A note on the Fundamental Theorem of Asset Pricing under model uncertainty

We show that the results of ArXiv:1305.6008 on the Fundamental Theorem of Asset Pricing and the super-hedging theorem can be extended to the case in which the options available for static hedging (\emph{hedging options}) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of \emph{robust no-arbitrage} which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are \emph{non-redundant}. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.Comment: Final version. To appear in Risk

Closed-Form Approximations for Spread Option Prices and Greeks

We develop a new closed-form approximation method for pricing spread options. Numerical analysis shows that our method is more accurate than existing analytical approximations. Our method is also extremely fast, with computing time more than two orders of magnitude shorter than one-dimensional numerical integration. We also develop closed-form approximations for the greeks of spread options. In addition, we analyze the price sensitivities of spread options and provide lower and upper bounds for digital spread options. Our method enables the accurate pricing of a bulk volume of spread options with different specifications in real time, which offers traders a potential edge in financial markets. The closed-form approximations of greeks serve as valuable tools in financial applications such as dynamic hedging and value-at-risk calculations.

Exchangeability type properties of asset prices

In this paper we analyse financial implications of exchangeability and similar properties of finite dimensional random vectors. We show how these properties are reflected in prices of some basket options in view of the well-known put-call symmetry property and the duality principle in option pricing. A particular attention is devoted to the case of asset prices driven by Levy processes. Based on this, concrete semi-static hedging techniques for multi-asset barrier options, such as certain weighted barrier spread options, weighted barrier swap options or weighted barrier quanto-swap options are suggested.Comment: The final version of the paper "Semi-static hedging under exchangeability type conditions". To appear in Advances in Applied Probabilit

Pricing index options by static hedging under finite liquidity

We develop a model for indifference pricing in derivatives markets where price quotes have bid-ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investor's beliefs, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimisation allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between super- and subhedging prices.Comment: 19 pages, 12 figure

Multivariate Hawkes-based Models in LOB: European, Spread and Basket Option Pricing

In this paper, we consider pricing of European options and spread options for Hawkes-based model for the limit order book. We introduce multivariate Hawkes process and the multivariable general compound Hawkes process. Exponential multivariate general compound Hawkes processes and limit theorems for them, namely, LLN and FCLT, are considered then. We also consider a special case of one-dimensional EMGCHP and its limit theorems. Option pricing with $1D$ EGCHP in LOB, hedging strategies, and numerical example are presented. We also introduce greeks calculations for those models. Margrabe's spread options valuations with Hawkes-based models for two assets and numerical example are presented. Also, Margrabe's spread option pricing with two $2D$ EMGCHP and numerical example are included. Basket options valuations with numerical example are included. We finally discuss the implied volatility and implied order flow. It reveals the relationship between stock volatility and the order flow in the limit order book system. In this way, the Hawkes-based model can provide more market forecast information than the classical Black-Scholes model

Closed-Form Approximations for Spread Option Prices and Greeks

We develop a new closed-form approximation method for pricing spread options. Numerical analysis shows that our method is more accurate than existing analytical approximations. Our method is also extremely fast, with computing time more than two orders of magnitude shorter than one-dimensional numerical integration. We also develop closed-form approximations for the greeks of spread options. In addition, we analyze the price sensitivities of spread options and provide lower and upper bounds for digital spread options. Our method enables the accurate pricing of a bulk volume of spread options with different specifications in real time, which offers traders a potential edge in financial markets. The closed-form approximations of greeks serve as valuable tools in financial applications such as dynamic hedging and value-at-risk calculations

PRICING AND HEDGING EUROPEAN OPTIONS ON FUTURES SPREADS USING THE BACHELIER SPREAD OPTION MODEL

The Bachelier model for pricing options on futures spreads (OFS) assumes changes in the underlying .futures prices and spread follow unrestricted arithmetic Brownian motion (UABM). The assumption of UABM allows for a convenient analytic solution for the price of an OFS. The same is not possible under the more traditional assumption of geometric Brownian motion (GBM). Given the additional complexity of methods for pricing and hedging OFS using GBM such as Monte Carlo simulation and binomial trees, it is worth investigating how results from the Bachelier model compare to these other methods. The Bachelier model is presented and then extended to price an OFS with three underlying commodities. Hedge parameters for both models are provided. Results indicate that for OFS with sufficiently low volatility, differences between the Bachelier model and methods assuming GBM are quite small.Marketing,

Closed-Form Approximations for Spread Option Prices and Greeks

We develop a new closed-form approximation method for pricing spread options. Numerical analysis shows that our method is more accurate than existing analytical approximations. Our method is also extremely fast, with computing time more than two orders of magnitude shorter than one-dimensional numerical integration. We also develop closed-form approximations for the greeks of spread options. In addition, we analyze the price sensitivities of spread options and provide lower and upper bounds for digital spread options. Our method enables the accurate pricing of a bulk volume of spread options with different specifications in real time, which offers traders a potential edge in financial markets. The closed-form approximations of greeks serve as valuable tools in financial applications such as dynamic hedging and value-at-risk calculations