BFC5915 Workshop 7 Suggested solutions 1BFC5915 Options, futures and risk managementWorkshop Suggested SolutionsQuestion 1Before we start, it is important to understand what the Black-Scholes formula does and doesnot apply to. It gives the fair value of a European call option written on a stock that does notpay dividends (or, at least, doesn’t pay dividends during the remaining life of the call option).It’s always a good practice to write down the key inputs that will be required to the variousformulae.S0 = $5.20 X = $4 T = 0.5 years r = 0.10 σ = 0.90a) Start by calculating d1 and d2, get probabilities from the tables, then plug into BlackScholes:( )[ ( ) ( ) ]0.8090 0.90 0.5 0.1726.ln 5.20 4.00 0.10 0.81 0.5 0.80900.90 0.511 ln 0.52 1120 21= – == –= + + = + + =d d Tr TSXTdσσσSince the probability tables only go to two decimal places, I will round these calcs totwo decimal places. Students who obtain probabilities using NORM.S.DIST in Excel(or their HP calculator) will have a slightly more accurate answer.N(0.81) = 0.7910 N(0.17) = 0.5675C E = (5.20 × 0.7910)- 4.00e -0.10×0.5(0.5675) = 1.95.The fair price for this call option is $1.95. An option’s value can be divided into intrinsicand time value components. The intrinsic value of a call is max(0, S – X). Whenever anoption is in-the-money, it will have some intrinsic value, in this case $1.20. And so longas there is some time left before the option expires, it will also have some time value,in this case $0.75. The time value reflects the likelihood that, in the remaining sixmonths before expiry, stock price can move even higher.BFC5915 Workshop 7 Suggested solutions 2b) Even before we complete the calculation, we know that the call option price will belower than in part a. There is a positive relation between call option price and the priceof the underlying share. The holder of a call option wants share price to rise far abovethe strike price. In part a, share price was $5.20, so the call was very much in the money.In part b, share price is only $3.80 (call option now out of the money). So the chancesof this call finishing in the money and providing a payoff are lower. This is reflected ina lower call option value.( )( )0.95.0.3202 0.32 0.37450.3162 0.32 0.62551 2== – – == =CEd Nd NGiven that this call option is out-of-the-money, the intrinsic value is zero. However, thecall is still worth $0.95 since there is some chance that, over the remaining six months,share price will move above the $4 exercise price so that the call is in-the-money andcan be exercised. Thus, the $0.95 is all time value.c) There is an inverse relationship between a call option’s exercise price and its value. Allelse equal, a call with X = $3 will have a greater value than a call with X = $4. This isbecause call options are profitable when share price rises above the exercise price.Clearly, this is more likely for a call with a lower exercise price. The value below($2.57) exceeds that in part a ($1.95).( )( )126.96.36.19947 0.62 0.73241.2611 1.26 0.89621 2== == =CEd Nd NThe total call premium ($2.57) can be divided into intrinsic value ($2.20) and time value($0.37).d) There is a positive relationship between time to maturity and option value (this is truefor both calls and puts). The reason is that a longer expiry date gives the option moretime to move into the money (“time is your friend”). So the call option in part d will bemore expensive than the call option in part a.( )( )2.21.0.0431 0.04 0.51600.8226 0.82 0.79391 2== == =CEd Nd NYou might ask: with more time to expiry, there is more time for the option to move inthe money, but there is also more time for the option to move out of the money. This isBFC5915 Workshop 7 Suggested solutions 3true. But options give the holder flexibility. If the former happens, great, we exercise itand make money. If the option moves out of the money, we just let it lapse. So we arenot overly concerned about the possibility that the extra time to expiry may see theoption move out of the money.Question 2Having calculated Black-Scholes prices for European call options, the price for thecorresponding put option is given by put-call parity. By ‘corresponding’, I mean a put writtenon the same stock, with the same exercise price, and the same time to maturity.Let’s be clear as to what the put-call parity formula works for. It prices a European put option,written on a stock that will not pay a dividend (over the remaining life of the option).a) PE = 1.95 – 5.20 + 4.00e-0.10×0.5 = 0.56.This put is out of the money (current share price of $5.20 is greater