Recent progresses in option pricing using ARCH processes for the underlying are summarized. The stylized facts are multiscale heteroscedasticity, fat-tailed distributions, time reversal asymmetry, and leverage. The process equations are based on a finite time increment, relative returns, fat-tailed innovations, and multiscale ARCH volatility. The European option price is the expected payoff in the physical measure P weighted by the change of measure dQ/dP, and an expansion in the process increme

The behavior of the implied volatility surface for European options was analyzed in details in [Zumbach and Fernandez, 2011] for prices computed with a new option pricing scheme based on the construction of the risk-neutral measure for realistic processes with a finite time increment. The resulting dynamics of the surface is static in the moneyness direction, and given by a volatility forecast in the time-to-maturity direction. This difference is the basis of a cross-product approximation of the

Long term investments in bonds offer known returns, but with risks corresponding to defaults of the underwriters. The excess return for a risky bond is measured by the spread between the expected yield and the risk-free rate. Similarly, the risk can be expressed in the form of a default yield, measuring the difference between the yield when no default occurs and the expected yield. For zero coupon bonds and for actual market data, the default yield is proportional to the probability of default p

Option pricing should be based on a realistic process for the underlying and on the construction of a risk-neutral measure as induced by a no-arbitrage replication strategy. This paper presents a realistic and complete, "first principles,' computation of option prices. The underlying is modeled by a long-memory ARCH process, with relative returns, fat-tailed innovations and multi-scale leverage. The process parameters are estimated on the SP500 stock index (in the physical P measure) and allows

The behavior of the implied volatility surface for European options was analyzed in details in [Zumbach and Fernandez, 2011] for prices computed with a new option pricing scheme based on the construction of the risk-neutral measure for realistic processes with a finite time increment. The resulting dynamics of the surface is static in the moneyness direction, and given by a volatility forecast in the time-to-maturity direction. This difference is the basis of a cross-product approximation of the

Long term investments in bonds offer known returns, but with risks corresponding to defaults of the underwriters. The excess return for a risky bond is measured by the spread between the expected yield and the risk-free rate. Similarly, the risk can be expressed in the form of a default yield, measuring the difference between the yield when no default occurs and the expected yield. For zero coupon bonds and for actual market data, the default yield is proportional to the probability of default p

Option pricing should be based on a realistic process for the underlying and on the construction of a risk-neutral measure as induced by a no-arbitrage replication strategy. This paper presents a realistic and complete, "first principles,' computation of option prices. The underlying is modeled by a long-memory ARCH process, with relative returns, fat-tailed innovations and multi-scale leverage. The process parameters are estimated on the SP500 stock index (in the physical P measure) and allows

A realistic ARCH process is set so as to duplicate for all practical purposes the properties of stock time series from 1 day to 1 year. The process includes heteroskedasticity with long memory, leverage, fat-tail innovations, relative return, price granularity, and holidays. Its adequacy to describe empirical data is controlled over a broad panel of statistics, including (robust L-statistics) skew, (robust) kurtosis, shape factor for the volatility distribution, and lagged correlations between c