Kind code of ref document : B. Ref document number : PI Country of ref document : BR. Country of ref document : US. A system for calculating a market value of an exotic option comprises an input means for receiving market and option contract input data ; a means for calculating a theoretical value of an exotic option from the input data; a means for calculating a market supplement adjustment to the theoretical value as a function of the expected stopping time of the exotic option; a means for applying the market supplement adjustment to the theoretical value to produce the market value; and an output means for outputting the calculated market value.
The system may also calculate bid and offer prices from the market value. A method of obtaining the market value of an exotic option and a method of obtaining bid and offer prices of an exotic option are also disclosed. The present invention relates to a method and system for pricing financial derivatives more specifically exotic options. Options are derivative securities whose values are a function of an underlying asset. The price of an underlying asset for immediate purchase is called the spot price.
A vanilla option on an underlying asset gives the buyer the right, but not the obligation, to buy Call or sell Put the underlying asset at the strike price. Where options are traded the price-maker prepares a bid price and an offer price.
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The bid price is the price at which the trader is willing to purchase the option and the offer price is the price at which the trader is willing to sell the option. The difference between the bid and offer prices is referred to as the bid-offer spread. In the early s Black and Scholes, and Merton, independently developed an option pricing model that is still in use today.
The BSM model, as it is commonly known, provides unique closed form solutions for the price of European vanilla options. BSM found that by constructing and dynamically maintaining an option replication portfolio consisting of assets whose prices are known, they could obtain a precise option price by exploiting the no-arbitrage condition.
The BSM model is limited in that it only values the convexity of the option delta with respect to the underlying asset price. Other crucial convexities in the real world are not priced by BSM models, such as vega and delta convexities to implied volatility.
While attempts have been made to derive a model which endogenously values all key convexities, price-makers prefer the pragmatic approach of adjusting the BSM implied volatility to make the model work in practice. These adjustments are called smile and skew and are defined by vega neutral butterflies and risk reversals respectively. A vega neutral butterfly is a trading strategy in which a strangle is purchased and a zero-delta straddle is sold, both with the same maturity date, such that the vega of the strategy starts at zero. A strangle is a trading strategy requiring the simultaneous purchase or sale of a Put option and a Call option, with identical face values and maturity dates but different strike prices, such that the delta of the strategy is equal to zero.
A zero-delta straddle is a trading strategy requiring the simultaneous purchase or sale of a Put option and a Call option, with identical face values, maturity dates and strike prices, such that the delta of the strategy is equal to zero. A risk reversal is a trading strategy in which a Call Put option is purchased and a Put Call option is sold, where both have identical deltas, maturity date and face value.
The BSM methodology has been applied to exotic as well as vanilla payoffs, to obtain the theoretical value of exotic options. For example, American binary options are amongst the most heavily traded exotic foreign exchange FX options. This is because in addition to being a popular product in their own right, they are also a crucial component of the popular reverse and regular barrier options.
In contrast to European vanilla options, American binary options terminate automatically if a touch level trades and they have discontinuous payoffs.
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A OT option obliges the writer to pay the buyer a fixed amount if the touch level trades in the market. The liability crystallises on the day the touch level trades, and is paid on the delivery date of the option. A DNT option obliges the writer to pay the buyer a fixed amount if the touch levels do not trade in the market. The touch levels are above and below the current spot exchange rate when the option is written and liability is crystallised at expiration and is paid on the delivery date of the option. Option risks are described by a set of partial derivatives commonly referred to as "the Greeks".
Option Greeks include: Delta: the amount that an option price will change given a small change in the price of the underlying asset. In otherwords it is the partial derivative of the option price which respect to the spot asset price; and - Vega: the amount that an option price will change given a small change in volatility. In otherwords it is the partial derivative of the option price with respect to volatility. Just as the BSM methodology must be modified to price vega and delta convexities of implied volatility for European vanilla options, the theoretical valuation of exotic options must also be adjusted to reflect the benefits or costs of these additional convexities of implied volatility.
Some of the approaches used in the FX market to-date to value exotic options are as follows: analytical methods - analogously to their European vanilla counterparts some exotic FX option price-makers take the BSM theoretical model as an accurate value of the American binary option' s delta convexity to the underlying exchange rate. They then value the market supplement to the theoretical value by calculating the value of vega convexity to implied volatility and delta convexity to implied volatility.
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The market supplement can be positive, negative or zero depending on spacial and temporal factors; recombinant trees - binomial and trinomial trees in one or two dimensions are constructed to approximate numerically the price of the American binary option for a sample of time and space; finite difference and finite element methods - in principle similar to trees but now forming a mesh of possible points in space and time.
These methods are common when parameterising implied volatility as local volatilities; Monte Carlo Simulation - simulations of the underlying exchange rate process are repeated manifold and a value of the American binary option is obtained for each exchange rate path.
These values are averaged and discounted. This method is common for stochastic volatility models and universal volatility models. The analytical method has been widely discredited, even though it has considerable intuitive appeal, because no one has been able to value a crucial risk correctly. International Patent Application No. The second problem is that it's model is dependent upon arbitrary constants. The present invention extends the analytical method of pricing derivatives to produce a model for determining market values and bid and offer prices of exotic options with increased accuracy and efficiency.
