Question

Suppose that the short rate is 4% and its real-world process is:

                        dr = 0.1(0.05 - r)dt + 0.01dz

While the risk-neutral process is:

                        dr = 0.1(0.11 - r)dt + 0.01dz

First Question:

What is the market price of interest rate risk?

Second Question: What is the expected return and volatility for a 5-year

zerocoupon bond in the risk-neutral world?

Third Question: What is the expected return and volatility for a 5-year zerocoupon bond in the real world?

Answer 1

1. Drift rate= 0.11-0.05=0.06% or 0.0006 that is difference between real world and risk-neutral process.

Volatility=0.1% that is 0.01.

Market rate of interest rate risk= 0.0006/0.01= -0.6.

2. Expected return in risk neutral world= risk free rate=4%

Volatility= (1-e^-0.1*5)/0.1

= (1-e^-0.5)/0.1

= (1-0.6065)/0.1

= 0.3935/0.1

= 3.935%

3. Price of the bond in risk neutral world= 0.04Dt-0.03935Dz

Bond prices are inversely linked to interest rates. When we move to the real world,

Price= {0.04+(-0.6*-0.03935)}dt- 0.03935dz

= (0.04+0.2361)dt-0.3935dz

= 0.6361dt-0.3935dz

This means that expected return on the bond increases from 4% to 0.6361 or 6.361% as we move from the risk neutral world to the real world.