FIN7029 Flashcards
Derivative
An instrument whose value depends on, or is derived from, the value of another asset.
3 major types: futures, forwards, options
Underlying asset
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.
Forwards contract
Agreement to buy or sell an asset at a certain future time for certain future price.
Futures contract
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).
Call option
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}
Put option
Option to sell a certain asset by a certain date for a certain price (strike price).
Arbitrage
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.
Futures prices
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
Futures prices - Assumptions
- 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
Futures for an investment asset (no income)
F_0 = S_0 e^{rT}
Asset with known income (for example, dividend
payment of a stock)
F_0 = (S_0 - I)e^{rT}
where I is the present value of income
European option
Can be exercised only at the end of its life
American option
Can be exercised at any time
Black-Scholes parameters
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
Black-Scholes model assumptions
- 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
Deterministic process
Where future states can be determined.
Stochastic process
Incorporates
randomness, making it impossible to precisely
determine.
Random walk - Markov
Over the next time step, our expected location depends only on where we are now, not where we have been.
Random walk - Martingale
Over the next time step, in terms of pure expectation we expect to remain where we are now.
Wiener process
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.
Properties of Wiener
W_T ~ N(0,T)
W_{t2} - W_{t1} ~ N(0,t2-t1)
Cov(W_{t1},W_{t2}) = min(t1,t2)
