A Moment Matching Approach to the Valuation of a Volume Weighted Average Price Option

Transcription

1 A Moment Matching Approach to the Valuation of a Volume Weighte Average Price Option Antony William Stace Department of Mathematics, University of Queenslan, Brisbane, Queenslan 472, Australia aws@maths.uq.eu.au Receive (December 13th 24) Revise (Day Month Year) In this paper we evelop a metho to fin the price of an option whose payoff epens on a volume weighte average price (VWAP). It assumes that the stock follows a geometric Brownian motion an that the rate of traes evolves as a mean reverting process. The price is obtaine by assuming that the VWAP at the final time has a lognormal istribution. The parameters of the approximating log normal istribution are obtaine by matching the first two moments of the volume weighte average price with a log normal process. A price is then easily obtaine when the market price of risk is a constant. The metho is fast an can easily be extene to price exotics such as lookbacks, barriers, an igitals which have a VWAP component. We concentrate on the price for calls, prices for puts can be obtaine in an analogous manner. Keywors: Option Pricing; Moment Matching; Volume Weighte Average Price. 1. Introuction A volume weighte average price (VWAP) is an average which gives more weight to perios of high traing than to perios of low traing in calculation of the average. Broker s aily performance is frequently measure against a VWAP an it is become increasingly popular for institutional investors to place buy an sell orers at a VWAP. This paper aresses the problem of fining the price of an option where the payoff is The maximum of the ifference between the final price of a stock an the VWAP an zero, i.e. max(s(t ) V W AP (T ), ) or The maximum of the ifference between the VWAP an a fixe strike an zero, i.e. max(v W AP (T ) K, ). This problem is certainly set in an incomplete market since there is no unerlying with which to hege the volume risk, an hence there is not a unique price. Any price which is obtaine will inclue a market price of volume risk which must be etermine from the market. 1

2 2 Antony William Stace The next section presents the efinition of a VWAP together with several options of interest. Following this, the moment matching is one, then some numeric examples are given of the price, finally conclusions an further work is presente. 2. Definitions an Notation 2.1. The VWAP price Denote the price of a stock at time t as S(t) an the number of traes of S(t) per unit time as U(t). Thus the turnover over the interval t 1, t 2 is t 2 t 1 S(s)U(s)s an the number of shares trae is t 2 t 1 U(s)s. Assume also that U(t) > for all t. The moel use for of S(t) an U(t) will be specifie later. The VWAP over the time interval [, t] is efine as I(t) = Y (t), with I() = S() (2.1) Z(t) where Y (t) = t S(s)U(s)s an Z(t) = t U(s)s, for the continuous case. We assume that t U(s)s, which is not unreasonable from a practical perspective. Note that this efinition reuces to the average use for the Asian option when U(t) is constant. This efinition is easily extene to the more practical iscrete case by iscretize the time interval [, t] into the intervals t =, t 1, t 2,..., t N = t an forming the average as I(t) = N S i U i i=, with I() = S(), (2.2) N U i i= where U i = U(t i ) an S i = S(t i ). Likewise we assume N U i. For the remainer of this work we eal with the continuous case. To give an iea of what the VWAP price looks like, Figure 1 shows a plot containing the stock price, the arithmetic average an the VWAP for several stocks. It is noticeable that there is little ifference between an arithmetic average price an VWAP, so it seems reasonable to assume that the volume risk is not altering the istribution of the VWAP significantly from the arithmetic average in normal market conitions. We can also see that for small times, both the arithmetic average price an VWAP are quite sensitive to changes in the stock. This sensitivity to the stock price eteriorates as the averaging time increases. i=

