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We consider a spectrally-negative Markov additive process as a model of a risk process in random environment. Following recent interest in alternative ruin concepts, we assume that ruin occurs when an independent Poissonian observer sees the process negative, where the observation rate may depend on the state of the environment. Using an approximation argument and spectral theory we establish an explicit formula for the resulting survival probabilities in this general setting. We also discuss an efficient evaluation of the involved quantities and provide a numerical illustration.

In this paper we introduce the concept of implied (il)liquidity of vanilla options. Implied liquidity is based on the fundamental theory of conic finance, in which the one-price model is abandoned and replaced by a two-price model giving bid and ask prices for traded assets. The pricing is done by making use of non-linear distorted expectations. We first recall a two parameter distortion function representing the notions of risk aversion and absence of gain enticement. After reviewing under the Black-Scholes setting the theory and numerics of the calculation of bid-ask prices under conic finance theory, we introduce the concept of implied liquidity. In a fixed market with no movement in the cone of acceptable risks and hence no change in liquidity as the market is then fixed, the bid ask spread moves around nonlinearly with maturity and/or volatility. Because the spread can move in a constant market with no change in liquidity, spread itself is not a perfect measure of liquidity. Implied liquidity can overcome this criticism. We illustrate the theory on SP500 and Dow Jones Index data. We show that for vanilla options we typically have for higher strikes (OTM) more implied illiquidity. We typically see not much term structure. Also, we perform a historical study, in which we clearly see a serious drying up of liquidity in the weeks post the Lehman bankruptcy. Next, we elaborate on stochastic liquidity behavior and potential liquidity contracts and modeling. Seen the evidence of changing liquidity in the recent past with a potentially very disruptive drying up of liquidity, these contracts could provide extra hedges for such circumstances. The above notion of implied liquidity leads toward a mean-reverting modeling of liquidity similar to stochastic volatility. We believe such stochastic liquidity modeling could be very useful in structured product pricing, delta-gamma-vega hedging studies and risk-management in general.

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