Derivates Pricing Flashcards
Futures are priced so that they cost 0 today.
When you enter into a futures contract, you’re agreeing to buy or sell an asset at a specified price (the futures price) at a future date. But:
You do not pay anything upfront (unlike with options).
There’s no cash exchanged at initiation of a standard futures contract.
This means the net present value of the contract must be zero at the start.
If the contract had value at inception, someone would instantly exploit it (arbitrage opportunity). That’s why exchanges and over-the-counter contracts settle at a fair price
Cox Rubenstein Model
The Cox-Ross-Rubinstein (1979) model is a discrete-time binomial option pricing model that:
Models stock prices as moving either up or down in each time step
Prices derivatives (like options) by using risk-neutral probabilities
Assumes a world where assets grow at the risk-free rate under these artificial “risk-neutral” probabilities
Key Insights Derivates Pricing
Calculate risk neutral prbability
CRR Key Assumptions
Constructing Pure securities?
You only make a profit when the stock ends exactly at 100. Hence, this mimics a pure security for state 100.
First Order And Optimality Conditions
What are Martingale Probabilities?
Martingale probabilities (also called risk-neutral probabilities) are artificial probabilities under which expected asset prices, discounted at the risk-free rate, become a martingale, meaning they follow a fair game — no drift or bias.
Martingale Probabilites Formula
Interpretation of Martingale Proba.
Equation Link to Euler
Key Takeaways
Key Insights
Derivatives are priced as the expected value of future payoffs, weighted by marginal utility-adjusted probabilities and discounted.
Martingale probabilities = utility-adjusted probabilities.
These formulas show the link between preferences (utility), prices, and risk-neutral probabilities.
Conclusions
No-arbitrage models and equilibrium models are equivalent to each other
The Euler equation is the starting point for the derivation of all existing capital market models
The Consumption CAPM considers assets with high payoffs in bad consumption states especially valuable
Arbitrage Pricing Theory (APT) is a special case of the Euler equation, too
Derivatives complete the market and thus make the solution for the stochastic discount factors unique
Pure securities and state prices enable replication and pricing of arbitrary state-contingent claims in a complete market
Risk-neutral valuation follows from state price theory, with state prices being the discounted Martingale probabilities