3 A Moment Matching Approach to the Valuation of a VWAP Option 3 95 IBM 4 DELL 9 35 $ 85 $ Day number Day number 35 NAB 5.5 Telstra $ $ Day number Day number Fig. 1. Stock Price, running Arithmetic an VWA prices from 1st January 23 until 31st December 23 Stock price=soli, Running arithmetic average = ashe, Running VWAP =ash/ot 2.2. Contracts of Interest Here we are intereste in European call options a with a floating strike, which have a payoff at expiry T of an fixe strike with payoff at the final time of It is these contracts which we price. V floating (T ) = (S T I(T )) +, (2.3) V fixe (T ) = (I(T ) K) +. (2.4) 3. The Moels We require moels for both the stock an volume processes. We assume that the stock evolves as the familiar geometric Brownian motion S(t) = µs(t)t + σs(t)w 1 (t), (3.1) a We use the notation (.) + to enote max(., )

4 4 Antony William Stace with µ being the rift of the stock, σ the variance an W 1 a Weiner process. A moel for volume is not quite as straight forwar to ecie on. We use a mean reverting process. We reason that the volume trae is riven by information. During normal perios the flow of information results in volume being at a level aroun U mean. When new information is receive the market reacts an the traing volume increases then some time later the volume settles back own to U mean. Having ecie that a mean reverting process is to be use for the volume intensity, we choose a log normal mean reverting process. The analysis which follows is also applicable for both Ornstein Uhlenbeck (OU) an Cox Ingersoll Ross (CIR) type mean reverting processes, in fact any process which has enough moments available to calculate the expectation an variance of the VWAP. We enote the volume intensity by U(t) an write it as U(t) = α(u mean U(t))t + βu(t)w 2 (t) (3.2) with α being the spee of mean reversion, U mean the long term average of the volume process, β the volatility of the volume process an W 2 a Weiner process. This is similar to [4] who use this moel to moel information flow which they proxy by number of traes. Assume that there is no correlation between the two Weiner processes W 1 an W 2. If this restriction is lifte then a moifie system of ifferential equations which escribe the expectation an variance of the VWAP is obtaine. We also assume that µ, σ, α, U mean an β are positive boune constants. A later moel might improve the moel for U(t) by introucing jumps into the moel. 4. The Approximation A partial ifferential equation approach to the valuation of (2.3) an (2.4), requires four state variables as well as time which leas to a numerically intractable solution, an further the bounary conitions of the partial ifferential equation are very ifficult to formulate. Monte Carlo can also be use to price (2.3) an (2.4) but has the raw back that it is slow. To obtain a numerically tractable problem, we approximate the istribution of the VWAP by a log normal process S. This is a common approximation use in the valuation of a Asian option. This has the avantage that it leas to a analytically tractable problem an at the very least gives a goo approximation to the price of (2.3) an (2.4). In making this approximation we have simplifie the problem of moeling the VWAP from I = I(t, S, U, Y, Z) to I = I(t, S). The process S(t) is given by S(t) = µ S(t)t + σ S(t) W (t), (4.1) with µ being the rift of the process, σ the variance an W (t) a Weiner process. It is the parameters µ an σ which we must fin. Once the parameters µ an σ have been obtaine, we can erive stanar partial ifferential equations to solve for the option

5 A Moment Matching Approach to the Valuation of a VWAP Option 5 price since the options (2.3) an (2.4) are now reuce to V fixe = V fixe (t, S) an V floating = V floating (t, S, S). 5. Matching The concept is simple, we want to match the first two moments of the VWAP given by (2.1) to the log normal process (4.1). This means that both the expectation an variance of the VWAP are require. To fin these we use the following approximations ( ) Y E E(Y ) Cov(Y, Z) Z E(Z) (E(Z)) 2 + E(Y ) V ar(z) (5.1) (E(Z)) 3 ( ) Y V ar Z ( ) 2 ( E(Y ) V ar(y ) E(Z) (E(Y )) 2 + V ar(z) Cov(Y, Z) 2 (E(Z)) 2 E(Y )E(Z) ), (5.2) see [5][p181]. The proof is easy an is base on a truncate Taylor s series expansion aroun (Y mean, Z mean ). The expectation an variance of (4.1) are well known an are E( S(t)) = S(t)e µt an (5.3) V ar( S(t)) = S(t) 2 e 2µt (e σ2t 1), (5.4) see for example [3][p95]. The iea of the metho is to obtain the variance an expectation of Y Z by the approximations in (5.1) an (5.2) an substitute these into (5.3) an (5.4) to fin µ an σ at any time. In orer to evaluate (5.1) an (5.2) a large number of expectations are require. The metho for fining the expectations is long an teious but relatively straight forwar, so only several of the expectations will be foun to emonstrate the technique. To start with, we fin E(S(t)) which we alreay know. To fin this we write own the stochastic ifferential equation for S(t) which is really short han for S(t) S() = S(t) = µs(t)t + σs(t)w 1 (t) (5.5) t µs(s)s + t σs(s)w 1 (s). (5.6) Taking expectation of (5.6) an using the property that the expectation of an Ito integral is we obtain E(S(t) S()) = E( = E( t t µs(s)s + t σs(s)w 1 (s) (5.7) µs(s)s). (5.8)

