more than the short call E2 At a high stock price: C1 – C2 diminishes. The longer call with E1 has the higher intrinsic value but the short call with E2 has the greater time value. Profit is lower with early close At a low stock price: The long call will have the greater time value. Profit is higher than holding to expiration. Which Holding period should an investor choose? If investor bet correctly, the longer the position is held the greater the profit. Longer holding periods gives P more time to move. If investor bet incorrectly, the shorter the HP the lower the loss, but S has less time to move Conclusion: It all depends Example: The following information is available to a trader: The stock is currently trading at $105 A call with E1 = 100, C1 = $7.0 Long Position A call with E2 = 110, C2 = $3.0 Short Position Bull money spread with call options: Total outlay $4, Max profit $6   S<100 100 < S < 110 S > 110 Long Call E1 S – 100 S – 100 Short Call E2 -(S – 110)   S – 100 10 You pay for the spread now with calls. 2. Bull money spread with Puts. Same underlying asset, same time to maturity. Long put E1 ( E1 < ST at t=0 ) Short put E2 ( E2 > ST at t=0 ) E1 < E2 Remember a trade is referred to as a “spread when it involves buying an option and selling an option on the same UA.   S E2 Long Put E1 E1 – S Short Put E2 -( E2 – S ) -( E2 – S )   E1 – E2 S – E2 You get paid now with puts There is a risk with early exercise Loss occurring before expiration has a higher PV than at expiration. Relationship between calls and puts in a Bull Money Spread   S E2 Call Spread S – E1 E2 – E1 Put Spread E1 – E2 S – E2 Difference X E2 – E1 E2 – E1 E2 – E1 Vc1 – Vc2 = Vp1 – Vp2 + X X = ( Vc1 – Vc2 ) – ( Vp1 – Vp2 ) Where Vc, Vp are the call and put values at expiration Arbitrage Relationship #2 Vc1 – Vc2 = Vp1 – Vp2 + E2 – E1 Where E2 – E1 is a riskless asset paying E2 – E1 at maturity Then today the following relationship should hold C1 – C2 = P1 – P2 + PV(E2 – E1) Arbitrage: You can only have 2 possible situations: Bull money spread where E1 < E2 1. C1 – C2 < P1 – P2 +( E2 – E1 ) eRF x t Long Call 1 Short Call 2 Short Put 1 Long Put 2 Borrow PV ( E2 – E1 ) 2. C1 – C2 > P1 – P2 +( E2 – E1 ) eRF x t Short Call 1 Long Call 2 Long Put 1 Short Put 2 Invest PV ( E2 – E1 ) Example: Price Call1 $10 100 Call 2 $5 110 Put 1 $3 100 Put2 $6 110 RF = 10% for the period PV (E2 – E1 ) = 10 / ( 1+ 0.1) = 9.09 C1 – C2 = P1 – P2 + ( E2 – E1 ) eRF x t 10 – 5 ≠ 3 – 6 + 9.09 5 < 6.09 Arbitrage exists To take advantage: Long Call 1 Long Put 2 Short Call 2 Short Put 1 Borrow $ 9.09 at the risk free rate In other words: C1 + P2 = P1 + C2 + ( E2 – E1 ) eRF x t Now Later Long Call 1 -10 Vc1 Short Call 1 (Vc2) Long Put 2 -6 Vp2 Short Put 1 (Vp1) Borrow 9.09 -10 1.09 Must be 0 Bear Money Spreads Long put E2 ( E2 > ST at t=0 ) Short put E1 ( E1 < ST at t=0 ) E1 < E2 Bear money spreads are the mirror image of a bull money spread. Bear money spreads limits the profits in bear markets to E2 – E1 Limits the loss in bull markets   S E2 Long Put E2 E2 – S E2 – S Short Put E1 -(E1 – S) Portfolio Value E2 – E1 E2 – S Profits: When S E P ; E1 < S0 ; E2 > S0 A collar is equivalent to a bull spread plus a RF bond paying E1 at expiration.   S E2 Long Stock Long Put E1 E1 – S Short Call E2 ( S – E2) Portfolio Value E1 E2 Profits: When S S0 profit on the stock is either E1 – S0– P +C or E2 – S0– P +C Note: E1 – S0– P +C is negative or a loss. In other words the potential gain or loss is limited. When E1 = S0 & E2 > S0 profit on the stock is either C – P or E2 – S0– P +C C – P is usually negative, loss is minimized Only in the middle range is there uncertainty Investors give up upward gains ( selling @ a max of E2 ) for the reassurance that the stock cannot be sold for less than E1. Collars are usually used with index options to protect portfolios not individual stocks. Notice that a collar is similar to bull money spread. In actuality a collar is equivalent to a bull money spread with calls plus a RF bond paying E1 at expiration. Holding Period: Before expiration we recoup more time value on the long put than we pay to buy back the short call. As S↓ the longer we hold the less we gain. As S ↑ the earlier we closeout the position the more time value we must pay for than we receive from selling the put. Because E2 > E1: As S↓ time value is greater on the long put As S ↑ time value is greater on the short call Butterfly Spread Combination of a bull and bear money spread Involves 3 exercise prices E1, E2 and E3. Where E2 is half way between E1 and E3. Call bull spread: Call Bear Spread: Long Call E1 Long Call E3 Short Call E2 Short Call E2 Put them together: Long Call E1 2 Short Calls E2 Long Call E3   S E3 Long Call E1 S – E1 S – E1 S – E1 2 short Calls E2 -2(S – E2) 2(S – E2) Long Call E3 S – E3 Portfolio value S – E1 2E2 – E1 – S 2E2 – E1 – E3 Profit: When S E3: 2E2 – E1 – E2 + 2C2 – C1 – C3 Two Break even points: St – E1 – C1 +2C2 – C3 = 0 → ST = E1+ C1 – 2 C2 + C3 In other words: A butterfly spread will be profitable if the S > E1 by at least C1 – 2C2 + C3 Butterfly spreads have a limited loss which occurs when ST moves away from E2 and a limited gain which peaks at ST = E2 Low risk transaction Straddles Long position in a call E Long position in a put E Both put and the call expire on the same day. Has 2 break even points. Holding both a put and a call with the same exercise price and expiration date, allows the investor to capitalize on stock price movement in either direction. Investors should invest in a straddle to profit from expected large swings in share prices in either direction. When would that be appropriate? Example: An investor is interested in investing in Pox Pharmaceutical which is currently trading at $25 a share. Pending an FDA approval, Pox share price is expected to rise sharply. If however approval is denied share price is expected to tumble. Available to the investor are : March Call; E = $25, C = $5 March Put; E = $25, P = $3 Long call, Long Put ; Same E, same maturity, total cost C + P   S<25 S>25 Long Call S – 25 Long Put 25 – S Portfolio Payoff 25 – S S – 25 Net-profit-loss diagram with breakeven points and maximum loss. 25 17 0 Stock Price 17 25 33 -8 This is called a straddle Maximum loss would be $8 Potential of unlimited gain on the up side Break even points are $17 and $33 Straddle tend to provide higher potential profit but also the greatest potential loss. For a given stock price, a straddle losses value as it nears expiration. This is due to loss of time value on both options. If FDA delays approval before expiration investor will lose C + P What happens if the options available to the investors are: March call, E = $30, C = $3 March Put, E = $20, P = $2 How would the investor’s strategy change?   S<20 2030 Long Put 20 20 – S Long Call 30 S – 30 Portfolio payoff 20 – S S – 30 20 15 0 15 20 30 35 Stock Price -5 This is called a Strangle. It involves a low max loss of $5, but a greater likelihood of occurrence.