Book 5: Derivatives, Alt. Inv, and Portfolio Management
The value of the fwd at any point in time t is Vt(T)
=St + PV (cost) - PV(benefit) - Fo(T) / (1+Rf)^T-1
Binomial Model
U= size of upmove D= size of downmove = 1/U pieU= probability of up move pieD= 1- pieU
In a one-period binomial model, the value of an option is best described as the present value of:
a probability-weighted average of two possible outcomes
out of the money CALL options
S-x <0
in the money PUT options
X-S >0
Put call FWD parity
derived with a forward contract rather than an underlying asset F0(T) / (1+Rf)^T + P0 = C0 + X / (1+Rf)^T P0 - C0 = [X-F0(T)] / (1+ Rf)^T
At expiration, the exercise value of a put option is:
positive if the underlying asset price is less than the exercise price. The exercise value of a put option is positive at expiration if the underlying asset price is less than the exercise price. Its exercise value is zero if the underlying asset price is greater than or equal to the exercise price. The exercise value of an option cannot be negative because the holder can allow it to expire unexercised. (LOS 48.d)
in the money CALL options
S-x>0
out of the money PUT options
X-S < 0
A synthetic European call option includes a short position in:
a risk-free bond. A synthetic European call option consists of a long position in the underlying asset, a long position in a European put option, and a short position in a risk-free bond (i.e., borrowing at the risk-free rate).
The intrinsic value of an option is equal to:
zero or the amount that it is in the money.
6 factors that determine option prices
1. Price of underlying asset 2. The exercise price 3. The risk free rate of interest 4. Volatility of the underlying 5. Time to expiration 6. Costs and benefits of holding the asset
No arbitrafe futures price
= S0 + PVo(Costs) - PV0(benefits) x (1+rf)^T = [S)- net cost of carry] x (1+Rf)^T
If an asset has both storage costs and benefits from holding the asset over the life of the fwd contract Fo(T)
= [S0 + PV0(cost) - PV0 (benefit)] (1+Rf)^T
Option premium
= intrinsic value + time value An options intrinsic value (to a buyer) is the amt of the payoff at expiration and is bounded by zero At any point during the life of an option, its value will typically be greater than its intrinsic value
Bond valued at the risk free rate
= risky bond + credit protection
Difference between a european and american options
A holder of an American option has the right to exercise prior to expiration while European options can only be exercised at expiration Prices will be the same unless the right to exercise prior to expiration has positive value
The spot price of an asset is $35 and the risk-free rate is 3%. If the net cost of carry for the asset over the next three months is $1 in present value terms, the no-arbitrage 3-month forward price is closest to:
F0(T) = [S0 − net cost of carry] × (1 + Rf)T = ($35 − $1) × (1.03)3/12 = $34.25
The spot price of an asset is 62 and the risk-free rate is 2.5%. If the net cost of carry for the asset over the next six months is −3 in present value terms, the no-arbitrage 6-month forward price is closest to:
F0(T) = [S0 − net cost of carry] × (1 + Rf)T = [62 − (−3)] × (1.025)6/12 = 65.81
A high water mark of £150 million was established two years ago for a British hedge fund. The end-of-year value before fees for last year was £140 million. This year's end-of-year value before fees is £155 million. The fund charges "2 and 20." Management fees are paid independently of incentive fees and are calculated on end-of-year values. What is the total fee paid this year?
Management fee is £155 million × 0.02 = £3.1 million. Incentive fee is (£155 million - £150 million) × 0.20 = £1.0 million. Total fee is £3.1 million + £1.0 million = £4.1 million. (LOS 50.d)
At the money CALL option
S=X
At the money put options
S=X
"2-and-20" denotes
a 2% management fee and a 20% incentive fee.
A fiduciary call is a portfolio that is made up of:
a call option and a bond that pays the exercise price of the call at option expiration.
At expiration, the exercise value of a call option is:
the greater of zero or the underlying asset price minus the exercise price. If the underlying asset price is greater than the exercise price of a call option, the value of the option is equal to the difference. If the underlying asset price is less than the exercise price, a call option expires with a value of zero.
Calculate the value of an option of a stock in 3 steps
1. Calculating the payoff of the option at maturity in both the up move and down move states 2. Calculating the expected value of the option in 1 year as the probability weighted average of the payoffs in each state 3. Discounting this expected value back to today at the risk free rate
A hedge fund with a 2 and 20 fee structure has a hard hurdle rate of 5%. If the incentive fee and management fee are calculated independently and the management fee is based on beginning-of-period asset values, an investor's net return over a period during which the gross value of the fund has increased 22% is closest to:
16.6% The management fee is 2% of the beginning asset value, which reduces an investor's gross return by 2% to 22 − 2 = 20%. The incentive fee is 20% of the excess gross return over the hurdle rate, or 0.20(0.22 - 0.05) = 3.4%. The investor return net of fees is 22% − 2% − 3.4% = 16.6%. (Study Session 17, Module 50.2, LOS 50.d)
Formula for time 0
F0(T)/S0 = (1+Rf)^T F0(T)= fwd contract price S0= asset T= settlement date of fwd contract
Which of the following will result from futures prices for a particular commodity being in contango?
Negative roll yield. A positive roll yield results from a backwardated market, whereas a negative yield is produced in a contango market. In backwardated (contango) markets, futures prices are lower (higher) than spot prices.
Which of the following most accurately states an example of replication in derivatives pricing?
Risky asset + derivative = risk-free asset. Replications of future payoffs, composed of a risky asset, a risk-free asset, and a derivative on the risky asset
In which of the following ways is an interest rate swap different from a series of forward rate agreements (FRAs)?
