Sources http://finance.bi.no/~bernt/gcc_prog/recipes/recipes/node7.html It's all Greek to me, Chris McMahon Futures; Jun 2007; 36, 7 http://www.quantnotes.com Put Call Parity THIS IS THE CALL-PUT PARITY PROOF Consider forming a portfolio by buying a put and selling a call option on the same underlying asset, at the same strike price E and same expiry time T If at expiration S = E then both the put and call are worthless If at expiration S > E then the put is worthless and The CALL costs us E S (owner of call will exercise), E-S is negative, shows we are losing money. If at expiration S < E then the call is worthless and The PUT pays us E S (we exercise put and earn money) Now, add the stock to the portfolio Thus the payoff will be E with certainty This is a risk free return If no arbitrage exists then this portfolio must cost Exe -rt PCS Ee rt OR PSCEe Excel spreadsheet on drive will calc both put and call. This method is preferred to the Black-Scholes put option calculation. We have a payoff of a known amount, E-S, although we do not know exactly what S will be. Then we add the stock S to our portfolio giving the above eq. WE are certain to get E, the exercise price, from this portfolio. This is not arbitrage, we've spent money to set this up. We basically made a loan where the person will pay us E in T periods. The exponent is the continuous time version of discounting. Buying a put, selling a call and adding a stock is the same as making a loan n amount E at risk free rate. P and C are the premiums on put and call. r is risk free rate, T is time (periods) to maturity. rt FNCE302: Investments Lecture 8 Advanced Options Page 1
Example Using our previous Apple example With S = $126.89 we found C = $18.22 (by using Balck-Scholes) We were told r = 0.167% and E = $130 And T = 10 P = C S + Ee -rt = $18.22 126.89 + 130e -0.00167 x 10 = $19.18 This is the price of the put, the premium on the put found by using put/call parity. Most put/call calculators will calculate the call value, use this method to get the put. Most of the above only applies to European options. American Option Early Exercise Recall the Apple example, but this time we will value the equivalent put, and assume the option is American Now at each node in the lattice we need to consider whether early exercise is optimal Strictly speaking you cannot use Black-Scholes with an American Option, there is no closed form Black-Scholes type solution for an American option. The reason is the fact that it may be optimal to exercise an American option early. If we use the risk-neutral exercise approach it is flexible enough to model early exercise opportunities. Idea is to use the same opportunity but when you calculate the value of the option at each node of the lattice you have to compare it (the solution) to the value if the option is exercised early. MUST COMPARE EACH POINT IN THE LATTICE TO THE VALUE IF YOU EXERCISED EARLY! FNCE302: Investments Lecture 8 Advanced Options Page 2
APPLE STOCK EXAMPLE, HOW TO CALCULATE THE VALUE OF AN AMERICAN OPTION. $126.89 IS THE OPENING STOCK PRICE. UP Tick = 11.4%, DOWN Tick = 1/11.4% Models a ten month period of how Apple's stock may move. 373.49 335.27 300.96 300.96 270.16 270.16 242.51 242.51 242.51 217.70 217.70 217.70 195.42 195.42 195.42 195.42 175.42 175.42 175.42 175.42 157.47 157.47 157.47 157.47 157.47 141.36 141.36 141.36 141.36 141.36 126.89 126.89 126.89 126.89 126.89 126.89 113.90 113.90 113.90 113.90 113.90 102.25 102.25 102.25 102.25 102.25 91.79 91.79 91.79 91.79 82.39 82.39 82.39 82.39 MUST DO ALL THREE OF THESE LATTICE TABLES 73.96 73.96 73.96 66.39 66.39 66.39 59.60 59.60 53.50 53.50 48.02 43.11 We will value a PUT OPTION, puts are far more likely to be exercised early. Methodology is the same for a call. (HERE WE DO NOT START AT RIGHT AND WORK BACK TO LEFT like we did on the premium chart!). Example: stock price is $373.49, PUT has $130 strike price, we would not exercise at this stock value (would not give away because stock price is more than strike), so we put a zero in the PUT lattice at the same position (blue circle). All points above on $130 on stock price lattice will have zero entries in put lattice. The point is, given any of these stock prices what would be the value of exercising the option at that time. Its an American option, we examine it and ask if it is worth exercising early. Red circle is an example of when we would exercise (red circle). The next step is to calculate the premium. (step 3 below) FNCE302: Investments Lecture 8 Advanced Options Page 3
