Digital Options. and d 1 = d 2 + σ τ, P int = e rτ[ KN( d 2) FN( d 1) ], with d 2 = ln(f/k) σ2 τ/2

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1 Digital Options The manager of a proprietary hedge fund studied the German yield curve and noticed that it used to be quite steep. At the time of the study, the overnight rate was approximately 3%. The one-month forward on the overnight rate is 3.50%. The proprietary trader had the following idea. Often at times the market thinks that one month hence the overnight rate is going to go higher to 3.50%. However, most of the time nothing happens to the overnight rate; it stays at 3%. The proprietary trader wanted to develop a low-premium, high-payout strategy to make huge profits himself if he was correct. On the other hand, the trader did not want to lose much in case he was wrong. The strategy was developed with digital options. The market determines the forward rate at 3.50%. The trader disagrees with the market and thinks the rate will stay down at 3%. Consider the following alternatives: 1 1. Engage in a short forward transaction. Although this is zero cost, it may be quite expensive if the trader turns out to be wrong and interest rates increase to 4%. 2. Buy an at-the-money put, struck at 3.50%. If interest rates stay at 3%, the trader is going to get 50 basis points times the notional amount. That s a very nice payout. However, there is a problem with this strategy, as the trader has to pay for an at-the-money European option for which the premium is quite expensive. In our case, the cost of the option is about basis points. So the trader will spend 14 to make To reduce the cost the trader can buy an out-of-the-money option, struck at 3.05%. Since this is an out-of-the-money put, it s going to be a lot cheaper. In our case, the cost of the out-of-the-money put is about 1.36 basis points. On the other hand, the payout of the option is going to be much lower. If interest rates are at 3%, the payout will be only 5 basis points. Now the trader has to spend 1.36 to make Buy a digital put option struck at 3.05%. The put pays 50 basis points if interest rates are lower than 3.05%. This option costs about 4.85 basis points. With the digital, the trader can spend 4.85 to make 50. To summarize, the structure is particularly attractive to option writers, for whom the digital structure means a known and limited loss in the event the option is exercised. For the purchaser, the major advantage of a digital option is that the option payoff is a known constant. This amount may be related to some fundamental amount related to the underlying hedging transaction. In addition, it overcomes the problem of a purchaser where the option expires ator slightly in-the-money and where the resultant payoff does not cover the cost of the option premium paid to purchase the option contract. 1 Calculations for European interest rate options make use of Black s model, which assumes that the futures price follows geometric Brownian motion. Under this model, the futures price can be viewed as a security providing a continuous dividend yield equal to r (i.e., q = r) and the price of a European interest rate put option is given by P int = e rτ[ KN( d 2) FN( d 1) ], with d 2 = ln(f/k) σ2 τ/2 σ τ and d 1 = d 2 + σ τ, where F is the forward rate, K the strike price, τ the time to expiration, r the risk-free rate, and σ the volatility. In this numerical example, we assume that the volatility is 35.33%. The reader should return to verify the price of the digital put option after reading Section 2. 1

2 1 Payoff Profile The simplest structure for a digital option is path-independent, whereby a call (resp. put) pays nothing if the underlying asset finishes below (resp. above) the strike price or pays a predetermined constant amount Q if the underlying asset finishes above (resp. below) the strike price. These options are also referred to as binary, cash-or-nothing, or all-or-nothing options. Q cf. K K 2 Valuation The valuation of the digital call option is extremely simple. The valuation formula is the present value of the area under the lognormal distribution curve to the right of the strike price. Therefore, the price of the digital call option is with C = e rτ E [ QI {ST >K}] = Qe rτ P{S T >K}=Qe rτ N(d 2 ) (1) d 2 = ln(s/k)+(r q σ2 /2)τ σ, τ where S is the spot price of the underlying, K the strike price, τ the time to expiration, r the risk-free rate, q the dividend yield, and σ the volatility. Example: Calculate the value of a derivative that pays off $100 in six months if the S&P 500 index is greater than 500 and zero otherwise. Assume that the current level of the index is 480, the risk-free rate is 8% pa, the dividend yield on the index is 3% pa, and the volatility of the index is 20%. Here, Q = 100, S = 480, K = 500, τ =0.5, r =0.08, q =0.03, and σ =0.2. With these values, we find that d 2 = , and N(d 2 )= Therefore, the derivative has a value of = $ Hedging A European option can be approximated by digital options. 2 replication of a digital option with European options. What is more interesting is the 2 It is possible to take a European option and to build steps to the top of the mountain, as illustrated below. We can have a series of digital options: One that pays a dollar struck at $100. Another that pays a dollar struck at $101. One more that pays a dollar struck at $102, etc. An alternative structure is more precise. It is created with One option paying $0.50 struck at $100. Another paying $0.50 struck at $ Yet another option paying $0.50 struck at $101, etc. We can make the steps as long or as small as we wish, and so we can make the replication as precise as possible. Along the same line, Pechlt [2] shows how digital options can be used as a means of valuing exotic options and constructing static hedges, illustrating the ideas with switch and corridor options. 2

