Top PDF Model risk quantification in option pricing

Model risk quantification in option pricing

Model risk quantification in option pricing

The likelihood for each parameter combination was calculated by assuming that the errors between model prices and market prices of options followed a flat-top Gaussian distribution. Using the calculated likelihoods, a weight was assigned to each model using the Akaike Information Criterion. The models could then be used to price various options, which combined with the model weights gave rise to price distributions of these options. An interesting result was that although multiple models gave approximately the same prices for vanilla options, they were still able to give differing prices for exotic options. Several risk measures were introduced to quantify the model risk with respect to the price distribution, with the regulatory technical standards in mind. Both empirical and simulated studies showed that up-and-out barrier options were more risky from a model risk perspective than arithmetic Asian call options and digital call options, at the studied levels of moneyness and barrier levels. It was also shown that the model risk measure in relative terms of option price increased quickly with moneyness K/S for call options far out of the money. In absolute terms the risk measure was lower for options out of the money compared to options in the money. The distribution of model weights over the different model classes seemed not to be very stable over time, however the risk measures themselves did not vary as heavily.
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A Copula-based Approach to Option Pricing and Risk Assessment

A Copula-based Approach to Option Pricing and Risk Assessment

Some concerns need to be addressed, however. Since option prices and VaR (or other risk measurements) are based on the MLE estimation of the model, we inevitably face the issue of model misspecification and parameter uncertainty. In particular, the validity of using the Plackett copula or the Frank copula needs further study. In recent years, this issue has started to attract some attention. For example, Fermanian and Scaillet (2003) propose a nonparametric estimation method for copula using a kernel-based approach. Chen and Fan (2006) use nonparametric marginal distributions and a parameterized copula to mitigate the inefficiency of the two-step estimation procedure. In addition, Chen, Fan and Patton (2003) develop two goodness-of-fit tests for copula models. To mitigate the problem of parameter uncertainty, one may use MCMC algorithms with some proper prior distributions.
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Option Pricing Model with Stochastic Exercise Price

Option Pricing Model with Stochastic Exercise Price

4. SUMMARY In this paper, we establish the option-pricing model when exercise price is random variable. Supposing that risk assets pay continuous dividend regarded as the func- tion of time. Assume that jump process is count process which is more general than Poisson process, it is established that the model of the stock pricing process is jump-diffusion process with continuous dividends. European option pricing formula and their parity are obtained when the jump distribution is lognormal.

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An Option Pricing Formula for the GARCH Diffusion Model

An Option Pricing Formula for the GARCH Diffusion Model

Furthermore, Nelson (1990) showed that under the GARCH diffusion model discrete time log-returns follow a GARCH(1,1) in mean ((GARCH(1,1)-M) process of Engle and Boller- slev (1986)). Hence, the nasty problem of making inference on continuous time parameters may be reduced to the inference on the GARCH(1,1)-M model; cf., for instance, Engle and Lee (1996) and Lewis (2000). This is an important advantage over other stochastic volatility models which lack of these simple estimators. In a Monte Carlo study we investigate inference results based on such an estimation procedure. Finally, the GARCH diffusion model is the ‘mean reverting’ extension of the Hull and White (1987) model where the variance process follows an uncorrelated log-normal process without drift. The GARCH diffusion model makes a marked improvement over the Hull and White model because the mean reverting drift gives stationary variance and log-return processes (cf. Genon-Catalot, Jeantheau and Laredo (2000)) and it can include the volatility risk premium in the variance process. By contrast, for the Hull and White model the analytical option pricing approximation is available only when the drift is equal to zero 5 . Furthermore, the mean reversion of the variance allows to approximate long maturity
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Option Pricing for a General Stock Model in Discrete Time

Option Pricing for a General Stock Model in Discrete Time

As there are no arbitrage opportunities in an efficient market, the seller of an option must find a risk neutral price. This thesis examines different characterizations of this option price. In the first characterization, the seller forms a hedging portfolio of shares of the stock and units of the bond at the time of the option’s sale so as to reduce his risk of losing money. Before the option matures, the present value stock price fluctuates in discrete time and, based on those changes, the seller alters the content of the portfolio at the end of each time period. The primal linear program captures the seller’s hedging activities. We use linear programming to explore the pricing of options for both the Trinomial Asset Pricing Model and the General Asset Pricing Model, allowing us to consider the pricing of any style of option.
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Quanto option pricing in the parsimonious Heston model

