Arbitrage, Stock Options, and Ethical Considerations in Finance
Arbitrage and Forward Exchange Rates
Suppose, as of September 16, it was possible to buy 1 Euro for 1.1341 US Dollars, one year USD LIBOR was 0.82615% (simple interest), and one year EURIBOR was 0.128% (also simple interest).
No-Arbitrage Dollar-Euro Exchange Rate
a. What is the no-arbitrage Dollar-Euro exchange rate, one year forward?
If you buy one Euro for 1.1341 US Dollars and invest it at 0.128% for one year, you get 1.00128 Euro. Alternatively, if you invest the 1.1341 US dollars for one year at US Dollar LIBOR, you get 1.1341 USD x 1.0082615 = 1.143469367 USD. If there is to be no arbitrage, the forward price of 1.00128 Euro must be 1.143469367 USD, so that the forward price of 1 Euro must be 1.142007597 USD (1.1420).
Arbitrage Strategy
b. Suppose the prevailing one year forward Dollar-Euro exchange rate was, in fact, less than the no-arbitrage rate. What arbitrage strategy could be employed to exploit this difference?
If the one year forward rate were less than the no-arbitrage strategy computed above, we could borrow 1 Euro, exchange it for 1.1341 US Dollars, invest this dollar amount at the US Dollar risk free rate to obtain 1.143469367 USD in one year’s time. At that point, we would discover that our USD would buy more than the 1.00128 Euro required to repay our Euro borrowing (Since the Dollar-Euro exchange rate is quoted as Euro/dollar, a lower rate means that it takes fewer Dollars to buy one Euro, so a given number of Dollars will buy more Euro.).
Hedging with Eurodollar Futures
c. Suppose that you work for a financial institution that has entered into a forward rate agreement to receive $100,000,000, in exchange for Euro, as per the rate computed above. Suppose, also, that your institution needs to hedge the present value of that $100,000,000 exposure in the Eurodollar futures market using the 3 month LIBOR futures contract. Part of the exposure arises from the possibility of fluctuation in 3 month US Dollar LIBOR between June 16, 2016 and September 16, 2016, an exposure that your institution decides to hedge with 3 month Eurodollar futures contracts. How many of which contracts should be bought or sold for this hedge?
If interest rates go up, the present value of the cashflow will decrease. The deliverable for the Eurodollar futures contract that expires on June 16, 2016 is a Eurodollar certificate of deposit requiring an investment of $1,000,000 on June 16 and maturing 3 months later. The futures price settles at 100% minus the 3 month LIBOR fixing on the maturity date, which means that the value of the future also decreases as interest rates go up. Therefore, a short position in the futures is the necessary hedge. Given that one contract hedges a $1,000,000 exposure, 100 contracts are required to hedge a $100,000,000 exposure.
Employee Stock Options: Valuation and Considerations
A company grants 1,000,000 options to its executives on November 1, 2014. The stock price on that date is $30 and the strike price of the options is also $30. The options last for 10 years and vest after three years. The company has issued similar at-the-money options for the last 10 years. The average time to exercise or expiry of these options is 4.5 years. The company therefore decides to use an expected life of 4.5 years. It estimates the long-term volatility of the stock price, using 5 years of historical data, to be 25%. The present value of dividends during the next 4.5 years is estimated to be $4. The 4.5-year zero-coupon risk-free interest rate is 5%. The option is therefore valued using the BSM with S0=30-4=26, K=30, r=5%, σ=25%, and T=4.5. The BSM gives the value of one option as $6.31. Hence, the income statement expense is 1,000,000×6.31, or $6,310,000.
Valuing Stock Options with Early Exercise Features
Suppose a company grants stock options that last 8 years and vest after 3 years. The stock price and strike price are both $40. The stock price volatility is 30%, the risk-free rate is 5%, and the company pays no dividends.
In this case, σ=0.3, Δt=2, and r=0.05.
