Derivatives

Question 1

Module 33.1, LOS 33.a

Describe Cash and Carry Arbitrage model

Answer 1

forward overpriced:

borrow money ⇒ buy (go long) the spot asset ⇒ go short the asset in the forward market

Question 2

Module 33.1, LOS 33.a

Describe Reverse Cash and Carry Arbitrage model

Answer 2

forward underpriced:

borrow asset ⇒ short (sell) spot asset ⇒ lend money ⇒ long (buy) forward

Question 3

Module 33.1, LOS 33.a

What are the maun CFA compounding convention for different instruments?

Answer 3

1) All LIBOR-based contracts such as FRAs, swaps, caps, floors, etc.
- 360 days per year and simple interest
- Multiply “r” by days/360
2) Equities, bonds, currencies and stock options:
- 365 days per year and periodic compound interest
- Raise (1 + r) to an exponent of days/365
3) Equity indexes:
- 365 days per year and continuous compounding
- Raise Euler’s number “e” to an exponent of “r” times days/365
4) Options on FRAs:
- 365 days per year and simple interest
- Multiply “r” by days/365

Question 4

Module 33.2, LOS 33.b

What is the no arbitrage price of an equity forward contract?

Answer 4

FP (of an equity security) = (S0 − PVD) × (1 + Rf)T

FP (of an equity security) = [S0 × (1 + Rf)T] − FVD

Question 5

Module 33.2, LOS 33.b

What is the no arbitrage price of an equity index forward contract?

Answer 5

FP(on an equity index) = S(0)×e^((Rcf−δc)×T)

Question 6

Module 33.3, LOS 33.d

What is the no arbitrage quoted price of fix bond forward contract with accrued interest?

Answer 6

QFP = FP/CF = [(full price)(1+Rf)^T−AI(T)−FVC]*(1/CF)

Question 7

Module 33.4, LOS 33.c

Where the value of an FRA comes from?

Answer 7

The interest savings on a loan to be made at the settlement date

Question 8

Module 33.4, LOS 33.c

when the value of an FRA is to be received ?

Answer 8

At the end of the loan

Question 9

Module 33.4, LOS 33.c

If the rate in the future is less than the FRA rate, does the short or the long pay?

Answer 9

The long is “obligated to borrow” at above-market rates and will have to make a payment to the short.

Question 10

Module 33.6, LOS 33.e

How to calculate fix rate of IRS?

Answer 10

SFR (periodic) = (1 − final discount factor)/sum of discount factors

Question 11

Module 33.6, LOS 33.e

When interest rates fall, who benefits - IRS fix rate payer or receiver?

Answer 11

Receiver benefits because he receives higher than the market rate

Question 12

Module 33.8, LOS 33.g

What is the equity swap value on a date?

Answer 12

Difference between index value and swap fix-side value

Question 13

Module 33.8, LOS 33.g

What is the one-for-another equity swap value on a date?

Answer 13

Difference in price for one stock - difference in price for another stock - no “pricing” swap at initiation

Question 14

Module 34.2, LOS 33.a, 33.b, 33.e

What is a fuduciary call?

Answer 14

Long call, plus an investment in a zero-coupon bond with a face value equal to the strike price

Question 15

Module 34.2, LOS 33.a, 33.b, 33.e

What is a protective put?

Answer 15

Long stock and long put

Question 16

Module 34.2, LOS 33.a, 33.b, 33.e

For which option is early exercise beneficial and why?

Answer 16

Deep-in-the-money put options could benefit from early exercise. For a deep-in-the money put option, the upside is limited (because the stock price cannot fall below zero). In such cases, the interest on intrinsic value can exceed the option’s time value.

Question 17

Module 34.4, LOS 34.c

How to value hedge ratio?

Answer 17

h = (Cu-Cd)/(Su-Sd)

Question 18

Module 34.6, LOS 34.f

What are the main BSM assumptions?

Answer 18

1) The underlying asset price follows a geometric Brownian motion process
2) Continuously compounded return is normally distributed
3) The yield on the underlying asset is constant
4) The volatility of the returns on the underlying asset is constant and known
5) The risk-free rate is constant and known
6) The options are European options
7) Markets are “frictionless.”

Question 19

Module 34.6, LOS 34.g

What are the BSM formulas for call and put options for non-dividend paying stocks?

Answer 19

1) call = S(0)N(d1) − e^(–rT)XN(d2)
2) put = e^(–rT)XN(–d2) − S(0)N(–d1)

d1 = (ln(S/X)+(r+σ^2/2)T)/(σ√T)
d2 = d1 - σ√T

Question 20

Module 34.6, LOS 34.g

In BSM, how to interpret N(d2)?

Answer 20

The risk-neutral probability that a call option will expire in the money

Question 21

Module 34.6, LOS 34.g

In BSM, what are the replicating portfolios?