6 6 Antony William Stace Then moving the expectation insie the integral we have E(S(t) S()) = t an finally ifferentiating with respect to time we obtain E(S(t)) t µe(s(s))s, (5.9) = µ (E(S(t))). (5.1) t (5.11) This is a simple ifferential equation, which can be solve together with the initial conition that E(S()) = S(). Solving this gives us the expecte result that E(S(t)) = S()e µt matching (5.3). To further illustrate the metho we o one more as an example, an then simply list the system of 19 ifferential equations in the Appenix. This system of equations can either be solve numerically, or a symbolic computer package such as Maple can be use. We nee the expectation of Y 2 (t), i.e. E(Y 2 (t)). To obtain this we appeal to Ito s Lemma an form (Y 2 (t)) = 2Y (t)y (t) (5.12) = 2Y (t)u(t)s(t)t. (5.13) Repeating the arguments above, we nee to solve the equation E(Y 2 (t)) t = 2E(Y (t)u(t)s(t)) (5.14) with the initial conition E(Y 2 ()) = Y 2 (). To solve (5.14), E(Y (t)u(t)s(t)) is require, so we repeat the above proceure for E(Y (t)u(t)s(t)) an the remaining expectations require. The full system of ifferential equations is given in the Appenix. Solving this system of ifferential equations means we can fin the expectation an variance of the VWAP at any time by substituting the necessary expectations foun from the system of orinary ifferential equations into (5.1) an (5.2). The solution of this type of orinary ifferential equation system is well known an is ominate by the positive eigenvalues. Fortunately we can fin the eigenvalues(with the help of Maple) of the orinary ifferential equation system, they are liste in Table 1. It is interesting to note that U mean oes not appear in the eigenvalues. From the eigenvalues in Table 1 we can see there are many positive eigenvalues an that they are all real values. The fact that there are many positive values is to be expecte since many of the expectations which the orinary ifferential equations represent are increasing functions with time, i.e. the expectation of the stock is a monotonically increasing function in time. To obtain a numeric solution the values of these eigenvalues shoul not be too big. µ an σ are small values, both typically between

7 A Moment Matching Approach to the Valuation of a VWAP Option 7 Eigenvalue Number of times occurring 2µ + σ 2 1 2µ 2α + σ 2 + β 2 1 2µ α + σ 2 1 µ 2α + β 2 1 β 2 2α 1 µ 3 µ α 3 α 3 5 Table 1. The eigenvalues of the orinary ifferential equation system. an.5 so these parameters o not pose a problem. The quantity 2α + β 2 occurs throughout the eigenvalues a number of times, this quantity shoul be small, or negative, so that numeric solutions can be obtaine, i.e. we ieally like to have 2α + β 2 otherwise the solution will blow up too fast. If this occurs, then the parameters shoul be rescale to enforce this conition. Now for any given time we can easily match the expectation an variance obtaine from the ifferential equations an approximations given by (5.1) an (5.2) to the log normal istribution of S(t) an obtain µ(t) an σ(t) since rewriting (5.3) an (5.4) we have µ(t) = 1 E( S(t)) log (5.15) t S() 1 σ(t) = t log V ar( S(t)) + (E( S(t))) 2 (E( S(t))) (5.16) 2 So now for a given time T we can now fin µ an σ which matches the final istribution of the VWAP to a lognormal istribution, these are given by µ(t ) an σ(t ). Which means we now have the parameters µ(t) an σ(t) for the process S(t) for any time. 6. Pricing In orer to fin a price for the fixe strike, we construct a portfolio consisting of two options V (t, S) an V 1 (t, S), an then from stanar no arbitrage arguments we obtain V t ( σ(t ) S) 2 2 V S + ( µ(t ) S λ(t, S) σ(t V ) S) rṽ =, (6.1) 2 S where λ(t, S) is the market price of S risk, an final conition V (T, S) = ( S(T ) K) +. An analytic solution to (6.1) exists when the market price of risk is constant,

