The Financial Econometrics of Option Markets

Transcription

1 of Option Markets Professor Vance L. Martin October 8th, 2013 October 8th, / 53

2 Outline of Workshop Day 1: 1. Introduction to options 2. Basic pricing ideas 3. Econometric interpretation to pricing 4. Speci cation of price dynamics 5. The Black-Scholes model 6. Eyeballing options data 7. Estimation (a) Using returns data (b) Using options data (c) Using returns and options data October 8th, / 53

3 Outline of Workshop Day 2: 1. Testing the Black-Scholes model (a) Biasedness (b) Heteroskedasticity (c) Smiles and smirks 2. Relaxing the assumptions of Black-Scholes (a) Distribution (b) Pricing skewness and kurtosis (c) Functional forms and nonparametrics (d) Time-varying volatility (GARCH, stochastic volatility, forward looking) (e) Jumps (Poisson, Hawkes) 3. Monte Carlo pricing methods 4. Application to Exchange options and contagion October 8th, / 53

4 Testing the Black-Scholes Model Consider the empirical model based on Black-Scholes c i = c BS i (σ) + u i where c i represents the observed price of the i th option and ci BS (σ) is the price of the i th option contract based on Black-Scholes with d i de ned as c i = p i Φ(d i ) k i e r i h i Φ(d i σ p h i ) d i = log(p i /k i ) + (r i σ2 )h i σ p h i The Black-Scholes model imposes very strong restrictions on the underlying process which can be tested using various inferential methods. Three tests of the model are now investigated. 1. Biasedness 2. Heteroskedasticity 3. Smiles and Smirks October 8th, / 53

5 Testing the Black-Scholes Model Biasedness The empirical model does not contain an intercept in the conditional mean suggesting that the conditional mean of the call option price equals the Black-Scholes price E [c i j p i ] = c BS i = p i Φ(d i ) k i e rh i Φ(d i σ p h i ) as E [u i ] = 0. This result is interpreted as the call option price is unbiased. To allow for the possibility of the observed option price being biased the empirical model is extended to include an intercept c i = β 0 + p i Φ(d i ) k i e rh i Φ(d i σ p h i ) + u i where β 0 is the unknown intercept. A test of the unbiasedness property is that the intercept is zero is based on the hypothesis β 0 = 0 This restriction can be tested using a standard t-test. October 8th, / 53

6 Testing the Black-Scholes Model Biasedness Example (Testing for Biasedness) The number of iterations for the algorithm to converge is 142 yielding a log-likelihood value of log L b θ = The parameter estimates are n o bθ = bσ 2 = , bω 2 = , bβ 0 = As se(bβ 0 ) = , a test of biasedness is given by the t-statistic tstat = = This constitutes signi cant bias and a rejection of Black-Scholes. October 8th, / 53

7 Testing the Black-Scholes Model Biasedness Example (Testing for Biasedness continued) Nonetheless, the volatility estimate is similar to when β 0 = 0 as the estimate of the annualised volatility is bσ = p = or 11.91% compared to 13.05% with β 0 = 0. October 8th, / 53

8 Testing the Black-Scholes Model Biasedness The biasedness property can be extended to allow for di erent biasedness for di erent contracts. This is achieved by including a dummy variable for di erent strike prices. As there is an intercept and there are 30 strike prices the regression equation is augmented to include just the rst 29 strike prices to circumvent the dummy variable trap. Alternatively, an equivalent approach is to include all 30 strike price dummy variables and exclude the intercept. October 8th, / 53

9 Testing the Black-Scholes Model Biasedness The empirical model is now expanded as c i = β j=1 β i D j,i + p i Φ(d i ) k i e rh i Φ(d i σ p h i ) + u i where D j,i, j = 1, 2,, 29, are dummy variables de ned as a dummy variable corresponding to the highest strike price of k = 550, is not needed as the equation contains an intercept term, thereby avoiding the 1 : k = 350 D 1,i = D 2,i =. D 29,i = 0 : otherwise 1 : k = : otherwise 1 : k = : otherwise October 8th, / 53

