Derivatives

Question 1

FPt = S0 * (1+Rf)^t

Answer 1

For an arbitrage opportunity of an:

1) overpriced asset - borrow money > buy the spot asset > short the asset in the forward market
2) underpriced asset – borrow asset > short the asset > lend the money > lend the forward

Question 2

No arbitrage price of a forward contract: Equity with a discrete dividend

Answer 2

FP = (S0 -PVD) * (1+Rf)^t

= (S0 * (1+Rf)^t) - FVD

Question 3

Value of a long position in a forward contract on a dividend paying stock

Answer 3

V(long position) = [St - PVDt] - [FP / (1+Rf)^T-t]
V(long) = V(short)

Taking S0 and removing PVD because the long position in a forward contract on a dividend paying stock.

Question 4

Price of Equity Index Forward Contracts

Answer 4

FP = S0 * e^(Rfc - deltac)*T

where,
Rfc = ln (1+Rf)

Question 5

No arbitrage forward price on coupon paying bond

Answer 5

FP = (S0-PVC) * (1+Rf)^T

= S0* (1+Rf)^T - FVC

Question 6

Value of forward contract prior to expiration

Answer 6

V(long position) = [St-PVCt] - [FP / (1+Rf)^T-t]
AND
FP = [(full price) * ( 1+Rf)^t - AIt - FVC]

where,
AI = Accrued Interest = (days since last coupon payment / days between coupon payments) * coupon amount

Full price = clean price + accrued interest

Question 7

Quoted Futures’ price

Answer 7

= FP/CF = [(full price) *(1+Rf)^t - AIt - FVC] * (1/CF)

Question 8

x by y FRA notation

Answer 8

contract expires in x months and the loan is settled in y months

The value of the FRA is the interest savings due to a lower rate, discounted by the time it takes

Question 9

Currency Forward Pricing

Answer 9

FP = S0 * [ (1+Rp)^T/(1+Rb)^T]

where,
Rp = price currency interest rate
Rb = base currency interest rate

Question 10

After initiation forward contract pricing

Answer 10

Vt = [(FPt - FP) * (contract size)] / (1+Rp)^T-t

Question 11

Value of a futures contract

Answer 11

= current futures price - mark to market price

Question 12

Discount factor for LIBOR rates(z)

Answer 12

z = 1/ [1 +(LIBOR * days/360)]

Question 13

Periodic swap rate

Answer 13

SFR = (1 - last discount rate) / sum of discount factors

For annual,
SFR(periodic) * # of settlement periods per year

Question 14

Interest Rate Swap

Answer 14

Value to payer = Sum(z) * (SFRnew-SFRold) * (days/360) * notional principal

Question 15

Equity Swaps

Answer 15

SFR(periodic) = (1 - last discount factor) / sum of discount factors

Question 16

Binomial Probability of an Up Move

Answer 16

P(u) = (1 + Rf - D) / (U-D)
P(d) = 1 - P(u)

Question 17

Put-Call Parity

Answer 17

Fiduciary call is equal to the value of a protective put
where, a fiduciary call is a long all plus investment in a zero coupon bond with the face value equal to the strike price.

S0 + P0 = C0 + PV(x)
C0 = S0 + P0 + PV(x)

Question 18

Hedge Ratio

Answer 18

C+ + C- / (S+ + S-) = the # of shares in an arbitrage trade

Question 19

Interest Rate Call Option Payoff

Answer 19

If the i.r. ^^, the value of the call option ^^ and there is a decrease in the value of the put option

If the i.r. decreases, the value of the call option decreases and there is an ^^ in the value of the put option.

Question 20

Interpretations of the Black Scholes Merton model

Answer 20

1. The BSM values the PV of the expected option payoff at expiration
2. Calls are a leveraged stock investment where N(d1) units of stock are purchased using (e^-rT * X * N(d2)) of borrowed funds.
3. N(d2) is interpretted as the risk neutral probability that a call option will expire in the money. N(-d2) or 1 - N(d2) is the risk neutral probability that a put option will expire in the money.

Question 21

Delta

Answer 21

A change in the asset prices vs. A change in the options’ price.

Put option delta increases from -e^-deltat to 0
Call option delta increases from 0 to e^-deltat

Question 22

Gamma

Answer 22

the rate of change in delta vs the change in stock prices

long calls and puts have a positive gamma

Gamma is highest at-the-money

Question 23

Vega

Answer 23

the sensitivity of options price changes to changes in the volatility of returns of the underlying asset.

Call and put options ^^ in price as volatility ^^

Vega gets larger as the options’ prices get closer at the money

Question 24

Rho

Answer 24

Measures the sensitivity of an options price to changes in the risk-free rate

Call options ^^ with an ^^ in Rf
Put options decrease with an ^^ in Rf

Question 25

Theta

Answer 25

The sensitivity of an options price to the passage of time

As time passes, the call options approaches maturity, decreasing its speculative value.

Question 26

Covered Call

Answer 26

Long stock & Short Call

Investment at inception = S0 - C0
Value at expiration = St - max(St-X,0)
Profit at expiration = min(X-St,0) - (S0-C0)
Max Gain = X - S0 - C0
Max Loss=S0-C0
breakeven point = S0-C0

Main point of a covered call is to generate income on the premiums

Question 27

Protective Put

Answer 27

Long stock and Long put
Use to minimize downside risk

Investment at Inception = S0+P0
Value at Expiration = Max(St,X)
Profit at Expiration = Max(St,X) - (S0+P0)
Max Gain = St - (S0+P0)
Max Loss = (S0-X) + P0
Breakeven Point = S0+P0
Policy Deductible = S0-X

Question 28

Bull Call Spread

Answer 28

Buy a call with a lower exercise price, Xl & subsidizing that purchase by selling a call with a higher exercise price, Xh

Question 29

Bear Call Spread

Answer 29

Selling a call with a low, Xl, and buy a call with a high Xh.

Question 30

Bear Put Spread

Answer 30

Buy a higher, Xh & sell a Xl.

Profit = max(0,Xh-St) - max(0,Xl-St)- Ph0 - Pl0
Max Profit = Xh - Xl - Ph0 -Pl0
Max Loss = S0 - Xl + (P0-C0)
Breakeven Price = S0 + (P0-C0)

Question 31

Straddle

Answer 31

Long call and long put with the same strike price on the same security
As if the investor is betting on higher levels of volatility

Profit = Max(0, St-X)
Max Profit = St-X - (C0+P0)
Max Loss = C0+P0
Breakeven Price = X - (C0+P0) OR X + (C0+P0)

Question 32

Annual Volatility of security X

Answer 32

252 trading days a year

Sigma = %changeP * SqRt(252/ trading days @ maturity)
where,
%changeP = (breakeven price - current price) / current price