BFC5915 Workshop 7 Suggested solutions 1 BFC5915 Options, futures and risk management Workshop Suggested Solutions Question 1 Before we start, it is important to understand what the Black-Scholes formula does and does not apply to. It gives the fair value of a European call option written on a stock that does not pay 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 various formulae. S0 = $5.20 X = $4 T = 0.5 years r = 0.10 σ = 0.90 a) Start by calculating d1 and d2, get probabilities from the tables, then plug into Black Scholes: ( ) [ ( ) ( ) ] 0.8090 0.90 0.5 0.1726. ln 5.20 4.00 0.10 0.81 0.5 0.8090 0.90 0.5 1 1 ln 0.5 2 1 12 0 2 1 = – = = – = + + = + + = d d T r T SX T d σ σ σ Since the probability tables only go to two decimal places, I will round these calcs to two 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) = 0.5675 C 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 intrinsic and time value components. The intrinsic value of a call is max(0, S – X). Whenever an option is in-the-money, it will have some intrinsic value, in this case $1.20. And so long as 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 six months before expiry, stock price can move even higher. BFC5915 Workshop 7 Suggested solutions 2 b) Even before we complete the calculation, we know that the call option price will be lower than in part a. There is a positive relation between call option price and the price of the underlying share. The holder of a call option wants share price to rise far above the 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 chances of this call finishing in the money and providing a payoff are lower. This is reflected in a lower call option value. ( ) ( ) 0.95. 0.3202 0.32 0.3745 0.3162 0.32 0.6255 1 2 = = – – = = = CE d N d N Given that this call option is out-of-the-money, the intrinsic value is zero. However, the call 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 and can 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. All else equal, a call with X = $3 will have a greater value than a call with X = $4. This is because 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). ( ) ( ) 2.57. 0.6247 0.62 0.7324 1.2611 1.26 0.8962 1 2 = = = = = CE d N d N The 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 true for both calls and puts). The reason is that a longer expiry date gives the option more time to move into the money (“time is your friend”). So the call option in part d will be more expensive than the call option in part a. ( ) ( ) 2.21. 0.0431 0.04 0.5160 0.8226 0.82 0.7939 1 2 = = = = = CE d N d N You might ask: with more time to expiry, there is more time for the option to move in the money, but there is also more time for the option to move out of the money. This is BFC5915 Workshop 7 Suggested solutions 3 true. But options give the holder flexibility. If the former happens, great, we exercise it and make money. If the option moves out of the money, we just let it lapse. So we are not overly concerned about the possibility that the extra time to expiry may see the option move out of the money. Question 2 Having calculated Black-Scholes prices for European call options, the price for the corresponding put option is given by put-call parity. By ‘corresponding’, I mean a put written on 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 intrinsic value 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 strike price. The lower the strike price, the more unlikely it is that share price will fall below it. 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. The reasons are the same as given in Question 1 part d. With more time before expiry, there is more time for share price to fall below the strike price and move the put into the money. BFC5915 Workshop 7 Suggested solutions 4 Question 3 This question examines the arbitrage opportunities which exist if the upper bounds are violated. The first task is to determine whether a bound has been violated. The second is to establish a trading strategy designed to capture an arbitrage profit. Third, prove that our trading strategy 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 cost today. 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 European or American) is the current share price. In this question, the upper bound is violated since 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 the right 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 thus providing a clue that the required strategy is to short the call (when something is overvalued, you want to short it). That’s the easy bit. The harder part is to know what else to do now to capture the arbitrage profit. With a short call, we may be forced to sell the shares, so we can protect ourselves by buying the shares today. Note also that, with options, there are two distinct possibilities at expiry: either they are in-the-money, or out-of-the-money. Hence, my payoff tables have a column for each possible scenario at expiry.
Cashflow at Maturity
ST < X
ST > X
Short call: rec premium Buy shares Invest surplus in bank
+ 9.00 – 8.50 – 0.50
Nil + ST 0.53 = 0.50 exp(0.06×1)
-(ST – 10) +ST 0.53 = 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 a profit of (ST – 10), so we will lose the same amount. The long share we purchased at Time 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 is positive. Thus, for zero upfront cost, we have a strategy that guarantees a positive payoff in the future – an arbitrage opportunity! BFC5915 Workshop 7 Suggested solutions 5 b) The upper bound on an American put option is the strike price X. Why? A put is a right to sell shares at the strike. The most money you can make with a long put is $X (this happens 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. We would short the put option (very happy to receive a premium that we know is too high). Examine the following table and recall that the American put can be exercised at any time up to and including maturity:
Cashflow at Maturity1
ST < X
ST > X
Short put rec premium Invest surplus in bank
+ 5.50 – 5.50
-(5.00 – ST) 5.84=5.50 exp(0.06×1)
Nil 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 and we pocket $5.84. If the put option finishes in the money (ST < X), it will be exercised against us. The counterparty will sell the share to us and we are forced to pay the $5 strike. We would then sell this share in the market at whatever the market price is (ST). The loss on the short put is –(5 – ST). The total cashflow is ST + 0.84. So we are guaranteed of getting a positive cashflow. If ST = $3, we get $3.84. If ST = $2, we get $2.84. Even if share price is zero, we still make $0.84. Therefore, regardless of whether ST < X or ST > X, the net cashflow at maturity is positive. Thus, for no upfront cost, we have a strategy which guarantees a positive payoff 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 6 Question 4 (*) This question examines violations of lower bounds. As with upper bounds, it would be rare to 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 rather than later, the trading by arbitrageurs would move option prices to the point where arbitrage opportunities vanish. Parts a and b below are absolutely trivial because the violations involve American-style options. Thus, we can capture the arbitrage profit immediately. Part c involves a European option, so the strategy is a little more difficult because we have to wait to expiry to exercise the option. a) The lower bound on an American call option is: [ ] [ ] 1.51. max 0,6.00 4.50 max 0, 0.05 3 / 52 0 ≥ ≥ – ≥ – – × – e C S Xe rT A . Note that this assumes that the company pays no dividends (and hence, the call option will not be exercised prior to maturity). If the option is trading at $1.20, this violates the lower bound. The fact that it is an American option makes it trivial to generate the arbitrage profit:
Enter long call option Immediately exercise right to buy shares Immediately 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 6.90 max 0, 0 ≥ ≥ – P ≥ X – S A Since the put option is trading at $0.35, this violates the lower bound.