10 Testing the Black-Scholes Model Biasedness Instead of constructing dummy variables for each strike price, dummy variables can be grouped together in terms of moneyness. A possible classi cation of moneyness is DUM_OUT j = DUM_AT j = DUM_IN j = ( ( ( p 1 : k < : otherwise 1 : 0.97 < p k < : otherwise 1 : 1.03 < p k 0 : otherwise If the model contains an intercept term then again it will only be necessary to include two of the moneyness variables. October 8th, / 53

11 Testing the Black-Scholes Model Biasedness Instead of just including dummy variables based on strike prices, dummy variables corresponding to other characteristics of option contracts can be used to augment the model. For example, dummy variables on the three types of maturities in the stock option data set would be de ned as 1 : hi = D MAY,i = 0 : otherwise 1 : hi = D JUNE,i = (1) 0 : otherwise 1 : hi = D SEPT,i = 0 : otherwise corresponding respectively to May, June and September option contracts. As before, only two of the three dummy variables need to be included if the model already contains an intercept term. October 8th, / 53

12 Testing the Black-Scholes Model Heteroskedasticity An important assumption of the empirical model is that the variance of the disturbance term ω 2, is constant across all contract types. To relax the assumption of homoskedasticity the dummy variables de ned for strikes and maturities in can be used to allow the disturbance variance to change over the sample. In the case of the maturity dummy variables, the pricing error disturbance term is speci ed as ω 2 i = exp (α 0 + α 1 D MAY,i + +α 2 D JUNE,i ) A joint test of the restrictions α 1 = α 2 = 0 is a test of homoskedasticity which can be performed using a likelihood ratio test for example. October 8th, / 53

13 Testing the Black-Scholes Model Smiles and Smirks In estimating the option price models the data consist of option contracts corresponding to 30 strike prices and 3 maturities, a total 90 di erent types of call option contracts. An important feature of the estimated model is that there is just a single estimate of the volatility parameter σ, regardless of the type of option contract. This is a restriction of the model which can be tested by reestimating the model for alternative strike prices and alternative maturities. October 8th, / 53

14 Testing the Black-Scholes Model Smiles and Smirks A test of the assumption that σ is invariant to strike prices k is to group the data for each k and reestimate σ for each group. 1. If the Black-Scholes model is consistent with the data then the estimates of σ for each group should not be statistically di erent from each other. 2. If the estimates σ are di erent across groups this is evidence against the Black-Scholes model. In practice there are two types of relationships between σ and k. 1. Volatility Smile - there is a U-shape relationship between σ and k, centered around the at-the-money options. Occurs in currency options. 2. Volatility Smirk or skew - there is an inverse relationship between σ and k, with volatility being relatively high for in-the-money options and low for out-of-the-money options. Occurs in stock options. October 8th, / 53

15 Testing the Black-Scholes Model Smiles and Smirks Example (Volatility Smirk in Stock Options) The model is estimated using option prices separately for each of the 30 groups of strike prices for the European call options written on the S&P500 stock index on April 4, The tted line is based on a nonparametric kernel regression. There is evidence of a smirk with estimates of σ ranging from 0.3 to 0.1.This result is evidence of misspeci cation IMVOL STRIKE October 8th, / 53

16 Relaxing the Black-Scholes Assumptions Evidence of volatility smiles and smirks in particular have led to the speci cation of more general models of asset prices. Some potential speci cations are: 1. Relaxing the assumption of lognormality and adopting a more general speci cation. 2. Relaxing the assumption of constant volatility. 3. Allow for additional e ects in the mean. 4. Allow for additional e ects in the variance of the pricing error (heteroskedasticity). October 8th, / 53

