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Conditional Asian options are recent market innovations, which offer cheaper and long-dated alternatives to regular Asian options. In contrast with payoffs from regular Asian options which are based on average asset prices, the payoffs from conditional Asian options are determined only by average prices above certain threshold. Due to the limited inclusion of prices, conditional Asian options further reduce the volatility in the payoffs than their regular counterparts and have been promoted in the market as viable hedging and risk management instruments for equity-linked life insurance products. There has been no previous academic literature on this subject and practitioners have only been known to price these products by simulations. We propose the first analytical approach to computing prices and deltas of conditional Asian options in comparison with regular Asian options. In the numerical examples, we put to the test some cost-benefit claims by practitioners. As a by-product, the work also presents some distributional properties of the occupation time and the time-integral of geometric Brownian motion during the occupation time.

We consider the problem of finding a consistent upper price bound for exotic options whose payoff depends on the stock price at two different predetermined time points (e.g. Asian option), given a finite number of observed call prices for these maturities. A model-free approach is used, only taking into account that the (discounted) stock price process is a martingale under the no-arbitrage condition. In case the payoff is directionally convex we obtain the worst case marginal pricing measures. The speed of convergence of the upper price bound is determined when the number of observed stock prices increases. We illustrate our findings with some numerical computations.

We derive a series expansion by Hermite polynomials for the price of an arithmetic Asian option. This requires the computation of moments and correlators of the underlying asset price which for a polynomial jump–diffusion process are given analytically; hence, no numerical simulation is required to evaluate the series. This allows to derive analytical expressions for the option Greeks. The weight function defining the Hermite polynomials is a Gaussian density with scale $b$. We find that the rate of convergence of the series depends on $b$, for which we prove a lower bound to guarantee convergence. Numerical examples show that the series expansion is accurate but unstable for initial values of the underlying process far from zero, mainly due to rounding errors.

This paper proposes a new hybrid algorithm to price the arithmetic Asian options under the geometric Brownian motion (GBM). The proposed algorithm is based on the control variate technique, such that the control variable is a combination of the barrier arithmetic Asian option and the geometric Asian option, which each one will be estimated by the importance sampling and the control variate techniques, respectively. Besides, we drive a conditional expectation for the estimator that it can reduce variance of simulations. The merits of the proposed algorithm for pricing arithmetic Asian options are illustrated by several examples.

This paper studies the effect of fractional volatility on path-dependent options, which are highly sensitive to the volatility structure of a targeted underlying asset process. To this end, we propose an approximation formula for average and barrier options when volatility follows a fractional Brownian motion. Furthermore, using the analytical formula, we investigate the impact of the Hurst index on option prices. Overall, our important finding is that when the maturity is short and speed of mean-reversion is slow, the impact of the Hurst index strongly influences the option prices and that is non-negligible. This is an important lesson for practitioners who uses standard Brownian motion.

This is a project on the use of the Monte Carlo scheme to price exotic options to be completed using Python. Both values of the Asian option and the Lookback option are calculated by using the Euler– Maruyama scheme for initially simulating the underlying stock price. In both cases, the following set of data is used to simulate underlying stock prices. Today’s stock price S0 = 100, strike price E = 100, Time to expiry (T − t) = 1 year, volatility sigma = 0.2, and constant risk-free interest rate r = 0.05.