In accordance with the present invention there is provided a method of obtaining the market value of an exotic option, comprising the steps of: providing market and option contract input data; calculating a theoretical value of the exotic option from the input data; calculating a market supplement adjustment to the theoretical value as a function of the expected stopping time of the exotic option; and applying the market supplement adjustment to the theoretical value to produce the market value. Typically the market value is used to calculate bid-offer prices. Preferably a bid-offer spread is calculated from the input data.
Preferably the bid-offer spread is also a function of the expected stopping time of the exotic option. Preferably the bid and offer prices are calculated as a function of the market value and the bid- offer spread. Typically an asymmetric slippage adjustment is calculated. Preferably the asymmetric slippage adjustment is calculated from the input data and a function of the expected stopping time of the exotic option. Preferably the bid and offer prices of the exotic option are calculated as a function of the market value, the bid- offer spread and the asymmetric slippage. In accordance with the present invention there is provided a method of obtaining bid and offer prices of an exotic option, comprising the steps of: providing market and option contract input data; calculating a theoretical value of the exotic option from the input data; calculating a market supplement adjustment to the theoretical value that incorporates the expected stopping time of the exotic option; calculating the bid-offer spread from the input data and a function of the expected stopping time of the exotic option; and calculating bid and offer prices of the exotic option as a function of the theoretical value, market supplement adjustment, and bid-offer spread.
Preferably adjusted bid-offer prices are calculated from an asymmetric slippage adjustment and the previously calculated bid-offer spread. Preferably the theoretical value is obtained by applying the no-arbitrage methods of Black-Scholes and Merton to exotic payoffs. Preferably in calculation of the theoretical value, whenever the theoretical value of an option is dependent on the solution of an infinite sum, a finite number of elements are summed to ensure at least a five digit accuracy.
Preferably the market supplement adjustment is a function of the input data only. Preferably the market supplement adjustment is calculated from a Convexity to Implied Volatility Adjustment and a Market Weight Adjustment. Preferably the vega neutral butterfly is identified using a term to maturity equal to the expected stopping time of the exotic option and a minimum delta. Preferably the minimum delta of the relevant vega neutral butterfly is chosen to match the delta of the touch level s at the expected stopping time.
Preferably when there are two asymmetric touch levels, the minimum absolute delta is selected for the vega neutral butterfly. Zeta fly is the different between the market value and the theoretical value of the relevant vega neutral butterfly. Preferably the risk reversal is identified using a term to maturity equal to the expected stopping time of the exotic option and the minimum delta.
Preferably the minimum delta of the equivalent risk reversal is chosen to match the delta of the touch level s at the expected stopping time.
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Preferably when there are two asymmetric touch levels, the minimum absolute delta is selected for the risk reversal. Zeta RR is the different between the market value and the theoretical value of the relevant risk reversal. Preferably the market weight adjustment is calculated from the expected stopping time of the exotic option and the nominal duration of the exotic option. Preferably the market supplement adjustment is calculated from a vega convexity value and a delta convexity value.
Preferably a mid-market value is calculated from the theoretical value and the value of the market supplement adjustment. Preferably the bid-offer spread is calculated such that it is independent of arbitrary constants and dependent only on the input data.
Preferably the Static Spread Adjustment includes a contribution from vega. Preferably the Static Spread Adjustment includes a contribution from the expected life of the option. Preferably the Dynamic Spread Adjustment includes a contribution from the expected life of the option. Preferably the bid-offer spread is supplemented by an asymmetric slippage component which has static and dynamic components.
Preferably bid and offer prices are calculated from the mid-market value and the supplemented bid-offer spread. Preferably the methods described above are computer implemented. According to another aspect of the present invention there is provided a system for calculating a market value of an exotic option comprising: input means for receiving market and option contract input data; means for calculating a theoretical value of an exotic option from the input data; means for calculating a market supplement adjustment to the theoretical value as a function of the expected stopping time of the exotic option; means for applying the market supplement adjustment to the theoretical value to produce the market value; and output means for outputting the calculated market value.
According to a further aspect of the present invention there is provided a system for obtaining bid and offer prices of an exotic option comprising: input means for receiving market and option contract input data; means for calculating a theoretical value of an exotic option from the input data; means for calculating a market supplement adjustment to the theoretical value that incorporates the expected stopping time of the exotic option; means for calculating a bid-offer spread from the input data as a function of the expected stopping time of the exotic option; means for calculating bid and offer prices of the exotic option as a function of the theoretical value, market supplement adjustment and bid-offer spread; and output means for outputting the calculated bid and offer prices.
According to another aspect of the present invention there is provided a computer program for controlling a computer to perform any one of the above mentioned methods. According to a further aspect of the present invention there is provided a computer program comprising instructions to operate a computer as one of the systems defined above. According to a further aspect of the present invention there is provided a computer readable storage medium comprising a computer program as defined above.
The market value of exotic options is almost always different to the theoretical value, and often by a substantial amount. Exotic option theoretical values are obtained by applying the no-arbitrage methods of Black- Scholes and Merton to exotic payoffs.
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However, it is common knowledge in the market that the BSM specification of uncertainty is a very limited approximation of reality. As a result, models have been developed to price other factors which are important to the market. At the core of the present invention is a unique weighting scheme for valuing factors not priced by BSM, which is a function of the option and the market only.