The Model of Lines for Option Pricing with Jumps

Transcription

1 The Model of Lines for Option Pricing with Jups Claudio Albanese, Sebastian Jaiungal and Ditri H Rubisov January 17, 21 Departent of Matheatics, University of Toronto Abstract This article reviews a pricing odel, suitable for variance-gaa jup processes, based on the ethod of lines The ethod accuracy is studied using European style calls as a benchark Ipleentation details for continuously and discretely onitored barrier options, and Aerican and Berudan options are given 1 Introduction Lattice odels introduced in [1] played an iportant role in extending the Black-Scholes odel [5] to a wide class of exotic payoff structures The eergence of jup odels for asset prices proposed in the works of Merton [23], Bates [3], [4], Madan et al [2] and Gean et al [14], stiulated interest in adequate generalizations and replaceents of lattice odels to accoodate fat-tailed return distributions In [1], we introduced a odel based on the ethod of lines, henceforth referred 1

2 to as the odel of lines, which is siple to ipleent, flexible and adits an intuitive financial interpretation In this article, a ore coplete description of the ethodology is supplied, and the solution techniques are elaborated on In this paper, the underlying price process is postulated to be a variance-gaa pure jup process, which was introduced and studied extensively by Madan, Carr, Chang, Gean, Seneta and Yor in the series of articles, [8], [19], [2], [21] The ethod of lines was first introduced in the financial literature by Carr [6] with the intent of pricing Aerican style options in the standard Black-Scholes odel Carr recognized that this ethod prices rando aturity contracts exactly, with aturities obeying the Erlang distribution for the n-th arrival of a Poisson process The variance-gaa pure jup process can be obtained fro the geoetric Brownian otion by eans of a stochastic tie change driven by a gaa distribution, which is a generalization of the Erlang distribution where the arrival nuber n is taken to be continuous To forulate the odel of lines, Carr s equations for randoized aturity contracts, found in [6], will be recast in a for that yields exact pricing forulas for particular cases of the variance-gaa process To accoodate the general variance-gaa odel, a Richardson extrapolation schee is proposed, and is deonstrated to work well with realistic odel paraeters While lattice odels are based on recobining trees in which both calendar tie and stock price are discretized, the odel of lines postualtes only the discretization of calendar tie As such, each key date in the odel corresponds to a continuous line for stock prices In ost situations, the solution of the equations for the pricing function along each line can be represented by siple polynoials with a finite nuber of ters Following Bates [3], consider easuring tie according to the ticks of a special financial clock, which runs at a speed proportional to the nuber of transactions per unit real tie According to the financial clock, transaction volue appears to be constant Furtherore, log-returns are found to be nearly norally distributed when easured with respect to the financial clock The odel of lines developed here can be viewed as follows: each line corresponds to a fixed date in calendar tie, while the corresponding duration in financial tie - proportional to trading volue - is rando and distributed as a Poisson exponential process The tie change over n-lines follows an Erlang distribution corresponding to the n th arrival of a Poisson process By forally taking the order n of the Erlang distribution to be a continuous variable, the arrival distribution becoes the gaa distribution Gaa distributions provide the tie change 2

3 function for the variance-gaa pure jup process Consequently, the odel of lines provides a ethod for pricing derivative clais in the variance gaa odel when the odel paraeters reduce the arrival distribution to an Erlang distribution As will be deonstrate later on, the general case can often be recovered with high precision by interpolation or extrapolation ethods The differential equations that arise in the odel of lines are siilar to the Black-Scholes equations, with the following iportant distinction: tie derivatives are replaced by finite differences, while derivatives with respect to stock price reain intact The pricing functions along each line are found to satisfy a syste of inhoogeneous ordinary differential-difference equations, which adit siple analytic solutions for ost options Although the situation is siilar to Carr s equations for Aerican style options with randoized aturity [6], there are iportant differences The key difference being that calendar tie in Carr s solution ust be reinterpreted as financial tie in the odel of lines Because trading occurs in calendar tie, not in financial tie, the re-interpretation breaks risk neutrality To restore risk neutrality, the stock price ust be scaled and the option price discounted fro one tie-step to the next Under this adjustent, the odel of lines reproduces the exact - up to negligible roundoff errors - prices of European style options in which the underlying follows a variance-gaa process The solution schee for European style puts and calls can easily be odified to exactly price barrier and Berudan options contingent on inforation on the lines only Path-dependent options requiring continuous onitoring, such as Aerican options and barrier options can also be priced efficiently However, in these cases, the odel of lines produces approxiate prices, as the exercise boundaries are assued to be piecewise constant between lines The odel of lines enjoys the sae calibration efficiencies and epirical explanatory power of the variance-gaa odel Nonetheless, it is still interesting to copare it with the better known stochastic volatility odels Diffusion odels where volatility is stochastic have been considered by a nuber of authors, including Hull and White [17], Wiggins [29], Scott [27], Melino and Turnbull [22], Heston [15], [16] Stochastic volatility odels based on GARCH, such as in Duan [13], have the added advantage that the postulated process for the underlying asset is well justified by historical tie series As in jup odels, the effect of stochastic volatility can forally be interpreted as inducing a rando tie change Furtherore, these 3

