These definitions are generated from the shared notation collection. Every entry is still draft until its wording, convention, source locator, and curriculum owner receive human review.
Ex-cash-flow value of the defaultable bond at a lattice node conditional on the issuer still being alive, before the subsequent survival and default branches; scheduled cash at the node has already been paid.
Alive-state conditioning keeps the bond’s own recovery branch separate from
the option contract’s zero payoff after a pre-exercise default.
Pricing-model probability that the issuer survives the next lattice period, conditional on being alive at the current node; a risk-neutral input, not an unconditional real-world forecast.
Multiplying conditional node probabilities along a realized path gives that
path’s modeled survival probability. State dependence prevents replacing all
node inputs with one unconditional number without an additional argument.
Alive-node value of a European bond option that is extinguished with zero option rebate by issuer default before exercise; a non-negative holder value conditional on the issuer being alive at the node.
The superscript identifies this lesson’s contractual default trigger. It is
not a market-price barrier option and carries no default rebate.
Coupon amount attributed to the interval from the previous coupon date through settlement under the stated day-count convention; a positive amount added to the clean price to obtain the dirty invoice price.
Accrued interest is convention-dependent. The introductory calculation uses
actual days elapsed divided by actual days in the surrounding coupon period;
it does not silently stand for every market day-count rule.
Promised amount paid to the bondholder on one scheduled payment date; a positive receipt for the bondholder, with default excluded in the bond lessons.
The promised bond cash flow CFkbond is the
coupon-payment on each scheduled date plus face-value on the
final date.
For n payments in the simplified model,
CFkbond=C+1{k=n}F
This is a promised-cash-flow description, not a default-adjusted expectation.
Fair dirty delivery price fixed at valuation time for delivery of the named bond at the forward date; positive delivery cash paid by the long under the lesson's no-arbitrage assumptions.
The bond forward price uses dirty cash units. Coupons whose record and payment
terms place them before delivery are income to the current bond owner, not the
forward buyer.
Number of scheduled coupon payments per year in the simplified regular bond, a positive integer.
The bond payment frequency mB is the number of scheduled coupon
payments per year in this simplified regular bond.
The subscript keeps it distinct from the general
compounding-frequency. The toy yield model later sets the two numerical
frequencies equal and states that assumption visibly.
NEEDS_SOURCE: verify the contractual and quotation convention before
review.
Present value of the simplified bond's promised payments at valuation time; the amount paid by the buyer, shown as a positive value in the bond lessons.
Bond price quoted without accrued interest under the stated settlement convention; a positive quoted price to the buyer before accrued interest is added.
The clean price excludes accrued interest. A quote per 100 face and a total
trade amount must be converted to the same units before they are compared.
Full cash or invoice price paid for the bond, equal to clean price plus accrued interest; a positive cash price paid by the buyer under the stated settlement convention.
The dirty price is the price basis used for cash settlement and present value
in the settlement lesson. On a coupon date with zero accrued interest it coincides with
the earlier simplified bond-price definition.
Bond price as a function of yield while promised positive fixed cash flows remain constant and only the yield varies.
The price-yield curve P(y) holds promised positive fixed cash flows constant
and evaluates bond-price across different
yield-to-maturity inputs.
In the toy model the curve slopes downward for non-negative yields and is not a
straight line. The price-yield lesson treats curvature qualitatively; duration and convexity
are separate later competencies.
Single nominal annual rate that reproduces the simplified bond price.
Yield to maturity y(mB) is the
single nominal annual rate,
compounded at the stated bond-payment-frequency, that reproduces the toy
bond-price from its promised cash flows.
In the bond lessons it appears in
P0=k=1∑n(1+y(mB)/mB)kCFkbond
It is not silently treated as the coupon rate, a spot rate, an effective annual
rate, a probability, or a guaranteed realized return.
NEEDS_SOURCE: verify exact textbook and official-guidance locators for the
quotation and interpretation before review.
Input year fraction that converts an annualized spread into the premium amount for one scheduled period; a positive model input.
The accrual year fraction αi is a direct input in this lesson. It
converts the annualized cds-contract-spread into a period amount.