17 Relaxing the Black-Scholes Assumptions Mixture of Lognormals Melick and Thomas (1997) specify a mixture of lognormal distributions. The empirical option pricing model is c i = αc BS 1,i (σ 1 ) + (1 α) c BS 1,i (σ 2 ), where BS (σ j ), j = 1, 2, is the Black-Scholes price c BS j,i = p i Φ(d j,i ) k i e rh i Φ(d j,i σ j p hi ) with constant volatility, σ j, j = 1, 2 and d j,i = log(p i /k i ) + (r i σ2 j )h i σ j p hi The parameter 0 α 1, is the mixing parameter which weights the two subordinate lognormal distributions. October 8th, / 53

18 Relaxing the Black-Scholes Assumptions Mixture of Lognormals Example (Estimating the Mixture Model) The number of iterations for the algorithm to converge is 76 yielding a log-likelihood value of log L b θ = The parameter estimates are n o bθ = bσ 2 1 = , bσ 2 1 = , bα = , bω 2 = The estimates of the volatilities of the two distributions, σ 2 1 and σ2 1, suggest that the two lognormal distributions have very di erent shapes. October 8th, / 53

19 Relaxing the Black-Scholes Assumptions Mixture of Lognormals Example (Estimating the Mixture Model) The estimate of α is suggesting that these two distributions have roughly equal weights. A test of equal weights given by the t-statistic tstat = = suggests a rejection of the hypothesis α = 0.5 at conventional signi cance levels. October 8th, / 53

20 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model The class of semi-nonparametric option pricing models discussed here are based on an augmentation of the normal returns density through the inclusion of higher order terms. Jarrow and Rudd (1982) were the rst to adopt this approach, which was implemented by Corrado and Su (1997) and Capelle-Blancard, Jurczenko and Maillet (2001). Let g (z t+h ) represent the true conditional price distribution of the standardised returns z t+h = log z t+h log z t r 1 2 σ2 h σ p h October 8th, / 53

21 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model Using an Edgeworth expansion of g (z t+h ) around the normal density gives g (z t+h ) = φ (z t+h ) + (κ 4 (G ) κ 4 (Φ)) 4! (κ 3 (G ) κ 3 (Φ)) 3! d 4 φ (z t+h ) dz 4 t+h d 3 φ (z t+h ) dz 3 t+h + ε (z t+h ) where ε (z t+h ) is an approximation error arising from the exclusion of higher order terms in the expansion and κ i is the i th cumulant of the associated distribution with κ 1 = 0 and κ 2 = 1 to standardize the distribution to have zero mean and unit variance. October 8th, / 53

22 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model Using the properties of the normal distribution z "1 3 t+h 3z t+h + γ 1 where g (z t+h ) = φ (z t+h ) +γ 2 z 4 t+h 6z 2 t+h # + ε (z t+h ) γ 1 = (κ 3 (G ) κ 3 (Φ)), γ 3! 2 = (κ 4 (G ) κ 4 (Φ)) 4! are the unknown parameters which capture respectively skewness and kurtosis. The expression shows that the di erence between the true density, g (z t+h ) and the normal density φ (z t+h ), is determined by the third and fourth moments. October 8th, / 53

23 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model As from the transformation of variable technique f (p t+h jp t ) = jjj g (z t+h ) where J is the Jacobian of the transformation from z t+h to p t+h, then z "1 3 t+h 3z t+h + γ 1 f (p t+h jp t ) = jjj φ (z t+h ) +γ 2 z 4 t+h 6z 2 t+h where for simplicity the approximation error is ignored. 6 # October 8th, / 53

24 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model Using this density to price options yields the Jarrow-Rudd option pricing model ci JR = ci BS + γ 1 Q 3,i + γ 2 Q 4,i where c BS i and is the Black-Scholes price for the i th contract and Q 3,i = e rh Z k (p t+h k) jjj z 3 t+h 3z t+h φdp t+h 6 Q 4,i = e rh Z k (p t+h k) jjj z 4 t+h 6zt+h φdp t+h 24 October 8th, / 53

