There are two ratings in a very simple world: non-default (ND) and defaultd. The real-world rating transition matrix per year is given by: $$P = \left[ \begin{array} { c c }
0.95 & 0.05 \
0 & 1
\end{array} \right]$$ i.e., the probability of defaulting when the current state is non-default is 0.05, and a defaulted bond never leaves that state and has zero recovery. The two-year zero-coupon risk-free rate is 4% (continuously-compounded). The price of a default-risk-bearing two-year $100 face value zero-coupon bond is $88. If the off-diagonal one-period transition probabilities in the real-world transition matrix are multiplied by a premium adjustment $\pi$ to get the risk-neutral transition matrix (as in the Jarrow-Lando-Turnbull model), then given the price of the two-year bond, what is the value of $\pi$ ?
A) 1.97
B) 2.00
C) 2.03
D) 2.10

Question

There are two ratings in a very simple world: non-default (ND) and defaultd. The real-world rating transition matrix per year is given by: $$P = \left[ \begin{array} { c c } 
0.95 & 0.05 \
0 & 1
\end{array} \right]$$ i.e., the probability of defaulting when the current state is non-default is 0.05, and a defaulted bond never leaves that state and has zero recovery. The two-year zero-coupon risk-free rate is 4% (continuously-compounded). The price of a default-risk-bearing two-year $100 face value zero-coupon bond is $88. If the off-diagonal one-period transition probabilities in the real-world transition matrix are multiplied by a premium adjustment $\pi$ to get the risk-neutral transition matrix (as in the Jarrow-Lando-Turnbull model), then given the price of the two-year bond, what is the value of $\pi$ ?
A) 1.97
B) 2.00
C) 2.03
D) 2.10

Quizplus · Accepted Answer

The price of a two-year zero-coupon bond in a risk-neutral world can be calculated using the risk-neutral transition matrix and the given bond price. The risk-neutral transition matrix adjusts the real-world probabilities to reflect the market's pricing of risk. The adjustment factor $\pi$ scales the off-diagonal elements of the transition matrix, which represent the transition from non-default to default. Given the bond's price of $88 and the risk-free rate of 4% (continuously compounded), we can calculate the expected payoff under the risk-neutral measure and solve for $\pi$ to match the bond's price. The correct adjustment factor that aligns with these conditions is 2.03, indicating how much the real-world probability of default must be scaled to match the market pricing under risk-neutral valuation.

There Are Two Ratings in a Very Simple World: Non-Default P=[0.950.0501]P = \left[ \begin{array} { c c } 0.95 & 0.05 \\0 & 1\end{array} \right]P=[0.950​0.051​]

There Are Two Ratings in a Very Simple World: Non-Default $P = \left[ \begin{array} { c c } 0.95 & 0.05 \\0 & 1\end{array} \right]$