For the lesson’s equal model-year periods, it is the arithmetic difference
between adjacent model times. This is not a claim about the calendar day-count
fraction of a market trade.
Annualized premium rate applied to notional and each stated accrual year fraction; a positive rate paid by the protection buyer in the simplified lesson.
The contractual spread s is the annualized premium rate in the lesson’s
simplified premium leg. Code uses decimal-per-year units, so 100 basis points
per year is represented as 0.01.
A contractual spread need not equal the par spread solved under a particular
valuation model after inception.
Conventional spread that is the input or the output of the lesson's simplified converter; it equals the zero-upfront par spread within the converter's implied flat-hazard model and need not equal the contract's fixed running coupon.
This lesson uses market-standard quote, abbreviated MSQ, for the
conventional spread that is the input or the output of its simplified converter. The
label is local to this lesson; it is not presented as universal market
terminology.
When the spread is the input, the converter treats sMSQ as a
zero-upfront par spread and infers one flat pricing-model hazard rate. When
the upfront is the input, it first infers that hazard rate and then returns the
corresponding zero-upfront spread. Thus MSQ equals par spread inside that one
converter model; it is not a second independent market observation once the
upfront and conversion conventions have been fixed.
The market gives the spread or upfront level. The conversion convention
gives the mapping between them. Neither role belongs to the fixed running
coupon that determines the actual premium cash flows.
Reference currency amount that scales the simplified premium and protection legs; a positive amount, not itself a signed leg cash flow.
The CDS notional N is the positive reference currency amount used by the
simplified lesson formulas. Premium and protection-leg magnitudes scale
linearly with it.
This draft does not assert a settlement mechanism or that the notional itself
is exchanged.
Contractual spread that makes the two positive leg magnitudes equal at valuation time with zero upfront amount, solved under the lesson's input curves, recovery, timing, and accrued-premium convention.
The par spread s⋆ is the contractual spread that makes the positive
premium-leg and protection-leg magnitudes equal at valuation time in the stated
model. It is calculated from the
cds-protection-leg-present-value magnitude and the
cds-premium-annuity under the same assumptions.
If the curve is an input, the par spread is an output. If a quoted spread is
the input of a converter, the same equality calibrates the curve instead. Within
one internally consistent converter model, the resulting market-standard quote
equals this par spread. It is distinct from the standard coupon that determines
the traded contract’s running cash flows.
Present-value coefficient that multiplies contractual spread and notional in the simplified premium leg; positive, and includes scheduled premiums and exact accrued premium under the lesson's default-time model.
The premium annuity A0prem is the positive per-unit-notional
coefficient multiplying the contractual spread. It always includes scheduled
survival-contingent premiums and exact expected premium accrued at modeled
default time.
The accrued term integrates the elapsed accrual fraction, discount factor, and
risk-neutral default density inside every period. The textbook half-period
method remains a comparison approximation, not the definition used by the
lesson’s exact flat-hazard engine.
Positive valuation-time magnitude of the simplified protection buyer's premium payments; its signed contribution to protection-buyer net value is negative.
The premium-leg present value PV0prem is shown as a positive
magnitude. In the lesson model it equals contractual spread times notional
times the selected cds-premium-annuity.
From the protection buyer’s signed perspective, this leg is paid and therefore
enters net value with a minus sign.
Signed protection-buyer value equal to protection-leg magnitude minus premium-leg magnitude; positive favors the protection buyer and negative favors the protection seller in the two-leg toy model.
The protection-buyer net present value PV0buyer is signed even
though both displayed leg values are positive magnitudes:
PV0buyer=PV0prot−PV0prem
This identity excludes upfront amounts, counterparty credit risk, collateral,
funding, and transaction costs.
Positive valuation-time magnitude of the simplified loss-given-default payment received by the protection buyer after a modeled default.
The protection-leg present value PV0prot is a positive magnitude
received by the protection buyer in the lesson’s one-default model. The
simplified payoff is loss-given-default times notional. Each interval
contribution integrates the risk-neutral default density and discounts from
the modeled default time inside that interval.
This draft does not specify actual contractual settlement, auction mechanics,
deliverables, or payment delays.