Working back to front 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.11 3.11 3.11 3.11 3.11 3.11 16.10 16.10 16.10 16.10 16.10 27.75 27.75 27.75 27.75 27.75 38.21 38.21 38.21 38.21 The probability of an up move, q, is.48. The probability of a down move is 1-.48=.52. The risk free rate is r f =16.7% V(U) = value of up branch, V(D) = value of down branch 47.61 47.61 47.61 47.61 56.04 56.04 56.04 63.61 63.61 63.61 70.40 70.40 76.50 76.50 81.98 86.89 Working back through the nodes by calculating the expected payoff then dividing by 1 + risk free rate. The value of any particular node is the maximum of the expected payoff and the exercise value. The winner is entered into the table and then the next node is calculated. MAX[ q* V ( U) (1 q)* V ( D).48* 47. 61.52* 63. 61 $47.92 1r 1.167 f, $56.04 ] So $47.92 is the value of the option if we continue to hold it and $56.04 is the value if we exercise early. So we earn more if we early exercise. Since we early exercised in this node that value, $56.04, the early exercise value, is used to calculate the value of the next node! Put option calc may be different from Call. (?) The entire MAX[, ] equation can be programmed into each node of an excel spreadsheet. You can reach a point where you have locked in so much gain there is too much risk of loss to continue to hold the asset. FNCE302: Investments Lecture 8 Advanced Options Page 4
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.12 0.00 0.00 0.88 0.22 0.00 0.00 2.89 1.59 0.43 0.00 6.46 4.75 2.87 0.84 0.00 11.57 9.78 7.69 5.13 1.61 17.95 16.34 14.47 12.19 9.12 3.11 23.92 22.46 20.79 18.76 16.10 31.02 29.94 28.81 27.75 27.75 39.05 38.51 38.21 38.21 47.61 47.61 47.61 47.61 56.04 56.04 56.04 63.61 63.61 63.61 70.40 70.40 76.50 76.50 81.98 86.89 Six Variables SIX VARIABLES THAT DETERMINE AN OPTION S PRICE The price of the underlying asset The strike price Time till expiration Volatility of the underlying asset Interest rates In the case of equity options dividends paid Dividends, if a srtock pays divends it will effect the value of the option At some point during the life of the option the stock price will climb. When a stock pays a dividend the price of the stock declines by some amount close to the dividend value, usually 80 cents on the dollar. So the value of an option is effected by the dividend. These items are know as the "Greeks", they are assigned Greek letters. We are interested in the relationship between an option value and the price of the underlying asset (first item above). We know some things, for example, with a call option if the price of the asset goes up the value of the call option goes up. We want to measure the effect and understand it. FNCE302: Investments Lecture 8 Advanced Options Page 5
Delta There is not necessarily a linear relationship between an option s value and the price of the underlying asset Each option has a DELTA that describes the theoretical relationship between the two 0 <= Delta <= 1, is expressed in a range between zero and 1.00 For example, an option with a delta of.90 would change in value by 90 cents for every dollar change in the underlying asset, approximately LONG CALLS or SHORT PUTS Delta of an option varies from zero (0) to one (1) (positive delta) Buy call, stock price goes up, call premium goes up write put, stock price up, premium on put goes down, therefore if I've written the put my position becomes more valuable. LONG (BUY) PUTS or SHORT (SELL) CALLS Delta varies between minus one (-1) and zero (0) if stock price goes up the value of my put goes down. Hence, a positive delta may be associated with a bullish option Long (bullish) futures position can be thought of as having a delta of 1 Short (bearish) futures position can be thought of as having a delta of -1 Futures positions can be thought of as having delta positions of +/- 1. Delta Example C S N d (CALL) 1 Back to Apple example Spot price = $126.89, Months to expiration = 10, Strike price = $130, Risk free rate = 0.167%, Monthly volatility = 11.7% Delta = N(d1) = N(0.1647) = 0.5654, this is saying that if the stock price goes up by $1 the value of the