25 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model Potential problem: the risk neutral probability distribution in is not constrained to be positive over the support of the density. Interestingly, this problem appears to have been ignored in the literature so far. The problem arises as the returns distribution is a function of cubic and quartic polynomials, which can, in general, yield negative values. The problem of negativity can be expected to be more severe when the Jarrow-Rudd approximation does not model the true distribution accurately, causing the polynomial terms to over-adjust, especially in the tails of the estimated distribution. October 8th, / 53

26 Relaxing the Black-Scholes Assumptions Pricing the Risk from Skewness and Kurtosis: Jarrow-Rudd Model To impose non-negativity on the underlying risk neutral probability distribution the semi-nonparametric density of Gallant and Tauchen (1989) is speci ed p (z t+h ) = φ (z t+h ) "1 + λ 1 z 3 t+h 3z t+h 6 zt+h 4 6zt+h 2 + λ + 3 # where the augmenting polynomial is now squared, forcing the probabilities to be greater than or equal to zero for all values of z t+h. Using this expression for the standardised returns yields the semi-nonparametric option pricing model which is referred to as the SNP option pricing model. October 8th, / 53

27 Relaxing the Black-Scholes Assumptions Nonparametric Pricing Based on Arti cial Neural Networks The Black-Scholes option price shows that there is a nonlinear relationship between the option price ct BS, and the remaining arguments, p t, k, h, r, σ. For more general option price models it is not always possible to derive an analytic expression for the price of the option. One way to proceed is to use nonparametric methods based on for example, arti cial neural networks (ANN). An example of an ANN model of option prices is given by where c i = α 0 + α 1 (p i k i ) + α 2 h i + α 3 L i + v i (2) L i = exp [ (γ 0 + γ 1 (p i k i ) + γ 2 h i )] and v i is a disturbance term. The function L i represents the arti cial neural network where the form of the ANN is the logistic squasher. October 8th, / 53

28 Relaxing the Black-Scholes Assumptions Nonparametric Pricing Based on Arti cial Neural Networks To estimate the parameters of the model in (2) an iterative maximum likelihood estimator can be used. Equivalently, the parameters can be chosen using MLE or even more simply a standard nonlinear least squares algorithm to minimise N vi 2 i=1 where N is the number of option contracts in the data set. The option prices are given by the tted values bc i = bα 0 + bα 1 (p i k i ) + bα 2 h i + bα 3 bl i where bl i = exp [ (bγ 0 + bγ 1 (p i k i ) + bγ 2 h i )] October 8th, / 53

29 Relaxing the Black-Scholes Assumptions Nonparametric Pricing Based on Arti cial Neural Networks Having estimated the ANN it is possible to compute various hedge parameters to be used in a risk management strategy. An estimate of the options s delta is given by where bl i = bc i p = bα 1 + bα 3 bγ 1 bl i exp [ (bγ 0 + bγ 1 (p i k i ) + bγ 2 h i )] (1 + exp [ (bγ 0 + bγ 1 (p i k i ) + bγ 2 h i )]) 2 which is the logistic density function. October 8th, / 53

30 Relaxing the Black-Scholes Assumptions Time-varying Volatility GARCH (Engle and Mustafa) A common method to allow for time-varying volatility is to use a GARCH speci cation. The model of asset prices over a day ( = 1/250) is log p t+1 log p t = r 1 2 σ2 t + p σ 2 t z t+1 σ 2 t+1 = α 0 + α 1 σ 2 t zt 2 + β 1 σ 2 t z t+1 N (0, 1) The variance is time-varying due to 1. Lagged squared shocks from the mean α 1 σ 2 t z2 t. Allows for memory in the variance to be one period. 2. Lagged conditional variance β 1 σ 2 t. Allows for memory in the variance to be longer than one period. But pricing options is more di cult as there is no closed form solution for the option price. Can use Monte Carlo methods. October 8th, / 53