4 odels are capable of explaining the skew of iplied volatilities of ediu and long dated options; however, intrinsic odel liitations are encountered with short dated options The observed steepness of iplied volatility skews cannot be justified by a odel where paths are continuous and volatility driven by a stationary process Models with state and tie dependent volatility, as in Rubinstein et al [18],[25], Deran and Kani [11], [12] and Stutzer [28], are ore effective with short dated options Unfortunately, this approach requires the introduction of a highly non-stationary process that requires frequent readjustents On the other hand, jup odels reproduce the skew of short and ediu dated options, while the predicted siles for longer dated clais are flatter than observed For a coparison aong these odels the reader is referred to the epirical studies [2], [26] The present authors believe that a odel which cobines both jups and stochastic volatility, possibly of the GARCH type, will perfor considerably better than either odels separately In a forthcoing paper, we deonstrate how such a synthesis can be ipleented by cobining the odel of lines with a two-level stochastic volatility process that gives rise to a recobining stochastic volatility tree It suffices to say that the odel of lines is not liited to pure jup odels, but rather, is an essential eleent of a ore elaborate pricing fraework The reainder of this paper is organized as follows: Section 2 is coposed of three parts: firstly, the variance gaa odel is reiviewed; secondly, the financial interpretation of the ethod of lines in ters of randoized aturity options as developed by Carr [6] and Carr and Faguet [7] is discussed; thirdly, the odel of lines is developed as a version of the ethod of lines that is appropriate for variance-gaa odels Section 3 contains the explicit solutions to the pricing probles for European options, continuously and discretely onitored barrier options, Aerican and Berudan options Our ipleentation of the Richardson extrapolation algorith is provided in Section 4 and Section 5 concludes the paper 2 The Variance Gaa Model The variance gaa odel introduced in [19], is an elegant extension of the standard geoetric Brownian otion process for stock prices In the variance-gaa odel, the log-returns on stock prices are postulated to follow a Brownian otion 4

5 Figure 1: Diagra showing several gaa process saple paths which describe financial tie Notice that these paths are quite different fro typical diffusion process paths Also, as ν tends towards zero the path becoes ore deterinistic not in calendar tie, but rather in financial tie, which flows faster or slower than real tie depending on arket activity see figure 1) Financial tie can in soe sense be thought of as following the trading volue as opposed to clock tick-tie This has the advantage that when trading volues rise, volatility increases and larger jups are ore likely to occur, as is found epirically To further odel financial tie, a secondary process is introduced, which perfors the tie change fro real-tie to financial tie This secondary process is assued to be a gaa process, which is essentially the continuous tie counterpart to the Poisson process To be specific, let S t denote the stock price process at tie t written as follows, ln St S ) = ωt X Γt;ν) θ; σ) 1) where X τ θ; σ) denotes a Brownian process with drift θ and volatility σ evaluated at tie τ; Γt; ν) denotes a gaa process with a ean rate of one and variance rate of ν evaluated at tie t; and ω is a factor necessary to aintain risk-neutrality Assuing the risk free rate r is constant in tie, ω is fixed as follows, le[s t ] = e rt S ω = r 1 1 ν ln θ 12 ) ) σ2 ν 2) The price of a European style option on that stock can be obtained by first conditioning on the financial tie given by 5

