Question

1. Answer three parts of this question. Your answers for each part should be no more than two pages long. a. Describe the assumed process for the evolution of the underlying asset price in the Black-Scholes optionpricing model. b. Derive the lower bound for a European put option written on a non-dividend paying stock and explain how you could make a riskless profit if this bound is violated. c. ‘American call options written on non-dividend paying stocks should never be exercised early’. Is this statement true? Prove your answer. d. Describe how a Collateralized Debt Obligation (CDO) is created. Why is this derivative popular with the banking sector? e. Assuming that the reference entity defaults after 2 years and 3 months, describe the cashflows arising from a 3-year CDS contract, with a notional principal of €80 million and a credit default spread of 160 basis points that is initiated today. Premium payments are made semi-annually in arrears.

Answer 1

a.The Black-Scholes formulation is used to estimate the fair value cost of a call option under a given set of conditions. The general idea behind the model is that an investor could perfectly hedge all option risk by buying and selling options over time. This “no arbitrage” solution implies that there is only one fair value option price, hence the solution of the Black-Scholes option price.

Model assumptions:

•European option

•No arbitrage

•Price evolution follows Weiner process (geometric random walk) with constant drift and variance

•No dividends

•Borrow and lend at the risk-free rate

•Buy, sell, and short any quantities or percentages

•No corresponding fees or costs

•No transaction costs

Put simply the Black–Scholes model of option pricing describes the following process: assuming that asset prices evolve according to a random process, and under a constant short-term interest rate, a market participant can construct a portfolio of assets (shares and risk-free bonds) that replicates the payoff profile of an option contract. Using the no-arbitrage rationale of asset pricing, the option price must be equal to the current price of the replicating portfolio, whose price is known.

When this approach is applied to bond options however, certain complications arise. The evolution of a bond price does not fit simply into the dynamic price process used to describe share price movements. For one thing, the price of a bond displays a pull-to-par effect, as it must reach 100 on maturity. Share prices do not have this constraint. In addition this par price constraint mean that the volatility of the bond price reduces as the bond approaches maturity. The basic Black–Scholes analysis therefore cannot be applied in the bond option market without modification. Another complication is the assumed constant level of the short rate. While this may not have much economic impact in terms of a share option, assuming that short rates are constant but that the bond price follows a random, dynamic process is contradictory. Therefore when applying the Black–Scholes analysis to bond options, analysts assume that interest rates, rather than bond prices, follow a random evolutionary path. This allows bond prices to converge to par, and bond price volatilities would reduce to zero, and interest rates would not be assumed to be constant. But what maturity interest rate is assumed to follow a stochastic process? In fact researchers know that interest rates along the term structure are closely correlated, and while certain models focus only on the short rate, it is increasingly common to find market participants making assumptions about the evolution of the entire term structure of interest rates when pricing bond options. This also requires users to make assumptions about the correlation of one interest rate to another, and to the term structure as a whole. It is clear therefore that for bond options a modification to the basic Black–Scholes analysis is required.

b.Options Arbitrage

            As derivative securities, options differ from futures in a very important respect. They represent rights rather than obligations � calls gives you the right to buy and puts gives you the right to sell. Consequently, a key feature of options is that the losses on an option position are limited to what you paid for the option, if you are a buyer. Since there is usually an underlying asset that is traded, you can, as with futures, construct positions that essentially are riskfree by combining options with the underlying asset.

Exercise Arbitrage

            The easiest arbitrage opportunities in the option market exist when options violate simple pricing bounds. No option, for instance, should sell for less than its exercise value. With a call option: Value of call > Value of Underlying Asset � Strike Price

With a put option: Value of put > Strike Price � Value of Underlying Asset

For instance, a call option with a strike price of $ 30 on a stock that is currently trading at $ 40 should never sell for less than $ 10. It it did, you could make an immediate profit by buying the call for less than $ 10 and exercising right away to make $ 10.

