Topic 2: Options

Question 0

What is the impact of zero volatility on prices

Answer 0

Zero volatility DOES NOT mean that prices are fixed.

It means that prices are known in advance

Question 1

Replicating portfolio

Answer 1

1. By trading in stock and cash you can replicate an option via construction of a replicating portfolio whose value mimics that of the option
2. The cost of the option is the t=0 value of this replicating portfolio
3. If you can replicate the option then you can hedge it by replicaitng its opposite
4. there are no probabilities involved in these calculations
5. American options can be priced in a binomial framework

Question 2

Impact of dividends on a stock’s price

Prepaid forward price

Answer 2

1. As stock goes from cum to ex, the price falls by this drop
2. Prepaid forward price: the current stock price So minus the PV of all dividends payable over the lifetime (T) of the option
3. If there are no dividends the prepaid forward price = the current stock price. Otherwise prepaid fwd price < So (owner of stock received dividends, owner of prepaid forward does not)

Question 3

Key points: stocks with dividends when using binomial tree

Answer 3

1. If the stock pays a dividend, then the stock price is the sum of the dividends + PV of the ex dividend value of the stock
2. u and d values are based on r, not (r - delta). This is like specifying delta = 0
3. There is an adjustment to sigma (vol)
4. u and d are applied at each node to the prepaid forward stockc price, then add back the PV of the divs to the new up/down price of the stock
5. PV of divs at each node is different
6. *** u and d process applies only to the prepaid forward price, not the dividend stream. Latter is pre-specified and not subject to volatility

Question 4

American options - key points

1. Cheaper or more expensive than European? Why?
2. Pricing methodology of American options
3. Difference between C (American call) and c (European call)
4. Difference between P and p

Answer 4

1. American options have more optionality (more choices) than European, therefore CANNOT be cheaper than equivalent European options
2. American options must be valued in binomial tree context because of the possibility of early exercise.
3. C vs c: can arise when asset pays cash spin off (eg dividend) - could be trigger for early exercise on C. Can arise if dividend is large.
4. P vs p: could arise if p value < K - S. eg, if interest rates are high, could be early trigger.

Question 5

Derivation of risk neutral probability

Answer 5

1. Probabilities are personal and idiosyncratic
2. p* and (1-p*) are probabilities
3. Use the risk free rate r in the equation
4. use risk neutral probabilities. This does not mean that all investors are risk neutral.
5. So is the sum of the expected cash flows of the stock, therefore So captures each investor’s determination of the fundamental value of So, a blended outcome. The current observable So blends individual expectations and therefore is NOT the result of one typical risk averse investor’s calculation.

Question 6

Option replication and risk neutral probabilities will lead to exactly the same option price – verify in equations

Answer 6

delta S + B = ……

C = ……

Question 7

Black Scholes limitations/assumptions

Answer 7

1. can only be used on European options
2. stock’s volatility assumed to be known and constant over the life of the option
3. Stock’s price changes smoothly (never gaps)
4. short term interest rate never changes
5. anyone can borrow or lend as much as he/she wants at a single interest rate
6. no trading costs
7. no tax impacts
8. no dividends
9. no takeovers or events to end option life early

Question 8

Black Scholes formula derivation - notes

1. cost of carry is reduced by:
2. N represents:
3. Strike price K; use of r
4. S and d (greek letter, div yld)
5. Ao = ?

Answer 8

1. cost of carry is reduced by div yield received, ie (r - d)
2. N represents normal distribution. Notice on puts and calls; N(d1) vs N(-d1): makes sense because of the symmetry of normal distribution
3. r is the risk free rate, and needs to go with K. Time value of money.
4. S and d go together
5. Ao is the cleansed price of the stock. Cleansed for dividend yield.

Question 9

Relationship between Black Scholes and Forwards

Answer 9

• with any long forward position, the value is the current price of the underlying stock less the PV of the price you’ve promised to pay at a forward date.
• note that forward is an obligation.
• Ao is So cleansed for divs, ie So EXP (-div yield x T)
• note black scholes equation, essentially breaks down to the forward ie Ao and K, each multiplied by weight.

Question 10

Derivation of Black Scholes:

sigma and T

Answer 10

• sigma and T: sigma SQRT T = key driver of time value of the option.

