Options Derivatives Notes
Assumption rate of stock price growth is exponential example AAPL
\[\frac{CoVariance(AAPL, SPY )}{Variance(SPY)}\]
1 Hello
First part Pricing
Basic securities stock bonds
pricing known pay offs - what is the value today? answer is deterministic
* pricing fixed cash flows
* spot interest rates
* forward rates
Similar principle can be used in stochastic payoffs? * options
assumptions * Pricing by No-arbitrage * Binomial trees model - there is a unique price for each random/nonrandom payoffs. (Reality there is a bid ask spread ) * Stochastic calulus, Ito rule, Brownian motion - Math tools to learn black scholes * Black-Scholes formula
Second part Hedging
Fixed income derivatives - bond market
2 3
Futures and forwards are linear function of basic securities/stocks
Swaps are linear function of basic securities. Swaps are just sequence of future contracts.
Options are nonlinear functions of stocks.
3 4
3.0.1 Forward Contract
F := forward price T := maturity date S(T) := spot (market) price at maturity
Forward price(F) is chosen so contract has 0 value today. Payoff at maturity * long := S(T) - F = spot - forward * short := F - S(T) = forward - spot
- meta-simplify fancy terms
- spot = current value
- forward = store bought price
3.0.2 Options
S(T) = current value K = strike
current value < strike implies 0 payoff
before maturity options are smooth nonlinear functions and they grow closer to looking like piecewise functions.
calls flipped vertically on it’s kink results in a put
long flipped horzontally on x-axis results in a short.
All Longs have negative premiums. (horizontal line below 0) All Shorts have positive premiums. (horizontal line above 0)
#5
swaps
#6
Exotic options * Asian options - value is average over period * Lookback option - value is max or min over period * Barrier option - value is binary, only pays if goes above a barrier (cheaper to buy but riskier) * Basket option - value depends on value of several assets
Writing an option - selling option Premium - price of option
#7
#10 Theoretically we can design any curve assuming we call options are traded for ALL(infinitely many) strike prices(unrealistic).
#21
4 45
Fundamental Theorem of asset pricing
5 Criticism of Econometrics
Assumptions made: Additivity + Linearlity
Does not account for: Unmeasurable non-additive and epistemologically unaccessible variables
Most Research is on Fixed Parameter Models.
