Chapter 11: Valuation of Derivatives Flashcards
Define an arbitrage opportunity
An arbitrage opportunity exists if we can set up a portfolio that:
a) has zero initial cost, implying long in some assets and short in others.
b) at some future time T: has 0 probability of loss, and greater than 0 probability of strictly positive profit.
Principle of no arbitrage
States that arbitrage opportunities do not exist.
Law of One price
States that any 2 securities or combination of securities that give exactly the same payments must have the same price.
Why does the law of one price follow from the principle of no arbitrage?
If this were not true, an arbitrage opportunity would exist because you could buy the cheaper and sell the more expensive of the two.
Derivative
Security / contract which
…. promises to make a payment
…. at a specified time in the future,
the amount of which
….. depends upon the behaviour of some underlying security
….. up to and including the time of the payment.
European option
An option that can only be exercised at expiry.
American option
An option that can be exercised at any date before its expiry.
in-the-money
Call: St > K
Put: St < K
out-of-the-money
Call: St < K
Put: St > K
at-the-money
Call: St = K
Put: St = K
Intrinsic value of a derivative
The value assuming expiry of the contract immediately, rather than at some time in the future.
For a call option: = max{St - K, 0}
The intrinsic value is:
- Positive if the option is in-the-money
- Zero if the option is out-of-the-money / at-the-money
Time value (option value)
The excess (or shortfall) of an option's premium over its intrinsic value. Premium refers to the option's CURRENT price, not the price originally paid.
The delta of a vanilla call option
𝛿v/𝛿s = Δ = Φ(d₁)
The delta of a vanilla put option
Δ = -Φ(-d₁) = Φ(d₁) - 1
Approximation for f(S, σ + δ)
f(S, σ + δ) ≈ f(S, σ) + δdf/dσ
What does it mean for at to be previsible?
It means that at is known, based on information up to but not including time t.
5 factors that effect the price of a European put option on a non dividend-paying share.
(1) The premium would decrease as the underlying share price increased.
(2) The premium would increase as the strike price increased.
(3) The premium would increase as the time to expiry increased.
(4) The premium would increase as the volatility of the underlying share
increased.
(5) The premium would decrease as interest rates increased.
Define a risk-Neutral Probability Measure
A risk-neutral (or Equivalent Martingale) probability measure, Q, is a probability measure that is
…. equivalent to the real-world probability measure, P,
… under which the discounted asset price is a martingale.
Explain what it means for a portfolio V(t) = ψt St + ϕt Bt to be self-financing
dV(t) = ψt dSt + ϕt dBt
At t + dt, there is no inflow or outflow of money necessary to make the value of the portfolio back up to V(t + dt).
Describe the concept of delta hedging
Delta-hedging is the creation of an instantaneously risk-free portfolio, consisting of:
- a short position in one option contract
- a position in delta units of the underlying asset.
Why is a delta-hedged portfolio risk-free (instantaneously)?
Since delta measures the rate of change in the option price wrt changes in the price of the underlying…
… movements in these 2 components of the portfolio will cancel out.
Why is a delta-hedged portfolio ONLY INSTANTANEOUSLY risk-free?
Since delta changes over time as the underlying security price changes, dynamic delta hedging requires regular rebalancing of the portfolio in order to maintain delta-neutrality.