6 the rando tie change, and then integrating over all financial ties with the appropriate density Suppose the pay-off of the option at tie T is φs T ), and denote the conditioned price of the option at current calendar tie by ps t, g), and the unconditioned price by P S t ) Then, and P S t ) = dg g T t ν 1 e g/ν Γ T t ν )ν T t ν ps t, g) 3) ps t, g) = e rt t) le[φ S T ) g] = e rt t) dx 2πσ2 g xθg) 2 e 2σ 2 g φe ωt t)x S t ) = e rt t)θ 1 2 σ2 )g P BS e ωt t) S t, g, θ 12 ) ) σ2, σ 4) where P BS S, g, r, σ) denotes the Black-Scholes price of the European option aturing in tie g, S denotes the spot, r the risk-free rate and σ the volatility For particular pay-offs puts and calls) it is possible to carry out the integral appearing in 3), the result can be expressed in ters of confluent hypergeoetric functions [2] However, in order to price path-dependent options, such as Aerican or barrier options, it is necessary to solve a difficult integro-differential equation It is conceivable that an eigenvalue decoposition for the pricing kernel could yield useful results; however, analytic tractability would suffer In this paper, a solution to the pricing proble which leads to exact analytic expressions consisting solely of exponentials and polynoials is presented Our solution uses the ethods of lines fraework, which is briefly discussed below The nuerical ethod of lines can be thought of as a diensional reduction of a partial differential equation to an ordinary differential equation Consider the Black-Scholes differential equation, τ rs S 1 } 2 σ2 S 2 S 2 P BS S, τ) = rp BS S, τ) 5) τ denotes tie to aturity) If tie is discretized, the tie derivative can be approxiated by a difference τ P BS S, τ) 1 τ P BSS, τ) P BS S, )) 6) 6

7 Figure 2: The standard ethod of lines yields exact prices for an option that atures at a rando tie which is distributed according to an Erlang distribution) and the partial differential equation reduces to a sequence of one-diensional ordinary differential equations This technique was used by Carr [6] and by Meyer and Van Der Hoek [24] to price Aerican options Clearly, the results obtained using the ethod of lines yields only an approxiate answer to the pricing proble Carr proposed a very suggestive interpretation of the ethod of lines, which iplies that this ethod prices rando aturity contracts exactly see figure 2) Starting fro this intuition, we now argue that a slight odification can render the ethod of lines exact for the variance-gaa odel To explore the connection between the ethod of lines and the variance-gaa odel, first note that the representation given in equation 4) for the conditioned variance-gaa price in ters of the Black-Scholes price can be used to obtain a PDE for the conditioned price If the tie to aturity is taken to be equal to ν the unconditioned price, given by 3), is siply the Laplace-Carson transforation of the conditioned price Consequently, as Carr deonstrated in [6], the PDE reduces to a sequence of ODE s These step will now be carried out explicitly The conditioned price can be easily shown to satisfy the following PDE, g D S ) ps t, g) = 7) li ps t, g) = e rν φe ων S t ) 8) g where the operator D S is defined as, D S 1 2 σ2 S 2 d SS θ 12 ) σ2 Sd S 9) 7

8 Applying the integral kernel appearing in the right hand side of 3) to the above PDE one finds, 1 ν [ ] P S t ) li ps t, g) D S P S t ) = 1) g Consequently, the unconditioned price of a European option aturing at tie t ν is given by the solution to the ordinary differential equation, 1 [ P St ) e rν φe ων S t ) ] D S P S t ) = 11) ν The first ter of this equation is the analog of the discretized tie in the usual ethod of lines fraework The ain difference between the standard ethod of lines and the one constructed here, is that both the stock price level and the option price ust be scaled fro line to line Furtherore, the variance-rate paraeter ν now has a natural interpretation as the tie-span between lines This procedure can be applied recursively to obtain the price of any pay-off with cash flows occurring only at integer ultiples of ν, and is given succinctly by the following differential-difference equations, D S P n) S) = 1 ν ) P n) S) e rν P n1) e ων S) 12) where P n) S) denotes the price function n tie-steps of size ν prior to aturity These equations ust be suppleented by appropriate boundary conditions see section 3) and a terinal tie condition at aturity, P ) S) = φ S) 13) The syste of equations 12) and 13) for what we ter the odel of lines There are several iportant features of the odel of lines that should be elaborated on Firstly, the drift which appears in 9) is not the risk-free rate This is because in financial tie the stock drifts according to θ rather than r Secondly, in the operator D S there is no constant ter, ie the ter rp in the usual Black-Scholes equation is issing On reflection it is clear that such a ter ust be absent because the discounting occurs in real tie and not financial tie Finally, once the price on one line is known, the price on the next line is deterined fro an option with a scaled spot and discounted price The 8

