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- Litterature Review

# Litterature Review

**Bachelier: A pioneer**

The long story of option pricing began in 1900 when Louis Bachelier developed the earliest known analytical valuation for standard options in his PhD thesis dissertation. Finding that stock price changes looked like a random walk process, Bachelier made the quite revolutionary assumption that stock prices follow an arithmetic brownian motion. While he was on the right track, his formula had clear drawbacks. Indeed, as emphasized by Merton (1973) and Smith (1976), it does not take into account any discounting and allows for negative stock prices and for option prices superior to the prices of the underlying securities, which lacks validity. However, due to the precociousness of his work, it took more than sixty years of research to propose any alternative option pricing models.

It is Sprenkle (1961) who first extended the work of Bachelier by switching to a geometric brownian motion (GBM) process for the stock price process. This adaptation did not receive much attention despite ruling out negative prices by assuming the log normality of returns. The reasons often put forth are the considerable number of parameters to estimate and the lack of information about how to do so. Three years later, Boness (1964) improved Sprenkle’s model by considering the time value of money. Samuelson (1965) quickly made the consideration that an option may have a different level of risk than the underlying stock, concluding that the use of the expected rate of return as a discount rate made by Boness was wrong.

**The Black-Scholes model and its limitations**

In 1973, Black and Scholes developed the first completely equilibrium option pricing model, which was going to become the greatest breakthrough in the pricing of stock options. A consequence that proved more influential was the realisation that by holding stock and risk-less debt, the option position could be hedged completely in a dynamic nature. The Black-Scholes model gave a serious impulse to the worldwide trading of options because it provided a widely suitable option pricing method.

Due to its success, more focus has been put upon the Black-Scholes model and its underlying assumptions.

Even though earlier empirical research had already started rejecting this simple hypothesis, the B&S model relies on the assumption that stock returns have a log-Normal distribution. Indeed, Mandelbrot (1963) and Fama (1965) found that stock returns exhibit excess kurtosis, suggesting that returns have a fat-tailed distribution.

Mandelbrot (1963) also documented what is commonly known as the “volatility clustering” and is perfectly explained by the author himself: “ (...) large changes tend to be followed by large changes and small changes tend to be followed by small changes (...).” This stylized fact clearly violates the independence of returns assumed in the B&S model.

Furthermore, Fama (1965) and Black (1976) noticed that large downward movements are generally more frequent than their upward counterparts. Statistically, this means that the stock return distribution is negatively skewed. Black (1976) further observed the existence of a negative correlation between stock prices and volatility, known as the “leverage effect”.

Additional studies such as Blatterg and Gonedes (1974) and MacBeth and Merville (1974) have also excluded the geometric brownian motion hypothesis by showing that stock returns are heteroskedastic.

The B&S formula has been even more questioned after the Black Monday at Wall Street since the probability of such an extreme event under the normal distribution is extremely low (less than 1.4 × 10^{−107}) . With investors fearing a reappearance as a result of this market crash, they began putting more value on deep OTM put options. Subsequently, those options were traded at a relatively higher price than ATM puts, ATM calls and OTM calls. Their volatilities were therefore higher resulting in a “volatility smile” that contradicts the B&S model under which the term structure of implied volatility is flat.

**Alternative models: Relaxing B&S assumptions**

Researchers have spared no effort to improve the BS model and reduce its biases by relaxing some of its assumptions. Subsequently, a variety of extensions have been advanced and are presented here below. While the GBM is a fairly acceptable approximation for price changes, it can be substantially improved. There are typically two ways the GBM assumption can be relaxed: by developing an alternative stochastic stock price process or by relaxing the assumption of constant volatility.

**Jump diffusion model**

Merton (1976), and Cox and Ross (1976) were the first to allow the stock to jump “up” or “down”, engendering a discontinuity in the stock price process. Using adequate parameters, Merton’s model was able to generate a lots of volatility smiles and skews. Particularly, choosing a negative mean for the jump process can readily capture short-term skews. Simultaneously, the model retains the undesirable independence property. Numerous studies on jump diffusion models have then been undertaken since that time.

**Stochastic volatility**

The heteroskedasticity in stock returns makes it very tempting to express the volatility as a stochastic process. Based on a body of work on stochastic volatility models (Scott (1987), Hull and White (1987), Stein and Stein (1991)), Heston (1993) advanced the first stochastic volatility model with a generalized solution. His model permits the capturing of essential features of stock markets, namely the leverage effect, the volatility clustering and the tail behavior of stock returns. However, it cannot yield realistic implied volatilities for short maturities.

**Stochastic volatility with jumps in stock price process**

From the two previous points, it seems clear that the way forward lies in associating a jump diffusion model with stochastic volatility (Bates (1996), Scott (1997)). By benefiting from the advantages of both the jumps in the stock price process and the stochastic volatility, those models seem more capable to match the market facts.

**Stochastic Volatility with jumps in both stock price and volatility processes**

Although models nesting both stochastic volatility and jumps have shown some success, Bates (2000) and Pan (2002) indicate that they are still incapable of fully capturing the empirical features of stock index options prices. Actually, the significant volatility smile of index option prices cannot be described solely based on the degree of volatility of volatility.

Several researchers have proposed to incorporate further jumps in the volatility process to amend the inaccurate descriptions of significant volatility smile (Duffie, Pan and Singleton (2000), and Eraker, Johannes and Polson (2003)). They provide evidence that this modeling is able to justify brutal and lasting market changes with upward movements in volatility. Nonetheless, those models cannot account for volatility spikes since they rely on the restrictive assumption of positive jumps in the volatility process. Research that followed then tried to mimic volatility spikes. Zerilli proposed Normal jumps in an innovative log-variance process that follows an Ornstein-Uhlenbeck process. Her results revealed that mimicking volatility spikes improved the option pricing model considerably.

**Stochastic volatility and stochastic interest rate**

Although a large body of literature has already suggested that interest rates may follow a random walk, none of the previous models have relaxed the assumption of a constant interest rate. Therefore, several researchers have started to develop models that allow the short-term interest rate to fluctuate randomly, namely Vasicek (1977), Cox, Ingersoll and Ross (1985) and Hull and White (1990). Although the inclusion of stochastic interest rate seems to have a limited effect on short-maturity option prices (Knoch (1992)), it may affect the accuracy when pricing long-maturity options and interest-rate derivatives (Bakshi et al. (2000), van Haastrecht (2010), Grzelak and Oosterlee (2011) and Kim, Yoon and Yu (2013)). Most of these models allow for both stochastic volatility and stochastic interest rates.

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