Fixed annualized rate used to determine the contract's running premium cash flows in the lesson's standard-coupon model; a positive rate paid by the protection buyer on surviving notional and as accrued premium after default.
The standard coupon cstd is the fixed annualized rate that
determines running premium cash flows in this lesson’s standardized contract.
It is converted from basis points per year to a decimal rate before
calculation.
The standard coupon is a contractual cash-flow input. It need not equal the
cds-market-standard-quote; a signed upfront amount balances the
difference at inception.
Time-zero cash amount that balances protection and fixed-coupon premium value under the lesson's pricing convention; positive means paid by the protection buyer and negative means received by the protection buyer.
The signed upfront amount U0 is exchanged at valuation time in this
lesson’s simplified conversion. A positive amount is paid by the protection
buyer and a negative amount is received by that buyer.
It equals protection-leg present value minus fixed-coupon premium-leg present
value under the same calibrated model inputs. Subtracting that signed amount
from the buyer’s pre-upfront value makes inception value zero.
Constant conditional default intensity used by the lesson's simplified exponential survival model; a risk-neutral pricing input conditional on survival to the current instant, not a cumulative probability.
The constant hazard rate λ is the simplified model’s conditional
default intensity per model-year under the explicitly stated probability
measure. The credit and CDS lessons use a risk-neutral pricing measure. It is
not a cumulative default probability and is not an interest rate.
The default time τ is random. The credit lessons state every survival and
default probability as the probability of an event about τ, and the CDS
lessons place each premium and protection payment by where τ falls in the
schedule.
Probability assigned by the model to first default during one stated time interval, conditional only through the input survival curve construction.
For two ordered endpoints, the interval default probability is the earlier
survival-probability minus the later survival probability. Lessons add
an explicit schedule index when they apply that subtraction.
This draft assumes at most one modeled default and a non-increasing input
survival curve.
Fraction of an explicitly stated reference amount not recovered under a deterministic recovery convention, relative to the same reference amount used by the recovery rate.
The loss-given-default fraction is the complement of the
recovery-rate:
LGD=1−R
The fraction has meaning only after the recovery base, timing, and perspective
have been stated. The one-period credit lesson uses par as that base; the CDS
lesson uses CDS notional and pays its simplified protection amount at exact
modeled default time.
Present value of one maturity payment that is par after survival and a fixed fraction of par after earlier default; a positive asset value to the holder in the one-period recovery-of-par-paid-at-maturity model.
The one-period recovery-of-par present value PV0RoP combines a
survival-state par payment and a default-state recovered-par payment, both paid
at the same scheduled maturity and discounted by the same input
discount-factor.
This definition does not cover recovery paid at default, recovery of market
value, coupons, multiple periods, or calibration.
Fraction of a stated reference amount recovered after a modeled default under an explicitly stated recovery convention; a non-negative fraction whose reference amount, payment timing, and settlement convention are set by the model that uses it.
The recovery rate R is a fraction between zero and one. It does not, by
itself, specify the amount to which recovery applies or when recovery is paid.
Those choices belong to the surrounding model.
The one-period credit lesson applies R to par paid at maturity. The CDS
lesson uses 1−R as its deterministic loss-given-default fraction and states
its exact modeled default-time payment separately. Neither use is an observed
recovery estimate or a universal contractual rule.
Probability, under the explicitly stated model measure, that no modeled default has occurred between valuation time and a stated future time.
The survival probability S(0,t) is the probability, under the explicitly
named model measure, that the modeled reference entity has not defaulted from
valuation-time through future time t.
It is a probability, not a discount-factor. A lesson must say whether
the probability is a pricing-model input or a real-world estimate; the draft
credit and CDS lessons use pricing-model probabilities as inputs.
Value at model time t of one unit of the asset or claim named as the derivative's underlying; a positive quoted value, while a position in the underlying carries its own signed quantity.
The underlying value St is the value at model time t of one unit of the
asset or claim referenced by a derivative contract. The lesson must state the
underlying, currency, price basis, and whether it distributes income.