31 Relaxing the Black-Scholes Assumptions Time-varying Volatility GARCH (Heston and Nandi) An alternative GARCH model to Engle and Mustafa that yields an analytical formula for pricing European options. The model of asset prices over a day ( = 1/250) is log p t+1 log p t = r + γσ 2 p t + σ 2 t z t+1 σ 2 t+1 = β 0 + β 1 σ 2 t + β 2 (z t β 3 σ t ) 2 z t+1 N (0, 1) Under risk neutrality the model becomes log p t+1 log p t = r 1 2 σ2 t + p σ 2 t z t+1 σ 2 t+1 = β 0 + β 1 σ 2 t + β 2 (z t β 3 σ t) 2 z t+1 N (0, 1) where the z t+1 = z t+1 + (λ + 1/2) σ t under the risk neutral measure A quasi closed form expression is now available to price options. October 8th, / 53

32 Relaxing the Black-Scholes Assumptions Time-varying Volatility Stochastic Volatility (Heston) A common method to allow for time-varying volatility is the stochastic volatility model. The model of asset prices over a day ( = 1/250) is log p t+1 log p t = z 2,t+1 r 1 2 σ2 t + p σ 2 t z 1,t+1 σ 2 t+1 σ 2 t = κ φ σ 2 p t + β σ 2 t z 2,t+1 z1,t ρ N, 0 ρ 1 This model introduces an additional disturbance term z 2,t, in contrast to GARCH volatility models. It is for this reason why stochastic volatility models tend to be more di cult to estimate. October 8th, / 53

33 Relaxing the Black-Scholes Assumptions Time-varying Volatility Stochastic Volatility (Heston) The volatility process is mean reverting to the long-run parameter φ, with the rate of conversion controlled by κ. The strength of the stochastic volatility is controlled by β, ie the parameter that controls the volatility of volatility. The volatility also displays the CIR speci cation with p β σ 2 t. (Model speci cation used by Heston (1993)). The two stochastic processes are allowed to be correlated with parameter ρ. Sometimes referred to as the leverage e ect. Heston derives European option prices for this model, although numerical procedures are needed to evaluate the option prices. October 8th, / 53

34 Relaxing the Black-Scholes Assumptions Time-varying Volatility Forward Looking Volatility (Engle and Rosenberg) The GARCH volatility speci cation is backward looking as σ 2 t is a function of past shocks. A speci cation that is a function of future shocks is σ t+hjt = exp (β 1 + β 2 ln (p t+h /p t )) This speci cation shows that conditional volatility is stochastic, as it is a function of the future return over the life of the option, ln (p t+h /p t ). Special cases 1. β 2 > 0 : the relationship between volatility and future return is positive. 2. β 2 < 0 : there is an inverse relationship between future return and volatility (leverage e ect). 3. β 2 = 0 : constant volatility speci cation which underlies the Black-Scholes model. October 8th, / 53

35 Relaxing the Black-Scholes Assumptions Time-varying Volatility Forward Looking Volatility (Engle and Rosenberg) A more natural speci cation to adopt in the context of option pricing as option prices are based on an evaluation of the future evolution of the underlying spot price. In common with a GARCH-type volatility speci cation, an additional error term is not introduced, with all randomness deriving from randomness in the asset price itself. This contrasts with a stochastic volatility model, in which the volatility process has its own random innovations. This speci cation has computational advantages compared with the GARCH and stochastic volatility speci cations. October 8th, / 53

36 Relaxing the Black-Scholes Assumptions Jump Models Poisson Processes The aim is to allow for infrequent jumps in asset prices. The rst form of jumps is based on a Poisson process. The model of asset prices over a day ( = 1/250) is log p t+1 log p t = r 1 z t+1 N (0, 1) dn t+1 Po (λ) 2 σ2 + p σ 2 z t+1 + J N t+1 where the size of the jump is J and N t+1 is a Poisson process with jump intensity parameter λ given by P ( N t+1 = 0) = 1 λ + o ( ) P ( N t+1 = 1) = λ + o ( ) P ( N t+1 > 1) = o (dt). The larger is λ, the more likely a jump will occur. October 8th, / 53

