In 1979, Cox, Ross, and Rubinstein showed that a simple up-or-down tree could reproduce the Black-Scholes price. That tree now prices callable bonds, convertibles, American options, and anything else with early exercise or path dependence.
A one-period model of default risk under risk-neutrality. Consider a one-period, zero-coupon bond with $1 face value, assuming interest rates are 0%, risk premiums are 0% (risk-neutral investors), and zero recovery on default. The bond's price then equals the probability of nondefault. If is the probability of total default, the value is the discounted expected payoff:
A price of $0.90 implies a 10% default probability; a price of $0.85 implies 15%. Because risk premiums are set to zero, this is a world of risk-neutrality.
Adding a default risk premium. Relax the zero-premium assumption. Compare a 1-year default-free bond yielding 5% with a 1-year defaultable bond yielding 7%. The 2% credit spread has two sources: default likelihood and risk-aversion.
Common mistakes
- Using the real-world probability of an up move as p. Candidates compute p by asking how likely the stock rises and plug in something like 0.6. The correct p is the risk-neutral (Q-measure) probability .
- Confusing a Q-measure with a P-measure. A Q-measure is a biased quasi-probability assumed under risk-neutrality, used by BDT and Black-Scholes. A P-measure is the real, unbiased statistical probability. Candidates treat the risk-neutral probability as the actual chance of an up move; it is not.
- Reading average compounded return as NPV. A two-period tree can show a negative average compounded (geometric) return while expected dollar value stays flat at NPV = 0. Candidates conclude the asset destroys wealth. In a risk-neutral, zero-rate world the dollars are correct; the compounded rate is the misleading figure.
Bottom line
- Binomial trees price up/down moves under Q-measure (risk-neutral) probabilities; a P-measure is a real, unbiased statistical probability, a Q-measure is a biased quasi-probability assumed under risk-neutrality.
- The risk-neutral up probability is , valid whenever d < 1 + r < u; never substitute the real-world chance of an up move.
- A one-period defaultable bond prices at the probability of nondefault; once a risk premium enters, the credit spread cannot separate default probability from risk-aversion.
- Convertibles value max(conversion, hold); callables value min(call price, hold) since the issuer holds the embedded call.
Exam shortcut
For tree questions, identify the method from the question stem: European + no early exercise is closed-form, anything American or callable is a tree. For factor model questions, check whether UMD (momentum) is listed; it separates Carhart from Fama-French. Remember: "Callable is min, Convertible is max" because the issuer caps upside and the holder captures upside.
The full lesson (about 5,617 words, 37 min read) adds 2 worked examples, all 7 common mistakes, a self-check, free in the app.
Learning objectives
- model types
- fi models intro
- bdt model
- credit risk economics
- structural model overview
- merton model
- kmv model
- reduced form models
- empirical credit models
- one period binomial
- multi period binomial
- tree prices formation
- convertible valuation
- callable bonds tree
- multifactor asset pricing
- fama french
- empirical mf challenges
- factor investing
- adaptive markets
- efficiently inefficient
- trend following
- divergence
- fundamental directional
- behavioral finance
- directional factors
- digital asset valuation
- pca statistical factors
- multifactor regression
- partial autocorrelations
- dynamic risk exposure
- changing correlation
- multifactor return approaches
- performance persistence
- rv overview
- statistical pairs equities
- pairs commodity spreads
- pairs rates fx
- rv market neutral risks
- depreciation tax shields
- tax deferral gains
- after tax comparisons
- transaction based indices
- appraisal based indices
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