Quantitative Finance

In this paper, Malliavin calculus is applied to arrive at exact formulas for the difference between the volatility swap strike and the zero vanna implied volatility for volatilities driven by fractional noise. To the best of our knowledge, our estimate is the first to derive the rigorous relationship between the zero vanna implied volatility and the volatility swap strike. In particular, we will see that the zero vanna implied volatility is a better approximation for the volatility swap strike than the ATMI.

Let X be a subset of L 1 that contains the space of simple random variables L and ρ:X→(−∞,∞] a dilatation monotone functional with the Fatou property. In this note, we show that ρ extends uniquely to a σ( L 1 ,L) lower semicontinuous and dilatation monotone functional ρ ¯ ¯ ¯ : L 1 →(−∞,∞] . Moreover, ρ ¯ ¯ ¯ preserves monotonicity, (quasi)convexity, and cash-additivity of ρ . Our findings complement recent extension results for quasiconvex law-invariant functionals proved in [17,20]. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 that retains the robust representations obtained in [4,6].

We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation ρ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each ρ<0 and m> 1 1− ρ 2 , the m -th moment of the stock price is infinite at each positive time.

An important but rarely-addressed option pricing question is how to choose appropriate strikes for implied volatility inputs when pricing more exotic multi-asset derivatives. By means of Malliavin Calculus we construct an optimal log-linear strikevconvention for exchange options under stochastic volatility models. This novel approach allows us to minimize the difference between the corresponding Margrabe computed price and the true option price. We show that this optimal convention does not depend on the specific stochastic volatility model chosen. Numerical examples are given which provide strong support to the new methodology.

We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our approach allows to remove the restrictive assumption of No Arbitrage of the Second Kind considered in [Bouchard, B., Deng, S. and Tan, X., Super-replication with proportional transaction cost under model uncertainty, Math. Fin., 29(3):837-860, 2019] and to show the duality under the more natural condition of No Strict Arbitrage. In addition, we extend the results to models with portfolio constraints.

Stochastic bridges are commonly used to impute missing data with a lower sampling rate to generate data with a higher sampling rate, while preserving key properties of the dynamics involved in an unbiased way. While the generation of Brownian bridges and Ornstein-Uhlenbeck bridges is well understood, unbiased generation of such stochastic bridges subject to a given extremum has been less explored in the literature. After a review of known results, we compare two algorithms for generating Brownian bridges constrained to a given extremum, one of which generalises to other diffusions. We further apply this to generate unbiased Ornstein-Uhlenbeck bridges and unconstrained processes, both constrained to a given extremum, along with more tractable numerical approximations of these algorithms. Finally, we consider the case of drift, and applications to geometric Brownian motions.

An open market is a subset of an entire equity market composed of a certain fixed number of top capitalization stocks. Though the number of stocks in the open market is fixed, the constituents of the market change over time as each company's rank by its market capitalization fluctuates. When one is allowed to invest also in the money market, the open market resembles the entire 'closed' equity market in the sense that the equivalence of market viability (lack of arbitrage) and the existence of numeraire portfolio (portfolio which cannot be outperformed) holds. When access to the money market is prohibited, some topics such as Capital Asset Pricing Model (CAPM), construction of functionally generated portfolios, and the concept of the universal portfolio are presented in the open market setting.

SREC markets are a relatively novel market-based system to incentivize the production of energy from solar means. A regulator imposes a floor on the amount of energy each regulated firm must generate from solar power in a given period and provides them with certificates for each generated MWh. Firms offset these certificates against the floor and pay a penalty for any lacking certificates. Certificates are tradable assets, allowing firms to purchase/sell them freely. In this work, we formulate a stochastic control problem for generating and trading in SREC markets from a regulated firm's perspective. We account for generation and trading costs, the impact both have on SREC prices, provide a characterization of the optimal strategy, and develop a numerical algorithm to solve this control problem. Through numerical experiments, we explore how a firm who acts optimally behaves under various conditions. We find that an optimal firm's generation and trading behaviour can be separated into various regimes, based on the marginal benefit of obtaining an additional SREC, and validate our theoretical characterization of the optimal strategy. We also conduct parameter sensitivity experiments and conduct comparisons of the optimal strategy to other candidate strategies.

We introduce a general framework for continuous-time betting markets, in which a bookmaker can dynamically control the prices of bets on outcomes of random events. In turn, the prices set by the bookmaker affect the rate or intensity of bets placed by gamblers. The bookmaker seeks a price process that maximizes his expected (utility of) terminal wealth. We obtain explicit solutions or characterizations to the bookmaker's optimal bookmaking problem in various interesting models.

This paper studies the infinite horizon optimal consumption with a path-dependent reference under the exponential utility. The performance is measured by the difference between the non-negative consumption rate and a fraction of the historical consumption maximum. The consumption running maximum process is chosen as an auxiliary state process that renders the value function two dimensional. The Hamilton-Jacobi-Bellman (HJB) equation can be heuristically expressed in a piecewise manner across different regions to take into account all constraints. By employing the dual transform and smooth-fit principle, some thresholds of the wealth variable are derived such that the classical solution to the HJB equation and the feedback optimal investment and consumption strategies can be obtained in the closed form in each region. The complete proof of the verification theorem is provided and numerical examples are presented to illustrate some financial implications.