This is the market's view on the reward that should be attached to the risk inherent in a specific financial security. So in short, it is the excess return that one would expect over the risk free rate for taking on exta risk.Sam

To incorporate the market price of risk in pricing instrument is essential when modelling quantities that are not directly tradable.In the "perfect" world of Black Scholes, in which you can always continuously and perfectly hedge your position, it is possible to build a portfolio that completely eliminates risk. In the Black Scholes world, you can always dynamically hedge your portfolio, and doing so you are eliminating risk totally, and the market price of risk is a quantity that does not need to appear in the model (or the differential equation that describes it).The situation changes radically if you are trading something that is not directly observable (e.g. volatility trading, interest rate products).Since volatilities (or interest rates) are not tradable, you will not be able to create a risk-free portfolio. In fact you will trade one or more assets that depends on that quantity, rather than the quantity itself (e.g. you want to take a position on an interest rate, you might want to trade bonds since their value depends on interest rates). The same happens for hedging, since you will need again to use another instrument similar to the original one to cover your position, but is not the underlying quantity.The result is that you will never have a portfolio that will completely eliminate risk, and as Sam said earlier, an agent will require a premium to balance his diminished utility function resulting from taking a risk,gc

Hi,I am interested in pricing an asset that depends upon 2 underlying non-traded assets. Should the prices of risk for the 2 underlying factors be made linearly dependent? For example, in equilibrium the price of risk scaled by asset vols should be equal across different assets.Any assitance or example models would be appreciated.

Can it also be measured from the expected return on holding variance swaps?

- ScilabGuru
**Posts:**297**Joined:**

QuoteOriginally posted by: maclachlHi,I am interested in pricing an asset that depends upon 2 underlying non-traded assets. Should the prices of risk for the 2 underlying factors be made linearly dependent? For example, in equilibrium the price of risk scaled by asset vols should be equal across different assets.Any assitance or example models would be appreciated.I am not sure that understood what do you mean by linear dependence of market prices of risk (ideally they are constant, right?). But In Hull, 5th edition, page 487 (chapter 21.2) you probably can find an answer for your question

Last edited by ScilabGuru on January 21st, 2004, 11:00 pm, edited 1 time in total.

Yes you are right - that is half the answer. More particularly I am interested in the relationship between lambda i and lambda j, etc.

- positivedefinite
**Posts:**9**Joined:**

Are you asking if the market price of risk of a composite of two assets is a linear combination of the respective prices of risk of two assets? The price of risk must be the same for every asset, since otherwise will create an arbitrage opportunity...., i.e. lambda_i = lambda_j = lamdba_composite.. Am I wrong here??

- Cuchulainn
**Posts:**62418**Joined:****Location:**Amsterdam-
**Contact:**

Does anyone know about the market price of risk in relation to the Heston 1993 model?thxDD

My derivation for Heston's process follows. I'd love to hear your comments and opinions. :The Heston model can be described by the equationsdS/S = m dt + sqrt(v) dWdv = a (b-v) dt + k sqrt(v) dB, corr(W,B)=rhoThe risk adjusted probability measure can be retrieved via Girsanov's theorem, by establishing that the Radon-Nikodym derivative is given byM_T = exp( -0.5 int_0^T h^2(t) dt - 0.5 int_0^T g^2(t) dt - int_0^T h(t) dW(t) - int_0^T g(t) dB(t) )Here, h(t) is the 'price of risk' associated with the asset level, while g(t) is the 'price of volatility risk'. Setting h(t)=(m-r)/sqrt(v) (the Sharpe ratio) is sufficient to make the discounted asset prices martingales, i.e. it is making the probability measure risk neutral. We are free to choose g(t) more or less as we wish (a feature of an incomplete market). Any choice of g(t) will change the volatility drift in the following way:a (b-v) -> a (b-v) + sqrt(v) ( rho k h(t) + sqrt(1-rho^2) g(t) )If we choose g(t) to be of the form lambda/sqrt(v), which is similar to the Sharpe ratio that we chose for h(t), the sqrt(v)'s above will cancel, and the new drift will becomea (b-v) + constant = a (b0-v)It will therefore appear, that under risk neutrality the long run volatility subsumes the volatility premium. Agents behave as if the volatility will revert to a higher long-run value, because they dislike volatility uncertainty. Even if lambda=0, there will be a residual volatility premium due to the correlation.KyriakosPS: The above derivation can be also found in my notes here.