9 discounting of the price is natural, and can be thought of as the spot price of the previous line The scaling of the spot itself can be understood fro the fact that, although the drift of the stock in financial tie is not equal to the risk free rate, risk neutrality ust still be enforced, hence across each line additional drifting ust be iposed Thus far, it was assued that the stock paid no dividends; if, however, dividends are paid continuously at a constant rate d, the risk-neutrality condition is effectively odified to, le[s t ] = e rd)t S 14) This iplies that ω ω d, while all other quantities reain intact The price of a continuously dividend paying stock then reduces to that of its non-dividend paying cousin with the above adjustent to ω Within the odel of lines fraework, it is also possible to incorporate dividends that are paid on the lines The ethodology is straight-forward, on the lines the spot ust be effectively reduced by the dividend pay-out, and aounts to the following alteration of the differential-difference equations 12) and 13), D S P n) S) = 1 ν ) P n) S) e rν P n1) e ων S d n )) 15) P ) S) = φ S) 16) where d n denotes the dividend paid on the n th line Extensions to situations with stochastic interest rates are also possible In this case the odel of lines would have to be written in the forward easure and the payoff paraeterized in ters of the forward price, which follows a artingale process Siilarly, extensions of Black s forulas for interest rate derivatives such as caps and swaptions can be obtained While this paper focuses on equity options, we are currently preparing articles on these and other extensions 9

10 3 Option Pricing in the Model of Lines The differential-difference equations 12), 13) are easier to solve if written in ters of the oneyness paraeter x t = lns t /K) 17) rather than the stock price It is also convenient to work with the scaled diensionless prices P n) x) defined so that P n) x) = K P n) x nων) 18) The differential equations for the scaled prices has the following siple for: D x P n) x) = erν ν P n1) x) 19) where, the differential operator D x is given by D x 1 2 σ2 d xx θd x 1 ν 2) and the final tie condition is now P ) x) = 1 φ x) 21) K Additional boundary conditions depending on the particulars of the contract, such as early exercise clauses for Aericans, ust also be included In the following sections, the above equations are used to derive closed for solutions for the price functions of a nuber of standard option contracts 31 European Options Consider a European put option struck at K and aturing at tie T The terinal boundary condition is expressed through the payoff function P ) x, K) = 1 K φx) 1 ex ) 22) 1

11 Notice that the coefficients of the syste of ODE s in 19) are constant in tie Furtherore, the boundary conditions do not depend on the tie change The general solution to our syste of equations on the lines n = 1, 2, therefore has the for n1 e dx where, the constant P n) x, K) = = a n) x, x > n1 e rnν e xnα e dx = b n) x, x < 23) α ων 24) has been introduced to lighten notations, and d ± are the positive and negative solutions of the characteristic polynoial of the differential operator D x, ie d ± = θ ± θ 2 2 ν σ2 σ 2 25) Most of the coefficients of the price function on the n th -line can be coputed using equation 19) and expressed in ters of the coefficients on the n 1) th -line The resulting recurrence relations to be solved backwards in tie are = γ a n1) 1 1) 2 σ 2 a n) 1 θ σ 2 d ) a n), 1 n 1 26) = γ b n1) 1 1) 2 σ 2 b n) 1 θ σ 2 d ) b n), 1 n 1 27) Here the discount factors, γ ± ν 1 e rναd± have been introduced and b n) n = a n) n The coefficients a n) and b n) are an exception, as the they are fixed by enforcing continuity of the pricing function and the delta ratio at the at-the-oney point x = 1 1 d d a n) b n) = e rnν e nα e nα a n) 1 b n) 1 28) The three equations 26), 27), 28)fully deterine the price of the European put option in ters of a finite cobination of eleentary functions It is quite surprising that although the underlying follows a variance-gaa process, the prices at integer ultiples of the variance rate can be expressed as siply as 23) 11

12 Figure 3: The iplied siles of the variance-gaa odel for aturities of 1, 2, 4, 8 and 16 weeks with paraeters: ν = 1 week, σ = 15%, θ = 2% and r = 5% Volatility siles for a variance-gaa process with σ = 15%, θ = 2%, and ν = 1 week are plotted in figure 3 The relative error between the iplied volatilities obtained using the exact prices in [2] and those obtained using the odel of lines were also calculated The largest relative error for the siles in figure 3 was found to be 1 3 % while the average relative error over the siles was found to be 1 5 % These negligible discrepancies are due to coputational round-off errors; there was also little difference in coputation tie between the two pricing schees Put options, the stock and a bond provide a spanning set of assets for all European style clais The price of European calls, C n) x, K), of the sae aturity, T, is obtained fro put-call parity, ie C n) x, K) = P n) x, K) Ke x e rt ) 29) Furtherore, the price F S) of ore general European style payoffs φs T ) can be reconstructed fro put option prices as 12

