Chapter 6 - Market Price of Risk

[welsh_owen]

Hi all,

I am working through the notes and have a couple of questions about the market price of risk (Chp 6, page 24).

When working in the risk-neutral world (i.e. expected growth rate of a risky asset = r (or some manipulation thereof - i.e. to take account of dividends etc.)) is the MPR effectively equal to zero? my thoughts are that this would mean an investor is indifferent to taking the risk or not as there is no additional reward for this.

If nothing else I was thinking this would tie up with the no arbitrage arguments used throughout the course.

Equivalent Martingale Measure (EMM)

The EMM on page 27 refers of Chp 6 refers only to the real-world probability measure (denoted P for most of the course). In Chapter 27 of the Hull text book - page 37 - this result appears to be also applied to the risk-neutral probability measure Q.

If the Numeraire is chosen to be the accumulated value of a risk-free cash bond would I be right in thinking this has zero volatility since it is wholly governed by a deterministic process (for basic cases at least)? Providing this is the case I am assuming that the MPR for the risky asset underlying the derivative pay-of and the bond must be the same (which is presumably 0 assuming that dividing by the volatility of g is not undefined for the MPR of the numeraire g).

Assuming this is the case would I be right in thinking the SDE on page 28 of chp 6 then becomes:

d(f/g) = (f/g)*Sigma(f) dz

This assumes that Sigma(g) is zero so can be ignored.

Thanks,
O

 

[Oxymoron]

If the Numeraire is chosen to be the accumulated value of a risk-free cash bond would I be right in thinking this has zero volatility since it is wholly governed by a deterministic process (for basic cases at least)? Providing this is the case I am assuming that the MPR for the risky asset underlying the derivative pay-of and the bond must be the same (which is presumably 0 assuming that dividing by the volatility of g is not undefined for the MPR of the numeraire g).

Yes, any numeraine that is perfectly deterministic has 0 market price of risk. Both the underlying and the derivative in this case will return the rate of return to be expected from the numeraine (like r if its cash account).

 

[manish.rex]

Hi all,

I am working through the notes and have a couple of questions about the market price of risk (Chp 6, page 24).

When working in the risk-neutral world (i.e. expected growth rate of a risky asset = r (or some manipulation thereof - i.e. to take account of dividends etc.)) is the MPR effectively equal to zero? my thoughts are that this would mean an investor is indifferent to taking the risk or not as there is no additional reward for this.

If nothing else I was thinking this would tie up with the no arbitrage arguments used throughout the course.

Equivalent Martingale Measure (EMM)
The EMM on page 27 refers of Chp 6 refers only to the real-world probability measure (denoted P for most of the course). In Chapter 27 of the Hull text book - page 37 - this result appears to be also applied to the risk-neutral probability measure Q.

If the Numeraire is chosen to be the accumulated value of a risk-free cash bond would I be right in thinking this has zero volatility since it is wholly governed by a deterministic process (for basic cases at least)? Providing this is the case I am assuming that the MPR for the risky asset underlying the derivative pay-of and the bond must be the same (which is presumably 0 assuming that dividing by the volatility of g is not undefined for the MPR of the numeraire g).

Assuming this is the case would I be right in thinking the SDE on page 28 of chp 6 then becomes:

d(f/g) = (f/g)*Sigma(f) dz

This assumes that Sigma(g) is zero so can be ignored.

Thanks,
O

From the chapter 27 , the MPR for any security is the volatility of the numeraire instruments. The choice of numeraire instrument is governmed by the the timing of pay-off from the original instrument. For a bond/stock, payoff is continuous over t, henve a cash bond is numeraire (0 volatility), for a forward rate agreement/caplet/floorlet, payoff is at later date, so numeraire is forward bond price(bond volatility), for a swaption, the payoffs are in annuity form, hence numeraire is a annuity factor(voliatlity here is the annuity volatility: LLM model).

 

[Foley]

Further to this post, is the definition of the equivalent martingale measure result in the notes correct? "Let g be the value of a numeraire asset with volatility sigma g. If f is the price of any security and the market price of risk is equal to sigma g, then f/g is a martingale under the real-world probability measure".

I take it this can only be correct if we assume the market price of risk in the real world is equal to sigma g?