In early 1970s one of the major breakthroughs in the pricing of stock options has been introduced known as the Black and Scholes formula. The model had a huge impact on the fields of pricing and hedging European style put and call options, due to its capability to compute their fair value by no arbitrage arguments. Specifically, this formula has been crucial to the development of financial engineering over the last decades and has been acknowledged as one of the most important works in academia. The fact that this formula is still widely used by practitioners emphasizes the magnitude of Black and Shcoles work, a work that granted them the Nobel prize in 1997.
Nowadays prices of options are mainly measured in terms of "implied volatility" which is derived from the parameter σ when the formula is used in reverse, based on observed option prices in the market. At this point the main drawback of the model arises. One of the assumptions of the model specifies that the implied volatilities stay constant, nevertheless it has been observed on the markets that this doesn't hold as the implied volatilities tend to change as the strike price changes (giving a "volatility smile") and also as the maturity changes (giving a "volatility term structure"). Moreover the assumption of the normally distributed returns has also been dropped due to the fact that market returns have shown to be leptokurtic.
Due to these inefficiencies of this model, several other extensions have been proposed. In the modern era of finance local volatility (Derman et al. (1994)), stochastic volatility (Heston (1993), Hull and White (1987)), stochastic volatility jump diffusion (Bates 1996), exponential Levy (Madan et al (1991)) are some of the models that have been introduced in an attempt to produce a better fit to the data.
Unlike other models that can fit better to the volatility smile curve (e.g. local volatility models), these particular models produce pricing biases but assume realistic assumptions for the underlying. From a hedging viewpoint, when using Black-Scholes model, traders must continuously change the volatility assumption in order to match the market prices something that drives the hedge change as well. Nevertheless, stochastic volatility models offer some order in this problem. Bakshi et al. (1997) suggested that the choice among models should be made after comparing their hedging performance. Thus they considered European options, developed an option pricing model (SVSI-J model) and compared it with other stochastic volatility, stochastic interest rates and jumps models. They concluded that models with stochastic volatility and random jumps provide good alternatives to the Black and Scholes formula due to the fact that they generate the best hedging performance.
The purpose of this work is to examine a similar problem for exotic options. Driven by the results of Bakshi et al we consider only models with stochastic volatility and jumps. From the family of stochastic volatility models we examine the Heston model and also consider the Bates model which is a stochastic volatility model with jumps (a combination of Heston's and Merton's models). ). Our main problem is that these particular exotic options are traded on the OTC market and thus we have no observations of their prices, making it difficult to measure the performance of this pricing model. We will apply the models by calibrating (find for the model parameters that replicate the observed prices) them to an implied volatility surface of market prices in order to obtain a time series of the model's parameters and see how well the model replicates the data. Then we consider on each day an exotic option and hedge it during its life time on the basis of greeks. Hedging will be done dynamically as static hedging (Derman et al, 1994) is difficult to apply in practise. At expiry or knock out of the option we observe the cumulative hedging error for that option. In this way, we collect all the cumulative hedging errors for options that started on different days. Finally, we compare the hedging errors for different options and hedging strategies.
Chapter 2
Literature Review
The first person to describe and model the stock price by a generalized Wiener process with a constant expected drift rate and volatility was Brachelier, who presented a stochastic analysis of the stock and option markets in his PhD thesis The theory of speculation" back in 1900. Despite the fact that he achieved to value options by no arbitrage arguments, the disadvantage of the model was the probability to generate negative stock prices. Samuelson (1965) proposed for the stock price the exponential of Bachelier's model:
,
where μ is the expected return of the stock, σ is the volatility of the stock price and Wt is a Wiener process. The model of the stock price behaviour is known as geometric Brownian motion and its stochastic differential equation is:
The formula above simply states that stock returns consist of a riskless drift μdt and normally distributed shocks σdWt. This model is known as Black & Scholes(BS) model (1973) because Black & Scholes developed methodology for pricing options by no arbitrage arguments.
Specifically Black & Scholes showed that the fair value of a European call option with strike price K and maturity T in a non dividend paying stock is given by the formula:
Where
And
The function N(x) denotes the cumulative probability distribution function for a standard normal distribution, So is the stock price at time zero, r is the risk-free interest rate, K is the strike price, T is the time to ïï¡ï´ïµï²ï©ï´ï¹ï€ ï¯ï¦ï€ ï´ï¨ï¥ï€ ï¯ï°ï´ï©ï¯ï®ï€ ï¡ï®ï¤ σ is the stock price volatility. The only parameter in the formula that is not directly observable from the market is σ. Nevertheless the formula is hardly ever used to price a European option due to its extreme liquidity and thus it is already priced by the market. It is important to indicate that unlike historical volatilities, implied volatilities illustrate the market's opinion about the volatility of a stock. In other words one can say that historical volatilities are a backward looking measure of market volatility contrary to the implied volatilities that are a forward looking measure. Today traders measure option prices in implied volatilities which are obtained by the reversion of the Black-Scholes formula.
One of the drawbacks of the Black and Scholes model is located in one of its assumptions. Specifically the model assumes that the underlying asset's price process is continuous and that the volatility is constant for different strikes and maturities. However when using real market data implied volatility seems to vary and when plotting it we obtain a rather convex curve known as the "volatility smile". Moreover the assumption of normally distributed stock returns contradicts with the real stock returns observed from the market that appear to have a leptokurtic (heavy tails and high peaks) and skewed distribution.
Hence viewing at these drawbacks we realize that there is room for improvement, for this several extensions have been proposed by researchers. Merton (1976) proposed a jump-diffusion model which introduced jumps in order to include the "crashes" that frequently take place in the markets. He considered that the stock prices follow exponential Lévy processes:
where is a Lévy process. This approach has enabled to model the sudden moves that are not contained in the Black and Scholes lognormal model.
Another approach considers volatility as a random variable generated by a stochastic process. This approach has been introduced to fill the assumption of the Black and Scholes model which requires volatility to a known function of time and the asset price.
where Vt is a stochastic process. This approach was introduced by Heston (1993) and is currently widely used for pricing options that are very sensitive to volatility such barrier options.
Moreover another approach considers the stock price to follow a diffusion process:
where the function σ determines the volatility at time t and price St.
In this work we examine the stochastic volatility model as proposed by Heston (1993) and the stochastic volatility plus jumps model proposed by Bates (1996) which, as previously indicated, are more suitable for pricing exotic options.
Models
The Heston model
Heston (1993) introduced a new technique to price in closed-form the price of a European call option when volatility varies stochastically. His model allows for arbitrary correlation between the spot asset return and the volatility. Heston introduced a stochastic volatility model where the stock price is generated by the following process:
where the volatility process is modeled by the Ornstein-Uhlenbeck process:
where and are correlated Wiener processes:
The parameter μ indicates the expected stock returns, θ stands for the volatility of volatility, ξ is the speed of mean reversion and η is the long run variance. It should be noted that the variance in the stochastic process of volatility is always positive and if 2κθ > σ2 then it cannot reach zero.
The Heston model covers the non-normal distribution of stock returns and takes into account the leverage effect (parameter Ï is typically negative as shown by empirical observations Cont (2001)) and that is why the the Wiener stochastic processes should be correlated.
Heston derived a closed form solution for the price of European plain vanilla calls on an asset under stochastic volatility. By applying the Itô lemma and Black-Scholes arbitrage arguments we obtain the Garman's partial differential equation (Heston 1993):
where λ is the market price of volatility risk.