call option will increase by approximately 57 cents. Check the results using the Apple example (increase stock price by $1) With S = $126.89 we found C = $18.22 Recalculating with S = $127.89 (a $1 increase) yields C = $18.79 (a 57 cent increase) Does a good job of estimating the change in price of a call option! Call option is a function of many things, in Black-Scholes world it is equal to: C(S,E, ) = SN(d 1 ) Eg -rt N(d 2 ) Now we want to take the partial derivative wrt S. The N(d) terms are functions of S, by coincidence the derivative wrt S is just N(d 1 ). But N(d 1 ) is also the cumulative probability of the z-score d 1. So with Black-Scholes we also get the price sensitivity of the underlying asset. FNCE302: Investments Lecture 8 Advanced Options Page 6
Delta for a Put For a put option So for the Apple example, for the put with the same characteristics Delta = 0.5654 1 = 0.4346 Thus for every increase in the price of the stock the call option will decline by about 43 cents. For a PUT option the delta is between -1 and zero. P N d 1 1 (PUT) S Delta Hedging Suppose I own 100,000 Apple shares (same characteristics as above) I do not want to sell my position However, I am concerned the stock price may decline in the short term If Apple stock falls by $1 I lose $100,000 (because I own 100,000 shares) SELL CALLS, a $1 decrease will net me approximately 56.54 cents So sell $100,000 / 0.5654 = 176,876 calls (sell this many calls to protect yourself against a $100,000 loss) (downside protection) Rather than selling the stock we will use options to hedge our position in the stock. Alternatively BUY PUTS A $1 decrease will net me approximately 43.46 cents So buy $100,000 / 0.4346 = 230,097 puts, would have to sell more puts than calls. Delta Hedge Adjusting Unfortunately, your delta hedge will need to adjusted over time Suppose one month from now Apple has declined to $116.89 Now for the call, assuming the risk free rate has remained the same, now T = 9 (we now only have 9 months left until expiration), and S = $116.89 (price of the stock has changed) We now recalculate N(d1) = 0.4663, what is the protection now? A further $1 decline will lead to an increase in value for our call delta hedge of $0.4663 x 176,876 = $82,473 So now our protection from hedging only protects the $100,000 up to a value of $82, 473. But a $1 decline in the price of the stock will still cost us $100,000, I am no longer fully hedged. So I have to adjust the hedge based on the current stock price. If underlying asset price changes you will have to examine and adjust your hedge position over time. The price of the underlying asset effects the delta hedge value. FNCE302: Investments Lecture 8 Advanced Options Page 7
How Does Delta Change With Strike (underlying asset value)? Heavy out of money heavy in the money Current position with delta of about 56 cents. As the stock price goes up and we go more in the money the delta approaches 1, similar for downside. This says that if we are heavy in the money, the heavier we are in a $1 change is stock price will effect the delta by approx. $1. Same for downside. Pricing Options on Dividend Paying Stocks We will consider Apple stock one more time Consider a put option with all parameters the same except The term is 3 months to expiration (changed for simplicity) The company is expected to pay a dividend of $5 per share This is paid immediately after the end of month 2 It is not state dependent, on tree node the price changes immediately after the calculation of the up and down value. this is due to payment of a dividend. The lattice is now not as symmetrical as before Here we look at the impact of dividends on option values. The dividend is not state dependent, if the stock price has been going up or down the dividend is still $5.00 per share. Lattice is not symmetrical! see next pae FNCE302: Investments Lecture 8 Advanced Options Page 8
