Some Numerical Methods For Determining The Implied Volatility In Option Pricing

We often come into contact with all kinds of financial products in our daily life,such as stock,foreign currency,fund and so on.In the beginning of this dissertation,we are going to introduce what are the financial derivatives.In short,a financial derivative can be defined as a financial instrument whose value depends on the values of other,more basic,underlying instruments.There are many kinds of derivatives,some of which are very popular,such as forward contract,futures contract and option.If we assume the underlying assets as stock,bond,or merchandise,there will be corresponding financial derivatives—the stock futures(option),the bond futures(option),the currency futures(option)to control the risk of these underlying variables.In the last several decades,derivatives have been becoming increasingly important in financial market.Futures and options are actively traded on many exchanges throughout the world.Many different types of forward contracts,swaps,options,and other derivatives are used by financial institutions,fund managers,and corporate treasurers in the over-the-counter market.Derivatives are added to bond issues,used in executive compensation plans,embedded in capital investment opportunities,used to transfer risks in mortgages from the original lenders to investors,and so on.Next,we will give a brief introduction to several specific derivatives.A forward contract is an agreement to buy or sell an asset at a certain future time for a certain price.It can be contrasted with a spot contract,which is an agreement to buy or sell an asset almost immediately.A forward contract is traded in the over-the-counter market,usually between two financial institutions or a financial institution and one of its clients.There will be risk for both sides of the contract.If the spot price is lower than the forward price,the market is called positive market.If the spot price is higher than the forward price,the market will be called negative market.The party agreeing to buy the underlying asset in the future assumes a long position,while the party agreeing to sell the asset in the future assumes a short position.The price agreed upon is called the delivery price or strike price,which is equal to the forward price at the time the contract is entered into.Just like the forward contract,a futures contract is also an agreement to buy or sell an asset at a certain future time for a certain price.However,futures contracts must be traded in the futures exchanges,which have made some standardized requirements in order to make sure the agreement will be carried out smoothly.What's more,the two sides of a futures contract may not know each other.In general,the payoff from a long position in a forward contract or a futures contract on one unit of an asset is S_T-K,where K is the delivery price,and S_Tis the spot price of the underlying asset at the maturity T.At the maturity T,the holder of the contract has to buy an asset whose price is S_Tfor K.Similarly,for the seller of the contract,the payoff in a forward contract or a futures contract on one unit of an asset is K-S_T.An option gives the holder the right to buy or sell the underlying asset by a certain date T for a certain price K.What's more,options can be traded both on exchanges and the over-the-counter market,which is different from the forward contract.There are two kinds of options—call option and put option.A call option gives the holder the right to buy the underlying asset by a certain date T for a certain price K.While a put option gives the holder the right to sell the underlying asset by a certain date T for a certain price K.European options and American options are two common kinds options.For European options,they can be exercised only on the maturity time T.However,for American options,they can be exercised at any time up to the expiration date.Therefore,American options give more choices to their holders,and the prices of American options are higher than European options with the same exercise time and the strike price.In financial market,most options are American options.It should be emphasized that an option gives the holder the right to do something.Of course the holder can choose not to exercise the right.But for the holders of forward con-tracts or futures contracts,they are obligated to buy or sell the underlying asset according to the contracts.What's more,it costs nothing to enter into a forward or futures contract,but there is a cost to have an option.In this dissertation,we denote the price of call option and put option by c and p,separately.From the introduction above,we can know that European option is more easier to be analyzed than American option since it can be exercised only on the maturity time.But,sometimes American option has same properties as European option.In this paper,we just take European option,especially European call option into consideration.How to price the option was the main problem at the early stage.Except for European option,there are other kinds of options,such as Asian option,Lookback option,Rainbow option and so on.Mathematicians and financial workers have developed several models,such as binomial tree model and stochastic process model,to solve this problem.As one of the most successful application of mathematics in economics,modern op-tion pricing theory plays an important role in economic study.Option is an useful and indispensable financial instrument.And there are exchanges for options trading special-ly.However,we still could not find an easy method to price an option,even for a basic European option.In 1973,Fischer Black and Myron Scholes derived the famous Black-Scholes option formula,extended by Robert Merton,which provided an elegant answer to the option pricing problem by identifying a relation between the value of the stock and its option.This discovery caused great repercussions in mathematics and finance.Moreover,in 1997,Myron Scholes and Robert Merton won the Nobel Prize in economics for their excellent work in finance.While the formula has been widely applied in option pricing problem,it is not an easy task to use this formula since the assumptions of this formula are so strong.We need to know several factors such as time,price of underlying instrumen-t,strike price,and volatility.Especially,the volatility is not directly observable,which raise some concerns about the practical suitability of the formula.But,as an easy and basic instrument,Black-Scholes formula is still used by many financial workers.In recent years,with the big progress in option pricing problem,more and more mathematicians shift their attention and energy to estimate the implied volatility in the Black-Scholes formula.Due to the structure of Black-Scholes formula,the implied volatil-ity cannot be found in closed form accurately but only through numerical approximation methods.There are many articles that supply numerical methods for estimating the im-plied volatility.In general,the investigations have been conducted in two directions.One is prag-matic,which aims to provide approximations that can easily calculated,while the other is theoretical,which explores the mathematical properties of the implied volatility.These methods both have advantages and disadvantages,and they have different applied situa-tion.In the Black-Scholes formula,Black and Scholes assumed the implied volatility is a constant.However,financial workers found that it is not reasonable to fix the volatility as a constant in financial market,which should be a variable relevant to time.Some math-ematicians use the asymptotic method to calculate the volatility aiming to find a good approximation formula for volatility.But the asymptotic method can have acceptable results only for at-the-money case.There will be very large relative error in other cas-es.Then,some researchers try to solve this problem using historic volatility method and matching option price method.But historic volatility method is inaccurate and matching option price method is unstable.Some mathematicians apply the Tikhonov regulariza-tion method and the maximum entropy method to finding the volatility in 1990s.These methods can have good approximation results under certain conditions.However,these methods also have some shortcomings.For instance,the Tikhonov regulation method will destroy the casual structure of the problem and make the discretized equation becoming a full matrix equation instead of a lower triangular matrix.And it is not an easy job to implement these numerical methods.In this dissertation,we are going to show other two numerical methods—a local regularization method and a numerical method for determin-ing the coefficient of a parabolic partial differential equation,with numerical examples to verify their accuracy and stability.In this dissertation,we show several numerical methods for determining the volatili-ty in option pricing problem.Firstly,we give a brief introduction to financial derivatives,focusing on European option.Then,we analyze two pricing models for European op-tion——binomial tree model and stochastic process model.Next,we derive the famous equation——Black-Scholes equation using several methods,such as dynamic replica-tion method,?-hedging principle,and risk neutral valuation method.After that,we use Black-Scholes formula,the numerical solution of PDE,and binomial tree model to deter-mine the price of the given European option with comparisons of the numerical results.In the last part,there is a lengthy analysis of how to find the volatility from the given data.In this dissertation,we use historic volatility method,matching option price method,closed form approximation,a local regularization method and numerical method for determining the coefficient of a parabolic PDE to finding the volatility?with several numerical ex-amples to verify the accuracy and stability.In the end,we conclude this dissertation with analysis of these methods and propose several modified schemes for the above methods.