8 8 Antony William Stace this solution is almost the same as the Black Scholes equation. In the floating strike case we again form a portfolio of two options V = V (t, S, S) an V 1 (t, S, S), an by no arbitrage arguments again we obtain V t (σ(t)s)2 2 V S 2 + ρσ(t)s σ(t ) S 2 V S S ( σ(t) S) 2 2 V S 2 +rs V S + ( µ(t ) S λ(t, S) σ(t V ) S) rv = (6.2) S with final conition V (T, S) = (S(T ) S(T )) +. See [2] or [7] for etails on constructing these partial ifferential equations. We can see that the rift of S(t) is present in both of these equations unlike the Black Scholes equation which oes not contain the rift of the stock. This is a consequence of the problem being set in an incomplete market with no unerlying in S(t) with which to hege with, this also leas to the market price of risk λ(t, S) being present in these equations. We know all the values in (6.1) an (6.2) except for the market price of risk λ(t, S). In reality the traer must look to the market to obtain λ(t, S). We can however place some constraints on the market price of risk. To begin, we can see that the VWAP price over the interval [, T ] is boune above by the maximum of the stock price over the interval, S max = max S t, an below by the minimum, t T S min = min S t, over the time interval [, T ], i.e. t T S min Y (T ) Z(T ) S max. (6.3) Using the bouns given by (6.3) upper an lower bouns for the floating strike contract at the final time are an (S(T ) S max ) + (S(T ) I(T )) + (S(T ) S min ) + (6.4) (S min K) + (I(T ) K) + (S max K) +, (6.5) see [6]. All of these bouns have analytic expressions. This means that for a given market price of risk we can solve (6.2) an (6.1), if the solution obtaine is not between those given in (6.4) an (6.5) then the market price of risk use is not vali. Depening on the form of the market price of risk there maybe no analytic solution. In such cases we will have to evaluate (6.1) an (6.2) via a numerical scheme such as finite ifferences or use Monte Carlo methos. 7. Numeric Example As an example, we emonstrate the metho on the system S(t) =.1S(t)t +.15S(t)W 1 (t) U(t) = 1(11 U(t))t + 2U(t)W 2 (t)

9 A Moment Matching Approach to the Valuation of a VWAP Option 9 over the time interval [,.5] with S = 11 an U = Fining µ(t) an σ(t) To price the option using this metho, we only nee the values of µ(t) an σ(t) at the final time, i.e. µ(t ) an σ(t ). But it is interesting to see how µ(t) an σ(t) evolve over time. First the orinary ifferential equations are solve for each time over the interval [,.5], an at each time we use the approximation to the mean an variance from (5.1) an (5.2) to obtain the values of µ(t) an σ(t). To benchmark our results to, we simulate the VWAP. To o this we iscretise the time from the start to the en of the contract into N partitions, i.e. t, t 1,..., t N an compute the value of S(t) an U(t) at each time t i. The exact solution of S(t) is use to evolve S(t) an Milstein s Scheme is use to evolve U(t), an S i+1 = S i exp((µ 1 2 σ2 )t + σ tz S i+1 ) U i+1 = U i + α(u mean U i )t + βu i tz U i β2 U i t((z U i+1 )2 1), where Zi+1 S an ZU i+1 are inepenent normal ranom numbers with mean an variance 1. Then the sums N Y j = S i U i an Z j = i= N i= are compute, so for each simulation j we fin the VWAP U i I j = Y j Z j. We repeat this M times. Finally we approximate the expectation of Y Z by ( ) Y E 1 M I i (7.1) Z M an variance by ( ) Y V ar 1 Z M M j=1 j=1 I 2 j 1 M M j=1 2 I j (7.2) We then use these values of the expectation an variance to fin µ(t) an σ(t) for the simulation which matches the log normal istribution using (5.15) an (5.16).