37 Relaxing the Black-Scholes Assumptions Jump Models Poisson Processes Note that the probabilities are a function of the time interval which in general will be small. This means that: 1. The occurrence of no jump is relatively high with probability of 1 λ. 2. The occurrence of one jump is relatively low with probability of λ. 3. The occurrence of more than one jump is e ectively zero. A problem with the Poisson process is that the jumps are independent. This is especially not the case during nancial crises where there tends to be a sequence of related jumps. October 8th, / 53

38 Relaxing the Black-Scholes Assumptions Jump Models Hawkes Processes One way to relax the assumption that jumps are independent is to use a Hawkes process. The model of asset prices over a day ( = 1/250) is log p t+1 log p t = r 1 2 σ2 + p σ 2 z t+1 + J N t+1 λ t+1 λ t = α 1 (λ λ t ) + β N t+1 z t+1 N (0, 1) N t+1 Po (λ t ) The jump intensity parameter is now time-varying as given by the second equation. October 8th, / 53

39 Relaxing the Black-Scholes Assumptions Jump Models Hawkes Processes An example of a simulated time series with jumps. October 8th, / 53

40 Monte Carlo Pricing Methods A feature of the option price solutions is that as the assumptions of the Black-Scholes model are relaxed, such as log-normality, constant volatility and no jumps, the calculation of the option price quickly becomes more di cult using analytical methods, if not impossible. In the latter situation numerical methods based on Monte Carlo simulations are available. Consider the 5 simulated time paths of the asset reported in the previous lecture, but reported here again for convenience P1 P2 P3 P4 P5 K October 8th, / 53

41 Monte Carlo Pricing Methods This Figure 40 was based on simulating the model log p t+h log p t = r 1 2 σ2 + p σ 2 hz t+h z t+h N (0, 1) where the interest rate is r = 0.05, the volatility is σ = 0.2 and = 1/250 represents daily movements. For a 6-month option (h = 0.5) with a strike price of k = 4.8, the Black-Scholes call option price was computed numerically as c t = exp ( rh) 1 N N max (p t+h k, 0) = i=1 with N = 5. October 8th, / 53

42 Monte Carlo Pricing Methods The e ect of increasing the number of simulation paths N, on the price of the option are summarised in following Table??. Number of Simulation Paths (N) Option Price Analytical October 8th, / 53

43 Monte Carlo Pricing Methods For comparison the Black-Scholes call option price based on the analytical solution is c t = p t Φ(d) ke rh Φ(d σ p h) = exp ( ) = where d = log(p t /k) + (r σ2 )h σ p h = log(5.00/4.80) + ( ) p 0.5 = while Φ(d) = Φ(0.5361) = and Φ(d σ p h) = Φ( p 0.5) = Φ(0.3947) = October 8th, / 53

44 Monte Carlo Pricing Methods The results show that as the number of simulation paths increases the numerical price becomes more accurate. When the number of sample paths increases to N = 200, the numerical price is , compared to the analytical price of , an error of less than 1%. Monte Carlo methods can be made more accurate through 1. Antithetic variates (resimulate the model changing the sign of the residuals) 2. Control variates (compute the bias between an analytical solution and its simulated value). Can price options where 1. Stochastic volatility 2. Jumping behaviour 3. Nonnormalities October 8th, / 53

45 Application to Exchange Options and Contagion Introduction Asset mispricing can be particularly signi cant during periods of nancial crises and contagion. Co-dependence structures across nancial markets change dramatically during periods of nancial turbulence that extends beyond the usual changes in volatilities and correlations. Additional crisis transmission channel operating through higher order co-moments of asset returns (see Fry, Martin and Tang (JBES, 2010)), proved to be signi cant during many nancial crises. Important implications for market participants engaged in the hedging of nancial risks and for nancial regulators seeking to manage risks across the nancial institutions October 8th, / 53

