Appendix 4A
Interest Rate Swap
The following diagram is from Larry Wall and John
Pringle, “Interest Rate Swaps” in Federal Reserve Bank of Atlanta, Financial
Derivatives, p. 73 Chart 2:
Widgets Unlimited borrows an amount in its own credit
market where it is charged a floating rate of LIBOR plus .5%. OneState Insurance
lends in its market at LIBOR plus .25%. Widgets contracts with DomBank to pay a
fixed rate of 9.5% and receive floating LIBOR in return. Widgets has been able
to convert its floating rate, the only one available in its market into a fixed
10% loan. OneState contracts with DomBank to receive 9.4% in return for paying
floating LIBOR in return. OneState has been able to convert the receipts from
its loan asset from LIBOR plus .25% to a fixed 9.65% return. DomBank earns a .1% spread, the difference
between the fixed rate it receives and the rate it pays as its compensation for
arranging the swap.
Appendix 4B
The Financial Zoo
The following are lists of some of the important option
and swap variations arising during and since the 1980s; these are by no means
all inclusive.
The basic call and put option on stocks gave rise to the
following innovations:[1]
Options on Bonds
Options on Interest Rate Swaps
Options on stock indices and stock index futures
Down and Out Call Options – identical to European call
except that the contract is cancelled if the stock price goes below a specified
lower boundary
Up and Out Put Options - the put counterpart for the down
and out call
Compound Options – options which allow the holder the
right to purchase or sell another option.
Bermudan Options – allows the holder to exercise the
option only on specific dates, usually used with fixed income instruments
Digital/Binary Options – pays a fixed amount if the option
expires in the money, regardless of the expiration price of the underlying
asset
Pay Later Options – no premium must be paid until
expiration; however the option must be exercised if the value of the underlying
asset is equal to or greater than the strike price
Delayed Option – allows the holder to receive another
option at expiration with strike price set equal to underlying asset price
Chooser Option – allows the holder to choose at
expiration whether the option is a call or a put at the given exercise price
Power Option – allows the holder the payoffs of a
standard option but with the value of the underlying security raised to some
power
Average Rate (Asian) Option – pays the difference between
the strike price and the average price of the asset over the option period
Look Back Option – allows the holder to purchase or sell
the underlying at the best price (maximum or minimum) over the option’s life
Ratchet Option – the strike price is reset to be equal to
the underlying asset price on a predetermined set of dates
Ladder Option – variation of ratchet but is reset only if
certain higher prices are reached
Barrier Option – after the initial strike price is set,
another level is established whereby if the underlying reaches that price the
option is cancelled
Rainbow Option – payoff is determined by the highest
price at expiration attained by two or more underlying assets
Spread Option – payoff is the difference in the prices of
two underlying assets
Quanto Option – payoff depends both on the underlying
price and the size of the exposure as a function of that underlying price
The basic swap types gave birth to the following
variants:[2]
Amortizing Swaps
Deferred Accelerated Cash Flow swaps
Deferred Starts, Spreadlock and Forward Swaps
Extendible and Concertina Swaps
Basis Swaps
Arrears Reset Swaps
Yield Curve Swaps
Index Differential swaps
Options on Swaps/Swaptions
Callable, Puttable and Extendable Swaps
Contingent Swaps
Rally Participation Swaps
Interest Rate Linked Swap Hybrids
Currency Linked Swap Hybrids
Cross Market Hybrids
Equity Swaps
Equity Linked Security Swaps
Commodity Swaps
Commodity Linked Security Swaps
Inflation Swaps
Appendix 4C
Collateralized
Mortgage Obligation
A CMO is
constructed as follows:
2. The SPE issues bonds to investors in
exchange for cash, which is used to purchase the portfolio of underlying
assets.
3. The bonds issued are in layers with different risk
characteristics called tranches. Senior tranches are paid from the cash flows
from the underlying assets before the junior tranches.
Losses are first borne by the junior tranches, and then by the senior tranches.
4. The payments are governed by a prepayment schedule.
The following example is given by Livingston.[3]
The first 25% of payments pay off the principal of tranche 1 and future
interest payments to tranche 1 are reduced. When the entire tranche 1 principal
is paid the tranche ceases to exist; the next 25% of principal payments is
allocated to tranche 2 etc. The low end is occupied by the Z bond, the lowest
priority class, which accrues interest until all other classes are repaid their
principal.
