Derivative securities

Question 1

Define the put-call parity formula:

1. For an asset
2. For a dividend paying asset
3. For a currency option
4. For a futures option

Answer 1

1. c + Ke^(-rT) = p + So
2. c + Ke^(-rT) = p + So - D
3. c + Ke^(-rT) = p + Soe^(-rfT)
4. c + Ke^(-rT) = p + Fo*e^(-rT)

Question 2

What are the advantages of a futures option?

Answer 2

1. May be easier to trade than underlying asset
2. Exercising option does not lead to delivery of underlying asset thus further postpone purchase
3. Underlying asset may not be available for trade
4. Futures options often entail lower transaction costs

Question 3

(Options)

Contrast american futures option with american spot options when there is a normal market and an inverted market

Answer 3

in a normal market the futures prices are higher than spot prices. i.e. you are already starting in the money. therefore you will obviously need to pay a premium amount. As such the call on a futures option will be worth more than on a spot option (a put holds the reverse of this)

In an inverted market the reverse of above is true

Question 4

What is the formula for the delta of a forward vs future?

what about a forward on a non-dividend paying asset?

Answer 4

Forward: e^(-qT)
Future: e^((r-q)T)
(REMEMBER YOU DONT GET RID OF THE Q, JUST CHANGE IT if indexfutures, currencyfutures, silverfutures!) (you change q for r when using option futures)
no div forward: the delta will just equal 1, because q in the above formula will be 0…

Question 5

What is the formula for the delta of an option (consider where underlying is currency, futures, index or the situation where there is no yield)

Answer 5

ΔCall = N(d1)*e^(-qT)
ΔPUT = (N(d1)- 1)*e^(-qT)

currency = replace q with rf
futures = replace q with r
no yield = remove the e^(-qT) component

Question 6

What happens to gamma as the delivery date draws nearer?

Answer 6

As the time to expiration draws nearer, the gamma ofat-the-moneyoptions increases while thegammaofin-the-moneyandout-of-the-moneyoptions decreases.

Question 7

Contrast the exposure of a covered call vs a naked put

Answer 7

They have the same exposure. Draw the graphs and you will see for yourself. the only difference is that the short call has unlimited upside exposure and the naked put has unlimited downside exposure

Question 8

Outline 4 strategies for hedging option positions

Answer 8

Naked strategy: Do nothing

Covered call: Buy underlying asset as well as use the option

Stop Loss strategy: Buy underlying asset when price exceeds strike and sell when price moves below strike. (doesn’t work well because unpredictable stock movements, and also cashflows take place at different points in time and must therefore be discounted, also purchases and sales cannot be made at exact strike price every time(in fact you will end up buying high and selling low!)).

Dynamic hedging: This is the process of maintaining delta neutrality by buying/selling the required number of underlying stock (or futures contracts) weekly, daily, hourly, etc. The cost of the hedge should reside around the Black-Scholes price. in fact the more often we rebalance the more likely we are to have a total cost equal to the Black-Scholes price.

Question 9

Contrast hedging of a forward contract Vs an option contract

Answer 9

The delta of an option is a function of time and the underlying stock price. Therefore hedging requires a dynamic approach.

The delta of a forward contract is 1, i.e. it is static. therefore, we can hedge and forget.

Question 10

Explain the need to use delta, gamma and vega neutrality when hedging

Answer 10

Delta = protects against small changes in underlying asset
Gamma = protects against large change sin underlying asset
Vega = protects against small changes in volatility (for example a small market disruption could lead to volatility changes for which we need protection)

Question 11

Contrast the advantages and disadvantages of a derivative exchange

Note: these can be reversed for the advantages/disadvantages of OTC contracts

Answer 11

Advantages:

1. No credit risk
2. Standardized contracts

Disadvantages:

1. Margin requirements
2. cannot be tailored to specific needs

Question 12

Outline the process of marking to market and the use of the margin account

Answer 12

marking to market is essentially just receiving the profit/loss from the futures position each day whilst also maintaining the margin account.

If the margin account drops below the maintenance margin then a margin call is made for the amount that brings the account back up to the initial margin.

Question 13

Contrast contango and backwardation

Answer 13

Contango (normal market) = futures prices trade higher than spot prices = upward sloping forwards curve = due to cost of carry and financing (asset may pay no interest, so holder should be rewarded for holding asset and not converting into money now which he could earn interest on) = usually where surplus of asset therefore low spot price because no need to worry about having asset

Backwardation = future prices lower than spot prices = due to convenience yield (i.e. there is a benefit to havingthe asset now) = usually a shortage of the asset so therefore beneficial to own now due to worry of future availability.

Question 14

Contrast futures and options

Answer 14

Options:

• Unlimited upside potential
• Limited loss potential
• Requires upfront investment

Futures:

• Unlimited loss potential
• Unlimited gain potential
• small upfront cost (only margin accounts)
• leverage to take large speculative positions

Question 15

When does a short hedgers position improve? what about a long hedger?

