cmt

1. Financial security -> value depends on underlying variable (is derived from it)

2. Used to transfer risk -> hedge risk (eliminate risk), speculate (Gaining a desired exposure without the need to physically buy the underlying), arbitrage

3. Traded on exchange sites or over the counter market (traders, banks etc contact each other directly) -> all derivative transaction must be registered

4. Futures/forewards with different maturities with different prices -> slope

   -> The shape is determined by the precise values of r and D (equity futures) or r, q, and C (commodity futures)

Arbitrage: strategy that takes advantage of two or more securities being mispriced to eachother

No-Arbitrage:

Law of one price:

Eqilibrium vs. No-Arbitrage pricing

Idea:

Difference:

Positions:

Payoff:

1. proof by contradiction

-> formular has to hold -> but we dont know what has to adjust (S0 or F0T) if over/underpriced

-> at maturity: F0T = ST

1. variation by replication

Degree of mispricing:

Is limited by the level of transaction costs of the market participant with the lowest transaction costs (usually banks etc)

-> wenn mispricing so dass sie arbitrage möglichkeit haben -> für die meisten noch kein arbitrage höherer kosten -> banken etc nutzen arbitrage aus -> preis passt sich wieder an

If A buys a futures/forewards contract from B, default risk arises

-> both could default before T

Soloution: clearing house

-> clearing house requires marking to market

Marking to market:

-> The loser pays (via the clearing house) the winner the day’s price change

-> Clearing House is still exposed to one day’s risk

Index futures:

-> ‘Asset‘ is an index not physically traded

• Value is determined as index level x contract multiplier

Can use Dividende yield:

Currency forewards:

Commodities are input factors for production

-> benifit from storing: benefit is called the “convenience yield“ denoted q

-> on the other hand storage costs: Storage costs CT from 0 to T

Cost of carry:

Treasury rates:

Rates on instruments issued by a government in its own currency

LIBOR rates:

rate of interest at which a bank is prepared to deposit money with another bank

Overnight rates:

rate at which banks lend or borrow funds from each other in overnight market

Compounding frequency:

Coupon bond:

The issuer promises to make periodic coupon payments and to repay the principal

Zero coupon bond:

is a coupon bond that does not pay a coupon

-> A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payo” only at time T

TSIR:

Yields of (default free) zero coupon bonds (zeros) with different maturities.

Discount rate:

-> To avoid arbitrage the discount factors must monotonically decline

Bootstrapping:

-> use coupon bonds to obtain the discount factors

Yield to maturity:

Foreward rate:

The forward rate is the future zero rate implied by today‘s term structure of interest rates

Determination of foreward rates:

The duration of a bond is a measure of interest rate risk

-> Diration of a ZB is equal to time to maturity T

Duration of a Portfolio:

-> The duration of a portfolio is simply the weighted average of the duration of the assets in the portfolio

$-Duration:

-> measures the absolute price change in $

-> can be useful because: for the Duration you divide by the assets value, which kan be 0 for derivatives

Limitations:

1. Duration approximates price changes by assuming a linear relationship. The bigger the change in interest rates, the bigger the approximation error

2. Duration assumes the term structure of interest rates shifts in parallel ways only

Like a bond but Instead of a fixed coupon, the FRN pays a floating rate -> not fixed coupon but payment orientates on marktzins z.b. SOFR

-> at maturity -> get principal back

-> usually positive spread (risk bounus) but can be negative for states

Problem:

-> SOFR of the future are unknown

Pricing of FRN:

-> The value of a FRN with a zero spread is its face value at every reset date

-> why: if invest in FRN with FV 100 you get 100 * marktzins for that year = investing 100 am Geldmarkt and getting 100 * marktzins for that year

-> law of one price: both should be worth 100

if spread not = 0:

If not at reset date:

Duration,

-> Shortly before the reset day, the value of the FRN is independent of changes in interest rates. Thus its duration is zero

-> at any point ε: duration is t - ε

agreement to exchange the cash flows on two instruments periodically for an agreed interval.