There are a number of common CMO variations. Some pools
have a planned amortization class with a fixed principal payment schedule that
must be met before other classes receive principal payments. Some have
principal only and interest only classes. As scheduled prepayments occur, the
holders of POs receive the added cash flows and the holders of IOs receive
reduced interest payments. With falling interest rates prepayments are faster
than anticipated and the PO value rises; with rising interest rates the value
of POs drops.[4]
Appendix 4D
Black-Scholes
Formula
Black and Scholes derived a partial differential
equation, now called the Black–Scholes equation, which governs the price of the
option over time. The derivation can be found in a number of works on option
and derivative pricing. The following gives the equation and solution for call
options the derivation of which is given in Briys and Bellalah at al. The model
variables are:
c: call price
S: underlying asset price
K: strike price
r: riskless interest rate
t: time
t*: option maturity date
σ: volatility of underlying asset
T = t* - t
The underlying assumptions are:
Option is European – option can be exercised only on its
expiration date (not before as in the case of an American type option)
Interest rate over the option lifetime is known
Underlying asset follows a random walk with variance
proportional to square root of the price
No dividends or distributions
No transactions costs
Short selling is allowed
Trading takes place continuously
Under these assumptions Black and Scholes obtained the
following partial differential equation:
[1/2]σ2S2 ∂2c(S,t) – rc (S,t) + ∂c(S,t)
+ rS ∂c(S,t) = 0
∂S2 ∂t ∂S
with boundary condition
c (S, t*) = max [0, St* - K]
A simple substitution transforms this into a form of the
heat transfer equation* of thermodynamics:
∂y = ∂2 y
∂t ∂S2
Applying the solution to the heat transfer equation Black
and Scholes obtain their famous equation:
c( S, T) = S N(d1) – K e-rT N(d2)
with
d1 = 1 [ln
(S/K) + (r + [1/2]σ2) T]
σ √T
d2 = d1 – σ √T
N(d) is the cumulative normal density function.[5]
* The heat transfer equation describes the change in
temperature with time along a rod or wire (one-dimensionally). Here y is the temperature,
S is the distance from a heat source along the rod and t is time. The process
is one of slow diffusion along the length so that the temperature is assumed to
change sufficiently slowly.
Appendix 4E
Modern Portfolio Theory
The theory is based on the following underlying
assumptions:
1. Risk of a portfolio is measured by the standard deviation
of returns.
2. Investors are risk averse.
3. Investors utility functions are concave and
increasing.
4. Investors are rational and seek to maximize their portfolio
return for a given level of risk.
Under these assumptions we have the following:[6]
Portfolio expected
return:
E(rP) = Ʃi xi E(ri)
where
xi: fraction of portfolio
invested in security i
ri: return on security i
Thus, the expected value of the portfolio return is a
weighted average of the individual securities expected returns.
Portfolio variance:
σ2(rP)
= Ʃi xi2
σi2 + Ʃi
Ʃj≠i xi
xj
σi σj ρij
The variance of the portfolio return is equal to the sum
of the variances of the individual securities returns (first summation) plus
the sum of the covariances of the individual securities returns (double
summation).
When the expected portfolio returns are plotted against
the risk, as measured by the standard deviation for a portfolio with no holdings of a risk-free asset
and with short selling permitted, a graph similar to the following is obtained.
All possible portfolios are contained on the inside of the backward bending
hyperbolic curve. Combinations along
the curve represent portfolios with lowest risk for each expected return.
The most efficient portfolios are on the upper part of the curve itself. Any
portfolio not on the upper part is inferior to another since these have the
highest returns for any specified level of risk.
The simplest form of the Capital Asset Pricing Model assumes
the existence of a risk free asset. It can be shown that the portfolio in the
efficient set with the highest value of the following ratio will be optimal:
[E(rP) – rF]
/ σ(rP)
where rF is the return on the risk free asset.
The point on the curve tangent to a line drawn from the
risk free return rF on the vertical axis will represent the optimal
portfolio.
[3] Livingston, Money and Capital
Markets, p. 342.
[4]
Ibid, pp. 343-45.
[5] Eric Briys,
at al. Options, Futures and Exotic Derivatives, New York, Wiley, 1998, pp.
92-95.
[6] For a full description see Robert Haugen, Modern Investment Theory, Englewood Cliffs, N.J., Prentice Hall, 1993, chapter 4.
No comments:
Post a Comment