than the $4 strike),so its intrinsic value is zero. Nonetheless, an out-of-the-money put still has time value.The entire $0.56 is time value reflecting the possibility that, over the remaining life,share price might fall below the strike and the put option will move into the money.b) PE = 0.95 – 3.80 + 4.00e-0.10×0.5 = 0.96.Now that share price ($3.80) is below the $4 strike, this put is in the money. It’s intrinsicvalue is $0.20 (it is 20c in the money). This means that its time value is $0.76.c) PE = 2.57 – 5.20 + 3.00e-0.10×0.5 = 0.22.The value of the put ($0.22) when strike price is $3 is much lower than in part a ($0.56)when strike price was $4. With a put option, you need share price to get below the strikeprice. The lower the strike price, the more unlikely it is that share price will fall belowit. Hence, there is a direct relationship between strike price and put value.d) PE = 2.21- 5.20 + 4.00e -0.10×0.75 = 0.72.Just like a call option, the put option value increases with more time to expiry. Thereasons are the same as given in Question 1 part d. With more time before expiry, thereis more time for share price to fall below the strike price and move the put into themoney.BFC5915 Workshop 7 Suggested solutions 4Question 3This question examines the arbitrage opportunities which exist if the upper bounds areviolated. The first task is to determine whether a bound has been violated. The second is toestablish a trading strategy designed to capture an arbitrage profit. Third, prove that our tradingstrategy does indeed produce an arbitrage profit.There are different ways to structure the tables to demonstrate an arbitrage profit (all are valid).My personal favourite is to devise a strategy to be implemented today that has zero setup costtoday. And then show that this strategy will give a certain/riskless positive payoff in the future.That is sufficient to prove that it is an arbitrage profit.a) The upper bound on the premium of a call option (regardless of whether it is Europeanor American) is the current share price. In this question, the upper bound is violatedsince the call premium ($9) exceeds the share price ($8.50).Think about it. Why would anyone pay $9 to get a long call option, that gives you theright to pay the $10 strike price to receive the underlying share? That would be crazy,since the underlying share itself only costs 8.50! Hence, the call is overpriced thusproviding a clue that the required strategy is to short the call (when something isovervalued, you want to short it).That’s the easy bit. The harder part is to know what else to do now to capture thearbitrage profit. With a short call, we may be forced to sell the shares, so we can protectourselves by buying the shares today.Note also that, with options, there are two distinct possibilities at expiry: either they arein-the-money, or out-of-the-money. Hence, my payoff tables have a column for eachpossible scenario at expiry.
Cashflow at Maturity
ST < X
ST > X
Short call: rec premiumBuy sharesInvest surplus in bank
+ 9.00– 8.50– 0.50
Nil+ ST0.53 = 0.50 exp(0.06×1)
-(ST – 10)+ST0.53 = 0.50 exp(0.06×1)
ST + 0.53
In the event that the call is exercised against us (ST > X), the long party will make aprofit of (ST – 10), so we will lose the same amount. The long share we purchased atTime 0 can now be sold for the market price ST. Overall, we pocket $10.53. Conversely,if the call finishes out-of-the-money, we pocket ST + 0.53. Even if share price is zero,we’re still ahead by $0.53.Therefore, regardless of whether ST < X or ST > X, the net cashflow at maturity ispositive. Thus, for zero upfront cost, we have a strategy that guarantees a positivepayoff in the future – an arbitrage opportunity!BFC5915 Workshop 7 Suggested solutions 5b) The upper bound on an American put option is the strike price X. Why? A put is a rightto sell shares at the strike. The most money you can make with a long put is $X (thishappens if share price drops to zero). Thus, a put option cannot trade for more than X.With strike of $5 and an option premium of $5.50, the upper bound is violated. Wewould short the put option (very happy to receive a premium that we know is toohigh).Examine the following table and recall that the American put can be exercised at anytime up to and including maturity:
Cashflow at Maturity1
ST < X
ST > X
Short put rec premiumInvest surplus in bank
+ 5.50– 5.50
-(5.00 – ST)5.84=5.50 exp(0.06×1)
ST + 0.84
