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5.1.2 Parameters illustrated with VMARKET experiments


[ SLIDE parameters - discounting - price of risk - drifts || VIDEO login]

Since the terminal value of the discount function at the maturity is simply $ P(T,T)=1$ , the parameters characterize either the forecast of the spot rate or the numerical method that will be examined later in sect.5.3.1. The financial parameters that are relevant in the applet are:

  • the lifetime or maturity date ($ T$ or RunTime) of a bond in years, e.g. 10 for a bond reaching its maturity in a decade,
  • the maximum volatility ( $ \sigma_\mathrm{max}$ or Volatility) of a bond estimated from historical values, e.g. 0.02 for a two percent volatility peak that will be reached after one third of the bond lifetime (5.1.1#fig.1),
  • the market price of risk ($ \lambda$ or MktPriceRsk) measuring the reward $ \lambda\sigma$ expected by the investors for taking an investment risk, e.g. -0.25 in a risk averse market with little appetite for risk. In the applet, the effect is further modulated by a cosine function reproducing ($ n$ or UserDouble) economic cycles during the lifetime of the bond,
  • the mean reversion target rate ($ b$ or MeanRevTarg) is the value towards which the spot rate returns to after a long time, e.g. 0.05 for a market with a 5% average rate,
  • the mean reversion velocity ($ a$ or MeanRevVelo) measures the speed of the process, e.g. 0.5 [1/year] for a mean reversion taking about $ 1/0.5=2$ years.
  • the spot rate ($ r$ or SpotRate) used to plot the term structure of the interest rates.
To visualize the evolution of a bond and the corresponding yield in a very simple case, the VMARKET applet below shows what happens in the absence of drifts (right hand side of 5.1.1#eq.5 equals zero) and without volatility.
VMARKET applet:  press Start/Stop to simulate the trivial case of a bond price, when the spot rate is fixed to a pre-determined value. The plots show the value of the discount function as a function of the spot rate (P[r] in black) for an increasing time to maturity t (Time on the top of the window, in years). Directly derived from that using (2.2.1#eq.1), two plots show the evolution of the yield curve (Y[r] in blue, for a fixed Time) and the term structure of the interest rates (Y[t] in grey, for a fixed SpotRate). The latter acquires a finite value and sweeps across the plot window over the time span of one simulation [0; RunTime] and is best viewed after rescaling with Display.

The discount function decreases exponentially backward in time $ P(t,T)=\exp(-r[T-t])$ as expected for a risk free investment (1.3#eq.6). The bond yield is equal to the spot rate $ Y(r)=r$ and the term structure of the interest rates is constant $ Y(t)=r$ .



Virtual market experiments: trivial bond
  1. Vary the length of the simulation domain MeshLength and, by clicking in the plot area, verify that bond yield is indeed equal to the spot rate.
  2. Modify the time to the maturity RunTime and verify that you have properly understand all three graphs that are plotted.

The second applet below illustrates the effect of a large volatility $ \sigma$ in the spot rate and accounts for the extra return investors expect from the market through the so-called market price of risk $ \lambda$ .

VMARKET applet:  press Start/Stop to study the effect of a large Volatility in the spot rate. The extra reward payed in a risk averse market is here modeled with the parameter MktPriceRsk. The plots show the value of the discount function as a function of the spot rate (P[r] in black) for an increasing time to maturity t (Time on the top of the window, in years). Directly derived from that using (2.2.1#eq.1), two plots show the evolution of the yield curve (Y[r] in blue, for a fixed Time) and the term structure of the interest rates (Y[t] in grey, for a fixed SpotRate). The latter acquires a finite value and sweeps across the plot window over the time span of one simulation [0; RunTime] and is best viewed after rescaling with Display.

Although this is not immediately apparent in the simulation, the main effect of the volatility is to reduce the curvature of the discount function $ P(r)$ by smearing out irregularities in the yield curves $ Y(r)$ , $ Y(t)$ : if the forecast rate changes rapidly, the yield curves do not follow immediately everywhere. The reward payed to the investor who accepts the risk associated with fluctuations in the spot rate is clearly visible, with an effective yield that increases with time for a positive value of the market price of risk $ \lambda$ .



Virtual market experiments: volatility and the market price of risk
  1. Vary the amount of volatility in the spot rate and observe how the effect evolves with time. Remember that 4% volatility is huge for credit markets!
  2. Change the value and the sign of the market price of risk $ \lambda$ associated in the applet with the MktPriceRsk parameter.
  3. Set UserDouble=2 to model two cycles in an economy. Try to identify when the forecasted rates are high; how is the corresponding yield?

The applet above illustrates the effect of evolving drifts in the forecast rates, here modeled with two economic cycles during the lifetime of the bond: recession $ \rightarrow$ cut rate $ \rightarrow$ over-heated economy $ \rightarrow$ rise rate... or rather the opposite when the time runs backward in the applet.

The third applet below finally illustrates the effect of a mean reversion, which accounts for the tendency of the forecasted rates to fall back to a long term average value.

VMARKET applet:  press Start/Stop to study the effect of a mean reversion process in the forecast of interest rates. A target rate of 6 per cent (parameter MeanRevTarg=0.06) is reached after approximatively 4 years when choosing the inverse for MeanRevVelo=0.25. The plots show the value of the discount function as a function of the spot rate (P[r] in black) for an increasing time to maturity t (Time on the top of the window, in years). Directly derived from that using (2.2.1#eq.1), two plots show the evolution of the yield curve (Y[r] in blue, for a fixed Time) and the term structure of the interest rates (Y[t] in grey, for a fixed SpotRate). The latter acquires a finite value and sweeps across the plot window over the time span of one simulation [0; RunTime] and is best viewed after rescaling with Display.

Observe how the yield rapidly drops for large values of the spot rate(on the right side of the plot), reflecting the reversion back to the long term average target.



Virtual market experiments: mean reversion
  1. Change the MeanRevVelo parameter $ a$ to modify the typical time scale for interest rates to revert back to the target level MeanRevTarg.
  2. Use all the forecasting parameters to approximate the price of a bond during 10 years reproducing the evolution of interest rates in your country.

Experimenting with the applet enables you to develop an intuitive understanding for the fundamental processes that characterize the credit market. The experiments also prepare you also for the inverse problem, where the term structure of the interest rates is known from the market (e.g. 2.2.2#fig.1) and the drift / volatility parameters are matched in order to extrapolate into the future (exercise 5.01).

SYLLABUS  Previous: 5.1.1 Term structure models  Up: 5.1 Discound bonds  Next: 5.2 Credit derivatives

      
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