Derivatives Flashcards
Derivatives
Two types of derivatives
-
Forward Commitments - Agreement between 2 parties to buy/sell an asset at a future date, at a price agreed on today
- Fowards
- Futures
- Swaps (effecitvely a series of forwards)
- Contingent Claims - Payoff dependent on a future event (eg options)
Derivatives
Forward Contracts
Difference between forwards and futures
Forwards are OTC, futures are listed
Futures settled via a clearinghouse, need daily settlement (mark to market) of margin with the futures exchange, no credit risk against other party.
Derivatives
5 benefits of derivs markets
- Price discovery (eg provide information about volatility)
- Market completeness - means any identifiable payoff can be achieved by trading instruments in the market
- Risk management
- Market efficiency
- Trading efficiency - lower costs
Derivatives
2 roles of arbitrage
- Facilitates the determination of prices
- Promotes market efficiency
Derivaites
Price vs Value
Value of forwards, futures and swaps starts at zero always.
So if a forward to deliver a $100 zero coupon bond trades at $97, the price was $97 but the value was $0.
If the zcpn at maturity is worth $97.25 the value is $0.25 and price is $0.
So basically trade in OTC style, “price” is the strike and “value” is the value of the forward/future/swap.
Derivatives
Forward Pricing
Formula for value of forward at expiration
Value of forward at initiation
Forward price at initiation
VT(T) = ST - F0(T)
where ST is underlying price at T, F0(T) is forward price
V0(T) = 0
F0(T) = S0(1+r)T
Assuming no divs, coupons, costs etc.
Derivatives
Forward Pricing
Determining the forward price
Simple Eqn: F0(T) = S0(1+r)T
In reality need to account for the benefits of holding § (divs, interest, convenience yield) and costs of holding ø.
F0(T) = S0(1+r)T - (§ - ø)(1+r)T
So deduct FV (not PV) of divs, coupons etc from the price, add back costs.
Derivatives
Forward Rate Agreement (FRA)
Definition
Relevant Dates
Basically a forward on an interest rate. The buyer/long is long the floating rate, they agree to borrow at the forward contract rate (ie strike) and lend at the underlying rate.
The earlier date is the settlement/expiration/delivery date, later date is the end of the forward period (ie determines the type of interest rate).
e.g. a 1 x 3 FRA could expire in 30 days at the 60 day libor rate (ie 60 day rate once 30 days are up).
Derivatives
Why do forward and future prices differ?
Explain relationship
(ignore credit)
If interest rates are known or constant they will be the same. However the daily settlement of futures means the cash flow profile for a future holder is different to a forward holder.
If rates are +VE correlated with prices, prefer to be long futures, since when spot price rises you receive cash on daily settlement and invest it at high rates (and inversely when spot prices fall can borrow at low rates to fund margin call.
So if rates +VE correl with spot price, futures prices > forward prices.
Derivatives
Options
Lower bound for option prices (american vs european)
Lower bound for an American option is the intrinsic value.
Lower bound for European options is the PV of the intrinsic value.
Derivatives
Options
Put-call parity
The two synthetic instruments
A fiduciary call (european call plus a bond with par value = strike price X) has the same value as,
a protective put (european put plus the underlying asset).
Both worth max(X,ST) at maturity.
c0 + X/(1+r)T = p0 + S0
Can use this to construct synthetic calls out of puts, bonds & underlying (and vice versa).
Derivatives
Options
Put-Call-Forward Parity
Can rearrange put-call parity eqn to get put-call-forward parity:
p0 + S0 = c0 + x/(1+r)T
p0 = c0 - [F0(T) - X]/(1+r)T
So to construct a synthetic forward, buy a call, short a put, and buy or sell bonds depending on the difference between X and F0(T).
If X > F0(T) need to buy bonds (puts ITM),
If X < F0(T) need to sell bonds (calls ITM).
Derivatives
Options - Binomial Valuation
Determine call price using 1 step binomial
Main problem is determining up/down probability (easy to calculate intrinsic value in up/down scenario, multiply by probabilities and discount back).
Up probability: π = (1 + r - d) / (u - d)
where r is RFR, u is S+/S0, d is S-/S0
In-advance swap
What payment is made on the first date?
In-advance swaps have first payment on day 1.
You need pay the relevant reset amounts from a normal swap, but DISCOUNTED by the libor fixing rate.
Off-market forward
Definition
Basically means the forward price doesn’t match what it should be (ie spot rate with growth rate applied).
May involve a cash payment at the start.
Alternatively swaps are a series of off-market forwards since individually they don’t price at par.