3 The dealer sells a digital option to a client and now needs to hedge himself. Consider a digital call option that pays $1 if the underlying is over $100 at expiration. Compare this digital option with a portfolio consisting of a long position in a call struck at $100 and a short position in a call struck at $101. Above $101 both positions pay out a dollar. Below $100 both positions result in zero. Between $100 and $101, the bull spread is not an exact hedge for the digital option (see diagram). $100 $101 An alternative hedge is to be long two European call options struck at $100 and short two European call options struck at $ This bull spread mirrors the digital option exactly except for the region between $100 and $ If the trader wants to be really aggressive, he can go long ten calls at $100 and short ten calls at $ This bull spread mirrors the digital option exactly except for the small region between $100 and $ In the general case, the hedge for a digital option struck at K which pays $1 is: Long 1/ε Europeans at point K. Short 1/ε Europeans at point K + ε. In the three hedging alternatives examined above, ε was $1, $0.50, and $0.10 respectively. 3 We can find ε so that the hedge is correct within a certain probability by constructing the bull spread. Note that: If S K, we are hedged. If S K + ε, we are hedged. If K<S<K+ε, we are not exactly hedged. Hence, we can choose ε so that P {K <S<K+ε}<1 α, where α is the desired probability of being hedged. Consider the following directive for exotic options traders: Digital calls are hedged with a bull spread with a 20-tick difference, unless volatility is lower than 10%, in which case we move to a 10-tick difference. 3 Digital options are probably one of the very first exotics that a lot of banks step into. A bank sells a digital and puts it on the books as a very aggressive bull spread, say as long 100 call options struck at $100 and short 100 call options truck at $ Since the bank already has the capability of pricing and hedging European options, it can replicate the digital with them. The payout of the digital and the bull spread is similar, so is the pricing. Delta and all other Greeks are pretty much the same too. Thus it s the most natural exotic to get into because the system can almost already handle it. 3

4 Besides the static hedging strategy outlined above, the dealer can also hedge his position dynamically by trading in the underlying asset. With this approach, we need to evaluate the delta and gamma of a digital (call) option. Assuming Q =1,wehave: = C S = e rτ n(d 2 ) d 2 S = e rτ n(d 2 ) Sσ, (2) τ Γ= S = e rτ n (d 2 ) (Sσ τ) 2 e rτ n(d 2 ) S 2 σ = e rτ d 1 n(d 2 ) τ S 2 σ 2 τ with d 1 = d 2 + σ τ. (3) Note that n (x) =dn(x)/dx = xn(x). The discontinuous nature of the digital option payoff creates peculiar hedging difficulties. Digital options are difficult to hedge near expiration because, around the strike, small moves in the underlying asset price can have very large effects on the value of the option. This reflects the fact that the absolute value of the delta can be large close to maturity. Additionally, the digital option delta may exhibit violent changes as the underlying price changes when the option is close to maturity (that is, the digital option is a high gamma instrument). 4 4 Variations A number of other digital option structures exist: Asset-or-nothing options, which pay the value of the underlying asset instead of paying a predetermined cash amount if exercised. Digital-gap options, where the payout is the sum defined by the underlying asset price minus a constant. Super shares (resp. step structures), which pay out the value of the underlying asset (resp. an agreed predetermined amount) only if at expiration the asset price falls in a predetermined range. One-touch options, which pay the predetermined amount if the option goes above or below during the option s life instead of at expiration. Path-dependent digitals, where the digital option incorporates barrier optionality. 5 asset-or-nothing super share step structure K K 1 K 2 K 1 K 2 4 For a visual on how the value, delta, and gamma of a digital option vary with the underlying asset price and time to expiration, numerical examples are provided in Barret [1]. The reader is encouraged to use the analytical formulas (1), (2), and (3) to reason the illustrated behaviors of those quantities. 5 Rubinstein and Reiner [3] describe a variety of barrier digital options and provide valuation formulas for these options. The interested reader is referred there. 4