46 Application to Exchange Options and Contagion Descriptive Statistics Noncrisis Subprime Great European Recession Debt US Mean Std dev Skewness France Mean Std dev Skewness Greece Mean Std dev Skewness October 8th, / 53

47 Application to Exchange Options and Contagion Descriptive Statistics (cont d) Statistics on coskewness Noncrisis Subprime Great European Recession Debt US (i = 1) France Greece France (i = 1) Greece US Greece (i = 1) France US October 8th, / 53

48 Application to Exchange Options and Contagion Objectives Investigate the implications of changes in the co-moments of the returns distribution that arise from contagion, in the context of pricing and hedging exchange options. Examine the potential size of mispricing from ignoring higher order co-moments of asset returns by comparing the simulated prices that allow for coskewness with the Black-Scholes price during periods of crises and contagion. Potential hedging losses from using the deltas that are derived based on the Black-Scholes price. October 8th, / 53

49 Application to Exchange Options and Contagion Approach Use exchange options to capture co-dependence. Noncrisis period is given by Black-Scholes where returns y i,t = log p i,t log p i,t 1, are bivariate normal Crisis period is represented by returns being distributed as bivariate generalized normal (GEN) f (r 1, r 2 ) = exp θ 1 r θ 2 r θ 3 r 1 r 2 + θ 4 r θ 5 r 2 1 r 2 +θ 6 r 1 r θ 7 r θ 8 r θ 9 r 4 2 η, where η is the normalising constant such that R R f (r 1, r 2 ) dr 1 r 2 = 1. To compute the option price assuming that returns are based on the generalized normal distribution, option prices are computed using Monte Carlo methods with N = simulation paths Ct GEN = exp [ rph] 1 N N max p1,t+h i i=1 p i 2,t+h, 0 October 8th, / 53

50 Application to Exchange Options and Contagion Generalized Joint Lognormal Price Distributions Associated with the bivariate GEN distribution of returns, there is the bivariate generalised lognormal distribution. Using the transformation of variable technique g (p 1,t, p 2,t ) = jjj f (ln p 1,t ln p 1,t 1, ln p 2,t ln p 2,t 1 ) where the Jacobian is jjj = 1 p 1,t p 2,t October 8th, / 53

51 Application to Exchange Options and Contagion Generalized Joint Lognormal Price Distributions The bivariate distribution of prices conditional on lagged prices is g (p 1,t, p 2,t j p 1,t 1, p 2,t 1 ) h = exp θ 1 (log p 1,t log p 1,t 1 ) 2 + θ 2 (log p 2,t log p 2,t 1 ) 2 +θ 3 (log p 1,t log p 1,t 1 ) (log p 2,t log p 2,t 1 ) +θ 4 (log p 1,t log p 1,t 1 ) 3 +θ 5 (log p 1,t log p 1,t 1 ) 2 (log p 2,t log p 2,t 1 ) +θ 6 (log p 1,t log p 1,t 1 ) (log p 2,t log p 2,t 1 ) 2 +θ 7 (log p 2,t log p 2,t 1 ) 3 + θ 8 (log p 1,t log p 1,t 1 ) 4 i +θ 9 (log p 2,t log p 2,t 1 ) 4 1 η p 1,t p 2,t October 8th, / 53

52 Application to Exchange Options and Contagion Generalized Joint Lognormal Price Distributions October 8th, / 53

53 Application to Exchange Options and Contagion E ect of Crisis and Contagion on Asset Pricing The following experiments are conducted. 1. E ect of mispricing of options when incorrectly using Black-Scholes when the true distribution is bivariate GEN. 2. E ect on delta hedging when assume the wrong distribution. 3. E ect on replicating risk free portfolios from incorrect choice of distribution. October 8th, / 53