Introduction • The calls and puts that we have examined so far are generically known as Plain Vanilla instruments, meaning that they are standardized, exchange traded options that basically everybody in the market completely understands. • They are not interesting, but may be useful. Much in the same way that plain vanilla ice cream may be good, but it is not terribly unusual or exciting. • We have already discussed why exchange-traded options would tend to be highly standardized. They may not, however, meet the needs of all market participants. Some may need highly specialized options. These options, known as exotic options trade in the OTC markets.
Introduction • So has the market evolved exotic (i.e. nonstandard) options? • Hull does a nice job of elucidating some reasons: • Because they may meet specific, and unusual, hedging needs of certain clients. • They may have tax, accounting or regulatory advantages over standard options. • They may allow a firm to more fully express its sentiments on certain aspects of the market. • Because there is more profit in them for investment banks. • Because they may allow an investment bank to sell an overpriced product to its less technically sophisticated clients.
Introduction • There are many types of exotic options, and more are created all of the time. • We will examine the following types of exotics: • Bermudian options • Forward start options • Compound options • Chooser options • Barrier options (knock-in and knock-out) • Binary options • Lookback options • Shout options • Asian options
Introduction • When modeling exotic options, one has to make a fundamental decision very early in the process: should you model the option in a continuous-time, Black-Scholes type of model, or in a binomial model. • Generally many exotic options are initially priced via a binomial model, and then at some point traders figure out a closed-form pricing model. • Sometimes, it turns out that no closed form solution is ever found. • Indeed, for certain highly path-dependent options, one cannot even work backwards in a lattice, instead one must use a Monte-Carlo method to value the option. • This is especially true for many interest rate derivatives.
Introduction • We will use some of both. • Hull tends to want to provide lots of closed-form continuous time options. • McDonald generally provides a richer discussion of the options themselves, but relegates his discussion of the pricing formulas to an appendix at the end of Chapter 14.
Bermuda Options • A Bermuda option is a generic name for an option that is in between an American and a European option. • Usually these are options that allow early exercise only on specific dates. • Sometimes they are options that do not allow early exercise until a specific date but then allow early exercise from that point on. These are said to contain a “lock out” period. • Sometimes they are standard options but have a strike price that changes, typically as a function of time. • Typically these are priced through a binomial model. • Example: A non-dividend paying stock is currently priced at $20, and you hold a put that allows early exercise in 2 months and in 4 months. The option expires in 6 months. Volatility is 30%, and r=5%. What is the value of this option?
Bermuda Options • We will use the CRR binomial model. First, we need to determine our lattice parameters: • We can then build the lattice itself. Since this is a rather long option, we will build the lattice in Excel.
Bermuda Options • The Bermuda option has a value of 1.47. • Compare this to a similar option with a lock-out provision for the first two months, which has a value of 1.49.
Forward Start Options • A forward start option is one in which the option is issued to you but “begins” at some point in the future. • These are widely used in employee/executive compensation schemes. • Generally you are granted the option today, but it does not become “active” until some date in the future, T1. Usually you would have to still be employed by the firm at time T1, or your option would not activate. • Usually these options are written such that the strike is determined to be “at the money” at time T1, that is, K=S1. • The option then expires at time T2. • So how do we value such an instrument?
Forward Start Options • The key to developing a pricing model is to realize that we know for certain that the option will be at the money at time T1. • Recall that we are assuming that other parameters, such as r and σ are constant. • The value of an at-the-money call option is proportional to the stock price. • To see this, notice the following example. Price a ½ year, at the money call option on a stock with volatility of .30, and with risk free rate of 5%. Price this option for different values of S0 (and hence of K).
Forward Start Options • Here is a list of Black-Scholes prices for S0 from 1 to 20. • Notice that the option price for S0>1 is just the S0=1 price multiplied by the ratio of the stock price to 1!
Forward Start Options • What this means is that for any given set of values for r, σ, and T, if an option is at the money, its value can be related to any other at-the-money option with the same r, σ, and T values. • Denote c0 as the Black-Scholes value of an at-the-money option at time 0, with maturity T = T2-T1, on the stock, that is, K=S0. • Since r, σ, at T do not change at time T1, only S1 and K, I know that the value of an at-the-money option at time T1 will be: c1=c0*(S1/S0) • Going back to the previous example, if S0 were $5, then c0=0.481744. Thus, if at time T1 S1=7, then we would expect c1=0.481744*(7/5)=0.481744*1.4 = 0.67444.