If the put finishes out-of-the-money (ST > X) , it will not be exercised against us andwe pocket $5.84.If the put option finishes in the money (ST < X), it will be exercised against us. Thecounterparty will sell the share to us and we are forced to pay the $5 strike. We wouldthen sell this share in the market at whatever the market price is (ST). The loss on theshort put is –(5 – ST). The total cashflow is ST + 0.84. So we are guaranteed of gettinga positive cashflow. If ST = $3, we get $3.84. If ST = $2, we get $2.84. Even if shareprice is zero, we still make $0.84.Therefore, regardless of whether ST < X or ST > X, the net cashflow at maturity ispositive. Thus, for no upfront cost, we have a strategy which guarantees a positivepayoff in the future – an arbitrage opportunity!1 Even though this table says ‘at maturity time T’, the calculations apply to whenever the put option is exercised.It is, after all, an American-style put.BFC5915 Workshop 7 Suggested solutions 6Question 4 (*)This question examines violations of lower bounds. As with upper bounds, it would be rareto see violations of this bound in practice. If traded option prices did violate a lower bound,traders would implement the strategies we use below to capture arbitrage profits. Sooner ratherthan later, the trading by arbitrageurs would move option prices to the point where arbitrageopportunities vanish.Parts a and b below are absolutely trivial because the violations involve American-styleoptions. Thus, we can capture the arbitrage profit immediately. Part c involves a Europeanoption, so the strategy is a little more difficult because we have to wait to expiry to exercise theoption.a) The lower bound on an American call option is:[ ][ ]1.51.max 0,6.00 4.50max 0,0.05 3 / 520≥≥ –≥ –– ×–eC S Xe rTA.Note that this assumes that the company pays no dividends (and hence, the call optionwill not be exercised prior to maturity). If the option is trading at $1.20, this violatesthe lower bound. The fact that it is an American option makes it trivial to generate thearbitrage profit:
Enter long call optionImmediately exercise right to buy sharesImmediately sell shares on market
– 1.20– 4.50+ 6.00
b) The lower bound on an American-style put option is:[ ][ ]0.60.max 0,7.50 6.90max 0, 0≥≥ –P ≥ X – SASince the put option is trading at $0.35, this violates the lower bound.
Enter long put option
Buy shares on marketImmediately exercise right to sellArbitrage profit
– 6.90+ 7.50+ 0.25
2 Note that this calculation ignores transactions costs like brokerage. In practice, we sometimes see apparentviolations on bounds, but after taking transactions costs into account, you cannot make money!BFC5915 Workshop 7 Suggested solutions 7c) The lower bound on a European-style call option is:[ ][ ]2.29.max 0,10 8max 0,0.05 9 /120≥≥ –≥ –– ×–eC S Xe rTEIf the call trades at $2, it is violating the lower bound. Since the call is European-style,we cannot immediately capture the arbitrage profit. However, we can set up a strategywhich costs nothing now and guarantees a positive payoff on maturity.The strategy requires buying the call, short selling the stock, and investing the surplusin the bank. Deciding to long the call is the easy part – it is underpriced so we certainlywant to buy it. Next, think about what happens if this call finishes in the money. Wewould exercise our right to buy the underlying share. With this in mind, we would shortsell the share today. If the option finishes in the money and we buy a share, we can usethat share to close out our short sale. Finally, the $8 that is left over upfront can beinvested to earn some interest.
Cashflow at Maturity
ST < X
ST > X
Long call optionShort sell sharesInvest surplus in bank
– 2.00+ 10.00– 8.00
Nil– ST+8.00 exp(0.05×9/12)
ST – 8.00-ST+8.00 exp(0.05×9/12)
8.31 – ST
Note that, if ST < X, the net cashflow is positive since ST < $8. If the call finishes in-themoney, we exercise our right to buy shares at strike $8 and out profit is (ST – 8). Wethen settle up our short sale by purchasing shares at the market price ST. Theaccumulated savings more than cover the cost of exercising the option, leaving a $0.31cashflow.nb: despite this question providing details on the standard deviation of the stock, thereis no need to calculate option price using Black-Scholes.BFC5915 Workshop 7 Suggested solutions 8Question 5a) With a call option, you want the share price to rise high above the strike price. So when theshare price rises, this is good for the holder of a call option. Hence the call option valueincreases. The opposite is true for put options (put options give a payoff when share priceis below the strike). So, as share price rises, this hurts the value of a put option.