Question 11

Interpreting calls and puts:

Answer 11

1. Use put-call parity. See that this holds. Otherwise arbitrage
2. May be asked in exam why call is worth more than put: identify if call is IN THE MONEY and therefore worth more. Compare PV (K) with Ao. DON’T compare S with K, you need to strip S.
3. Calculating the forward is worthwhile. F = A - PV (K). If F is positive, call is in the money. If F is negative, put is in the money

Question 12

Calculating FX options - tips

Answer 12

1. ALWAYS work out the commodity currency. This will usually be signalled by a 1 in front of it.
2. div yld (d) will be the interest rate that gives a return on the COMMODITY currency. (eg if commodity = AUD, use the AUD risk free rate as the div yld)
3. Watch the volatility. Volatility of the terms and commodity currencies are not interchangeable

Question 13

Value of an American option:

Answer 13

Value of Am option = max(discounted expected value, immediate exercise value)

Question 14

Different asset classes with cash spin offs

• identify the div yld (greek letter d)
• futures (may be in exam)
• currency

Answer 14

1. identify the commodity
2. d measures the benefit of holding that commodity
3. use d to cleanse the value of So.
4. eg:
   stock indices; use cc rate of div yld.
   futures and forwards: use risk free rate for d. (d = r because benefit is the return on dollars not spent, ie no upfront payment of cash. Futures are just like forwards because there are no flows)
   options on currency: rate of return is the interest on the commodity currency

Question 15

Options on currency

Answer 15

rate of return measured by the commodity currency. Carry trade, replicates the underlying long FX forward contract, reflects comparison of:

• invest one unit of the commodity at rate r(c)
• sell one unit for So units of the term currency then invest those units at terms currency rate r(t); then repatriate proceeds back to commodity currency once the investment matures

In PRICING currency options the relevant risk free rate of interest is the rate on the TERMS currency

Question 16

Implied Volatility

Answer 16

• on chart, plot sigma against K
• in the money calls have higher volatility
• shorter dated calls have higher implied volatility than longer dated calls
• consider volatility smirk, smile & frown

Question 17

Implied Volatility:

- volatility smirk

Answer 17

• caused by the distribution of the underlying. Equity markets - up stairs down ladders.
• Distribution of rate of return has negative skew: greater probability of a large negative rate of return
• option makers charge more for low strike options and this manifests into higher implied volatility (low strike put suddenly comes in the money as price plunges)

Question 18

implied volatility

Answer 18

Implied volatility tells you the market forecast of volatility (SD of log returns)

• implied volatility depends on the strike of the option (very high and very low K = higher implied volatility)
• implied volatility varies with the term of the option.

Question 19

Puts and calls with the same strike and time to expiration must have xxxxx implied volatility

Answer 19

musrt have the same implied volatility

Question 20

Using implied volatility

Answer 20

• if there is no market observable price, use implied vol to generate a price consistent with traded options
• option prices are often traded in vol terms rather than dollar price
• volatility skew - how well a model works, eg see if implied volatility is the same for all strike prices
• cannot use implied vol to determine if an option price is correct; but implied vol does tell us the market’s assessment of volatility
• for a given expiration month, volatility decreases as K increases = volatility skew

Question 21

Model risk results in:

Answer 21

• selling a security too low or buying too high
• inappropriate hedging trades (loss / profits)
• over estimating profits so that excessive bonuses are paid
• unproductive allocation of resources due to the above
• some may game the system for personal advantage
• loss of reputation (P/L)

Question 22

Manage valuation risks

Answer 22

1. Valuation reserves: reserves are for uncertainty concerning current valuation, not for uncertainty concerning future market variation.
2. Position limits: quantum of trading in markets exposed to valuation risk is constrained.

Question 23

Accounting: FAS 157 (US) or AASB 7 (Aus): disclose information about valuations;

1. define 3 levels
2. valuations as at:

Answer 23

1. Level 1: Assets that have observable market prices are valued at that market price
2. Level 2: Assets with no observable market price are valued using models - models have inputs that are observable market prices
3. Level 3: Assets valued using models, but inputs are NOT observable in the market

Valuations are on the basis of immediate sale rather than long term or intrinsic value

Question 24

How to calculate expected value?

What rate should be used in discounting?

Answer 24

1. Calculate expected value: use ordinary probability and the required rate of return for a particular individual, which incorporates their risk premium
2. Or: use the risk neutral probability p* and the risk free rate of return.

Question 25

Will the option price be different if different probabilities and discount rates are used?

Answer 25

No. Options are priced relative to the current underlying stock price, and all investor specific attitude towards risk expectations are embedded in So.

Question 26

Currency forward

Answer 26

F = So EXP ^ (r - divyld)T

Question 27

CUrrency option:

1. note when converting currency
2. Calculate future value on an AUD premium

Answer 27

1. convert at the STRIKE price not the spot price as the option gives the right to buy/sell at the strike
2. When calculating future value on an AUD amount use the AUD interest rate