Pricing Options on Dividend Paying Stocks Price Lattice The stock price still goes up or down at first period. At second period nothing has changed, if we compare these prices to the large lattice a few pages back they are the same up to time 2. At time 2+ the we have expanded the tree, the value is now dependent on the path. In original lattice the value was 157, in this lattice it is 152. In original lattice it was 126, in this one it is 121. In original its 102, in this one 97. These stock prices are lower by $5 because after we calculate the up or down movement we then immediately pay $5 per share. Calculate the up and down values then subtract off the value of the dividend from those values. We break the tree into 3 sub-trees because everything has decreased by $5 the nodes no longer come back together. Consider $136.87 in column 3, it has a certain path it took to get there. But because we paid the lump sum of $5 the nodes no longer have the common matching point. In the old tree if we had got to $102 and gone up we would have been at $113. If we had got to $126 and gone down we would have been at $113. In this world we can no longer do that. $97 up path leads to $108. $121 down leads to $109. Nodes no longer meet up, more complicated. COLUMN 3 IS AN INTERMEDIATE POINT BETWEEN 2+ AND 2. Now path dependent. Must be concerned with how you got to a particular point. 373.49 Stock Price Lattice 335.27 300.96 300.96 270.16 270.16 242.51 242.51 242.51 217.70 217.70 217.70 195.42 195.42 195.42 195.42 175.42 175.42 175.42 175.42 157.47 157.47 157.47 157.47 157.47 141.36 141.36 141.36 141.36 141.36 126.89 126.89 126.89 126.89 126.89 126.89 113.90 113.90 113.90 113.90 113.90 102.25 102.25 102.25 102.25 102.25 91.79 91.79 91.79 91.79 82.39 82.39 82.39 82.39 73.96 73.96 73.96 66.39 66.39 66.39 59.60 59.60 53.50 53.50 48.02 43.11 FNCE302: Investments Lecture 8 Advanced Options Page 9
Value of Put American Option: more likely to exercise put options on dividend paying stocks, more likely to exercise put immediately after dividend paid. Exercise price is still $130 so at time 3 I will compare the values in the table to $130. We see that column 3 values 21.67, 42.70, and 20.58 are all in the money. Now we have to take these values back to each previous node. Since it is an American option at the $32.53 we must compare to see weather early exercise is optimal, we would use the MAX[, ] formula. Is early exercise optimal? $97.25 at a $130 strike price, it is beneficial to exercise the option immediately following the dividend rather than continuing to hold the option. We find this with the MAX[, ] calculation. We are far more likely early exercise put options on dividend paying stocks. With put options after dividends are paid the stock price drops, therefore more likely to be in the valuable region of the put option. For that reason exercising an option immediately following a dividend is far more likely with a put option. He may post these spreadsheets but they are not flexible, not easily used for another situation. Exotic Options Barrier Lookback Digital/Binary option Bermudan options Buyer has the right to exercise at a set (always discretely spaced) number of times We will look at currency options later. These are not the only exotic options. Bermudan options are kind of in-between American options and European options. A Bermudan Option can be exercised at set points in time, discrete 9as opposed to American option which is a continuous time). This is a relatively small difference, the others are more complicated. FNCE302: Investments Lecture 8 Advanced Options Page 10
Barrier Option 1. Up-and-Out Spot price starts below the barrier level and has to move up for the option to be knocked out. When you move above the barrier the option is destroyed. 2. Down-and-Out Spot price starts above the barrier level and has to move down for the option to become null and void. Below the barrier and the option is destroyed. 3. Up-and-In Spot price starts below the barrier level and has to move up for the option to become activated. Start below barrier level (this is not the exercise price) and the asset value must go above the barrier level before the option comes to life. EXAMPLE: barrier option on Apple stock, exercise price $130, barrier of $140. The option doesn't become alive until the stock price goes above $140. At that point it becomes alive and is just a regular boring option. If the stock price does not surpass the barrier price it never comes to life. It is still alive if it rises above barrier and then comes back down below barrier. 4. Down-and-In Spot price starts above the barrier level and has to move down for the option to become activated. Must drop through a barrier before it comes to life. Four basic varieties. Sell price for an up-and-out starts below a barrier level (this is not the strike price). There is an additional value called the barrier level. The option is not live until the stock price moves beyond the barrier level. Useful if you are worried about catastrophic events and want protection, you want the protection all the way down if the really bad thing happens. Adding the barrier levels makes the barrier options much cheaper to buy due to lower probability that it will come to life. example nest page FNCE302: Investments Lecture 8 Advanced Options Page 11
