Pricing options in incomplete equity markets via the instantaneous Sharpe ratio

Abstract

We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. First, we apply our method to price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and Saá-Requejo (2000) and of Björk and Slinko (2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar (2000).

Relative Hedging of Systematic Mortality Risk

“…This correspondence between the standard deviation premium principle using the instantaneous Sharpe ratio and the good deal bounds was observed in Bayraktar and Young (2008). In particular, when γ SR = 0, H is equal to the expectation under the minimal martingale measure Q and is given by the right-hand-side of (40).…”

“…In this section we will obtain the cost, denoted by H, of delivering f instantaneously in the pricing framework of Bayraktar and Young (2008), Young (2008) and Bayraktar and Young (2007a). This framework assumes that the insurer requires compensation for the mortality risk it assumes in the form of a pre-specified Sharpe ratio, which is a continuous version of the standard-deviation premium principle.…”

Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities

“…Our work is this section is motivated by similar results of Bayraktar and Young (2008); see the remark at the end of their Section 2.5. It is straightforward to show that one can write (2.16) as…”

“…Also, in the limit as the number of contracts approaches infinity, the resulting value can be represented as an expectation with respect to an equivalent martingale measure. Because of these properties, we anticipate that our valuation methodology will prove useful in pricing risks in other incomplete markets; for example, see Bayraktar and Young (2008).…”

“…Specifically, we assume that the insurance company chooses a hedging strategy that minimizes the local variance, as in Schweizer (2001a), then picks a target ratio α of net expected return to standard deviation, and finally determines the corresponding value for any given mortality-contingent claim that leads to this pre-determined α. See Bayraktar and Young (2008) for further discussion of this methodology from the standpoint of the mean-variance efficient frontier.…”