5 Example: Combine a long position in an asset-or-nothing struck at K 1 and a short position in an asset-or-nothing struck at K 2. Effectively, we created an option that entitles its holder to receive the stock if it ends in the range K 1 to K 2. Otherwise, the holder receives nothing. The payout diagram for the super share is illustrated on the previous page. Suppose the dealer buys a stock. The dealer now has a stock that he purchased and paid for. At the same time, the dealer sells super shares to his various clients. Client A buys an option and pays a premium for it. One year later, client A is entitled to receive the stock if it ends above zero and below or equal to $10. Client B is going to get this stock if it ends above $10 and below or equal to $20. Client C will receive it if it ends above $20 and below or equal to $30. Client D will receive it if it ends above $30 and below or equal to $40. And so on. Each client has purchased a super share. Taken together, all the super shares add up to one position in the share. In the past, that is actually how some dealers made money. They bought a share, and then they sold all these little individual super shares. The total premiums the dealers received were more than the current stock price and were more than they paid for the share. On the expiration date, the dealers would just pass the stock certificate to that particular client whose super share expired in the money. In the meantime the dealers collected the dividends paid out on the share and kept it themselves. Example: Consider a long position in a digital option struck at $90 combined with a short position in a digital option struck at $110. The holder of this position will receive $100 if the stock is above $90 but below $110. The payout diagram of the step structure is shown on the previous page. How do you price it? The price is basically the probability that the underlying will end up between $90 and $110, discounted to today. With a spot price of $100, time to expiration of 1 year, risk-free rate of 2.25%, dividend yield of 3%, and volatility of 20%, the price comes out to about $ Summary Digital options represent a unique class of exotic options which focus on varying the payout profile of options. This alteration of the payoff pattern of options creates unique transactional hedging opportunities, since it allows writers a known cost in the event that the option expires in-the-money. Similarly, it allows option purchasers to nominate a particular payoff related to underlying hedging requirement. It is not too difficult to price and hedge digital options unless they are very near expiration and very near the strike. Digital options have proved to be among the most enduring and popular types of exotic options and have been incorporated in a variety of securities and derivative transactions. In addition, the concept of digital options provides the basis of contingent-premium options. 6 6 In fact, digital options are rarely used alone. They can also be integrated into structured off-balance sheet products such as swap with embedded leverage (Spel also called a range, time, accrual or corridor swap), or in the form of range notes. These strategies are illustrated in Barret [1]. 5

6 6 Problems 1. Calculate the value in dollars of a derivative that pays off 10,000 in one year provided that the dollar-sterling exchange rate is greater than at that time. The current exchange rate is The dollar and sterling interest rates are 4% and 8% pa respectively. The volatility of the exchange rate is 12% pa. 2. In Section 3, we considered the following hedge for a digital call option struck at K which pays $1: Long 1/ε Europeans at point K. Short 1/ε Europeans at point K + ε. (a) In this case, if the bull spread did not match the digital, then it underhedged it. (i) Construct a more symmetrical hedge for the digital option. (ii) Construct a bull spread that will assure the dealer of overhedging. (b) Use the bull spread to show that the digital call option should be priced at C = e rτ N(d 2 ). 3. By considering bull spread hedges similar to Problem 2, or by direct evaluation, or otherwise, show that (a) the price of a digital put option is P = e rτ N( d 2 ). (b) the price of a asset-or-nothing call option is C a = Se qτ N(d 1 ), where d 1 = d 2 +σ τ. (c) the price of a asset-or-nothing put option is P a = Se qτ N( d 1 ). 4. Derive the Greeks (i.e., delta, gamma, theta time derivative, and vega volatility derivative of option price) of the digital call option from the pricing formula C = e rτ N(d 2 ). How do these Greeks compare with those for the European call option, whose pricing formula is C std = Se qτ N(d 1 ) Ke rτ N(d 2 )? 5. In Section 4 we considered a step structure which pays $100 if the stock is above $90 but below $110 at expiration. Investigate the sensitivity of the step structure s value, delta and gamma to changes in the underlying asset price S and time to expiration τ. In particular, consider 80 S 120 (dollars) and 0 τ 1 (year). Summarize your findings with charts (as examples, see charts 6 8 of Barret [1]). References [1] Barret, G. (1995). The Binary Option Mechanism. Corporate Finance Risk Management Yearbook 1995 (May 95), [2] Pechlt, A. (1995). Classified Information. Risk, 8 (Jun 95), Also in Over the Rainbow: Developments in Exotic Options and Complex Swaps, Chapter 8 (pp 71 74). [3] Rubinstein, M. and E. Reiner (1991). Unscrambling the Binary Code. Risk, 4 (Oct 91),