EXAMPLE Use the Apple example again, but now suppose the option is a CALL BARRIER UP-AND-IN OPTION Barrier level $300, Exercise price $130. Does not come to life unless / until the stock price exceeds $300. Otherwise the same as the other options. Consider the previous lattice, same as before, only the red region is active. 0 1 2 3 4 5 6 7 8 9 10 The path taken to get to a certain point is important here. Consider the price of $157.47 in column 10, it is in the money but it will never be exercised. Why? Because there is no path that allows me to land at that point AND goes through a value of $300 or more required to bring the option to life. That point is in the money but is never activated. On the other hand we can get to the top entry in column 9 through the top entry in column 8 which is greater than $300. We will have to modify the way we calculate things because there are lots of ways of getting to this node but one one where the option becomes active after passing through a value greater than the barrier price. It is in the money but only after passing through that point. Other points will be in the money with even more paths passing through the barrier price. 373.49 335.27 300.96 300.96 270.16 270.16 242.51 242.51 242.51 217.70 217.70 217.70 195.42 195.42 195.42 195.42 175.42 175.42 175.42 175.42 157.47 157.47 157.47 157.47 157.47 141.36 141.36 141.36 141.36 141.36 126.89 126.89 126.89 126.89 126.89 126.89 113.90 113.90 113.90 113.90 113.90 102.25 102.25 102.25 102.25 102.25 91.79 91.79 91.79 91.79 82.39 82.39 82.39 82.39 73.96 73.96 73.96 66.39 66.39 66.39 59.60 59.60 53.50 53.50 48.02 43.11 This region is in the money but barrier is $300 The different ways we can get to a node will impact the value of the option! So how do we find the value of the option? Below is a simple example. FNCE302: Investments Lecture 8 Advanced Options Page 12
Example Value = S - E Prob UUUUUUUUUU 243.39 0.48 10 10 ways 9 U and 1 D 170.96 10 x 0.48 9 x 0.52 UUUUUUUUDD 112.51 0.48 8 x 0.52 2 10 9 8 2 ($243.99 x 0.48 + $170. 96 x 10 x 0.48 x 0.52 + $ 112. 51 x 0.48 x 0.52 ) Call premium = $1.36 1.0016710 This calculation is the points we pass through times the probability of getting to that point. For example, $170.96 has 9 ups (.48 9 ) and 1 down (.58) and 10 ways of getting there. U=Up, D=Down. UUU means up 3 times, DUD means down-up-down. Value of the option: Strike Exercise Price = 373-130 = 243.39 What is the probability of getting to a particular point? Probability of going up once is.48, probability of going up twice is.48*.48=.2304 Up 3 times is.48 3. up i and down n-i = n! i! ni! where the order is unimportant meaning you can do the downs 1 st of 5 th or 8 th, order is unimportant to the probability calc. In the above example n=10, i=1, COMBIN(10,1)=10. [Excel: =COMBIN()] i can be the ups or downs. So you can see in situations where there are very few possibilities for an option to come to life they will have to sell very cheaply. These options are OTC so the price is negotiated. WHENEVER THE PRICING IS PATH DEPENDENT YOU MUST CALCULATE THE PROBABILITIES. Lookback Option Path dependent option where the option owner has the right to buy (sell) the underlying instrument at its lowest (highest) price over some preceding period In our example, to keep things simple, suppose the lookback period is 1 month One period earlier we are one of 10 nodes Allows you to sell or exercise the option at the highest price hit (if it is a call option) over some pre-specified period. So the exercise is not dependent on the final price. No buyers remorse, no wishing "I had exercised 3 months ago because the stock price was higher". Highest price within the pre-specified period is used as the strike price to calculate the exercise to maturity. It "LOOKS BACK" over the prior prices to find the most benefit. These are very valuable, more so then regular options. More expensive. FNCE302: Investments Lecture 8 Advanced Options Page 13
