Why Term Structure Models Use Short Rates or State Prices
Summary
The document explains why term structure models often specify short-rate or state-price dynamics instead of assigning separate price processes to bonds at each maturity. Independently modeling bond prices can produce inconsistencies that permit arbitrage. Modeling a short rate under the risk-neutral measure gives a coherent way to derive prices across maturities while enforcing no-arbitrage relationships.
It describes short-rate and state-price approaches as closely related formulations that can yield the same models, with the choice often driven by convenience. It also notes that short-rate modeling is not the only option: a market model can specify dynamics for multiple forward rates and their correlations. The explanation is introductory and does not derive pricing equations or compare the assumptions and practical tradeoffs of particular models.
Key ideas
- Independent bond price processes can fail to satisfy no-arbitrage relationships across maturities.
- Risk-neutral short-rate dynamics provide a framework for deriving consistent bond prices.
- Short-rate and state-price formulations can describe closely related models.
- Market models offer an alternative by specifying correlated forward-rate dynamics.
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Full text
# Term Structure Modelling - Why model the state prices and not an asset or rate # Term Structure Modelling - Why model the state prices and not an asset or rate When modelling stocks we specify the model in terms of the dynamics of the stock itself (e.g. in Black-Scholes, Heston and SABR - often denoted $S$). However, as I am reading about Term Structure Models many sources start with the dynamics of the short rate (often denoted $r$) or state price vector (often denoted $X$). Why are we using this the approach as opposed to just model the bond prices directly? And what is the difference between modelling short rates and state prices? ## Answer by fes (score 1) https://quant.stackexchange.com/a/73296 If you specify separate price processes for bonds with different maturities, the resulting model is typically not arbitrage free. Solving the bond prices by instead specifying a short rate process under the risk neutral measure is a convenient way to guarantee that there are no arbitrage opportunities. How about the difference between modelling short rates under the risk neutral measure and specifying a process for state prices or the stochastic discount factor? Typically there is not much difference as the same short rate models could be derived either way. It is more about which approach is more convenient. Note that specifying a short rate process, however, is not the only way to model bonds. For example the so called Market Model instead specifies a separate process for different forward rates and assumes these processes are correlated.
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