Quanto option pricing in the parsimonious Heston model

The main idea is to adjust the correlation ρ in W (t) such, that the empirical correlation of generated paths (measured as in definition 3) matches the historical correlation ρ emp . The simulations can be done e.g. using a simple Euler–scheme with full truncation in case of negative variance (see [Lord et al., 2006]). The paths have to be generated under the physical measure P , as ρ emp comes from historical data 2 . However, the calibration routine for the single Heston model gives us the risk–neutral parameters only. We therefore have to suitably transform the parameters for a model representation under the physical measure.
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The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model

p and φ[a; n, p] = the probability that the sum of n random variables which equal 1 with probability p and 0 with probability 1−p will be ≥ a. These formulas imply that C is the discounted expected value of the call’s terminal payoff that would occur in a risk-neutral world.

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A closed-form GARCH option pricing model

A closed-form GARCH option pricing model

errors for the three years are 9.4%, 7.9% and 11.58% respectively. As expected, they are higher than their in-sample counterparts. In order to compute out-of-sample prices under the updated version of the BS model, we estimate an implied volatility on the previous trading day (prior to every Wednesday) by minimizing the sum of squared errors between model and market option prices and use the estimated implied volatility to price options every Wednesday. By doing this we allow the BS model to have extra flexibility, while the GARCH model is still subject to its time invariant parameters being held constant at the in-sample estimates. The results reported in Tables 2 (a,b,c) for the years 92,93 and 94 clearly show that GARCH maintains its pricing superiority over the BS model even in an out-of-sample context. The average percentage pricing errors under the updated BS are 13.42%, 16.66% and 16.07% and exceed the percentage pricing errors under the GARCH model, by anywhere between 27 percent and 52 percent. Therefore, it is not the case that the GARCH model is superior just because it has more parameters. Its superior performance stems from its ability to generate the appropriate skewness through the correlation between returns and volatility in the risk neutral distribution of the underlying asset’s returns. The superior out-of-sample performance of the GARCH model is especially encouraging in the context of the results reported in Dumas, Fleming and Whaley (1996) who find that time varying volatility models based on the implied binomial trees of Derman and Kani (1994), Dupiere (1994), and Rubinstein (1994) under perform the BS model in out-of-sample tests.
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An empirical model of volatility of returns and option pricing

An empirical model of volatility of returns and option pricing

The no-arbitrage argument assumes that the portfolio is kept globally risk free via dynamic rebalancing. The delta hedge portfolio is instantaneously risk free, but has finite risk over finite time intervals Δ t unless continuous time updating/rebalancing is accomplished to within observational error. However, an agent cannot afford to update too often (this would be quite expensive due to trading fees), and this introduces errors that in turn produce risk. This risk is recognized by traders, who do not use the risk free interest rate for r’ in (23) and (24) (where r’ determines µ ’(t) and therefore r), but use instead an expected asset return r’ that exceeds r o by a few percentage points. The reason for this choice is also theoretically clear: why bother to construct a hedge that must be dynamically balanced, very frequently updated, merely to get the same rate of return r o that a
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A Stochastic Volatility Model With Realized Measures for Option Pricing

A Stochastic Volatility Model With Realized Measures for Option Pricing

1 plots the skewness (left panel) and excess kurtosis (right panel) associated to the six models RV-LHARG, SV-LHARG, SV-ARG, SV-LARG, SV-HARG , and CGARCH under the risk-neutral measure Q. It confirms that SV-ARG and SV-HARG are not designed to replicate the negative skewness. For RV-LHARG and for all SV models with no heteroge- neous volatility structure, the level of both skewness and kurtosis is moderate and rapidly declines toward zero. The picture is significantly different for SV-LHARG and CGARCH . Among the SV class of models, SV-LHARG reaches the highest (lowest) levels of excess kurtosis (skewness). On the other hand, CGARCH is the model with the maxima level of excess kurtosis. This fact will have important consequences on the pricing performances. We expect the best performance of the CGARCH for long-term options in in-sample tests. However, the very high level of persistence can lead to systematic over-pricing of long-term options, whenever the model miss-fits the short-term level of volatility – as it may happen out-of-sample.
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A Nonparametric Option Pricing Model Using Higher Moments

