# Implicit relation between risk and reward

I want to differentiate w.r.t. $$\sigma^2$$ the following equation
$$u'(Y)\mu$$ + $$\frac{u''(Y)}{2}(\sigma^2 + \mu^2) = 0$$
where we can consider $$\mu$$(reward) as an implicit function of $$\sigma^2$$(risk) of small bets
u'(Y)$$\frac{du}{d\sigma^2} +\frac{u''(Y)}{2} + \mu u''(Y) \frac{du}{d\sigma^2}=0$$
Hence,
$$u'(\sigma^2)=\frac{u''(Y)}{2(u'(Y) + \mu u''(Y)}$$
$$u'(\sigma^2)=\frac{u''(Y)}{2(u'(Y) + \mu(\sigma^2) u''(Y)}$$
My another question is what does 1st derivative and 2nd derivative of Utility function imply?

My second question is as follows:

Consider a financial market with one single period, with interest rate r and one stock S. Suppose that $$S_0 =1$$ and, for n=1, $$S_1$$ can take two different values: 2, 1/2.

For which values of r the market is viable? viable means (free of arbitrage opportunities).

What if $$S_1$$ can also take the value 1.

Solution provided.

We want to calculate the values of r such that there is an arbitrage opportunity. We take a portfolio with zero initial value $$V_0=0$$ Then we invest the amount q in the stock without risk, we have to invest −q in the risky stock (q can be negative or positive). We calculate the value of this portfolio in time 2.

$$V_1(\omega_1)=q(r-1)$$.......(1)

$$V_1(\omega_2)=q(r+1/2)$$........(2)

So, if r > 1 there is an arbitrage oppportunity taking q positive (money in the bank account and short position in the risky stock) and if r < −1/2 we have an arbitrage opportunity with q positive (borrowing money and investing in the risky stock). The situation does not change if $$S_1$$ take the value 1.

I want to know,Now if r<-1/2 how there is an arbitrage opportunity?Would any member explain me in his reply?

That means to have a viable market, $$r$$ must be between -0.5 To 1

• Please don't ask two questions in one but post separate questions. – LocalVolatility Mar 8 at 11:12

$$\mu = u(\sigma^2)$$ diffentiating $$diff(u(\sigma^2)^2) = 2u(\sigma^2)u'(\sigma^2) =2\mu(\sigma^2)u'(\sigma^2)$$