- Cuchulainn
**Posts:**62418**Joined:****Location:**Amsterdam-
**Contact:**

Sammus and I are working on Heston from PDE/FDM perspective and we are using Heston 1993 as well.Eventually we shall need all the parameters, hence our interest.

Last edited by Cuchulainn on February 24th, 2005, 11:00 pm, edited 1 time in total.

I always thought that the price of risk was the difference between the price at which a trader would buy the product and the price which your model gives you. This would have to be adjusted if the trader took a bigger cut due to a trade being imperfectly understood/unhedgable.Can we imagine a theoretical "Perfect Trader", being an algorithm which given a portfolio is able to continuosly hedge the risk and guarentee a return of the risk free rate?In which case, the price of risk for a particular product/portfolio would be the difference between the value given by your favourate mathematical model (using whatever assumptions you like) and the value at which the "perfect trader" would accept the product.I realise that his does not give a numerical answer, but it could be a starting point. The market price of risk then reflects any and all parameters which we have not incorporated into our valuation model, but traders do observe in the market.

QuoteOriginally posted by: RezMy derivation for Heston's process follows. I'd love to hear your comments and opinions. :The Heston model can be described by the equationsdS/S = m dt + sqrt(v) dWdv = a (b-v) dt + k sqrt(v) dB, corr(W,B)=rhoThe risk adjusted probability measure can be retrieved via Girsanov's theorem, by establishing that the Radon-Nikodym derivative is given byM_T = exp( -0.5 int_0^T h^2(t) dt - 0.5 int_0^T g^2(t) dt - int_0^T h(t) dW(t) - int_0^T g(t) dB(t) )Here, h(t) is the 'price of risk' associated with the asset level, while g(t) is the 'price of volatility risk'. Setting h(t)=(m-r)/sqrt(v) (the Sharpe ratio) is sufficient to make the discounted asset prices martingales, i.e. it is making the probability measure risk neutral. We are free to choose g(t) more or less as we wish (a feature of an incomplete market). Any choice of g(t) will change the volatility drift in the following way:a (b-v) -> a (b-v) + sqrt(v) ( rho k h(t) + sqrt(1-rho^2) g(t) )If we choose g(t) to be of the form lambda/sqrt(v), which is similar to the Sharpe ratio that we chose for h(t), the sqrt(v)'s above will cancel, and the new drift will becomea (b-v) + constant = a (b0-v)It will therefore appear, that under risk neutrality the long run volatility subsumes the volatility premium. Agents behave as if the volatility will revert to a higher long-run value, because they dislike volatility uncertainty. Even if lambda=0, there will be a residual volatility premium due to the correlation.KyriakosPS: The above derivation can be also found in my notes here.It seems to me that the market prices of risk for the state variables (say, SV and stock price in the Heston mdoel) can be found by writing down the drift of an contingent claim under the 'physical' measure and the risk-neutral measure, respectively. A visual comparsion of the two drift will immediately spot the extra terms which are the compensation for taking the risk of the corresponding state variables, hence they are the market prices of risk.Rgds,

Just two quick questions:1. why is your radon-nikodym derivative (i.e., M_T) multiplication of two radon-nikodym derivatives for S(t) and v(t)?2. how did you derive this?: a (b-v) -> a (b-v) + sqrt(v) ( rho k h(t) + sqrt(1-rho^2) g(t) ) - this somehow looks like the premium for volatility risk though Thanks in advance!

GZIP: On