Forward Start Options • This is nice because it means that all we have to do is to take determine the risk-neutral expected value for this option and discount it at the risk-free rate, i.e.: • Of course, S0 and c0 are known with certainty at time 0, and the expected stock price at time T1 under risk-neutrality is simply • Plugging in and simplifying yields:
Compound Options • A compound option is simply an option to buy an option. There are four types of these options: • Call on a call. • Call on a put. • Put on a call. • Put on a put. • There will be two sets of strikes and maturities. The first set, K1 and T1, are the strike and time at which you can exercise the option on the option, and K2 and T2 are the strike and time at which you can exercise the option on the underlying. • Of course, you can also be long or short any of these options, so there are really a total of 8 potential positions.
Compound Options • Now lets think just for a minute about what these options really allow you to do. • Call on a call: you have the right at time T1 to pay K1 for an option that will allow you to buy the underlying stock at time T2 at price K2. • Call on a put: you have the right at time T1 to pay K1 for an option that will allow you to sell the underlying stock at time T2 for price K2. • Put on a call: you have the right at time T1 to sell for price K1 an option that will allow you to take a short position in a call on the stock at strike K2 and with maturity T2. • Put on a call: you have the right at time T1 to sell for price K1 an option that will allow you to take a short position in a put on the stock at strike K2 and with maturity T2.
Compound Options • There are closed form solutions available for these options. For example, the value of a European Call on a Call is: • M is the cumulative bivariate normal distribution. S* is the asset price at time T1 for which the option price at time T1 equals K1.
Compound Options • Even though a closed-form solution exists, I find it quite instructive to examine a call on a call with the binomial model, just so that we can fully illustrate how this instrument behaves. • Additionally, the binomial has the advantage of being relatively easy to implement and does not require the bivariate binomial distribution. • Let’s go back to the base case we have been working. We have a stock with S0=20, r=.05, σ=.30. Let’s assume that the option on which the option will be written is a call with strike of $20 which expires at time T2=6 months. Let’s further assume that we have a call on that call that expires in three months, and that its strike is $2. Both options are European. • Thus, S0=20, K1=2, T1=.25, K2=20, and T2=.50
Compound Options • From the spreadsheet we can see that the option on the underlying would be worth $1.86 at time 0, but that the option on the option would be worth $0.69. • If we were to value a put on that call, it would be worth $0.81. • One could just as easily determine the value of a call on a put or a put on a put in the same manner.
Chooser Options • A chooser option is frequently referred to as an as you like it option. • This option gives you the right to declare at time T1 whether the option is a put or a call with maturity at time T2. • At time T1 the option value is the max(c,p). • We can use put-call parity to determine a valuation formula if the chooser option is based on European options and they have the same maturity and strike price. • Let’s redefine the T1 value as:
Chooser Options • We can verify this easily enough. Let’s assume that we have a three month chooser option on a stock currently priced at 20, with σ=.30, T2-T1=.25, and with K=20, and r=.05, and q=0. • From Black-Scholes, the value of a call with strike K and maturity T2=.50, (and the above parameters) is: 1.93. • The number of put options to buy is e0(.25)=1. • The strike for the put is 20e-(.05)(.25)=19.75. • The Black-Scholes value for the put, therefore is: $0.95. • Thus, the total value of the chooser is 1.93+(1)0.95 = 2.88. • When we build this through the binomial. • Even with only a 1 month time step, we still get a value for the chooser of 2.91.
Barrier Options • Barrier options are options that have a payout that is dependent not only on the terminal stock price, but also depend upon whether the stock attains some “barrier” during the life of the option. Two general kinds: • Knock-in options: The option comes into being only if the stock reaches a given barrier during its life. • Knock-out options: The option ceases to exist if the stock reaches a given barrier during the options life. • Some examples: • Down-and-outcall. This is a call with strike K that ceases to exist if the asset price reaches the barrier level H, where S0>H. • Down-and-in call. This is call with strike K that comes into existence only if the stock price reaches the barrier level.
Barrier Options • Assume for a moment that you held two portfolios: • Portfolio A: One call with strike K. • Portfolio B: One down and in call with strike K and barrier H. One down and out call with strike K and barrier H. • At maturity there are two potential states of the world, let’s compare the portfolio values in each: • The stock remained above the barrier at all times (state 1). • A: Max(0,ST-K). • B: Down and In: 0 (never activated). Down and out: max(0,ST-K) • The stock at some point touched the barrier (state 2). • A: Max(0,ST-K) • B: Down and In: max(0,ST-K) Down and Out: 0 (died when stock touched barrier).
