When valuing options in the real world using the binomial model we usually use multi period lattices. Explain the assumptions you would make in constructing a multi period lattice.
- In each small time slice, h, the up down behaviour of the stock is the same
- In each small time slice h, the behaviour of the stock price is independent of its behaviour in the previous time slice
Reasons why credit spread might be high
- zero coupon bond
- high volatility
- long tenor
- high probability of default
In the context of Black Scholes, the expression N(d1) can be interpreted as:
- the sensitivity of a call option to a one unit change in the underlying, assuming no dividends.
- N(d1) is the delta of a call option assuming no dividends. Delta is interpreted as the sensitivity of an option to changes in the underlyng
In the context of Black Scholes, the expression N(d2) can be interpreted as
- the risk neutral probability that the call will be exercised
- N(-d2) is the risk neutral probability that a put will be exercised
When can Black Scholes be used for valuing American options?
- BS is designed for European options
- Can be used where American and European options are identical.
- This occurs for CALL options where there are NO UNDERLYING CASHFLOWS
What is the impact of borrowing penalties (ie large difference between borrowing & lending rate) on call options
Impact: the price on calls and puts will generally be higher because call option requires little funding, whereas using a levered position requires funding at the penalty rate. Therefore retail investors would pay more for a call option. If call options are overpriced relative to fair value, then put-call parity would mean arbitrage drives up the value of puts (sell call, buy forward, buy put)
Explain the situation where a portfolio can have positive vega but negative gamma
- For an individual option vega and gamma must both be positive or must both be negative
- For a portfolio it is possible to have different signs if we are short short term options (negative gamma) and long in long term options (positive vega.
- Portfolio with positive vega benefits from an increase in volatility
- Portfolio with negative gamma will lose money following a price gap
Consider stock with vol of 45% and current value of $10. Which has the greatest VaR over 10 day horizon.
a) long one stock
b) long call on the stock with a strike of $10 expiring in one year
c) short call on the stock with a strike of $10 expiring in 1 year
d) short call on the stock with strike of $9 expiring in 1 year
Ans: a) long one stock.
At the money options have a delta of 0.5 whereas a stock position has a delta of around 1. So the long stock has greater sensitivity to changes in the stock price which implies greater VaR
Middle managers at a large financial institution have each been allocated a significant holding of restricted shares in the financial institution. Which statement is true?
a) the share allocation results in inefficient risk bearing
b) the share allocation encourages more entrepreneurial risk taking behaviour
c) allocating the same number of at the money options to each employee would be preferable from the perspective of the employee
d) all of the above
Ans: a) the share allocation results in inefficient risk bearing
Allocating same number of options to each employee is not desirable from the perspective of employees as the value of the options is much lower than the value of the equivalent number of shares.
Result of share allocations on middle managers:
- force middle managers to take additional exposure to the company share price which is not well diversified.
- This is inefficient, external shareholders are generally diversified and therefore have a lower riks premium for bearing the risk
- managers with large undiversified share holdings may choose to avoid risky projects with positive NPV because they are concerned about personal risk profile
- = “underinvestment problem”
Speculative grade securities tend to (4) (when compared with investment grade)
Speculative grade securities tend to:
- have more volatile ratings than IG
- lower Gini coefficients (Gini coefficient = quant measure of ratings performance, ie ability of ratings to predict result)
- Shorter time to default
Multi choice: Economic exposure = Translation exposure = Transaction exposure = Commodity exposure =
Economic exposure = exchange rates can impact firms cashflows, market share and value
Translation exposure = accounting derived changed in owners equity that occur as need to translate foreign currency statements into single currency
Transaction exposure = Sensitivity of contractual CFs to a change in exchange rate. Buy/sell goods, repatriate earnings, interest pmts associated
Commodity exposure =
Vega
- change in option price for change in vol
- increase in vol -> increase price of option
- vega is greatest for ATM calls & puts, and for options with moderate to short times to expiry
- vega is positive for both bought calls and bought puts
Monte Carlo simulation
- Monte Carlo depends on risk neutral valuation, does not work with actual probabilities
- Uniform distribution = each outcome equally probable
- Can be used with any distribution(?)