13 indicated in [9], where it is shown that a static replication arguent leads to the pricing forula, F n) S) = 32 Barrier Options φs)e rn t φ S)[C n, K) P n), K)] S φ K)P n) log S K, K) dk S φ K)C n) log S K, K) dk 3) As an exaple of a barrier option, consider a pay-at-expiry, down-and-out put with barrier H such that h = ln H K > 31) and rebate R = ρk 32) Since the boundary value is constant and the option price depends on the probability of breaching the barrier, but not on the tie when it happens, the boundary condition does not depend on the financial tie change, just as in the case of European options Thus, the price of the barrier instruent ust satisfy one of the following conditions, e rnν ρ ; x nα h pay-at-expiry P n) x, K, H) = ρ ; x nα h pay-at-hit 33) where α was defined in 24) Notice that in ters of the oneyness paraeter x, the boundary oves upwards if α is positive and downwards otherwise This is a consequence of equation 18) The sign of α largely depends on the size of the skewness paraeter θ, and can be explained as follows: for a sall kurtosis paraeter recall that ν was estiated in [2] to be 2) equation 2) leads to the approxiation α r θ 1 2 σ2 )ν; consequently, if θ r 1 2 σ2 then α > For typical equity options, the iplied volatility sile is negatively skewed iplying a negative θ paraeter, this in turn iplies that α is typically positive Nonetheless, it is entirely possible for α to be negative; as such, both signs of α will be discussed here If α is positive, then the boundary condition oves in towards x = and eventually crosses it; while if α is negative the boundary condition continually oves away fro x = In the forer case, the solution contains at ost three regions, the 13

14 Figure 4: The oving boundary and solution regions Figure 5: The oving boundary and solution regions for a down and out barrier option with negative α Notice that the nuber of regions constantly increases for a down and out barrier option with positive α Notice that the nuber of regions is at ost three and reduces to two when the boundary oves across the x = line region when the spot is above the strike, between the strike and the boundary, and below the boundary When the boundary collides with x =, the solution then contains only two regions see figure 4) However, with α <, the nuber of regions constantly increases see figure 5) In either case, the boundary is stationary in the original variables, even though it oves in P paraeterization The pricing schee for α > will now be presented Before the oving boundary crosses x =, the price function can 14

15 be broken up into three regions and can be written as a series, n1 e dx a n,) x, x > = P n) x, K, H) = e rnν e xnα e rnν ρ n1 = e dx a n,1) } e dx b n,1) x, x n1) x, x < x n1) 34) where the shifting barrier boundaries, x i), have been introduced, x i) i 1)α h 35) The a n,i) coefficients satisfy the recurrence relations given in 26) while the b n,i) coefficients satisfy the recurrence relations given in 27) Once again, the coefficients with = cannot be obtained fro the recurrence relations alone, and instead ust be found by iposing the continuity of the scaled price and its derivative at x =, and continuity at the boundary x = x n1), d d d e xn1) d e xn1) d a n,) a n,1) b n,1) = c n) c n) c n) 1 36) where, c n) i = e rnν e nα, i = e h e rnν R 1) n1 =1 e dxn1) a n,1) } e dxn1) b n,1) x n1)), i = 1 37) c n) = e nα a n,) 1 a n,1) 1 b n,1) 1 38) Equation 36) is the barrier analog of equation 28) for Europeans 15

16 The oving boundary crosses the strike when n h α, the solution then consists of only two regions, P n) x, K, H) = n1 e dx e rnν ρ = a n,) x, x > x n1), x x n1) 39) Of course, the a n,) coefficients still satisfy the recurrence relations given in 26), while the = coefficient is obtained by enforcing continuity at the boundary x = x n1), n1 a n,) = e dxn1) rnν ρ =1 e dxn1) a n,) x n1)) 4) This concludes the discussion of the α > scenario If however, α <, the nuber of regions on the n th -line is equal to n 2 as can be seen fro figure 5 In this case the solution is given by, n1 e dx = e rnν e xnα e rnν ρ P n) x, K, H) = e rnν ρ a n,) x, x > n1 = n1 = n1 = e dx a n,2) e dx a n,1) e dx a n,n1) } e dx b n,1) x, x 1) < x } e dx b n,2) x, x 2) < x x 1) } e dx b n,n1) x, x n1) < x x n) 41) e rnν ρ, x x n1) As usual, the a n,i) coefficients satisfy the recurrence relation 26) while the b n,i) coefficients satisfy the recurrence relation 27) Enforcing continuity in the pricing function P and its derivative at each region switch except for the last, where only continuity is forced, allows the = coefficients to be calculated The linear syste of equations can be copactly 16