Barrier Options • We can see, therefore, that the two portfolios have the same terminal values in all states of the world, so they must be equal. Denote c as the value of the normal call option at time 0, cdi as the value of the down and in option at time 0 and cdo as the value of the down and out option at time 0. Then, c = cdi + cdo • So we really only need to determine the value of one of these barriers to determine the value of the other at time 0. • Which of the two that we will choose to solve depends upon the relationship between K and H.
Barrier Options • If H<=K, we value the down and in call with this modified Black-Scholes equation: • On the other hand, if H>=K, then value the down and out call:
Barrier Options • An up-and-out call is a regular call that ceases to exist if the asset price reached the barrier level, H, while an up-and-in call is one that does not come into existence unless the asset price hits the barrier. • There are, of course, similar barrier options for puts. • Hull and McDonald both supply you with exact pricing formulas. • On an exam I would expect you to know what these barrier options are, and be able to work through a binomial example of them.
Barrier Options • These can also be valued through a Binomial lattice, but its trickier than it might at first appear. • Let’s value a down and out option through the lattice we have been using this evening. • Usually its easiest to explicitly solve for one of the values and then to use the relationship between a normal call and the barrier option to determine the other barrier’s value. Let’s begin with the down_and_out call since valuing it requires a relatively minor change to your boundary conditions:
Barrier Options • To value a down_and_in call – i.e. one where the call only activates upon hitting the lower value – you again just adjust your standard boundary conditions. • You simply have to realize that at a node where the stock price is at or below the barrier, then the value of the barrier option is exactly the same as the value of a standard call at that node. • At nodes where the stock price is above the barrier value, then the value of the option is equal to the expected present value of the barrier option at the next time step!
Barrier Options • To actually implement a down_and_in binomial pricing model, you really have to track two options through the lattice: a standard European option, and then the down and in option. • At a node where the barrier it touched (or lower), you set the value of your DI option to equal the European option (in this way you take into account the payoffs when the DI is active later, but above the barrier.) If the stock is above the barrier, you set the option equal to the discounted expected value of the future values of the DI option.
Binary Options • A Binary Option is basically a standard option, but one in which the payout is altered such that it only pays a fixed amount if the option ends up in the money. • In a cash or nothing binary, the amount is a lump sum, denoted Q. • In the standard Black-Scholes equation, N(d2) is the (risk-neutral) probability of the stock price being greater than strike price at time T. Thus the value of the Binary option, with strike price K, is given by:
Binary Options • For a cash or nothing put, again with payoff of Q dollars if the option ends in the money, the value is: • Another type of Binary Option is the asset or nothing call where the payoff is the asset if the stock ends in the money (this is essentially a European call, but where you do not have to pay the strike!). The same basic formula holds, just substitute in the (risk-neutral) expected value of the asset for Q (i.e. Q = S0(r-q)), and simplify the equation:
Binary Options • Similarly, an asset or nothing put pays off the asset price if it ends up in the money, its value is given by: • Example: A stock is currently priced at $20, has volatility of 30%, and pays no dividend. You hold an asset or nothing call with a strike of 22, that expires in 6 months. Assuming the risk free rate is 5%, what is the value of this option? • $6.62.
Binary Options • Obviously it is a trivial modification of the standard call or put to implement these in a binomial model. In fact, all you really have to do is modify the terminal boundary conditions for a European call or put – otherwise they are priced exactly the same in a binomial!
Lookback Options • A Lookback option is one in which the payout is a function of both the terminal stock value and the maximum or minimum value the stock achieves during the life of the option. • For a European-style lookback call, the payoff is the greater of zero or the difference between the final asset price and the minimum value of the asset during the life of the instrument: clookback=max(0,ST-Sminimum) • For a European-style lookback put, the payoff is the greater of zero or the difference between the maximum value the asset reaches during the life of the option and the final asset price.
Lookback Options • There are valuation formulas for lookbacks. • Note that Smin is the minimum price achieved at the time of valuation – at time t=0, then Smin=S0.
