In the current markets, options with different strikes or maturities are usually priced with different implied volatilities. This stylized fact, which is.
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- What is a foreign exchange option?
- Derivatives - Volatility Surfaces
- Chapter 6 All about Volatility | The Derivatives Academy
Bachelier's work was the starting point of a major study by Kolmogorov in While Bachelier was on the right track, his formula had clear drawbacks. It did not take into account any discounting and allowed 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 60 years of research to propose any alternative option pricing models.
It is Sprenkle in 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.
In , Samuelson 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. In , 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 realization 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 Black-Scholes model relies on the assumption that stock returns have a log-Normal distribution. Indeed, Mandelbrot and Fama found that stock returns exhibit excess kurtosis, suggesting that returns have a fat-tailed distribution. This stylized fact clearly violates the independence of returns assumed in the Black-Scholes model. Furthermore, Fama and Black noticed that large downward movements are generally more frequent than their upward counterparts.
Statistically, this means that the stock return distribution is negatively skewed. Additional studies such as Blattberg and Gonedes , MacBeth and Merville have also excluded the GBM hypothesis by showing that stock returns are heteroskedastic. In other words, the variance of aggregate stock returns changes over time.
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With investors fearing a reappearance as a result of this market crash, they began putting more value on deep OTM put options. 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 been undertaken since that time.
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 , Hull and White , Stein and Stein , Heston 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. Bates and Scott have associated a jump diffusion model with stochastic volatility. 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. Although models nesting both stochastic volatility and jumps have shown some success, Bates and Pan 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 inaccurate descriptions of significant volatility smile.
Duffie, Pan and Singleton models volatility as an affine process that can jump up violently and can justify brutal and lasting market changes with upward movements in volatility. Nonetheless, their model cannot subsequently jump down as observed in the data. I will only name Professor Zerilli as she was my teacher back in and it is the only one I remember to be honest :.
In , 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. We have already spoken about the Black-Scholes model but we will further derive the Black-Scholes equation in this section. A simple explanation of its meaning is that we expect it to have zero growth. Our option price is expected to grow at same rate as bank account and growth of each cancels out in the given process.
This is what it means to be a martingale. We do not expect change over time so we have zero expected growth. This translates to the discounted price having a zero drift term.
What is a foreign exchange option?
This is the Black-Scholes equation. It is a partial differential equation PDE describing the evolution of the option price as a function of the current stock price and the current time. The equation does not change if we vary the payoff function of the derivative.
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- Foreign Exchange (FX) Options!
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However, the associated boundary conditions, which are required to solve the equation do vary! The above implies that two stocks with the same volatility but different drifts will have the same option prices. The pricing of any derivative must be done in the risk-neutral measure in order to avoir arbitrage. Under this measure, we have seen that the drift changed and was independent of the drift of the stock.
Financially, this reflects the fact that the hedging strategy ensures that the underlying drift of the stock is balanced against the drift of the option. The drifts are balanced since drift reflect the risk premium demanded by investors to account for uncertainty and that uncertainty has been hedged away. I am sharing some of it with you here below. Exotic equity derivatives usually require a more sophisticated model than the Black-Scholes model. The most popular alternative model is a local volatility model LocVol , which is the only complete consistent volatility model.
Complete: it allows hedging based only on the underlying. Consistent: it does not contain a contradiction. LocVol models try to stay close to the Black-Scholes model by introducing more flexibility into the volatility. LocVol models offer a way of capturing the implied skew without introducing additional sources of randomness; the only source of which is the underlying asset's price that is modeled as a random variable.
In LocVol models, the volatility is a deterministic function of the asset's level.
In the Black-Scholes model, the asset's price is modeled as a log-normal random variable, which means that the asset's log-returns are normally distributed. However, the fact that we have a skew is the market telling us that the asset's log-returns have an implied distribution that is not Normal.
Derivatives - Volatility Surfaces
LocVol is still a one-factor model and it also allows for risk-neutral dynamics, which means that, like Black-Scholes, the model is still preference free from the financial point. The LocVol model is the simplest one to account for skew and offers a consistent structure for pricing options. Well, as we said, the presence of skew is the market telling us that the asset's log-returns are not normally distributed. In fact, the market is implying some distribution. If we are given a set of vanilla options prices for a fixed maturity across strikes, can we find a distribution that corresponds to these prices?
In other words, can we find a distribution for the asset price so that if we used such distribution to price vanilla options on this asset, it would give the same options prices as the ones seen in the market?
Chapter 6 All about Volatility | The Derivatives Academy
YES , theoretically, there is a way to find the distribution LocVol model which corresponds exactly to all vanilla prices taken from the skew. In fact, LocVol extends beyond skew and can also capture term structure. It can therefore theoretically supply us with a model that gives the exact same prices for vanillas taken from a whole implied volatility surface.