17 represented as follows: d d d y 1) d y 1) y 1) d y 1) y 1) d y 1) y 2) d y 2) y 1) d y 1) y 2) d y 2) y 2) d y 2) y 2) d y 2) y n1) d y n1) y n1) d y n1) y n1) d y n1) y n) y n1) d y n1) y n) a n,) a n,1) b n,1) a n,n1) b n,n1) = c n) c n) c n) 1 c 1 n) c n) n c n n) c n) n1 42) where, y i) ± = e xi) d ± 43) and the right hand side for the sooth pasting conditions of the price function are given by, e rnν e nα, i = c n) i = ρ 1)e rnν e hnα) n1 =1 n1 =1 y i) a n,i1) y n1) a n,n1) ) a n,i) n1 =1 y n1) y 1) a n,2) y 1) b n,2) y i) b n,i1) ) a n,1) )} b n,1) h), i = 1 )} b n,i) x i)), 2 i n } b n,n1) x n1)), i = n 1 44) 17

18 while the right hand side for the sooth pasting conditions of the delta are as follows, e nα a n,) 1 a n,1) 1 b n,1) 1, i = c in) = e hnα) n1 =1 y 1) d h) a n,2) y 1) d h) b n,2) ) a n,1) )} b n,1) h) 1, i = 1 45) n1 =1 y i) d x i)) a n,i1) y i) d x i)) b n,i1) ) a n,i) )} b n,i) x i)) 1, 2 i n The evaluation algorith for down-and-out puts adits an easy generalization to the case of barriers that are piecewise constant between the lines In that case, the boundary value h for each tie-step n is replaced with h n The case when the barrier condition is discretely onitored on the lines can also be accoodated In this situation, the price on each line is first evaluated assuing the absence of the barrier on that line This eans that the last region in the pricing forula 34), 39), and 41), are replaced by a series siilar to that in the pricing forula for European options 23) However, before using this solution to evaluate the price on the next line, the pricing function needs to be truncated at the barrier value on the current line As a result, the pricing function ay not be continuous at the boundary of this last region, as should be expected for a discretely onitored barrier 33 Aerican Options To price Aerican options the sae ethods that apply in the barrier case can be utilized As entioned in the introduction, the odel of lines applied to the Aerican option pricing proble does not yield exact prices; rather, the prices obtained in this section are those of a piecewise constant barrier option, where the holder of the option is allowed to adjust the level of barrier over every tie-interval With this in ind, the ain difference between the barrier option discussed in the previous section and the Aerican option priced here, is that the oving boundary ust now be located using the optionality clause, in addition to assuring that the sooth pasting requireents are satisfied An additional sooth pasting requireent is obtained 18

19 Figure 6: The boundary for an Aerican option with positive α The nuber of solution regions at first increases, but it soon begins to decrease, and eventually reduces Figure 7: The boundary for an Aerican option with negative α The nuber of regions in this scenario is constantly increasing to two regions by axiizing the options value; the standard arguents iply that axiizing the option s price is equivalent to forcing the hedge ratio at the exercise boundary to be equal to negative one As in the case of European and barrier options, the boundary conditions are the sae as those for the Black-Scholes partial differential equation Intuitively, the optial exercise boundary is expected to decrease as tie to aturity lengthens However, since the price of the option is obtained fro the scaled prices by shifting its arguent by α see equation 18)), two situations corresponding to the sign of α can occur If α > the relevant boundary for the scaled prices P x) decreases initially and eventually starts to increase and ultiately crosses the strike level This situation is illustrated in figure 6 On the other hand, when α, the boundary will constantly decrease, forcing the nuber of regions to increase as tie to aturity increases as depicted in figure 7 In either case the price on the n th -line can be written in a general for, assuing that the nuber of regions is b n 2, as 19

20 follows, P n) x) = n1 e dx = e rnν e xnα a n,) x, x > n1 = e rn1)ν e xnα e dx a n,1) n1 = e dx a n,2) } e dx b n,1) x, x n,1) < x < } e dx b n,2) x, x n,2) < x < x n,1) 46) e rnbn1)ν e xnα n1 = e dx a n,bn) } e dx b n,bn) x, x n,bn) < x < x n,bn1) 1 e xnα, x < x n,bn) With this ansatz, the position of the optial exercise boundary on the n th -line is given by S n) ex = S e xn,bn) nα The coefficients a n,i), satisfy the recurrence relation 26), while the b n,i) coefficients satisfy the recurrence relation 27) Of course, the recurrence relations do not deterine the = coefficients, rather they are obtained by enforcing continuity in the price and the delta at the end of every interval Also the optial exercise boundary ust be solved for In the case of negative α, the nuber of regions is increasing, and as is clear fro figure 7, x n,i) = x n1,i) for i = 1,, b n1 and the nuber of regions between the strike level and the exercise boundary is b n = n Consequently, all the regions for the new line with the exception of the new optial exercise point) are known fro the previous line The syste of 2n 1) equations which deterines the new exercise point x n,bn) in addition to the = coefficients, is very siilar to the barrier case, only one ore equation, which enforces continuity in the delta, ust be added, and the syste is now 2