Signed current value of an existing forward from the named counterparty's perspective; positive to the long when the current forward price exceeds the contract's fixed delivery price under the lesson model.
The forward contract value Vtfwd is a signed present value. A
new fair forward starts with zero value, but an existing forward can gain or
lose value when the current forward price changes.
Contractual currency amount per unit of underlying paid by the long at delivery and received by the short.
The delivery price Kfwd is fixed when a forward contract is
formed. It is not the contract’s current value and generally does not change
when market forward prices later move.
Future model time when the forward counterparties exchange the underlying and delivery payment; a contract date shared by the long and short, not the underlying asset's maturity.
The forward delivery time Tfwd is when the long pays the
contractual delivery price and receives the underlying. It is distinct from a
bond maturity or an option expiry.
Delivery price that would give a newly struck forward for the stated delivery time zero current value; a quoted contract rate rather than a cash amount received at quotation time.
The forward price Ft,Tfwd is
the delivery price that makes a new forward maturing at
Tfwd worth zero at time t under the
stated carry model. It differs from the fixed delivery price of an older
contract.
Valuation-time value of deterministic cash income paid by the underlying after valuation and before the contract horizon, forward delivery or the matched option expiry. It is subtracted from spot value because the forward or option position does not receive that income.
Only income paid before the horizon enters I0; income paid later belongs to
whoever holds the underlying then.
One-period discount factor at a lattice node: currency at the node per one unit of currency at either successor node one step later.
Backward induction multiplies the local expectation of the successor values by
di,j. In a lattice with issuer default, the same factor discounts from an
alive node to the next lattice time.
Pricing probability assigned to the up successor, conditional on the current node; in a lattice with issuer default, also conditional on survival to the next lattice time.
The down successor receives the complementary weight 1−qi,j. The weight
is a pricing probability of the lattice, not a forecast of the next state.
Claim value at time row i and state node j obtained by one-period backward induction from its successor nodes, conditional on reaching that node under the input pricing lattice.
A lattice node value is conditional on the node’s modeled state. The indices
are labels, not currency amounts or probabilities.
Integer label for one state node within a time row of a finite recombining lattice; a bookkeeping label under the stated successor ordering, not a probability or state value.
The state-node index j distinguishes nodes within one lattice row. The next
row’s successor indices are interpreted under the module’s declared ordering.
Time from the valuation date to one scheduled cash-flow event; a time coordinate, not a calendar date or payment index.
The payment time tk is the time in years from valuation-time to
cash flow k.
The subscript k labels a row in an ordered schedule. The value tk is a
time coordinate; later lessons will distinguish it from an actual calendar
date and its year-fraction convention.
Gives model pricing weights under which discounted traded prices satisfy the martingale condition; it prices payoffs relative to a stated numeraire and is not a forecast of actual event frequencies.
The risk-neutral probability measure Q gives the scenario weights
used by the stated no-arbitrage pricing model. It is equivalent to the
real-world measure P — the two agree on which outcomes are
possible — but assigns those outcomes different weights.
What holds under Q. Relative to the chosen numeraire (here the
cash account), every discounted traded price is a Q-martingale: its
value today equals the Q-weighted expectation of its discounted
future value. A claim’s time-zero price is therefore the expectation, taken
under Q, of its discounted payoff; when the discount factor is
deterministic it factors out of that expectation. The superscript in
EQ records that the weights are Q‘s.
Changing from P to Q reweights the modeled outcomes; it
does not change the payoff in any scenario. The name does not mean that
outcomes are risk-free, that volatility vanishes, or that all investors are
indifferent to risk.
Amount received or paid at one event from the stated holder perspective; positive means received and negative means paid by that holder.
The signed cash flow CFk is the amount exchanged at
payment-timetk, measured from one explicitly named perspective.
This playground uses positive amounts for receipts and negative amounts for
payments by the stated holder. Changing perspective reverses every sign; it
does not change the contract’s dates or absolute amounts.
Signed amount delivered by a claim at the stated future horizon, before its outcome is known; positive means received and negative means paid by the claim holder.
The terminal random payoff XT is the signed
amount delivered by a claim at the stated future horizon before the outcome is
known.
