Chap 6 - Derivatives and Risk-Neutral Valuation Flashcards
What is traded in the organized markets ?
Organized markets:
* Products: Futures, options * Standard contracts * Exchange rules * Clearing house
What is a derivative product ?
Definition :
* Financial instrument whose value depends on the price of an underlying asset
(real or financial instrument).
Contract types:
* Forward
contracts
* Options
* Swaps
* FRAs
* Etc
What is traded in the over the Counter (OTC) markets ?
Over the Counter (OTC) markets :
* Products: Forwards, swaps, exotic options * Non-standard contracts between two
parties
* No clearing house * 70% of transactions
What is a foward ?
Definition :
The investor undertakes to buy (sell) a specified quantity of the underlying at a specified
price on a specified date.
Foward price = S0e^rt
How it works :
* Over-the-counter (OTC) contracts
* No exchange before maturity
* Forward price set at t=0 so that contract value = 0
* Delivery or cash settlement (more common)
What is the formula for a foward price with dividends ?
Forward price in
discrete compounding : F = (S0 - I)e^rt
I = Present value of dividends
What is the formula for the continuous forward price ?
F = S0^e(r-q)t
q = Average annual dividend rate, under continuous compunding
What are some disavantages of using forwards ?
Disadvantages of Forward :
- Difficult to exit a Forward position (customized contract)
- Impossible to cancel the contract
- Possible to offset the position with a 2nd contract, but difficult to do
- Credit risk
- Delivery risk
Solution: Futures contracts
What is a future ?
Definition :
* The investor undertakes to buy (sell) a specified quantity of the underlying at a specified
price on a specified date.
Future price = S0e^rt
How it works
* Standardized contract on an organized market
* Contract usually closed before maturity
* Marking to Market (MTM)
What are some advantages of using futures and itโs biggest disadvantage ?
Advantages of Futures :
Market standardization and clearing house:
* Increases contract liquidity (entry and exit)
* Makes it easy to compare prices
* Reduces default risk (Marking to Market, margin replenishment)
Disadvantage of futures :
* No customized contracts
What is a Initial margin ?
Initial margin :
- Future market participants must have collateral to take long or short
positions. - Represents a percentage of the asset purchase price
What is a Maintenance margin ?
Maintenance margin :
- Minimum margin to be maintained
- If the value of collateral falls below the minimum margin - Margin Call
What are the two types of options ?
1- Call
* the right, but not the obligation, to buy the underlying at a given price during a
predetermined period.
2- Put
* the right, but not the obligation, to sell the underlying at a given price during a
predetermined period.
What is the formula of the payout from a call ?
Call = Max( ST-K, 0 )
What is the formula of the payout from a put ?
Put = Max( K-ST,0)
What are the Black and Scholes formulas ?
In certanity : C = S0 - Ke^-rt
Uncertanity : C= S0*N(d1) - Ke^-rt * N(d2)
With N(d1) and N(d2) representing probabilistic adjustments for uncertainty
What is Put-Call parity ?
Put-Call parity
Letโs assume we have the following two portfolios:
* Portfolio A = A European call and a quantity of money equivalent to ๐พ๐(โrt)
* Portfolio B = A European put and a stock
Both portfolios are worth at expiration:
max( ๐T,๐พ)
Both portfolios should have the
same value today:
๐ + ๐พ๐(โrt) = ๐ + ๐0
Example of how to determine a bond price to prevent arbritage :
Nine-month riskless securities trade for $97,000, and 12-month riskless securities sell for P (both with $100,000 face values and zero coupons). A forward
contract on a three-month, riskless, zero-coupon bond, with a $100,000 face value and a delivery of nine months, specifies a forward price of $99,000. What is the arbitrage-free price of the 12-month zero-coupon security (i.e., P)? The 12-month bond offers a ratio of terminal wealth to investment of ($100,000/P). The nine-month bond reinvested for three months using the forward contract offers a 12-month wealth ratio of ($100,000/$97,000) ($100,000/$99,000).
Setting the two wealth ratios equal and solving for P generates P=$96,030.
The 12-month bond must sell for $96,030 to prevent arbitrage.
In a risk-neutral world (in which investors do not require risk premiums for bearing risks), the forward price will be driven toward equaling the expected spot price because any other relationship would allow trading that offered abnormal expected return (note that due to the assumption of risk neutrality there would be no
concern regarding risk).
True
What is a reference rate and some examples ? (US and EU)
A reference rate is a market rate specified in contracts such as a forward contract that fluctuates with market conditions and drives the magnitude and direction
of cash settlements. (LIBOR, Euribor, SOFR)
The term notional principal is used to indicate that the principal amount is not actually exchanged, but rather serves to scale the size of the rate-related payments.
True
Application on how to find the FRA rate on a perfect market:
FTโt = [(T ร rT) โ (t ร rt)] / (T โ t)
A three-year riskless security trades at a yield of 3.4%, whereas a forward contract on a two-year riskless security that settles in three years trades at a forward rate of 2.4%. Assuming that the rates are continuously compounded, what is the no-arbitrage yield of a five-year riskless security? Inserting 3.4% as the shorter-term rate in Equation 6.2 and 2.4% as the left side of Equation 6.2, the longer-term rate, RT, can be solved as 3.0%, noting that T=5 and t=3. Note that earning 3.0% for five years (15%) is equal to the sum of earning 3.4% for three years (10.2%) and 2.4% for two years (4.8%).
The rates may be summed due to the assumption of continuous compounding.
When actual market prices deviate from arbitrage-free prices,
investors may use skill-based strategies that attempt to earn superior profits by anticipating that relative prices will tend to revert toward their arbitrage-free levels. Relative value hedge fund strategies (discussed in Chapter 17) are examples of such
strategies.
True That
What is a commodity swap ?
A commodity swap is a portfolio of commodity forwards. Typically, the settlement times are equally spaced. For example, an oil refinery might regularly need to purchase crude oil. Rather than bear the risk of fluctuating oil prices, the refinery may decide to lock in the purchase price of the oil by entering various forward contracts to purchase the oil at prespecified prices (i.e., to swap cash for oil). Instead of entering into a series of separate forward contracts, the refinery may enter into a single swap that calls for quarterly or monthly exchanges through time at prices set at the initiation of the swap.
In summary, for forward contracts on financial securities, the slope and curvature of the term structure of forward prices (the forward curve) are driven entirely by the relationship between the underlying securityโs dividend yield and the riskless interest rate (both of which may vary in T). The forward curve will be flat when r=q, upward sloping when r>q, and downward sloping when q>r.
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