Answer 21

1) Calls can be thought of as a leveraged stock investment where N(d1) units of stock are purchased using e^(–rT)XN(d2) of borrowed funds.
2) A portfolio that replicates a put option consists of a long position in N(–d2) bonds and a short position in N(–d1) stocks.

Question 22

Module 34.6, LOS 34.h

What are the BSM formulas for call and put options for dividend paying stocks?

Answer 22

1) call = S(0)e^(–δT)N(d1) − e^(–rT)XN(d2)
2) put = e^(–rT)XN(–d2) − S(0)e^(–δT)N(–d1)

d1 = (ln(S/X)+(r-δ+σ^2/2)T)/(σ√T)
d2 = d1 - σ√T

Question 23

Module 34.6, LOS 34.h

What are the BSM formulas for call and put options on currencies?

Answer 23

1) call = S(0)e^(–r(B or F)T)N(d1) − e^(–r(P or D)T)XN(d2)
2) put = e^(–r(P or D)T)XN(–d2) − S(0)e^(–r(B or F)T)N(–d1)

where B - base (foreign rate)
P - price (domestic rate)

Question 24

Module 34.6, LOS 34.h

What are the B model formulas for call and put options on futures?

Answer 24

1) call = F(T)e^(–rT)N(d1) − e^(–rT)XN(d2)
2) put = e^(–rT)XN(–d2) − F(T)e^(–rT)N(–d1)

d1 = (ln(F/X)+(σ^2/2)T)/(σ√T)
d2 = d1 - σ√T

Question 25

Module 34.6, LOS 34.j

What is the B model formula for call option on FRA?

Answer 25

call = AP*e^(–AP)(FRAmn*N(d1) − XN(d2))*notional
AP = actual/365

Question 26

Module 34.6, LOS 34.j

What is the B model formula for payer swaption?

Answer 26

pay = (AP) PVA [SFR N(d1) − X N(d2)] NP
pay = value of the payer swaption

AP = 1/# of settlement periods per year in the underlying swap
SFR = current market swap fixed rate
X = exercise rate specified in the payer swaption
NP = notional principal of the underlying swap

d1 = (ln(SFR/X)+(σ^2/2)T)/(σ√T)
d2 = d1 - σ√T

Question 27

Module 34.7, LOS 34.k

What does options delta show, what is the relation for calls and put?

Answer 27

1) Delta - option price in relation to underlying asset price
- call delta is positive
- put delta is negative
∆C ≈ e–δTN(d1) × ∆S
∆P ≈ –e–δTN(–d1) × ∆S
2) Delta is the slope of the prior-to-expiration curve

Question 28

Module 34.7, LOS 34.k

What does options gamma show, what is the relation for calls and put?

Answer 28

Gamma measures the rate of change in delta as the underlying stock price changes.
∆C ≈ call delta × ∆S + ½ gamma × ∆S2
∆P ≈ put delta × ∆S + ½ gamma × ∆S2

Question 29

Module 34.7, LOS 34.k

What does options vega show, what is the relation for calls and put?

Answer 29

Vega measures the sensitivity of the option price to changes in the volatility of returns on the underlying asset

The higher volatility - the more puts and calls are valuable.
Vega gets larger as the option gets closer to being at-the-money

Question 30

Module 34.7, LOS 34.k

What does options rho show, what is the relation for calls and put?

Answer 30

Rho measures the sensitivity of the option price to changes in the risk-free rate

Relation for calls is positive, for puts - negative

Question 31

Module 34.7, LOS 34.k

What does options theta show, what is the relation for calls and put?

Answer 31

Theta measures the sensitivity of option price to the passage of time
Theta in most cases <0 (except for deep-in-the-money puts)
For calls with passage of time value decreases, for puts - increases

Question 32

Module 34.7, LOS 34.l

What is delta hedge and how to define number of options needed?

Answer 32

1) long position in a stock + short position in a call option so that the value of the portfolio does not change as the stock price changes
# of short call options needed to delta hedge=
# of shares hedged / delta of call option
2) long position in a stock + long position in a put option 
# of long put options needed to delta hedge=
- # of shares hedged / delta of put option

Question 33

Module 34.7, LOS 34.l

Why is delta hedge called dynamic?

Answer 33

The delta-neutral portfolio must be continually rebalanced to maintain the hedge - since delta changes over time

Question 34

Module 34.7, LOS 34.m

What is gamma risk in delta hedging context?

Answer 34

Gamma risk is therefore the risk that the stock price might abruptly “jump,” leaving an otherwise delta-hedged portfolio unhedged

Question 35

Module 34.7, LOS 34.n

What is implied volatility in options pricing?

Answer 35

Volatility derived from BSM model given all other inputs. No closed form solution exists -> iterative process