Nonlinear Valuation under Collateral, Credit Risk and Funding Costs: A Numerical Case Study Extending Black-Scholes

We develop an arbitrage-free framework for consistent valuation of derivative trades with collateralization, counterparty credit gap risk, and funding costs, following the approach first proposed by Pallavicini and co-authors in 2011. Based on the risk-neutral pricing principle, we derive a general pricing equation where Credit, Debit, Liquidity and Funding Valuation Adjustments (CVA, DVA, LVA and FVA) are introduced by simply modifying the payout cash-flows of the deal. Funding costs and specific close-out procedures at default break the bilateral nature of the deal price and render the valuation problem a non-linear and recursive one. CVA and FVA are in general not really additive adjustments, and the risk for double counting is concrete. We introduce a new adjustment, called a Non-linearity Valuation Adjustment (NVA), to address double-counting. The theoretical risk free rate disappears from our final equations. The framework can be tailored also to CCP trading under initial and variation margins, as explained in detail in Brigo and Pallavicini (2014). In particular, we allow for asymmetric collateral and funding rates, replacement close-out and re-hypothecation. The valuation equation takes the form of a backward stochastic differential equation or semi-linear partial differential equation, and can be cast as a set of iterative equations that can be solved by least-squares Monte Carlo. We propose such a simulation algorithm in a case study involving a generalization of the benchmark model of Black and Scholes for option pricing. Our numerical results confirm that funding risk has a non-trivial impact on the deal price, and that double counting matters too. We conclude the article with an analysis of large scale implications of non-linearity of the pricing equations.Comment: An updated version of this report will appear in the volume: Veronesi, P. (Editor), \Handbook in Fixed-Income Securities, Wiley, 201

The 2011 European short sale ban on financial stocks: a cure or a curse? : [version 31 july 2013]

Did the August 2011 European short sale bans on financial stocks accomplish their goals? In order to answer this question, we use stock options’ implied volatility skews to proxy for investors’ risk aversion. We find that on ban announcement day, risk aversion levels rose for all stocks but more so for the banned financial stocks. The banned stocks’ volatility skews remained elevated during the ban but dropped for the other unbanned stocks. We show that it is the imposition of the ban itself that led to the increase in risk aversion rather than other causes such as information flow, options trading volumes, or stock specific factors. Substitution effects were minimal, as banned stocks’ put trading volumes and put-call ratios declined during the ban. We argue that although the ban succeeded in curbing further selling pressure on financial stocks by redirecting trading activity towards index options, this result came at the cost of increased risk aversion and some degree of market failure

Can the Market Add and Subtract? Mispricing in Tech Stock Carve-Outs

Recent equity carve-outs in US technology stocks appear to violate a basic premise of financial theory: identical assets have identical prices. In our 1998-2000 sample, holders of a share of company A are expected to receive x shares of company B, but the price of A is less than x times the price of B. A prominent example involves 3Com and Palm. Arbitrage does not eliminate these blatant mispricing due to short sale constraints, so that B is overpriced but expensive or impossible to sell short. Evidence from options prices shows that shorting costs are extremely high, eliminating exploitable arbitrage opportunities.

Option Investor Rationality Revisited

Do option investors rationally exercise their options? Numerous studies report evidence of irrational behavior. In this paper, we pay careful attention to intraday option quotes and reach the opposite conclusion. An exercise boundary violation (EBV) occurs when the best bid price for an American option is below the option’s intrinsic value. Far from being unusual, we show that EBVs occur very frequently. Under these conditions, the rational response of an investor liquidating an option is to exercise the option rather than sell it. Empirically, we find that the likelihood of early exercise is strongly influenced by the existence and duration of EBVs. Not only do these results reverse standard theory on American option valuation and optimal exercise strategy, but they also suggest that the ability to avoid selling at an EBV price creates an additional source of value for American options that is unrelated, and in addition to, dividend payments. This additional value may help explain why American options appear overpriced relative to European options

The new risk management: the good, the bad, and the ugly

At one time, risk management was limited to insurance and the avoidance of lawsuits and accidents. The new risk management also includes using tools developed for pricing financial options for the management of financial risks within the firm. Trading in financial markets based on these tools can insulate companies from the risk of changes in interest rates, input prices, or currency fluctuations. In this article Philip H. Dybvig and William J. Marshall introduce the new risk management and the policy choices firms should be considering.Management ; Risk

Model-Independent Pricing of Asian Options via Optimal Martingale Transport

In this article we discuss the problem of calculating optimal model-independent (robust) bounds for the price of Asian options with discrete and continuous averaging. We will give geometric characterisations of the maximising and the minimising pricing model for certain types of Asian options in discrete and continuous time. In discrete time the problem is reduced to finding the optimal martingale transport for the cost function $|x+y|$. In the continuous time case we consider the cases with one and two given marginals. We describe the maximising models in both of these cases as well as the minimising model in the one-marginal case and relate the two-marginals case to the discrete time problem with two marginals