A Nonparametric Option Pricing Model Using Higher Moments

(e) skewness of log-returns, and (f) kurtosis of log-returns. Each scenario will consist of 100 simulated return series data, with each series data having 1260 return periods, equivalent to five-year’s worth of data.For the risk-free interest rate, three cases are assumed: the 4-week US Treasury Bills secondary market interest rate for the low rate case, the 3-year US Treasury Bond interest rate for the middle rate case, and the 20-year Treasury Bond interest rate for the high rate case. The rates used are based on the of Governors of the Federal Reserve System (2015) for the date of 31 March 2015. These rates are, respectively, 0.05% p.a., 0.89% p.a. compounded semiannually, and 2.31% p.a. compounded semiannually. For moneyness, the term K ∗
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Martingale option pricing

Martingale option pricing

We show that our earlier generalization of the Black-Scholes partial differential equation (pde) for variable diffusion coefficients is equivalent to a Martingale in the risk neutral discounted stock price. Previously, the equivalence of Black-Scholes to a Martingale was proven for the case of the Gaussian returns model by Harrison and Kreps, but we prove it for much a much larger class of returns models where the returns diffusion coefficient depends irreducibly on both returns x and time t. That option prices blow up if fat tails in logarithmic returns x are
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Empirical study on the existence of Tuned Risk Aversion in option pricing

Empirical study on the existence of Tuned Risk Aversion in option pricing

We looked at patterns of implied distortion and found profiles similar to the implied volatility skew. We tested two alternative modeling setups where we kept volatility constant and captured the complete probability distribution ad- justment, normally provided by changing σ of the underlying, by solely applying distortion γ. Within this modeling setup we were able to fit all bid and ask prices of different strikes in a model with a single volatility for the underlying. Also implied distortion patterns under TRA showed a significantly flatter pro- file compared to sdc and therefore TRA provides a way to model a functional relationship between time to maturity and distortion, while at the same remain consistent and prevent spreads from blowing up. TRA provides a very promis- ing framework into finding a uniform model for derivative pricing that is able to price with a single volatility for the underlying and a feasible way to relate risk aversion and time to maturity.
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Risk preference based option pricing in a fractional Brownian market

Risk preference based option pricing in a fractional Brownian market

The advantages of a transition to a preference based pricing approach will turn out to be the following: The use of conditional expectation in its traditional sense will make it possible to point out the problems arising in valuation models when dealing with path-dependent processes. Moreover, advances in stochastic analysis will be used to plausibly illustrate the features of fractional Brownian motion and to make fractional option pricing comparable to the classical Brownian model. Especially, the consequences of the existence of long-range-dependence on option pricing should be clarified.
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A GARCH Option Pricing Model with Filtered Historical Simulation

A GARCH Option Pricing Model with Filtered Historical Simulation

volatilities can account for excess out-of-the-money put prices and provide an adequate pricing of the downside market risk. To assess the time stability of SPD per unit probabilities, Figure 7 shows estimates of M t,t+τ on four consecutive Wednesdays in July 2002, July 2003 and July 2004, and for time to maturity τ closest to 100 days. Similar qualitative findings are observed for other periods and time to maturities. SPD per unit probability under both FHS and Gaussian innovations tends to change slowly through time, although our FHS method provides more stable estimates. 24 This finding is consistent with aggregate risk aversion (absolute slope of SPD per unit probability) changing slowly through time at a given wealth level. For example, Rosenberg and Engle (2002) document that empirical risk aversions change with the business cycle in a counter-cyclical way, increasing (decreasing) during contractions (expansions) of the economy.
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PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL

