We propose a Fundamental Theorem of Asset Pricing and a Super‐Replication Theorem in a model‐independent framework. We prove these theorems in the setting of finite, discrete time and a market ...consisting of a risky asset S as well as options written on this risky asset. As a technical condition, we assume the existence of a traded option with a superlinearly growing payoff‐function, e.g., a power option. This condition is not needed when sufficiently many vanilla options maturing at the horizon T are traded in the market.
We consider the problem of finding a model‐free upper bound on the price of a forward start straddle with payoff . The bound depends on the prices of vanilla call and put options with maturities T1 ...and T2, but does not rely on any modeling assumptions concerning the dynamics of the underlying. The bound can be enforced by a super‐replicating strategy involving puts, calls, and a forward transaction. We find an upper bound, and a model which is consistent with T1 and T2 vanilla option prices for which the model‐based price of the straddle is equal to the upper bound. This proves that the bound is best possible. For lognormal marginals we show that the upper bound is at most 30% higher than the Black–Scholes price. The problem can be recast as finding the solution to a Skorokhod embedding problem with nontrivial initial law so as to maximize .
In this work, we introduce the notion of fully incomplete markets. We prove that for these markets, the super‐replication price coincides with the model‐free super‐replication price. Namely, the ...knowledge of the model does not reduce the super‐replication price. We provide two families of fully incomplete models: stochastic volatility models and rough volatility models. Moreover, we give several computational examples. Our approach is purely probabilistic.
We study super-replication of European contingent claims in an illiquid market with insider information. Illiquidity is captured by quadratic transaction costs and insider information is modeled by ...an investor who can peek into the future. Our main result describes the scaling limit of the super-replication prices when the number of trading periods increases to infinity. Moreover, the scaling limit gives us the asymptotic value of being an insider.
Market delay and G-expectations Dolinsky, Yan; Zouari, Jonathan
Stochastic processes and their applications,
February 2020, 2020-02-00, Letnik:
130, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We study super-replication of contingent claims in markets with delayed filtration. The first result in this paper reveals that in the Black–Scholes model with constant delay the super-replication ...price is prohibitively costly and leads to trivial buy-and-hold strategies. Our second result says that the scaling limit of super-replication prices for binomial models with a fixed number of times of delay H is equal to the G-expectation with volatility uncertainty interval 0,σH+1.
We extend the super-replication theorem in a dynamic setting, both in the numéraire-based as well as in the numéraire-free setting. For this purpose, we generalize the notion of admissible ...strategies. In particular, we obtain a well-defined super-replication price process, which is right-continuous under some regularity assumptions.
LIQUIDITY IN A BINOMIAL MARKET Gökay, Selim; Soner, Halil Mete
Mathematical finance,
April 2012, Letnik:
22, Številka:
2
Journal Article
Recenzirano
We study the binomial version of the illiquid market model introduced by Çetin, Jarrow, and Protter for continuous time and develop efficient numerical methods for its analysis. In particular, we ...characterize the liquidity premium that results from the model. In Çetin, Jarrow, and Protter, the arbitrage free price of a European option traded in this illiquid market is equal to the classical value. However, the corresponding hedge does not exist and the price is obtained only in L2‐approximating sense. Çetin, Soner, and Touzi investigated the super‐replication problem using the same supply curve model but under some restrictions on the trading strategies. They showed that the super‐replicating cost differs from the Black–Scholes value of the claim, thus proving the existence of liquidity premium. In this paper, we study the super‐replication problem in discrete time but with no assumptions on the portfolio process. We recover the same liquidity premium as in the continuous‐time limit. This is an independent justification of the restrictions introduced in Çetin, Soner, and Touzi. Moreover, we also propose an algorithm to calculate the option’s price for a binomial market.
This paper studies arbitrage-free financial markets with bid-ask spreads whose super-hedging prices are submodular. The submodular assumption on the super-hedging price, or the supermodularity ...usually assumed on utility functions, is the formal expression of perfect complementarity, which dates back to Fisher, Pareto, and Edgeworth, according to Samuelson (J Econ Lit 12:1255–1289, 1974). Our main contribution provides several characterizations of financial markets with frictions that are submodular as a consequence of a more general study of submodular pricing rules. First, a market is submodular if and only if its super-hedging price is a Choquet integral and if and only if its set of risk-neutral probabilities is representable as the core of a submodular non-additive probability that is uniquely defined, called risk-neutral capacity. Second, a market is representable by its risk neutral capacity if and only if it is equivalent to a market, only composed of bid-ask event securities.
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