Its realized value may differ across scenarios. Positive amounts are receipts
and negative amounts are payments from the stated claim-holder perspective.
Future model time when a European option's exercise decision and payoff are determined; a contract date shared by holder and writer, distinct from a bond maturity.
The option expiry Topt is the only exercise time for a European
option. The underlying asset can continue beyond that time.
Contractual price per unit of underlying used to determine the option's exercise payoff; a positive contractual amount, with payoff signs depending on call or put and holder or writer perspective.
The option strike K is the contractual price used in the call or put payoff
at expiry. It is fixed by the contract and is not the option premium.
Operator returning the average of a random quantity, each outcome weighted by its probability under a stated measure; a superscript names that measure when more than one is in play.
Pprobability measure(dimensionless probability weights between zero and one)
The rigorous definition is the formula shown for this entry: the expectation of
an integrable random variable is its integral against the probability measure
of the model’s outcome space, taken over the whole sample space.
When the outcomes form a finite partition and the random value is constant on
each event, that integral collapses to the familiar weighted sum — each value
times its probability, added over the events — which is how the early lessons
in this course compute it.
The operator is linear, is order-preserving, returns a constant unchanged, and
obeys the tower property under iterated conditioning. Its result carries the
units of the quantity being averaged; discounting, when needed, is a separate
step.
A superscript on the operator names the measure when the choice matters — a
“Q” for the risk-neutral measure, a “P” for the real-world one.
Assigns modeled probabilities intended to describe actual-world event likelihoods; used for forecasting and statistical statements under the stated real-world model.
The real-world probability measure P assigns modeled probabilities
intended to describe actual-world event likelihoods.
It is the measure used when the question is a forecast rather than an
arbitrage-consistent price. A model’s probabilities under P need
not equal its pricing weights under the
risk-neutral-probability-measure.
Its reciprocal is called a discount factor. Keeping the two concepts separate
prevents the prototype’s compounding factor from being mislabeled as its
inverse.
Value at future model time t of one currency unit paid at later model time T under the stated deterministic curve, with t strictly before T.
The forward discount factor compares the values of the same later payment at
two different model times. This deterministic-curve definition does not claim
that future discount factors are known in a stochastic-rate model.
The glyph Z keeps this future-to-future factor distinct from the shared
valuation-time discount-factor glyph D; no existing glyph is rebound.
Ex-cash-flow value of the defaultable bond at a lattice node conditional on the issuer still being alive, before the subsequent survival and default branches; scheduled cash at the node has already been paid.
Pricing-model probability that the issuer survives the next lattice period, conditional on being alive at the current node; a risk-neutral input, not an unconditional real-world forecast.
Alive-node value of a European bond option that is extinguished with zero option rebate by issuer default before exercise; a non-negative holder value conditional on the issuer being alive at the node.
Units: stated currency per option at the node time
Coupon amount attributed to the interval from the previous coupon date through settlement under the stated day-count convention; a positive amount added to the clean price to obtain the dirty invoice price.
Promised amount paid to the bondholder on one scheduled payment date; a positive receipt for the bondholder, with default excluded in the bond lessons.
Fair dirty delivery price fixed at valuation time for delivery of the named bond at the forward date; positive delivery cash paid by the long under the lesson's no-arbitrage assumptions.
Units: stated currency at forward delivery per bond
Present value of the simplified bond's promised payments at valuation time; the amount paid by the buyer, shown as a positive value in the bond lessons.
Bond price quoted without accrued interest under the stated settlement convention; a positive quoted price to the buyer before accrued interest is added.
Full cash or invoice price paid for the bond, equal to clean price plus accrued interest; a positive cash price paid by the buyer under the stated settlement convention.
Annualized premium rate applied to notional and each stated accrual year fraction; a positive rate paid by the protection buyer in the simplified lesson.
Units: decimal per year in calculations; basis points per year when explicitly quoted
Conventional spread that is the input or the output of the lesson's simplified converter; it equals the zero-upfront par spread within the converter's implied flat-hazard model and need not equal the contract's fixed running coupon.