21 non-linear due to the new exercise point, d d d y 1) d y 1) y 1) d y 1) y 1) d y 1) y 2) d y 2) y 1) d y 1) y 2) d y 2) y 2) d y 2) y 2) d y 2) y n1) d y n1) y n1) d y n1) y n1) d y n1) y n) d y n) y n1) d y n1) y n) d y n) a n,) a n,1) b n,1) a n,n) b n,n) = c n) c n) c n) 1 c 1 n) c n) n c n n) 47) where, y i) ± = e xn,i) d ± 48) and the right hand side for the sooth pasting conditions of the price function are given by, c n) i = e rnν e nα, i = e rni)ν e rni1)ν n1 =1 y i) a n,i1) y i) b n,i1) ) a n,i) )} b n,i) x i)), 1 i n 49) 21

22 while the right hand side for the sooth pasting conditions of the delta are as follows, e nα a n,) 1 a n,1) 1 b n,1) 1, i = c in) = n d x i)) ) a n,i1) a n,i) =1 y i) y i) d x i)) b n,i1) )} b n,i) x i)) 1, 1 i n 5) also, a n,n1) = b n,n1) This is the coplete solution, and although the syste is non-linear, each guess of x n,bn) renders it linear In the case of positive α, the boundary should first be assued to lie below the old boundary x n,bn) < x n1,bn1) ) If no solution of syste 47) exists, then the assuption is false The botto ost region ust then be deleted fro the solution assuption and the syste ust be solved once again If no solution still exists, delete yet another region and so on until a solution is found Once the nuber of regions reduces to two, there will always be an optial solution 4 Extrapolation Techniques Although the tie-step dictates the ν paraeter in the VG odel, it is possible to use the odel of lines to obtain good approxiations to the variance-gaa prices when ν is different fro t Just as Carr [6] deonstrated that Richardson extrapolation to ν = reproduces the Black-Scholes value in the usual ethod of lines, we propose to use an extrapolation schee to obtain the prices of options for ν t and in particular for ν > t Figure 8 shows the exact iplied Black- Scholes volatilities σ BS ν) for one-onth European options with various strike levels as a function of the paraeter ν Quadratic polynoials in ln ν were used to fit the first three points at ν = t = 1, 2 and 4 weeks, and extrapolated to the fourth point at ν = 8 weeks, ie σ BS ν) = A A 1 log ν A 2 log ν) 2 51) The fitted curves in figure 8 visually deonstrate the quality of the approxiation The absolute error easured in ters of the iplied Black-Scholes volatility for at-the-oney instruents was found to be negligible The largest errors appeared 22

23 Figure 8: Extrapolation of the iplied volatility of one onth European options for a VG odel with ν = 8 weeks using a fit to volatilities obtained with ν = t = 1, 2 and 4 weeks The odel paraeters were σ = 15%, θ = 2% and r = 5% and the spot was taken to be $1 in the out-of-the oney option struck at 8% of the spot, for which the absolute error in iplied volatility was 9% Our conclusion is that extrapolation allows for the pricing of variance-gaa odels with realistic paraeter choices using the odel of lines, albeit the prices thus obtained are approxiate Just as in the case of European options, it is possible to use extrapolation to obtain the prices when ν t for exotic options In figure 9 the boundary of a Berudan option which can be exercised every 8 weeks is plotted as a function of ν = t = 1, 2, 4 and 8 weeks The black dots in figure 9 for the predicted 8 week boundary obtained by extrapolation fro the first three points The extrapolation is based on a fit to a quadratic polynoial of ln ν to the first three boundaries The errors obtained by this extrapolation ethod are inial with a axiu absolute error of $18 for the longest aturity option The at-the-oney prices fitted to a linear function of ν are displayed in figure 1 The errors are once again negligible with the longest aturity option being underpriced by $2 23

