AIM: Relate the difference between true and risk-neutral probabilities to interest rate drift.

42.1 Assume a binomial interest rate tree where the six-month rate either jumps up 50 basis points or jumps down 50 basis points. The real-world (or “true”) probabilities are 50% and 50% for each up- and down-state. If the interest rate drift is +20 basis point under risk-neutral probabilities, what is the risk-neutral probability of an up-state (p)?

* a. 50%
* b. 60%
* c. 70%
* d. 80%

42.2 Which of the following is MOST essential to the argument that contingent claims can be valued with risk-neutral pricing?

* a. The no-arbitrage price is invariant to investor risk preferences
* b. The no-arbitrage price accounts for investor risk preferences
* c. Investors are risk-neutral in the imaginary world
* d. The discount rate impounds investor risk aversion

42.3 In regard to the valuation of contingent claims (derivatives) by risk-neutral pricing, each of the following is true EXCEPT for:

* a. Expected discounted value will equal arbitrage price
* b. We must assume the growth (return) on the underlying equals the risk-free rate
* c. The risk-free discount rate is appropriate
* d. The derivative price is the same in the imaginary world (risk-neutral investors) and the real world

Answers:

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Business Graph
AIMS: Using replicating portfolios develop and use an arbitrage argument to price a call option on a zero-coupon security. In addition: Explain why the option cannot be properly priced using expected discounted values. Explain the role of up-state and down-state probabilities in the option valuation.
Questions:

40.1 Assume the market six-month and one-year spot rates are 2.0% and 2.2%, respectively. Assume, per Tuckman’s two-step binomial interest rate tree (i.e., each step is six months), that the six months from now the six-month rate will be either 2.5% (+0.5%) or 2.0% (-0.5%) with equal probability. If a bond’s face value is $1,000, what is the market price of the bond (note: Tuckman assumes semi-annual compounding)?

* a. $968.45
* b. $964.63
* c. $978.36
* d. $982.12

40.2 What are the risk-neutral probabilities?

* a. p = 90.1% and 1-p = 9.9%
* b. p = 9.9% and 1-p = 90.1%
* c. p = 80.1% and 1-p = 19.9%
* d. p = 19.9% and 1-p = 80.1%

40.3 Use a replicating portfolio to determine the price of a call option, that matures in six months, to purchase the $1,000 face value bond at a strike price of $990. What is the market price of the call option?

* a. $0.25
* b. $1.25
* c. $3.25
* d. $9.25

40.4 What is the discounted expected value of the call option?

* a. $0.97
* b. $1.27
* c. $1.97
* d. $2.37

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AIM: Define market efficiency, identify the three forms of market efficiency, and discuss the link between efficiency and the CAPM.
Questions:

30.1 If the STRONG-FORM of efficient market hypothesis (EMH) is true, then which investor group can produce sustained alpha?

* a. Fundamental analysts
* b. Stock exchange specialists
* c. Corporate insiders
* d. No groups

30.2 Which is the best test of WEAK-FORM EMH?

* a. Autocorrelation of security returns
* b. Price-to-book ratio in Fama-French three factor model
* c. Returns for professional security analysts and money manager
* d. Event studies

30.3 Which is the best test of SEMISTRONG-form EMH (conditional on a priori acceptance of weak-form)?

* a. Autocorrelation of security returns
b. Price-to-book ratio in Fama-French three factor model
* c. Returns for professional security analysts and money manager
* d. Performance of algorithmic trading rule(s) based on market data; e.g., block trades

30.4 Which is the best test of STRONG-form EMH (conditional on a priori acceptance the semi-strong form)?

* a. Autocorrelation of security returns
* b. Price-to-book ratio in Fama-French three factor model
* c. Returns for professional security analysts and money manager
* d. PEG (price-to-earnings growth) ratio

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AIM: Derive the basic equilibrium formula for pricing commodity forwards and futures

The forward price is equal to the expected spot price in the future, but discounted to the present.