Answer 15

Short hedger = when basis strengthens unexpectedly

Long hedger = when basis weakens unexpectedly

Question 16

How to minimise basis risk?

Answer 16

Choose a contract with delivery as close to as possible (but after) the delivery month of the underlying asset

(when cross hedging is necessary) choose the instrument whose futures price is most highly correlated with the asset price

Question 17

When hedging an equity portfolio, overall what is the outcome in respect to the return achieved?

Answer 17

the return achieved will be aprox equal to the risk free rate (i.e. hedging creates risk free outcome)

Question 18

When hedging an equity portfolio how do we change the beta to a desired level?

Answer 18

(B-B*) x P/F = N

Long to increase Beta
Short to decrease Beta

Question 19

Outline the process of cash and carry arbitrage

Answer 19

Where the Market quoted price is greater than the theoretical forward price the market has overpriced the derivative.

1. Borrow cash at risk free rate and buy the underlying asset + take short position in the derivative
2. Receive dividends and invest at risk free rate at each date
3. Sell the underlying asset at the locked in price under the derivative + repay loan + receive dividends invested at risk free rate
4. Profit equal to the amount by which the market overpriced the derivative should be achieved

Question 20

How do we value a forward contract?

What is the difference for futures?

Answer 20

Remember value at time 0 is always 0.
Super easy:

Just calculate the difference between Ft and Fo and discount by appropriate factor back to time 0.

Consider if long or short position taken initially to determine if Ft or Fo is negative, etc.

(Ft-Fo)*e^-r(T-t) (long position)

For futures we use no discounting as they can be closed out at any point in time (exchange traded)

Question 21

Explain the arbitrage strategies for a foreign currency futures/forwards contract where

1. market< theoretical
2. Market > theoretical

Answer 21

1. Borrow foreign currency at rf, convert to the domestic currency (St) and invest at r + Enter a long position in a forward contract to buy the foreign currency (Ft) at maturity and repay the foreign currency loan
2. Borrow domestic currency at r, convert to the foreign currency (St) and invest at rf + Enter a short position in a forward contract to sell the foreign currency (Ft) (convert to the domestic currency) at
   maturity and repay the domestic currency loan

Question 22

Contrast the arbitrage approach where for an index the:

1. market is < theoretical
2. market is > theoretical

Answer 22

1. Market < theoretical = long position in derivative + short sell underlying stock + invest proceeds
2. Market > theoretical = short position in derivative + buy underlying stocks

Question 23

How do we determine if an arbitrage opportunity is possible for an option with bounds?

How do we determine the minimum amount of profit to be made quickly?

If one of the bounds is violated what do we do to make a riskless profit?

Answer 23

Use the upper and lower bounds of the option to see if c or p violates the bounds.

profit = The amount by which the bounds are violated will be the minimum profit to be made.

outcome depends on which bound + whether put or call. However! what seems to work is to analyse if its overpriced (sell) or underpriced (buy) and to make sure you take a secondary action to set you up to follow through with the first option (for example if you buy a put you will want to buy the stock too, so borrow the cash to do so…)

Question 24

What is the put-call-parity essentially saying?

Answer 24

That if you havetwo portfolios:

1. a call option and the PV of the strike (i.e. cash to execute the option upon delivery)
2. a put option and the underlying stock.

Then the value of each portfolio is the same, therefore they must be worth the same today.

Question 25

How do we use put call parity to determine an arbitrage? what approach do we take?

Answer 25

Simply complete the put-call-parity formula.

If one side is less and one side is more then buy the undervalued side and sell the overvalued side.

You will end up with a position in two options. You will execute the one that lands in the money and ignore the one that does not

Question 26

Contrast American option prices with European option prices (what is the one exception?)

Answer 26

American options will always be worth more due to early exercise premium…

However, an American call paying no dividends is never good to exercise early and therefore has the same price as a European option (C = c)

Question 27

What two things must you remember about the formula sheet and the bounds for options?

Answer 27

1. All of the lower bounds are actually a max(x,y,0) function
2. all of the american bounds are the same, except for the put bounds where you remove any e^() component (for currency and futures remove e^() component for all bounds.

Question 28

What arbitrage strategy do we use if the lower bound of a call option is violated?

what if the upper bound is violated?

Answer 28

Call is undervalued, therefore.

1. Long the call option
2. short sell underlying stock
3. Invest cash

Question 29

Contrast a covered call with a protective put

Answer 29

Covered call = When you have a short call position so you buy the underlying stock too = final position of a short put

Protective put = long position in asset + long position in put = final position of long call

Question 30

What are two tips to accurately depict a bear spread and a bull spread?

Answer 30

Just remember that one of the options will align with the curvature of the desired spread + this option will be the one on the outer side of the spread.

Question 31

How do we establish a butterfly spread with calls? when would we want this?