10 1 Antony William Stace Approximating µ Time.9 Approximating σ Time Fig. 2. Solving for µ(t) an σ(t) by both simulation an solving the orinary ifferential equations. Soli - results from simulations, Dashe - results from orinary ifferential equations simulations an time split into 1 3 intervals.(s =.1St +.15SW 1, U = 11(1 U)t + 2UW 2, S() = 11, U() = 8). The results obtaine for µ(t) an σ(t) from both the simulation an the orinary ifferential equations are shown in Figure 2. We can see in Figure 2 that µ(t) an σ(t) obtaine from the moment matching approach qualitatively matches that obtaine from simulation an is especially goo for times longer than.2. The approximation to µ(t) an σ(t) for small time is not as goo as larger time. Finally we plot the probability ensity function of the VWAP istribution at the final time obtaine from simulation in Figure 3. We also plot the corresponing lognormal probability ensity function obtaine by solving the orinary ifferential equations an also from matching the mean an variance of the simulation to the log normal istribution. We can see that the log normal approximation to the VWAP matches the empirical probability ensity function well an that there is little ifference between the lognormal fit obtaine from the orinary ifferential equations an simulation. It was notice that as σ increase, the approximating probability ensity function eteriorate which is expecte since the log normal

11 A Moment Matching Approach to the Valuation of a VWAP Option 11 approximation to the arithmetic average use in the pricing of the Asian option works well only for small σ..6 Log Normal Fit From Emprical Log Normal Fit From ODEs Empricial VWAP Fig. 3. Empirical verse fitte probability ensity function at the final time for the VWAP. Note how well the value obtaine for µ(t) an σ(t) by simulation matches those obtaine from solving the system of orinary ifferential equations simulations an time split into 1 4 intervals.(s =.1St +.15SW 1, U = 11(1 U)t + 2UW 2, S() = 11, U() = 8) Solving the Partial Differential Equation Now that µ an σ have been obtaine, we move onto solving the partial ifferential equations. If we assume that λ is a constant then there is a close form solution to the fixe strike option since the partial ifferential equation essentially becomes the Black Scholes equation. This solution is given by V fixe () = e (r µ(t )+σ(t )λ)t S()Φ( 1 ) Ke rt Φ( 2 ) (7.3) where 1 = 2 + σ(t ) T an 2 = log(s()/k) + ( µ(t ) σ(t )λ 1 2 σ2 (T ))T σ(t ) T

12 12 Antony William Stace where Φ( ) is the cumulative normal istribution function, see [1]. Solutions to the floating strike option, an when the fixe strike has a constant market price of risk, must be solve numerically using Monte Carlo or the Finite Difference methos Fixe Strike Using (7.3) a typical solution to the price is foun in Figure 4. Note that the bouns of the VWAP option are those given by solving (6.5). We can see how the value of λ shifts the solution up an own relative to the Black Scholes call an the European fixe strike Asian call price Bouns λ= 1 λ= 1 BS price Asian Call 2 Price S Fig. 4. Typical solution of the fixe strike VWAP option.(s =.1St+.15SW 1, U = 11(1 U)t + 2UW 2, S() = 11, U() = 8) 8. Extensions This technique can be easily incorporate into a variance reuction scheme as a control variate when Monte Carlo is use to value options which are functions of a VWAP. The pricing of exotics such as lookbacks, barriers an igitals which are base on a VWAP can also be easily price. The heging however woul possibly be ifficult.