The FRAs that replicate an interest rate swap may be off-market contracts. An interest rate swap may be replicated by a series of off-market FRAs (i.e., FRAs with nonzero values at initiation), if their present values sum to zero at initiation. The fixed rate is known at initiation for either an interest rate swap or a series of FRAs. Parties to both FRAs and interest rate swaps may agree to off-market prices at initiation.
An investor writes a put option with an exercise price of $40 when the stock price is $42. The option premium is $1. At expiration the stock price is $37. The investor will realize:
a loss of $2. Because the stock price at expiration is less than the exercise price, the buyer of the put option will exercise it against the writer. The writer will have to pay $40 for the stock and can only sell it for $37 in the market. However, the put writer collected the $1 premium for writing the option, which reduces the net loss to $2.
A net benefit from holding the underlying asset of a forward contract will:
decrease the no-arbitrage forward price at initiation. Compared to an underlying asset with no net holding cost or benefit, a net benefit from holding the underlying asset will decrease the no-arbitrage forward price at initiation and the value of a forward contract during its life. Holding costs and benefits have no effect on the value of a forward contract at expiration.
The price of an out-of-the-money option is:
equal to its time value.
An option's intrinsic value is equal to the amount the option is:
in the money, and the time value is the market value minus the intrinsic value.
Compared to an asset with no net cost of carry, holding costs that are greater than benefits:
increase the no-arbitrage price of the forward contract.
It is possible to profit from arbitrage when there are no costs or benefits to holding the underlying asset and the forward contract price is:
less than the future value of the spot price. An opportunity for arbitrage exists if the forward price is not equal to the future value of the spot price, compounded at the risk-free rate over the period of the forward contract.
The type of real estate index that most likely exhibits sample selection bias is:
repeat sales index. A repeat sales index includes prices of properties that have recently sold. Because these properties may not be representative of overall property values (may be biased toward properties that have declined or increased the most in value of the period), there is the risk of sample selection bias. An appraisal index or a REIT index is generally constructed for a sample of representative properties or REIT property pools. (Study Session 17, Module 50.1, LOS 50.e)
An American call option is most likely to be exercised early when:
the underlying asset pays dividends.
`Springfield Fund of Funds invests in two hedge funds, DXS and REF funds. Springfield initially invested $50.0 million in DXS and $100.0 million in REF. After one year, DXS and REF were valued at $55.5 million and $104.5 million, respectively, net of both hedge fund management fees and incentive fees. Springfield Fund of Funds charges 1.0% management fee based on assets under management at the beginning of the year and a 10.0% incentive fee independent of management fees. The annual net return for Springfield Fund of Funds is closest to:
5.0% Management fee = $150.0 × 1.0% = $1.5 million Net value at end of year after hedge fund fees = $55.5 + $104.5 = $160.0 million Incentive fee = ($160.0 - $150.0) × 10% = $1.0 million Total fees = $1.5 + $1.0 = $2.5 million Net of fees: $160.0 - $2.5 = $157.5 million Net return = ($157.5 / $150.0) - 1 = 5.0% (Study Session 17, Module 50.2, LOS 50.d)
A Hong Kong hedge fund was valued at HK$400 million last year. At year's end the value before fees was HK$480 million. The fund charges 2 and 20. Management fees are calculated on end-of-year values. Incentive fees are independent of management fees and calculated using no hurdle rate. The previous year the fund's net return was 2.5%. The annualized return for the last two years is closest to:
7.9% Management fee is HK$480 million × 0.02 = HK$9.6 million. Incentive fee is (HK$480 million - HK$400 million) × 0.20 = HK$16.0 million. Total fee is HK$9.6 million + HK$16.0 million = HK$25.6 million. Net of fee: HK$480.0 - HK$25.6 = HK$454.4 million Net return: (HK$454.4 / HK$400.00) - 1 = 13.6% Two year annualized return is (1.136 × 1.025)1/2 - 1 = 7.9% (Study Session 17, Module 50.2, LOS 50.d)
net cost of carry
= PV (benefits of holding the asset) - PV(costs of holding the asset)
F0(T)
= S0 x (1+RF)^T = no arbitrage price of the fwd contract
Vt(T)
=St-F0(T)/(1+ Rf)^T-1 At settlement, t = T so that T - t = 0 (there is no time left on the contract). The discounting term is (1 + Rf)0 = 1 and the payoff to a long forward is ST − F0(T), the difference between the spot price of the asset at expiration and the price of the forward contract.
Bidco Corporation common stock has a market value of $30.00. Which statement about put and call options available on Bidco common is most accurate?
A put with a strike price of $35.00 is in-the-money. A put is in-the-money when its exercise price is higher than the market value of the underlying asset. A put with a $35.00 strike price allows the trader to sell 100 shares of stock for $35.00 per share, which is $5.00 higher than the prevailing market value. This gives the put a value, hence, it is in-the-money. For a call to be in-the-money, its strike price would have to be lower than the market value of the underlying common stock, allowing the trader to purchase 100 shares at a price below the prevailing market value. At-the-money is when the strike price and asset market value are equal. A put with a strike price of $20.00 does not have intrinsic value because it is below the $30 price of the stock. It does have time value meaning it is worth something because there is the possibility the put will come into the money before it expires.
Which of the following will increase the value of a call option?
An increase in volatility. Increased volatility of the underlying asset increases both put values and call values. A higher exercise price or an increase in cash flows on the underlying asset decrease the value of a call option.
Put-Call Parity
Based on the payoffs of two portfolio combinations, a fiduciary call and protective put S + p = c + X / (1+Rf)^T * Must be european style and puts/calls must have the same exercise price and time to expiration for these relations to hold