Answer 31

1 long call low strike
1 long call high strike
2 short call mid strike

use this in a low volatility (stable) market.. alternatively use a reverse butterfly spread in a high volatility market

Question 32

How do we establish a straddle (2 ways)

Answer 32

Straddle Long: Long put and Long call = useful for high volatility = requires initial premium

Straddle Short: Short call and short put = useful for low volatility = earn premium income in low volatility market

Question 33

How do we establish a strip and a strap?

Answer 33

Strip = 2 long put and 1 long call = favour high volatility leaning towards a bearish market

Strap = 2 long call and 1 long put = favour high volatility leaning towards a bullish market

Question 34

What is the order of short volatility strategies?

Answer 34

Straddle, Strangle, butterfly

Largest peak to smallest peak…

Question 35

How do we establish a strangle?

Answer 35

Same as with straddles, however we use different strikes for the two options

Question 36

How do we use the payoff table for spreads?

Answer 36

Create ranges around your k values + determine value of each option within corresponding ranges + add premium = payoff

Question 37

What do we need to assert no arbitrage arguments?

when using no arbitrage arguments how do we adapt for if there is a dividend being paid?

Answer 37

We need a portfolio consisting of Delta long assets and 1 short put or call option (since we are short this is why we minus the value of fu or fd)

Dividend being paid:

1. Calculate the delta by multiplying by e^(-qT)
2. Calculate the terminal portfolio value by multiplying the stock price by e^(qT)

Question 38

How do we calculate the standard deviation of a stocks daily return?

Answer 38

= volatility * SQRT(1/252)

We can use this to determine the expected change in the stock per day

Question 39

What are the five assumptions of Black-scholes? (LDRTA)

Answer 39

1. The stock price has a lognormal distribution with µ and σ constant
2. The stock pays no dividends during the life of the option
3. Investors can borrow or lend at the same constant risk-free rate r
4. There are no transaction costs or taxes
5. There are no arbitrage opportunities and Security trading is continuous

Question 40

Outline the four properties of the Black-Scholes model

Answer 40

1. N(d1) = delta of option
2. N(d2) : The risk-neutral probability that the call will be exercised (ends up in-the-money)
3. As S0 becomes very large, c → S − Ke−rT and p → 0
4. As S0 becomes very small, c → 0 and p → Ke−rT
   − S

Question 41

What are two causes of stock volatility?

Answer 41

1. new information

2. trading

Question 42

What is implied volatility?

Answer 42

implied volatility = The implied volatility of an option is the volatility for which the Black-Scholes price equals the market option price

Question 43

How do we alter the Black-Scholes formula if we have dividends expressed in $ and not as a yield?

Answer 43

Simple switch out So with the difference between the stock price and PV of the dividends = So*

Question 44

How do we approximate the American call using the Black Scholes model?

Answer 44

This is actually nice and simple.

Firstly calculate the european call/put as per normal (remember it must have dividends for a call. if it is a non-dividend call option then it is never exercised early therefore C = c).

Next calculate the same thing, however T becomes the time of the last dividend and we also do not include the value of the last dividend in D when calculating So* (refer to ETQ 20,b for an example)

Question 45

Does the expected rate of return on a share effect the option price?

Does the strike effect stock volatility

Answer 45

Nope

Nope

Question 46

When we use (N-d1) and produce a positive answer what do we need to remember about the delta of a put?

Answer 46

the delta of a put is always negative, so take the negative!

Question 47

Can the black-scholes only be used to price European options?

Answer 47

Nope! an american call with no dividends has C = c. Also we can approximate a dividend paying call by taking the max of the european call and the european call maturing just before the final ex-dividend date.

Question 48

How does an arbitrage strategy using the put-call parity differ for option futures?

Answer 48

We need to remember here that futures are at play

1. If we sell the future we receive no cash for this, because futures are settled daily. Rather we will use the futures option to close out our position on the future to make a gain of $x. (the difference between the strike and Fo)

Question 49

Outline the features of an interest rate swap

Outline the features of a currency swap

Answer 49

Interest Rate swap:
1. No initial exchange of principal
2. Netting of cashflows
3. Cashflows determined in advance but paid in arrears
(this is why you use the previous Libor each time)

Currency swap:

1. principal exchanged at start and finish
2. no netting of cashflows (i.e. exchanged in full)

Question 50

What is one argument against interest rate swaps?

Answer 50

‘maturity’ mismatch of the rates. i.e. the fixed rates may be quoted as fixed for 1 year, however the floating rates may be set for only 6 months after which they may skyrocket or plunge leading to the swap effect drastically changing (benefiting one party and hindering the other)

Question 51

How do we value the floating leg of an interest rate swap?

Answer 51

Simply take the notional + the floating payment and discount the whole amount from the date of the next floating payment

Question 52

For a delta neutral portfolio what does a stationary stock price vs a wildly swinging stock price mean for a portfolio with a positive gamma and a portfolio with a negative gamma?

Answer 52

Positive gamma = portfolio loses value if stationary stock | Gains value if wildly swinging

Negative gamma = reverse of above