13 A Moment Matching Approach to the Valuation of a VWAP Option Conclusions an Further Work We have emonstrate a fast metho to approximate the price of an option which is a function of the VWAP for both fixe an floating strikes. The metho can easily be extene to price exotics. The approximation works well for values of σ up to about.2. Even when σ is not in this range the metho can be use as a control variate in a Monte Carlo scheme. A natural extension to this work is to match higher moments of the VWAP istribution to another process, perhaps to a shifte lognormal istribution. As well as this a better approximation to the variance an mean of the VWAP at the final time can be use. This work is part of a larger work ealing with the pricing an heging of options base on a VWAP. Further work inclues a practical heging strategy an more accurate pricing methos for these options. 1. Appenix This appenix lists in full all the ifferential equations which nee to be solve to get the expectations. Throughout it was assume that the correlation between W 1 an W 2 is. t E(S) = µe(s), E(S ) = S t E(U) = α(u mean E(U)), E(U ) = U t E(Z) = E(U), E(Z ) = t E(SU) = (µ α)e(su) + αu meane(s), E(S U ) = S U t E(Y ) = E(SU), E(Y ) = t E(Y 2 ) = 2E(Y SU), E(Y 2 ) = t E(Y SU) = E(S2 U 2 ) + αu mean E(Y S) + (µ α)e(y SU), E(Y S U ) = t E(S2 U 2 ) = (2µ 2α + σ 2 + β 2 )E(S 2 U 2 ) + 2αU mean E(US 2 ), E(S 2 U 2 ) = S2 U 2 t E(S2 U) = (2µ α + σ 2 )E(S 2 U) + αu mean E(S 2 ), E(S 2 U ) = S 2 U t E(S2 ) = (2µ + σ 2 )E(S 2 ), E(S 2 ) = S2 t E(Y S) = µe(y S) + E(S2 U), E(Y S ) = t E(Y U) = E(U 2 S) αe(y U) + αu mean E(Y ), E(Y U ) = t E(Y Z) = E(Y U) + E(ZSU), E(Y Z ) = t E(U 2 S) = 2αU mean E(US) + (µ 2α + β 2 )E(U 2 S), E(U 2 S ) = U 2 S t E(ZSU) = αe(zs) + (µ α)e(zsu) + E(SU 2 ), E(Z S U ) = t E(ZS) = E(SU) + µe(zs), E(Z S ) = t E(Z2 ) = 2E(ZU), E(Z) 2 = t E(ZU) = αu meane(z) αe(zu) + E(U 2 ), E(Z U ) = t E(U 2 ) = 2αU mean E(U) + (β 2 2α)E(U 2 ), E(U 2 ) = U 2

14 14 Antony William Stace Acknowlegments The author wishes to thank the University of Queenslan an the Australian Research Council Centre of Excellence for Mathematics an Statistics of Complex Systems for financial support. References [1] Fre Espen Benth. Option Theory with Stochastic Analysis An Introuction to Mathematical Finance. Springer, 24. [2] Tomas Björk. Arbitrage Theory in Continuous Time, Secon Eition. Oxfor University Press, 24. [3] Paul Glasserman. Monte Carlo Methos in Financial Engineering. Springer, 24. [4] S. Howison an D. Lamper. Traing Volume an Stochastic Volatility. OCIAM/MFG Working Paper, 21. [5] Alexaner M. Moo, Franklin A. Graybill, an Duane C. Boes. Introuction To The Theory Of Statistics, Thir Eition. McGraw-Hill, [6] Antony Stace. Bouns For The Price of an Option Base on a Volume Weighte Average Price. Maths Department, University of Queenslan, Working Paper, 24. [7] Paul Wilmott. Paul Wilmott on Quantitative Finance, 2 Volume Set. John Wiley & Sons, 2.