In fact, you can tighten these bounds for call options, if you are willing to create a portfolio of the underlying asset and the option and hold it through the option�s expiration.  The bounds then become:

With a call option: Value of call > Value of Underlying Asset � Present value of Strike Price

With a put option: Value of put > Present value of Strike Price � Value of Underlying Asset

Too see why, consider the call option in the previous example. Assume that you have one year to expiration and that the riskless interest rate is 10%.

Present value of Strike Price = $ 30/1.10 = $27.27

Lower Bound on call value = $ 40 - $27.27 = $12.73

The call has to trade for more than $12.73. What would happen if it traded for less, say $ 12? You would buy the call for $ 12, sell short a share of stock for $ 40 and invest the net proceeds of $ 28 ($40 � 12) at the riskless rate of 10%. Consider what happens a year from now:

If the stock price > 30: You first collect the proceeds from the riskless investment ($28(1.10) =$30.80), exercise the option (buy the share at $ 30) and cover your short sale. You will then get to keep the difference of $0.80.

If the stock price < 30: You collect the proceeds from the riskless investment ($30.80), but a share in the open market for the prevailing price then (which is less than $30) and keep the difference.

In other words, you invest nothing today and are guaranteed a positive payoff in the future. You could construct a similar example with puts.

            The arbitrage bounds work best for non-dividend paying stocks and for options that can be exercised only at expiration (European options). Most options in the real world can be exercised only at expiration (American options) and are on stocks that pay dividends. Even with these options, though, you should not see short term options trading violating these bounds by large margins, partly because exercise is so rare even with listed American options and dividends tend to be small. As options become long term and dividends become larger and more uncertain, you may very well find options that violate these pricing bounds, but you may not be able to profit off them.

c.NO,American option, especially if it’s a non-dividend paying stock.

The option has intrinsic value and time value. The intrinsic value of the option is always greater than 0. Along with that the cash has time value, so you would rather delay paying the strike price by exercising it as late as possible. You could use that money to earn interest.

So, a positive intrinsic value plus time value implies that you are better off selling the option rather than exercising it early. This is true for a non-dividend paying stock.

However, for a dividend paying stock, the only time it may pay to exercise a call option is the day before the stock goes ex-dividend, and only if the dividend minus the cost of carry is less than the corresponding Put. By exercising, the option holder may forego the time value but will make up from the dividend received. We have used the word ‘may’ because the dividend may not be high enough to justify the early exercise.

d.A collateralized debt obligation (CDO) is a complex structured finance product that is backed by a pool of loans and other assets and sold to institutional investors. A CDO is a particular type of derivative because, as its name implies, its value is derived from another underlying asset. These assets become the collateral if the loan defaults.

To create a CDO, investment banks gather cash flow-generating assets—such as mortgages, bonds, and other types of debt—and repackage them into discrete classes, or tranches based on the level of credit risk assumed by the investor.

The tranches of CDOs are named to reflect their risk profiles; for example, senior debt, mezzanine debt, and junior debt—pictured in the sample below along with their Standard and Poor's (S&P) credit ratings. But the actual structure varies depending on the individual product.

Collateralized debt obligations are complicated, and numerous professionals have a hand in creating them:

• Securities firms, who approve the selection of collateral, structure the notes into tranches and sell them to investors
• CDO managers, who select the collateral and often manage the CDO portfolios
• Rating agencies, who assess the CDOs and assign them credit ratings
• Financial guarantors, who promise to reimburse investors for any losses on the CDO tranches in exchange for premium payments
• Investors such as pension funds and hedge funds

Advantages of securitization – Depository banks had incentive to "securitize" loans they originated—often in the form of CDO securities—because this removes the loans from their books. The transfer of these loans (along with related risk) to security-buying investors in return for cash replenishes the banks' capital. This enabled them to remain in compliance with capital requirement laws while lending again and generating additional origination fees.