- Possible to simulate for large tails and changing volatility
- Can be used with any distribution - log normal, normal etc. This is an assumption, note for model risk
Disaster myopia
- people underestimate the probability of adverse events if no adverse events have occurred in the recent past.
- Form of over optimism
- people might choose to take on risky investments without an adequate reward, without thoroughly assessing the risks and without a contingency plan.
Tendency for people to fall for a story =
- easy to understand and follow a story, rational thinking takes more work
Self attribution bias =
- ## attribute good performance to self, bad performance to circumstance
Multi choice: Monte carlo simulation:
a) makes heavy use of computer power
b) uses statistical notions of sampling
c) can deal with highly non linear risk situations
d) all of the above
(d) all of the above
Merton Default Model
a) higher asset volatility means higher PD
b) higher dividend yield means lower PD
c) asset volatility is not important
d) lower asset volatility means a higher PD
e) cannot tell from information provided.
Merton Default Model
ANS: a) higher asset volatility means higher PD
b) higher dividend yield means lower PD (NO)
c) asset volatility is not important (NO)
d) lower asset volatility means a higher PD (NO)
e) cannot tell from information provided.
In the binomial option pricing model;
a) The stock price is for a firm with an all equity capital structure
b) The net cost of carry is always positive
c) Volatility is a constant throughout
d) People’s attitudes to risk don’t matter
d) People’s attitudes to risk don’t matter
Assumption of risk neutrality and the use of the risk free rate
Option replication and risk neutral probabilities will lead to exactly the same option price
A multinational sets up manufacturing faciltiies in the same country that it sells the product but which isn’t in balance sheet currency.
a) company faces flat demand curve
b) company has no FX exposure
c) company will seek to maximise profit, at least in theory
d) company’s commodity exposure is minimised
e) company can sell its product cheaper
a) company faces flat demand curve (Not necessarily; flat demand curve is when the market is highly competitive)
b) company has no FX exposure (no - still needs to translate currency back to balance sheet currency)
c) company will seek to maximise profit, at least in theory (yes…)
d) company’s commodity exposure is minimised
e) company can sell its product cheaper
Consider an American put on a publicly traded stock:
a) early exercise is less likely if option is deep ITM
b) early exercise is less likely if time to expiry is short
c) early exercise is more likely if interest rates are high
d) early exercise will never happen if the stock pays a dividend while the option is still alive
For America Puts: Exercise put if
- deep ITM
- high interest (to start earning interest immediately)
- low vol
c) early exercise is more likely if interest rates are high
Note, early exercise for calls is more likely if it occurs before the ex div date and is deep ITM
Consensus provision risk is:
a) The tendency for people to fall for a story
b) An example of self attribution bias
c) The tendency for people to focus on recent events when making decisions
d) AN example of incentive conflict
= d) AN example of incentive conflict (people agree too readily; perceive they will not be rewarded for questioning a strategy or bringing ill tidings)
a) The tendency for people to fall for a story (NO - this is
b) An example of self attribution bias
c) The tendency for people to focus on recent events when making decisions
A European put pn a non dividend paying stock is very close to expiry and is currently ATM. This option would exhibit:
a) Low gamma, low vega
b) High gamma, low vega
c) Low gamma, high vega
d) High gamma, high vega
= d) High gamma, high vega
Note: Vega highest when ATM
Gamma: deep ITM, delta approaches 1 for call and -1 for put; and gamma is zero. Liewise, deep OTM has gamma = 0. Gamma is high when ATM. Slope of delta is changing.
a) Low gamma, low vega
b) High gamma, low vega
c) Low gamma, high vega