24 Figure 9: The boundary of a Berudan option which can be exercised every 8 weeks plotted for several values of ν = t The black dots show the boundary with ν = 8 weeks obtained by extrapolation using the first three boundaries The odel paraeters were σ = 15%, θ = 2% and r = 5% and the spot was taken to be $1 Figure 1: The prices of the at-the-oney options whose boundaries are shown in figure 9 The lines indicate a fit to the first three prices extrapolated to the fourth 24

25 5 Conclusions We proposed a new pricing odel, coined the odel of lines, which is appropriate for both path-dependent and pathindependent options when the underlying asset follows a variance gaa process The odel of lines for jup processes is siilar to the ethod of lines for Aerican options in the Black-Scholes odel However, contrary to the nuerical approxiation schee, the odel of lines produced exact prices for a large class of options, the would be errors were reinterpreted as the effect of the jup coponent in the process for the underlying The new ethodology was applied to several pricing probles for European, Aerican and barrier options, and the ethod was found to be both nuerically efficient and siple to ipleent 6 Acknowledgeents CA and SJ would like to thank the Natural Science and Engineering Research Council of Canada for financial support We thank Peter Carr for valuable coents References [1] C Albanese, S Jaiungal, and D H Rubisov Juping in line to appear in Risk Magazine, February, 21 [2] G Bakshi, C Cao, and Z Chen Epirical perforance of alternative option pricing odels Journal of Finance, 52:23 249, 1997 [3] DS Bates The crash of 87: Was it expected? the evidence fro options arkets Journal of Finance, 46:19 144, 1991 [4] DS Bates Jups and stochastic volatility: exchange rate processes iplicit in deutscheark options Review of Financial Studies, 9:69 18,

26 [5] F Black and M Scholes The pricing of options and corporate liabilities The Journal of Political Econoy, 81: , 1973 [6] P Carr Randoization of the aerican put Review of Financial Studies, 11: , 1998 [7] P Carr and D Faguet Fast accurate valuation of aerican options Working Paper, 1994 [8] P Carr, H Gean, DB Madan, and M Yor The fine structure of asset returns: An epirical investigation Working Paper, 2 [9] P Carr and D Madan Towards a theory of volatility trading Morgan Stanley working paper [1] J Cox and A Ross The valuation of options for alternative stochastic processes Journal of Financial Econoics, 3: , 1976 [11] E Deran and I Kani Riding on a sile Risk, 7:32 39, 1994 [12] E Deran and I Kani Stochastic iplied trees: Arbitrage pricing with stochastic ter and strike structure of volatility International Journal of Theoretical and Applied Finance, 3:7 22, 1998 [13] JC Duan Cracking the sile Risk, 9:55 59, 1996 [14] H Gean and T Ane Stochastic subordination Risk, Septeber, 1996 [15] S Heston A closed-for solution for options with stochastic volatility with applications to bond and currency options The Review of Financial Studies, 6: , 1993 [16] S Heston Invisible paraeters in options prices Journal of Finance, 48: , 1993 [17] J Hull and A White The pricing of options on assets with stochastic volatility Journal of Finance, 42:281 3,

27 [18] JC Jackwerth and M Rubinstein Recovering probability distributions fro option prices Journal of Finance, pages , 1996 [19] D Madan and E Seneta The variance-gaa odel for share arket returns Journal of Business, 63: , 199 [2] DB Madan, P Carr, and EC Chang The variance gaa process and option pricing European Financial Review, 2:79 15, 1998 [21] DB Madan and F Milne Option pricing with vg artingale coponents Matheatical Finance, 1:39 55, 1991 [22] A Melino and S Turnbull Pricing foreign currency options with stochastic volatility Journal of Econoetrics, 45: , 199 [23] R Merton Option pricing when underlying stock returns are discontinuous Journal of Financial Econoics, 3: , 1976 [24] GH Meyer and J Van Der Hoek The evaluation of aerican options with the ethod of lines Advances In Futures and Options Research, 9: , 1997 [25] M Rubinstein Iplied binoial trees Journal of Finance, 49:455 48, 1978 [26] M Rubinstein Non-paraetric tests of alternative option pricing odels Journal of Finance, 4:455 48, 1985 [27] L Scott Option pricing when the variance changes randoly, theory, estiation and an application Journal of Financial and Quantitative Analysis, 22: , 1987 [28] M Stutzer A siple non-paraeteric approach to derivative security valuation Journal of Finance, 51: , 1996 [29] JB Wiggins Option values under stochastic volatility: theory and epirical estiates Journal of Financial Econoics, 19: ,