The probability on the up branches is 0.5158 and the probability on the down branches is 0.4842. There are three nodes where early exercise could be desirable: D, G, and H. We assume that the probabilities that the holder will choose to exercise at nodes D, G, and H have been estimated as 40%, 80%, and 30%, respectively. We suppose that the probability of an employee leaving the company during each time step is 5% (2.5% per year). For the purposes of the calculation, it is assumed that employees always leave at the end of a time period. If an employee leaves the company before an option has vested or when the option is out of the money, the option is forfeited. In other cases the option must be exercised immediately. The value of the option at the final nodes is its intrinsic value. At node H there is a 30% chance that the employee will choose to exercise the option. In cases where the employee does not choose to exercise, there is a 5% chance that the employee leaves the company and has to exercise. The total probability of exercise is therefore 0.3+0.7×0.05=0.335. If the option is exercised, its value is 61.14-40=21.14. If it is not exercised, its value is
The value of the option at node H is therefore 0.335×21.14+0.665×24.95=23.67 The value at node G is similarly 0.81×102.83+0.19×106.64=103.56 We now move on to the nodes at time 4 years. At node F the option is clearly worth zero. At node E there is a 5% chance that the employee will forfeit the option because he or she leaves the company and a 95% chance that the option will be retained. In the latter case the option is worth
The option is therefore worth 0.95×11.05=10.49. At node D there is a 0.43 probability that the option will be exercised and a 0.57 chance that it will be retained. The value of the option is 56.44. Consider next the initial node and the nodes at time 2 years. The option has not vested at these nodes. There is a 5% chance that the option will be forfeited and a 95% chance that it will be retained for a further 2 years. The valuation of the option at the initial node is 14.97. (This compares with a valuation of 17.98 for a regular option.)
Backdating Stock Options and Ethical Considerations
Backdating: Suppose that a company decides to grant at-the-money options to its executives on April 30 when the stock price is $50. If the stock price was $42 on April 3, it is tempting to behave as if those the options were granted on April 3 and use a strike price of $42. This is legal provided that the company reports the options as $8 in the money on the date when the decision to grant the options is made, April 30. But it is illegal for the company to report the options as at-the-money and granted on April 3. The value on April 3 of an option with a strike price of $42 is much less than its value on April 30. Shareholders are misled about the true cost of the decision to grant options if the company reports the options as granted on April 3.
Nature of Employee Stock Options
Nature: Employee stock options are call options issued by a company on its own stock. They are often at-the-money at the time of issue. They often last as long as 10 years.
Features of Employee Stock Options
Feature: There is a vesting period during which options cannot be exercised. When employees leave during the vesting period options are forfeited. When employees leave after the vesting period in-the-money options are exercised immediately and out of the money options are forfeited. Employees are not permitted to sell options. When options are exercised the company issues new shares.
Exercise Decision
Exercise Decision: To realize cash from an employee stock option the employee must exercise the options and sell the underlying shares. Even when the underlying stock pays no dividend an employee stock option (unlike a regular call option) is often exercised early.
Dilution
Dilution: Employee stock options are liable to dilute the interests of shareholders because new shares are bought at below market price. However this dilution takes place at the time the market hears that the options have been granted. It does not take place at the time the options are exercised.
Black-Scholes Model (BSM)
BSM: After carefully constructing a portfolio, Π, to be instantaneously riskless, the assumption of no arbitrage implies that such a portfolio must earn the riskless rate of return over the infinitesimal time interval, dt. In symbols, the change in the value of the portfolio, dΠ, must be rΠdt, thus justifying the equation dΠ=rΠdt. The BS equation follows upon writing out this equation in terms of the derivative and the underlying.
Ethical and Legal Obligations in Financial Markets
Ethically and legally, Tony Conti was required to submit RaboBank’s version of LIBOR without any regard to how it might benefit any trades that RaboBank had on its books (Though his submissions could, and, in fact, should depend on how much RaboBank needed to borrow in the interbank market as part of its day-to-day operations.). If he did take RaboBank’s trades into consideration – even once, and even a little bit – then he was guilty as charged. Nevertheless, over the time period in question, he received over 90 requests to benefit specific trades by raising or lowering his submission from what it would otherwise be. Upon receipt of the first request, Tony was placed in a very difficult situation, especially given that his boss, Anthony Allen, had explicitly instructed Tony to keep the requestors happy.
CFA ethics guidelines require CFA’s to require requests to manipulate markets to be reported either to one’s supervisor or to the appropriate compliance unit. Clearly, it wouldn’t do to report the problem to Allen! Furthermore, any report made to compliance almost certainly would have been discovered by Allen, probably placing Tony’s job on the line, and certainly eliminating any possibility for bonuses or future advancement. On the other hand, complying with manipulation requests was illegal, and that there was a real risk of discovery, given that they were delivered via Bloomberg Chat, and that many others were involved with the scheme.
Tony could quit or ask to be transferred to another job within RaboBank (Ex-traders sometimes become risk managers, for example.). However, such transfer requests are not necessarily granted, and, when they are, they almost certainly require a good word from one’s current boss. Similarly, finding another job is not always easy. In both cases, making the necessary arrangements could require considerable time, especially since there aren’t many jobs that pay the large sums that Conti was making as a trader. Conti, with a wife and three kids, presumably had a lifestyle that depended on a trader’s income.