Lookback Options • Example: A stock is priced at $50, with a volatility of 40%, and a dividend yield of 0. The risk free rate is 10%. How much is a Lookback call on this stock worth? • $8.04 • What if the option had originally been a 6 month option, and the lowest value the stock had reached to date were $45, what would the Lookback call be worth today (again it still has three months to maturity)? • $9.04. • Essentially in a lookback option, the strike price is set to be the appropriate extreme observed over the life of the option.
Shout Options • A shout option is one in which the long party can “shout” at the short party one time during the life of the option, which sets a sort of lower payoff level. • At maturity, the holder receives either the intrinsic value at the time of the shout, or the payoff to a usual European call/put (depending upon the type of option that it is.) • Assume that you “shout” at time tau, then your payout at maturity would be (for a call): • Essentially a “shout” allows you lock in a payout without forcing you to give up your ability to earn a higher payout if the price increases.
Shout Options • Consider if you had a “shout” option with 6 months to maturity. The stock price is $20, and the strike is $22. Let’s say that after 1 month the stock price had risen to $30. You could “shout” at that point and you would guarantee that you would earn at least (30-22)=$8 at maturity. • If subsequently the stock price fell to $28 at maturity, you would still earn a payout based on your “shouted” stock price of $30, so you would get $8. • If, however, the stock price rose to finish at $40, you would receive (40-22) = $18. • Notice that you can write the terminal payout as:
Shout Options • Shout options are very similar to American options, and they are normally valued through a lattice type arrangement. • You can value them as you work backwards: you simply compare at each node the value of the option if you “shout” or not at that node, and assume the option is worth the more valuable of the two. • This is somewhat trickier that it may at first appear. Note that you have to determine the value of the option if you shout at that point – not the intrinsic value of the option. This means you have to go back to the end of the lattice and revalue the option knowing that you shouted at that node! • This is not too difficult to do in software, since you can simply call a function recursively, but it is challenging to illustrate graphically.
Shout Options • To get a sense of what I mean by this, consider a simple example. Assume you have a call shout option on a stock with a price of $20, 3 months to maturity, volatility of 30%, a risk free rate of 5%, and a time step of 1 month. The strike is 22. • From the CRR binomial model, the relevant parameters are: u=1.090, d=.917, and p = .5024 • The lattice this generates is:
Shout Options: 25.93 23.78 21.81 21.81 20 20 18.34 18.34 16.82 16.82
Shout Options • So now we can begin to value the option. We start at the terminal time step and apply the standard conditions of max(ST-K,0).
Shout Options 25.93 Max(0,25.93-22)= 3.93 23.78 21.81 0 21.81 20 20 18.34 0 18.34 16.82 16.82 0
Shout Options • We can then begin to work backwards through the lattice. At each node of the lattice we assume that the option value is the greater of two conditional values: • The value of the option if we do not shout at this node. • The value of the option if we do shout at this node. • Calculating the first one is relatively simple, just determine the discounted expected value of the option at the next time step – the procedure we normally use when valuing an option. • The next slide shows this value for each node at time step 2:
Shout Options 25.93 3.93 23.78 (.502*3.93)e-.05/12 =1.98 21.81 0 21.81 20 0 20 18.34 0 18.34 16.82 0 16.82 0
Shout Options • The second step is a little trickier, we have to determine the value of the option if we shout at this node. • This will not matter at nodes (2,0) and (2,1), since the stock price is less than the strike, we would not shout here, and the value is 0. • At node (2,2), however, if we shout, we have to figure out the value of the option under the assumption that we shouted at this node. • If we shout at (2,2), the minimum payout would be (23.78-22)=$1.78 at each of the nodes that we can still reach on level 3. • This won’t affect node (3,3), since the payout is higher, but it would affect node (3,2), where St=1.81, and were the payout was originally 0.
Shout Options 25.93 3.93 3.93 – stays the same! 23.78 1.98 no shout (.502*3.93+.498(1.78))e-.05/12 =2.87 with Shout 21.81 0 1.78 – because of the shout! 21.81 20 0 20 18.34 0 18.34 16.82 0 16.82 0 Determine the value of the option assuming that we shouted at node (2,2).
Shout Options • We then assume then replace the value of the option at that node with the $2.87 value we calculated.
Shout Options 25.93 3.93 23.78 2.87 21.81 0 21.81 20 0 20 18.34 0 18.34 16.82 0 16.82 0
Shout Options • We can then work on time step 1. • In this case we are fortunate in that the stock price at both nodes is below the strike of $22, so we can simply work our way backwards as we normally would.