Enter long put option
Buy shares on market Immediately exercise right to sell Arbitrage profit
– 6.90 + 7.50 + 0.25
2 Note that this calculation ignores transactions costs like brokerage. In practice, we sometimes see apparent violations on bounds, but after taking transactions costs into account, you cannot make money! BFC5915 Workshop 7 Suggested solutions 7 c) The lower bound on a European-style call option is: [ ] [ ] 2.29. max 0,10 8 max 0, 0.05 9 /12 0 ≥ ≥ – ≥ – – × – e C S Xe rT E If 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 strategy which costs nothing now and guarantees a positive payoff on maturity. The strategy requires buying the call, short selling the stock, and investing the surplus in the bank. Deciding to long the call is the easy part – it is underpriced so we certainly want to buy it. Next, think about what happens if this call finishes in the money. We would exercise our right to buy the underlying share. With this in mind, we would short sell the share today. If the option finishes in the money and we buy a share, we can use that share to close out our short sale. Finally, the $8 that is left over upfront can be invested to earn some interest.
Cashflow at Maturity
ST < X
ST > X
Long call option Short sell shares Invest 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). We then settle up our short sale by purchasing shares at the market price ST. The accumulated savings more than cover the cost of exercising the option, leaving a $0.31 cashflow. nb: despite this question providing details on the standard deviation of the stock, there is no need to calculate option price using Black-Scholes. BFC5915 Workshop 7 Suggested solutions 8 Question 5 a) With a call option, you want the share price to rise high above the strike price. So when the share price rises, this is good for the holder of a call option. Hence the call option value increases. The opposite is true for put options (put options give a payoff when share price is 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 higher strike price are less valuable. The higher the strike, the less likely it is for share price to rise above the strike and get the call option in the money. Conversely, for put options, the higher the strike, the more likely it is that share price will be below the strike meaning the put is in 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 want share price falling below the strike. The more time there is before the option expires, the more time there is for share price to move in a favourable direction for you. Hence, the higher the option value. Time is your friend!
Time to expiry (in years)
Call option value
Put option value
BFC5915 Workshop 7 Suggested solutions 9 d) Volatility is a very important influence on option prices. To get a payoff on the option, you need share price to move in a favourable direction (up for calls, and down for puts). The more volatile the underlying share, the better your chances of this happening. Hence, the value of both call and put options is positively related to volatility.
Volatility (σ) of AAPL returns
Call option value
Put option value
e) The riskfree rate of interest does factor into option values, but its influence is minor. For example, if the riskfree rate doubles from say 2% to 4%, there is a modest change in option values. In contrast, we saw above that if volatility doubled, or time to expiry doubled, there is a large impact on option values.
Riskfree rate of interest
Call option value
Put option value
Question 6 a) Pricing an American call with 6 step binomial tree: C=1.2735 b) Pricing an American call with 10 step binomial tree: C=1.2812. The 10 step tree is displayed below. The 10 step tree is the biggest tree displayed in my version of the software. BFC5915 Workshop 7 Suggested solutions 10 You can see the nodes where exercise is optimal. The cell entries are in red. It is clear from the tree that the only time it is optimal to exercise the American call option on a non-dividend paying 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 stock early. We showed that in Lecture 7. That means that an American-style call option and a 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 the 18-step binomial price. The binomial price converges to the BSM but oscillates, first above and then below etc. g) The binomial price converges to the BSM price for American call options. Question 7 One really important point to note, is that the BSM model is not able to price an American put option, because it is unable to take account of the early exercise possibility in the analytical approach that is used. (In mathematical terms we have the situation where we would have to BFC5915 Workshop 7 Suggested solutions 11 solve the partial differential equations, with free boundary conditions. In the European option case the boundary conditions are fixed – at expiry). a) Pricing an American put with 6 step binomial tree: P= 0.7956 b) Pricing an American put with 10 step binomial tree: P=0.8048. The 10 step tree is displayed below. The 10 step tree is the biggest tree displayed in my version of the software. You can see the nodes where early exercise is optimal. The cell entries are in red. One way to identify these nodes is to say the ddddd node. That is the node reached by five successive down moves in the tree. Another node where early exercise is optimal is ddddddu node. That is, the node 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-step binomial tree. Explanations could include: estimated volatility of 40% is too high – a BFC5915 Workshop 7 Suggested solutions 12 lower volatility would give a lower option price. The price in the market is too low and represents an arbitrage opportunity? Can you think of other explanaions? e) BSM price of European put option is 0.7874 f) BSM cannot price the early exercise flexibility of an American put option. The American-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 can take account of the early exercise flexibility by replacing the fair (arbitrage-free) value with the early exercise vale at the nodes in the tree where it is optimal to exercise the option early.