Non storability
AIM: Explain the effect non‐storability has on electricity prices

Because electricity cannot (mostly) be stored, the forward market provides “invaluable price discovery.” Price changes largely reflect “[consensus] changes in the expected future spot price.

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A spread strategy is a position with two or more options of the same type (i.e., two or more calls; or, two or more puts).

… bull spread (type of vertical spread)

buy (long) a call option and sell (short) a call option on the same stock (and same expiration) but with a higher strike price. In this example, long call (strike = $20, premium = $1.99) + short call at higher strike (strike = $23, premium = $0.83)
Features of bull spread:

• Net debit but outlook is bullish

… Bear spread (type of vertical spread)

Buy (long) a call option call option and sell (short) a call option on the same stock (and same expiration) but with a lower stock price. In this example, bear spread: long put (strike = $23, premium = $2.93) + short put at lower strike (strike = $20, premium = $1.20)
Features of bear spread:

• Net debit but outlook is bearish

… Butterfly spread (sideway strategy)

Buy a call option at low strike price K1, buy a call option with high strike price K3, and sell two call options at strike price K2 halfway between K1 and K2. In this example, the butterfly spread: Long call (strike @ $18, premium = $3.21), long call (strike @ $22, premium = $1.13 ), short two calls (strike @ $20, premium = $1.99)

Features of butterfly spread:

• Expects low volatility (range-bound), Capped risk

… Calendar spread

In a calendar spread, the options have the same strike price but different expiration dates. The calendar spread can be created with calls or puts.
Two calls: sell a call option with strike price K1 and buy a call option with same strike price K1 but with a longer maturity term
Two puts: sell a put option with strike price K1 and buy a put option with same strike price K1 but with a longer maturity term

Short call with 1 year maturity (strike = $20, premium = $1.99) +Long call with 1.25 year maturity (strike = $20, premium = $2.27)

… Diagonal spread

In a diagonal spread, both the expiration date and the strike price of the calls are different.

… Box spread

A box spread is a combination of a bull call spread with strike prices K1 and K2 and a bear put spread with the same two strike prices. The payoff from a box spread is always K2 – K1.
The value of the box spread is always the present value of its payoff or (K2-K1)*EXP(-rT).

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AIM: Define and contrast exotic derivatives and plain vanilla derivatives.
Questions:
09.01. In comparison to a plain vanilla derivative, each of the the following is true of an EXOTIC derivative EXCEPT for (most likely answer):
a. BETTER hedge effectiveness
b. BETTER liquidity
c. LOWER basis risk
d. HIGHER (more) counterparty risk
09.02 Which is not an EXOTIC (according to Hull)?
a. Compound option
b. Bull call spread
c. Executive stock option (ESO) with four (4) year vesting
d. Option indexed to S&P 500
Answers:
09.01. In comparison to a plain vanilla derivative, each of the the following is true of an EXOTIC derivative EXCEPT for (most likely answer):
a. BETTER hedge effectiveness
b. BETTER liquidity
c. LOWER basis risk
d. HIGHER (more) counterparty risk
b. BETTER liquidity
The benefit of nonstandard terms is a better hedge but the price is typically less liquidity.
The essential difference between vanilla and exotic is similar to the difference between forward and futures: a vanilla derivative has standard terms which enables exchange trading; the exotic has non-standard terms which often requires OTC. As such, the primary (typical) trade-off is between liquidity and basis risk. A vanilla instrument will tend to have higher liquidity due to standardized terms but the exotic can be customized to LOWER basis risk (and thus giving BETTER hedge effectiveness).
09.02 Which is not an EXOTIC (according to Hull)?
a. Compound option
b. Bull call spread
c. Executive stock option (ESO) with four (4) year vesting
d. Option indexed to S&P 500
b. Bull call spread
The bull call spread is a TRADING strategy (long call plus short call with higher strike price): combinations of vanilla options are still vanilla. In regard to (c) and (d), these are American options with non-standard features.

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