-> Most swaps (90%) are interest rate swaps (IRS) -> IRS exchanges fixed for variable interest rates

Swaps market:

• unregulated

• Only OTC

• anonymous

Commercial use:

-> Banks lend at fixed rates to long-term borrowers/mortgagors but pay variable rates to short-term lenders.

-> manage risk by swapping either its fixed rate assets for variable rate ones, or its variable rate liabilities for fixed

Example:

IRS swap ist wie:

1. long in FRN (zahlt 100 am Anfang und bekommt immer die SORF + am Ende die 100 zurück)

2. Short in Bond (bekommt 100 am Anfang und zahlt immer fixen Zins + am Ende die 100 zurück)

-> buyer is long the FRN and short the bond

-> zusammen: wie ein IRS swap

At initiation, an IRS has a value of zero. However, as time passes, this will, in general, not hold during the life of the swap.

-> To find the value of the swap, one can use the bond/FRN decomposition.

-> If the swap rate increased, the value of the swap is positive for the buyer (fixed payer), and negative for the seller.

The swap has (at initiation) a value of zero

-> only the -duration can be obtained

-> This can be done by using the above decomposition, e.g. interpreting the swap as a portfolio of a coupon bond and a FRN

Im gegensatz zu futures/forewards: no obligation

-> The buyer is long the option, the seller is short the option

Notation:

The price of an option is also called ‘premium‘

• The price of a European call is denoted C

• The price of a European put is denoted P

• The specified price to buy or sell the asset is called the strike

price or the exercise price and is usually denoted K or X

Call option

-> has a price

-> buyer has to pay price and can then decide to buy for strike price at maturity

-> will buy if strike price under market price

Put option

-> has a price

-> buyer has to pay price and can then decide to sell for strike price at maturity

-> will sell if strike price over market price

General:

• Payoff of a call long:

Max(ST - K; 0)

• Payoff of a call short:

- Max(ST - K; 0)

• Payoff of a put long:

Max(K - ST; 0)

• Payoff of a put short:

- Max(K - ST; 0)

It has to hold:

Upper bound for european calls:

C < S0

-> der call price can not be as expensive as the stock itself

Lower bound for european calls:

C > S0 - K(1 + r)^-T

Lower boung Pus:

P > K1 + r)^-T - S0

American vs european option:

-> The only (but important) di”erence is that American options can be exercised early

-> American option cannot be worth less than European option!

-> NEVER optimal to excess an american call early (mom dividend paying)

Two states:

1. stock goes up by factor u

2. stock goes down by factor d

-> to determine call value: reproduce option wirh stock and loan

example:

Number of stocks to buy:

Numer of stocks and amount to borrow:

Cost of portfolio C0:

Methoden:

1. C0 = Delta * S0 + B (Replikation -> Option mit stock und borrowed money nachbilden)

2. C0 = qCu … siehe letztes Bild

-> für 1.:

-> A risk neutral investor only requires an expected return of r for every (risky) portfolio

2 periods:

Solve backwards:

-> (1 + r) immer hoch die periode also bei laufzeit 1 jahr und 2 schritte -> 1/2

Important: Delta and B change over time!

-> similar to black scholes merten if n -> infinity (30 steps upwards)

-> Let sigma denote the (annual) volatility of the underlying

-> The number of steps in the tree is denoted by n, T is the time to maturity

n -> infinity (indefinitely many time steps):

-> Black-Scholes formular

-> option value is independent of the expected return µ

-> Relevant factors: S, r, K, T, sigma -> Except of sigma, everything is known

For american Put:

-> delta des underlying ist immer = 1

Hedging

-> bank sells x call options

-> Delta hedge: Buy delta shares of every option sold

-> However, delta is not constant, i.e. The hedging portfolio must be rebalanced trough time

Gamma:

-> das gamma des underlying ist immer = 0

Theta:

Vega:

Rho:

-> Traders usually ensure that their portfolios are delta-neutral at least once a day

-> Whenever the opportunity arises, they improve gamma and vega

-> Delta can be changed by taking a position in the underlying asset

-> To adjust gamma and vega it is necessary to take a position in an option or other derivative