PRICING EXOTIC OPTION UNDER STOCHASTIC VOLATILITY MODEL

Exotic options are called “customer tailored options” or “special purpose option” because they are fl exible to be tailored to the specifi c needs of investors. Strategies based on exotic options are often employed to hedge the specifi c risk exposures from the fi nancial markets. Because exotic options are more effi cient and less expensive than their standard counterparts, they are playing a signifi cant hedging role in cost effective ways. Unlike the vanilla call and put options, exotic options are either variation on the payoff patterns of plain vanilla options or they are totally different kinds of derivatives with other features. While simple vanilla call and put options are traded in the exchanges, most of the exotic options are traded in the over-the-counter markets.
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A Nonparametric Option Pricing Model Using Higher Moments

A Nonparametric Option Pricing Model Using Higher Moments

of the log-returns, (e) skewness of log-returns, and (f) kurtosis of log-returns. Each scenario will consist of 100 simulated return series data, with each series data having 1260 return periods, equivalent to five-year’s worth of data. For the risk-free interest rate, three cases are assumed: the 4-week US Trea- sury Bills secondary market interest rate for the low rate case, the 3-year US Treasury Bond interest rate for the middle rate case, and the 20-year Treasury Bond interest rate for the high rate case. The rates used are based on [16] for the date of 31 March 2015. These rates are, respectively, 0.05% p.a., 0.89% p.a. compounded semiannually, and 2.31% p.a. compounded semiannually.
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A GARCH option pricing model with filtered historical simulation

A GARCH option pricing model with filtered historical simulation

There is a general consensus that asset returns exhibit variances that change through time. GARCH models are a popular choice to model these changing variances, as is well documented in financial literature. However, the success of GARCH in modeling historical return variances only partially extends to option pricing. Duan (1995) and Heston and Nandi (2000) among others assume normal return innovations and parametric risk premiums in order to derive GARCH-type pricing models. Their models consider historical and pricing (i.e., risk neutral) return dynamics in a unified framework. Unfortunately, they also imply that, up to the risk premium, conditional volatilities of historical and pricing distributions are governed by the same model parameters. Empirical studies, for instance by Chernov and Ghysels (2000) and Christoffersen and Jacobs (2004), show that this restriction leads to rather poor pricing and hedging performances. The reason is that changing volatility in real markets makes the perfect replication argument of Black and Scholes (1973) invalid. Markets are then incomplete in the sense that perfect replication of contingent claims using only the underlying asset and a riskless bond is impossible. Consequently, an investor would not necessarily price the option as if the distribution of its return had a different drift but unchanged volatility. Of course markets become complete if a sufficient (possibly infinite) number of contingent claims are available. In this case, a well-defined pricing density exists. 1
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Option Pricing: Real and Risk-Neutral Distributions

Option Pricing: Real and Risk-Neutral Distributions

is derived in terms of the prices of the risk free asset and the stock or index on which options are written. Then any derivative is uniquely priced in terms of the risk free rate and the stock or index price. The natural extension of the single period binomial model is the widely used multiperiod binomial model developed by Cox and Ross (1976), Cox, Ross and Rubinstein (1979), and Rendleman and Bartter (1979). The stock price evolves on a multi-stage binomial tree over the life of the option so that the stock price assumes a wide range of values. Yet the market is complete because in each subperiod there are only two states. An option can be hedged or replicated on the binomial tree by adjusting the amounts held in the stock and the risk free asset at each stage of the binomial process. This type of trading is called dynamic trading and renders the market dynamically
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Strategic option pricing

Strategic option pricing

Abstract : In this paper an extension of the well-known binomial approach to option pricing is presented. The classical question is: What is the price of an option on the risky asset? The traditional answer is obtained with the help of a replicating portfolio by ruling out arbitrage. Instead a two-person game from the Nash equilibrium of which the option price can be derived is formulated. Consequently both the underlying as- set’s price at expiration and the price of the option on this asset are endogenously de- termined. The option price derived this way turns out, however, to be identical to the classical no-arbitrage option price of the binomial model if the expiration-date prices of the underlying asset and the corresponding risk-neutral probability are properly adjusted according to the Nash equilibrium data of the game.
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