0 1 2 3 4 5 6 7 8 9 10 We can only get to $242.51 AND 373.49 pass through a $300 via 8 ups and 2 335.27 down. (UUU example last page) 300.96 300.96 270.16 270.16 242.51 242.51 242.51 217.70 217.70 217.70 195.42 195.42 195.42 195.42 175.42 175.42 175.42 175.42 157.47 157.47 157.47 157.47 157.47 141.36 141.36 141.36 141.36 141.36 126.89 126.89 126.89 126.89 126.89 126.89 113.90 113.90 113.90 113.90 113.90 102.25 102.25 102.25 102.25 102.25 91.79 91.79 91.79 91.79 82.39 82.39 82.39 82.39 73.96 73.96 73.96 66.39 66.39 66.39 59.60 59.60 53.50 53.50 48.02 43.11 Look-back Example: (using above lattice). Our otion allows us to "look back" one period. Your path determines what the high value you pasted was but in this example we can only look back one period. Pick the highest price within the look back period your option allows along the path you took to get to a particular point. Say I am at 335.27 in month 9, only one way of getting here UP 9 times,.48 9. Get to look back one period. Must consider that you can be out-of-the-money at a particular point but because of the path the stock price took getting there and the number of look-back periods allowed in yur option, you can still be at an advantageous point. LOOK-BACK Example, we get 1 period lookback Consider the top node at month 9 Only one path UUUUUUUUU Probability 0.48 9 After this, the path could go up (0.48) or down (0.52) If up Exercise Price = S E = $243.49 If down Exercise Price = S E = $335.27 130 = $205.27 (continued next page ) FNCE302: Investments Lecture 8 Advanced Options Page 14
Consider the second to top node at month 9 There are nine paths 7 U and 2 D Probability 0.48 7 x 0.52 2 x (9!/(2! x (9 2)!)) After this, the path could go up (0.48) or down (0.52) If up Exercise Price = S E = $112.51 If down Exercise Price = S E = $217.70 130 = $87.70 Consider the next to top node at month 9 There are nine paths 8 U and 1 D Probability 9 x 0.48 8 x 0.52 1 After this, the path could go up (0.48) or down (0.52) If up Exercise Price = S E = $170.92 If down Exercise Price = S E = $270.16 130 = $140.16 Now we gather the information for all the potential ways of exercising these options: Payoff Probability U D # Paths U D U D Payoff 9 0 1 243.49 205.27 0.0006 0.0007 0.30 8 1 9 170.96 140.16 0.0063 0.0069 2.04 7 2 36 112.51 87.70 0.0274 0.0297 5.69 6 3 84 65.42 45.42 0.0693 0.0751 7.95 5 4 126 27.47 11.36 0.1127 0.1221 4.48 Sum = 20.47 Number of paths from factorial combinations equation. Premium = $20.47/1.00167 10 = $20.13 Discounting the result at the risk-free rate (.167%) for 10 periods This option is worth $20.13, selling for almost $2 more than the comparable European option. The extra $2 is the benefit of looking back 1 period. FNCE302: Investments Lecture 8 Advanced Options Page 15
Binary Option Comparison of a binary versus standard vanilla option Identical except for payout profile If the underlying instrument moves "in the money", a binary will pay a fixed amount Where used Useful when magnitude of event is difficult to measure Weather for example Exactly the same as an ordinary option but instead of calculating the payout as being the strike price minus the exercise price for the call version of the exercise price minus strike price for the put version it is an all or nothing prospect. With a binary option if it ends up in the money at expiration you get a payoff that is not dependent on how "in the money" it is. So I may have a binary option over the Apple stock, I would have to specify what the payout is going to be but the idea is if it is in the money then the payout is some dollar value which is fixed. All of nothing. Only two possible payoffs, some lump sum or zero. Why use these? Used when the magnitude of the event is difficult to measure. Could be a weather event such as a hurricane, hard to say how bad it was but easy to say if it happened. Binary in these situations because easy to agree hurricane happened but ha=rd to agree on intensity. Palladium Market uses these. Difficult to find out the final traded price. Easy to asses a range. FNCE302: Investments Lecture 8 Advanced Options Page 16