Lecture 3 title slide (slide 1)
reminder of the three equations in three unknowns; pricers know the $2.97 price; want to "get hedge", i.e., to find the hedge parameters x and y; enough to find x (slide 2)
finding x by manipulating the equations (slide 3)
finding x by manipulating the graphic (slides 4-5)
the difference of a generic portfolio (slide 6)
the difference of a portfolio determines how many Euros it has (slide 7)
same difference as a 50 Euro portfolio implies 50 Euros (slide 8)
general form: if the difference of a portfolio agrees with that of N units of the risky asset, then the portfolio contains exactly N units of the risky asset; all difference comes from the risky asset, none from the risk-free asset (slide 9)
the hedge parameter x is a difference quotient; in the limit, it's a rate of change (slide 10)
going from x to y; reminder of the graphic (slides 11-12)
the graphic with the values of x, y and ? (slide 13)
Cathy tries to undercut Alice; Alice's response (slide 14)
computation of the cash flow from Alice's response (slide 15-17)
the concept of arbitrage; Cathy's undercutting Alice introduces arbitrage (slide 18)
the value of the hedging portfolio is the value of the option, i.e., the value of the derivative, an illustration with Cathy and Fred (slides 19-21)
the "Delta" of the option, for Cathy and Fred; no hedging at the end of the option; beginning of rouding error analysis (slides 22-24)
end of rounding error analysis (slide 25)
the risk-free (bank loan) side of the hedge, an illustration with Cathy and Fred (slide 26)
remainder of lecture: more practice pricing and hedging; obligation derivatives; put-call parity; forwards and futures; an option whose hedging strategy involves millions of adjustments (slide 27)
second act of Lecture 3 title slide (slide 28)
the template for Gail and Harry, with no hedge parameters filled in; calculation of the Delta of the option at one of the nodes; start of analysis of rounding error (slide 29)
end of analysis of rounding error (slide 30)
the other hedging parameter at the same node; solving the "equation of the node" (slides 31-32)
hedging parameters are calculated node-by-node, and the ordering of the nodes is completely arbitrary; by constrast, underlying values are calculated forward in time, while derivative values are calculated backward in time (slide 33)
the template filled in with values to high accuracy; the payoff function connects the underlying to the derivative (slide 34)
Irwin's three month plight seeking to *sell*; Gail sells the option; put options vs. call options; strike price or exercise price; the term of the option (slide 35)
the payoff function for Irwin's put option; comparison with the payoff function for Harry's call option (slide 36)
the risk-neutral probabilities (slide 37)
Gail template for Irwin's put option (slide 38)
Jack's plight; Gail sells this option, too; "obligation derivative" (a.k.a. "off-market forward"); promised underlying; receivable dollars or strike price; payoff function (slide 39)
the template for the obligation derivative; the Delta is constant (slide 40)
the other hedge parameter depends only on the month, grows by the risk-free factor in all scenarios (slide 41)
simple description of the hedging strategy; complete markets; completeness implies that, if you find a hedging strategy that works, you know it's the only one that works (slide 42)
for an obligation derivative, you can hedge the promised underlying and the receivable dollars separately; this hedge works, so it's the only one that works; obligation derivative price is inital price of underlying minus present value of receivables (slide 43)
remainder of lecture: put-call parity; forwards and futures; an option whose hedging strategy involves millions of adjustments (slide 44)
third act of Lecture 3 title slide (slide 45)
the reproducing equation; the payoff functions f,g,h for Harry, Irwin and Jack (put, call, obligation derivative); f-g=h by the reproducing equation (slide 46)
the initial nodes for Jack, Harry and Irwin; call price minus put price is obligation derivative price; similar result for hedging parameters (slide 47)
obligation derivatives are easy to price by completeness, call price minus put price is equal to inital price of underlying minus present value of receivables; assumptions(slide 48)
put-call parity (slide 49)
the three equalities (slide 50)
failure of put-call parity yields model-independent arbitrage (slide 51)
pricing of forwards; forward is a special kind of obligation derivative and so can be priced by market completeness; a mispriced forward yields immediate model-independent arbitrage (slide 52)
futures; margin requirements and margin calls (slide 53)
Kyle's plight; a call option with millions of hedging adjustments; saved by the Central Limit Theorem