Units: decimal per year in calculations; basis points per year when explicitly quoted
Contractual spread that makes the two positive leg magnitudes equal at valuation time with zero upfront amount, solved under the lesson's input curves, recovery, timing, and accrued-premium convention.
Units: decimal per year in calculations; basis points per year when explicitly quoted
Present-value coefficient that multiplies contractual spread and notional in the simplified premium leg; positive, and includes scheduled premiums and exact accrued premium under the lesson's default-time model.
Units: model-years of present value per unit notional
Positive valuation-time magnitude of the simplified protection buyer's premium payments; its signed contribution to protection-buyer net value is negative.
Signed protection-buyer value equal to protection-leg magnitude minus premium-leg magnitude; positive favors the protection buyer and negative favors the protection seller in the two-leg toy model.
Fixed annualized rate used to determine the contract's running premium cash flows in the lesson's standard-coupon model; a positive rate paid by the protection buyer on surviving notional and as accrued premium after default.
Units: decimal per year in calculations; basis points per year when explicitly quoted
Time-zero cash amount that balances protection and fixed-coupon premium value under the lesson's pricing convention; positive means paid by the protection buyer and negative means received by the protection buyer.
Constant conditional default intensity used by the lesson's simplified exponential survival model; a risk-neutral pricing input conditional on survival to the current instant, not a cumulative probability.
Fraction of an explicitly stated reference amount not recovered under a deterministic recovery convention, relative to the same reference amount used by the recovery rate.
Present value of one maturity payment that is par after survival and a fixed fraction of par after earlier default; a positive asset value to the holder in the one-period recovery-of-par-paid-at-maturity model.
Fraction of a stated reference amount recovered after a modeled default under an explicitly stated recovery convention; a non-negative fraction whose reference amount, payment timing, and settlement convention are set by the model that uses it.
Value at model time t of one unit of the asset or claim named as the derivative's underlying; a positive quoted value, while a position in the underlying carries its own signed quantity.
Units: stated currency per unit of underlying at model time
Signed current value of an existing forward from the named counterparty's perspective; positive to the long when the current forward price exceeds the contract's fixed delivery price under the lesson model.
Future model time when the forward counterparties exchange the underlying and delivery payment; a contract date shared by the long and short, not the underlying asset's maturity.
Delivery price that would give a newly struck forward for the stated delivery time zero current value; a quoted contract rate rather than a cash amount received at quotation time.
Units: delivery-time currency per unit of underlying
Valuation-time value of deterministic cash income paid by the underlying after valuation and before the contract horizon, forward delivery or the matched option expiry. It is subtracted from spot value because the forward or option position does not receive that income.
Units: stated currency at valuation time per unit of underlying
Pricing probability assigned to the up successor, conditional on the current node; in a lattice with issuer default, also conditional on survival to the next lattice time.
Claim value at time row i and state node j obtained by one-period backward induction from its successor nodes, conditional on reaching that node under the input pricing lattice.
Integer label for one state node within a time row of a finite recombining lattice; a bookkeeping label under the stated successor ordering, not a probability or state value.
Gives model pricing weights under which discounted traded prices satisfy the martingale condition; it prices payoffs relative to a stated numeraire and is not a forecast of actual event frequencies.
Units: dimensionless probability weights between zero and one
Signed amount delivered by a claim at the stated future horizon, before its outcome is known; positive means received and negative means paid by the claim holder.
Future model time when a European option's exercise decision and payoff are determined; a contract date shared by holder and writer, distinct from a bond maturity.
Contractual price per unit of underlying used to determine the option's exercise payoff; a positive contractual amount, with payoff signs depending on call or put and holder or writer perspective.
Units: expiry-time currency per unit of underlying
Operator returning the average of a random quantity, each outcome weighted by its probability under a stated measure; a superscript names that measure when more than one is in play.
E[X]=∫ΩX(ω)dP(ω)
Symbols
Xrandom variable
Ωsample space
ωoutcome
Pprobability measure(dimensionless probability weights between zero and one)
Assigns modeled probabilities intended to describe actual-world event likelihoods; used for forecasting and statistical statements under the stated real-world model.
Units: dimensionless probability weights between zero and one