AAPL share price
Call option value
Put option value
b) Using the same logic as part (a), it follows that – all else equal – call options with a higherstrike price are less valuable. The higher the strike, the less likely it is for share price to riseabove the strike and get the call option in the money. Conversely, for put options, the higherthe strike, the more likely it is that share price will be below the strike meaning the put isin the money.
Option strike price
Call option value
Put option value
c) With call options, you want share price rising above the strike. With put options, you wantshare price falling below the strike. The more time there is before the option expires, themore time there is for share price to move in a favourable direction for you. Hence, thehigher the option value. Time is your friend!
Time to expiry (in years)
Call option value
Put option value
BFC5915 Workshop 7 Suggested solutions 9d) Volatility is a very important influence on option prices. To get a payoff on the option, youneed share price to move in a favourable direction (up for calls, and down for puts). Themore volatile the underlying share, the better your chances of this happening. Hence, thevalue of both call and put options is positively related to volatility.
Volatility (σ) ofAAPL returns
e) The riskfree rate of interest does factor into option values, but its influence is minor. Forexample, if the riskfree rate doubles from say 2% to 4%, there is a modest change in optionvalues. In contrast, we saw above that if volatility doubled, or time to expiry doubled, thereis a large impact on option values.
Riskfree rateof interest
Question 6a) Pricing an American call with 6 step binomial tree: C=1.2735b) Pricing an American call with 10 step binomial tree: C=1.2812. The 10 step tree isdisplayed below. The 10 step tree is the biggest tree displayed in my version of thesoftware.BFC5915 Workshop 7 Suggested solutions 10You can see the nodes where exercise is optimal. The cell entries are in red. It is clear from thetree that the only time it is optimal to exercise the American call option on a non-dividendpaying stock is at expiry of the option.c) Price of call option with 18 step tree is 1.2844.d) BSM price of European call option is 1.2796.e) It is never optimal to exercise an American call option on a non-dividend paying stockearly. We showed that in Lecture 7. That means that an American-style call option anda European-style call option on a non-dividend paying stock will have the same price.f) Interesting that the binomial price with 12 steps is closer to the analytic price than the18-step binomial price. The binomial price converges to the BSM but oscillates, firstabove and then below etc.g) The binomial price converges to the BSM price for American call options.Question 7One really important point to note, is that the BSM model is not able to price an American putoption, because it is unable to take account of the early exercise possibility in the analyticalapproach that is used. (In mathematical terms we have the situation where we would have toBFC5915 Workshop 7 Suggested solutions 11solve the partial differential equations, with free boundary conditions. In the European optioncase the boundary conditions are fixed – at expiry).a) Pricing an American put with 6 step binomial tree: P= 0.7956b) Pricing an American put with 10 step binomial tree: P=0.8048. The 10 step tree isdisplayed below. The 10 step tree is the biggest tree displayed in my version of thesoftware.You can see the nodes where early exercise is optimal. The cell entries are in red. One way toidentify these nodes is to say the ddddd node. That is the node reached by five successive downmoves in the tree. Another node where early exercise is optimal is ddddddu node. That is, thenode reached by six successive down moves followed by one up move in the tree, etc.c) Price of put option with 18 step tree is 0.8088.d) Price observed in the market is $0.75, lower than the $0.81 derived from an 18-stepbinomial tree. Explanations could include: estimated volatility of 40% is too high – aBFC5915 Workshop 7 Suggested solutions 12lower volatility would give a lower option price. The price in the market is too low andrepresents an arbitrage opportunity? Can you think of other explanaions?e) BSM price of European put option is 0.7874f) BSM cannot price the early exercise flexibility of an American put option. TheAmerican-style put option is worth more than the European-style put option.g) The binomial model provides a method to price American put options, because it cantake account of the early exercise flexibility by replacing the fair (arbitrage-free) valuewith the early exercise vale at the nodes in the tree where it is optimal to exercise theoption early.
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