Bond pricing

An example of a two-factor RF model could e.g. be enforced by a separate modeling of bond prices for corporate bonds and default spreads. 

Finally, we obtain the risk-neutral bond price dynamics... 

Later on, we need to change the Gog) bond price proeess from the risk-neutral to the appropriate forward measure in order to compute the price of a bond option by summing over the single risk-neutral exercise probabilities. 

Now, together with the (log) bond price dynamics (5.5) we have... 

Starting from the risk-neutral bond price dynamics (5.4), we derive the well known closed-form solution for the price of a zero-coupon bond option. Thus, as shown in section (2.1) the price of a call option on a discount bond is given by... 

Hence, together with our (log) bond price dynamics under the 7b-forward measure (5.9) we obtain the dynamics of the new variable X(t) as follows... 

Then, plugging the (log) bond price d5mamics (5.9) under the To-forward measure in (5.15) and collecting the deterministic and stochastic terms leads to... 

Note that we are now able to rewrite the eharacteristic function of the (log) bond price X (Tq, T)) in terms of the transform 0,(z) as follows... 

For the term structure of discount bond prices we use the Bloomberg market data of US-treasury STRIPS (09.09,2005). Not traded bond prices are derived from a cubic spline data interpolation between existing STRIPS. 

At last, looking at the density function in Tq we see that the approximated pdf coming from the EE (black line) perfectly fits the empirical distribution coming from a MC simulation of the underlying bond price dynamics (figure (5.2.S7 and S8)). Recapitulating, we found that the lEE performs more accurate than an adequate MC simulation study, by consuming only a fractional amount of computational time. 

Thus, going forward we only have to adapt our pricing framework to the new multi-factor dynamics of the arbitrage-free bond price given by... 

Again, applying ltd s lemma to the bond price dynamics P e) = e leads to... 

Hence, the absence of arbitrage opportunities implies tbat the drift term of the bond price dynamics equals the risk-free interest rate. This implies... 

Equivalent to section (5.1), we first have to change the measure of the (log) bond price process, before we are able to compute the option prices. Therefore, we transform the (log) bond price dynamics... 

Armed with the (log) bond price process (6.10) under the 7b-measure, we can derive a formula for the price of an option on a discount bond. This new solution extends the well known formula (5.27) by an additional correlation function c T,V) implied by the RF dW t,T). 

As aforementioned, it can be shown that the expectation Yi zm ) is determined by an exponential affine form fulfilling the martingale condition. Now, postulating a multiple-Field term structure model the bond price dynamics are given by... 

In recent works Collin-Dufresne and Goldstein [18], Heiddari and Wu [36], Jarrow, Li, and Zhao [45] and Li, Zhao [54] have extended the HJM approach to a framework, where either the volatility of forward rates, or the volatility of bond prices is driven by a subordinated stochastic process. One major implication of these new type of models is an additional source of uncertainty driving the volatility. This implies the existence of an additional market price of risk. Intuitively, this market price of risk cannot be hedged only by bonds. As a result of this, we have a new class of models causing incomplete bond markets ... 

In other words, assuming a complete market stochastic volatility model implies that the short rate is modeled directly, while the traded asset (bond) has to be derived. Therefore, only the direct modeling of the bond price dynamics, together with stochastic volatility leads to an incomplete market model analog to the stochastic volatility models of equity markets". ... 

Along the lines of HIM it can be shown that the existence of an arbitrage-free setup implies that the drift ji t,T) is fully determined by the volatility function cr (f, T) and the stochastic state variable V (f). Applying Ito s lemma to the bond price... 

Han [34] uses a model that is a special case of our model framework, but postulating bond price dynamics that are uncorrelated with the subordinated stochastic volatility process. 

Again, we need to transform the process for the (log) bond price dynamics dXit, T) from the risk-neutral measure Q to the forward measure Tq. Thus, following section (5.1) we derive a measure transformation specially adapted to the additional innovation of the stochastic volatility. The bond price can be computed by integrating from t to 7b via... 

Note that the impact of this correlation effect is not in contradiction to the results found by Bakshi, Cao and Chen [5], Nandi [62] and Schobel and Zhu [69] for equity options. They found higher option prices given positive correlations and vice verca. On the other hand, we have a risk-neutral bond price process, where the source of uncertainty is negatively assigned (see e.g. (7.2)). Thus, assuming a USV bond model with negative correlated Brownian motions is the fixed income market analog of a stochastic volatility equity market model, with positive correlated sources of uncertainty. ... 

Again, our well known exponential affine approach could be applied to compute the bond prices via FRFT or lEE. 

Heath D, Jarrow R, Morton A (1992) Bond Pricing and the Term Structure of Interest Rates A New Methodology for Contingent Claims Evaluation. Econometrica 60 77-105. 

However, a critical issue on this spread measure is how the asset swap has been stmctured. ASW measure works very well when bond prices trade at or near to par. Most corporate bonds trade with price away from the par (as in this case), thus making the ASW an inaccurate spread measure. If the bond trades at premium, the ASW spread will overestimate the level of credit risk conversely, if the bond trades at discount the ASW spread will underestimate the level of credit risk. Therefore, in the case of HERIM and TKAAV, the ASW spread overestimates the credit risk associated with the bonds because both trade significantly at premium. 

Bond Price Swap Price Swap Rate( Redeu-uti ji (i)... 

Models that seek to value options or describe a yield curve also describe the dynamics of asset price changes. The same process is said to apply to changes in share prices, bond prices, interest rates and exchange rates. The process by which prices and interest rates evolve over time is known as a stochastic process, and this is a fundamental concept in finance theory. Essentially, a stochastic process is a time series of random variables. Generally, the random variables in a stochastic process are related in a non-random manner, and so therefore we can capture them in a probability density function. A good introduction is given in Neftci (1996), and following his approach we very briefly summarise the main features here. 

A conventional bond has an expected return of 5.875% and a standard deviation of 1 2.50% per annum. The initial price of the bond is 100. From Equation (2.20), the dynamics of the bond price are given by ... 

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