FIN7029

Question 1

Derivative

Answer 1

An instrument whose value depends on, or is derived from, the value of another asset.

3 major types: futures, forwards, options

Question 2

Underlying asset

Answer 2

An investment term that refers to the real financial asset or security that a financial derivative is based on. Thus, the value of the underlying asset drives the value of the financial derivative.

Question 3

Forwards contract

Answer 3

Agreement to buy or sell an asset at a certain future time for certain future price.

Question 4

Futures contract

Answer 4

Agreement to buy or sell an asset at a certain future time for certain future price that is traded on an exchange.
Futures contract is a standard contract.
At expiry, the value of the contract is the difference between:
-The forward price (F) agreed in advance,
-The current spot price (S).

Question 5

Call option

Answer 5

Option to buy a certain asset by a certain date for a certain price (strike price).

Value:
f = DF(T) * E[max(S-K,0)]
where DF(T) is the discount factor, DF(T) = e^{-rT}

Question 6

Put option

Answer 6

Option to sell a certain asset by a certain date for a certain price (strike price).

Question 7

Arbitrage

Answer 7

Ability to make risk-free profit.
Occurs in markets when 2 equivalent products have different prices. In markets where everything is freely trade-able - buy the cheap one, sell the expensive one.
In an efficient market, there should be no arbitrage opportunities.

Question 8

Arbitrage Implications

Answer 8

Law of one price: identical products should have the same
price now.
* Why? If not, arbitrage opportunities exist and will be exploited until the market corrects itself.
* Corollary: products which provide the same future payoff must have the same value today.
* Good models should not permit arbitrage!
* Modelling assumption: arbitrage opportunities do not exist
(or they will be quickly eliminated)

Question 9

Arbitrage Limits

Answer 9

Arbitrage opportunities do occasionally exist
* Market frictions mean that not all price anomalies can be exploited
* Such frictions may be small for at least some (large) market players

Question 10

Futures prices

Answer 10

F_0 = S_0 e^{rT}

T = time until delivery date in a contract,
S_0 = price of underlying asset today (spot price),
F_0 = futures price today,
r = risk-free rate
e = eulers number

Question 11

Futures prices - Assumptions

Answer 11

• no transaction costs/ bid-ask spread
• the same tax rate on all trading profits
• borrow/lend money at same rate
• underlying asset has no costs (e.g storage) or rewards (e.g dividends)
• ability to take long/short positions

Question 12

Futures for an investment asset (no income)

Answer 12

F_0 = S_0 e^{rT}

Question 13

Asset with known income (for example, dividend
payment of a stock)

Answer 13

F_0 = (S_0 - I)e^{rT}

where I is the present value of income

Question 14

European option

Answer 14

Can be exercised only at the end of its life

Question 15

American option

Answer 15

Can be exercised at any time

Question 16

Black-Scholes parameters

Answer 16

S = spot price,
k = strike price,
r = continuous risk-free rate
q = continuous divided yield
T = time to expiry
sigma = volatility
N(x) = standard normal cumulative distribution

Question 17

Black-Scholes model assumptions

Answer 17

• no arbitrage
• rational investors
• frictionless market - no trading costs, taxes, spreads
• infinite divisibility of assets
• ability to lend/borrow at risk-free rate
• continuous trading
• price of underlying follows a lognormal distribution

Question 18

Deterministic process

Answer 18

Where future states can be determined.

Question 19

Stochastic process

Answer 19

Incorporates
randomness, making it impossible to precisely
determine.

Question 20

Random walk - Markov

Answer 20

Over the next time step, our expected location depends only on where we are now, not where we have been.

Question 21

Random walk - Martingale

Answer 21

Over the next time step, in terms of pure expectation we expect to remain where we are now.

Question 22

Wiener process

Answer 22

A stochastic process W_t is a Wiener process if:
1) W_0 = 0,
2) W_t is continuous,
3) For any 0<t1<t2 the change in value is normally distributed:
W_{t2} - W{t1} ~ N(0, t2-t1)
4) For any 0<t1<t2<t3<t4 the increments W_{t4} - W_{t3} and W_{t2} - W_{t1} are independent.

Question 23

Properties of Wiener

Answer 23

W_T ~ N(0,T)
W_{t2} - W_{t1} ~ N(0,t2-t1)
Cov(W_{t1},W_{t2}) = min(t1,t2)

Question 24

Binomial Tree vs BSM

Answer 24

• Binomial Model: Used for discrete time steps, suitable for
  American options (which may be exercised early).
• Black Scholes Merton Model: A continuous-time model, best for
  European options without early exercise.
• The binomial model converges to the Black-Scholes price as the
  number of steps increases.

Question 25

Shortcuts to Improve Convergence

Answer 25

▫ Control variate technique ▫ Richardson extrapolation ▫ Adaptive mesh techniques

Question 26

Yield Curves

Answer 26

* A yield curve represents the relationship between interest rates (or yields) and different maturities of fixed-income securities (such as bonds or swaps). * Constructed from benchmark instruments such as: – Cash rates (up to 1 year): Short-term interest rates set by central banks or interbank lending markets. – Futures/FRAs (Forward Rate Agreements): Contracts that lock in future interest rates. * Cash rates readily imply discount factors * But coupon bearing instruments do not * Bootstrapping is the process of determining discount factors that are consistent (i.e. arbitrage free) with the prices of instruments comprising the yield curve

Question 27

CRM - Structural Models

Answer 27

– Based on debtor’s ability to pay – Links the firm’s assets and liabilities (i.e. financial structure) – Default set at a particular reference point

Question 28

CRM - Reduced Form Models

Answer 28

– Use economic and financial measures as predictors – Using stochastic processes and hazard rate modelling

Question 29

CRM - Intensity Models

Answer 29

– Special type of reduced form model – Default modelled as unpredictable Poisson-like jump process – Parameterized by intensity which drives time to switch state and move to default

Question 30

Merton Model

Answer 30

* A structural model that links debt and equity * In general, a firm’s Total Assets = Equity (owned by shareholders) + Debt (held by bond holders) * Merton’s model can easily be understood from a few key insights: – A company with liabilities exceeding assets is in difficulty! – Shareholders have potentially unlimited upside but limited downside. – Bond holders have limited upside (and limited downside).

Question 31

Merton Model Strengths

Answer 31

– Connects default with capital structure – Can be estimated solely from market data – Provides a default probability estimate

Question 32

Merton Model Weaknesses

Answer 32

– Default can only happen at maturity, which is unrealistic – Assume constant asset volatility and risk-free interest rate – Ignores macroeconomic factors – Business cycle is not explicitly incorporated

Question 33

Reduced Form Model

Answer 33

* A simple reduced form model assumes a two-state world: default and non-default * Default is considered an absorbing state (there is no way back) and happens unpredictably.