A Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option

A Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option Antony Stace Department of Mathematics and MASCOS University of Queensland 15th October 2004 AUSTRALIAN RESEARCH COUNCIL Centre of Excellence for Mathematics and Statistics of Complex Systems

Plan of talk What an option is

Plan of talk What an option is What an Asian Option is and more importantly what a Volume Weighted Average Price is Option

Plan of talk What an option is What an Asian Option is and more importantly what a Volume Weighted Average Price is Option Future work

Plan of talk What an option is What an Asian Option is and more importantly what a Volume Weighted Average Price is Option Future work Questions

Options, the basics Gives the holder the right to something, without the obligation.

Options, the basics Gives the holder the right to something, without the obligation. European Call Option: The right, but not obligation, to buy a share at a specified price(the strike) and time.

Options, the basics Gives the holder the right to something, without the obligation. European Call Option: The right, but not obligation, to buy a share at a specified price(the strike) and time. Used to manage risk. Used in currencies markets, commodities such as oil, electriricy...

Options, the basics Gives the holder the right to something, without the obligation. European Call Option: The right, but not obligation, to buy a share at a specified price(the strike) and time. Used to manage risk. Used in currencies markets, commodities such as oil, electriricy... Managed funds/superannuation funds use puts to protect against stock declines(i hope!).

Options, the basics Gives the holder the right to something, without the obligation. European Call Option: The right, but not obligation, to buy a share at a specified price(the strike) and time. Used to manage risk. Used in currencies markets, commodities such as oil, electriricy... Managed funds/superannuation funds use puts to protect against stock declines(i hope!). There are many different types of options, European, American, Asian, Bermudan, Australian, Lookback, Barrier, Spread, Options on Options...and the list continues to grow all the time as people want new products to manage their risk.

European Call, V T = max(s T K, 0) The strike price, K, is $105

European Call, V T = max(s T K, 0) The strike price, K, is $105 Option Value $ 40 35 30 25 20 15 10 5 0 5 50 60 70 80 90 100 110 120 130 Stock Price $

European Call, V T = max(s T K, 0) The strike price, K, is $105 40 35 Time to expiry=1 Final Payoff Option Value $ 30 25 20 15 10 5 0 5 50 60 70 80 90 100 110 120 130 Stock Price $

European Call, V T = max(s T K, 0) The strike price, K, is $105 40 35 Time to expiry=0.5 Final Payoff Option Value $ 30 25 20 15 10 5 0 5 50 60 70 80 90 100 110 120 130 Stock Price $

European Call, V T = max(s T K, 0) The strike price, K, is $105 40 35 Time to expiry=0.3 Final Payoff Option Value $ 30 25 20 15 10 5 0 5 50 60 70 80 90 100 110 120 130 Stock Price $

European Call, V T = max(s T K, 0) The strike price, K, is $105 40 35 Time to expiry=0.1 Final Payoff Option Value $ 30 25 20 15 10 5 0 5 50 60 70 80 90 100 110 120 130 Stock Price $

European Call, V T = max(s T K, 0) The strike price, K, is $105 Stock Price 180 170 160 150 140 130 120 110 100 90 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

European Call, V T = max(s T K, 0) Closed form solution published by Black and Scholes in 1973 C = S 0 N(d 1 ) Ke rt N(d 2 ) with d 1 = ln(s 0/K) + (r + σ 2 /2)T σ T, d 2 = ln(s 0/K) + (r σ 2 /2)T σ T where K is the strike price, S 0 is the price of the share at time 0, σ is the share s volatility,t the time to expiry and N( ) is the cumulative probability function.

European Call, V T = max(s T K, 0) Closed form solution published by Black and Scholes in 1973 C = S 0 N(d 1 ) Ke rt N(d 2 ) with d 1 = ln(s 0/K) + (r + σ 2 /2)T σ T, d 2 = ln(s 0/K) + (r σ 2 /2)T σ T where K is the strike price, S 0 is the price of the share at time 0, σ is the share s volatility,t the time to expiry and N( ) is the cumulative probability function. Assumes stock evolves as Geometric Brownian motion, ds = µsdt + σsdw (Log normal)

European Call, V T = max(s T K, 0) Closed form solution published by Black and Scholes in 1973 C = S 0 N(d 1 ) Ke rt N(d 2 ) with d 1 = ln(s 0/K) + (r + σ 2 /2)T σ T, d 2 = ln(s 0/K) + (r σ 2 /2)T σ T where K is the strike price, S 0 is the price of the share at time 0, σ is the share s volatility,t the time to expiry and N( ) is the cumulative probability function. Assumes stock evolves as Geometric Brownian motion, ds = µsdt + σsdw (Log normal) The solution, remarkably, does not contain drift of the stock

Running Average - 1 t t 0 S νdν 110 105 Stock Price Running Average 100 95 Price 90 85 80 75 70 65 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Running Average - 1 t t 0 S νdν 130 120 Stock Price Running Average 110 100 Price 90 80 70 60 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Running Average - 1 t t 0 S νdν 120 115 Stock Price Running Average Price 110 105 100 95 90 85 80 75 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Running Average - 1 t t 0 S νdν 120 115 Stock Price Running Average 110 Price 105 100 95 90 85 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Running Average - 1 t t 0 S νdν 110 105 Stock Price Running Average 100 Price 95 90 85 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Running Average - 1 t t 0 S νdν 160 150 Stock Price Running Average 140 130 Price 120 110 100 90 80 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time

Asian Option,V T = ( 1 T ) + T S 0 vdv k Similar to my option is most like this one Cheaper than vanilla call or put. At the money it is about half the cost of a European. In fact volatility is about σ 3. Price is simply r = r 1 σ2 2 (r 6 ) and σ = σ 3 into the BS for the price of a Geometric Asian Option Very popular in currency and commodity markets

A Volume Weighted Average Price Assigns more weight to periods of heavy trading, than light trading

A Volume Weighted Average Price Assigns more weight to periods of heavy trading, than light trading Example: Suppose a stock trades at $10 today and there are 100 trades, tomorrow it trades at $100 and there is 1 trade. The volume weighted average price is $10 100+$100 1 100+1 = $10.89 while a arithmetic weighted average price is $10+$100 2 = $55.00.

A Volume Weighted Average Price Assigns more weight to periods of heavy trading, than light trading Example: Suppose a stock trades at $10 today and there are 100 trades, tomorrow it trades at $100 and there is 1 trade. The volume weighted average price is $10 100+$100 1 100+1 = $10.89 while a arithmetic weighted average price is $10+$100 2 = $55.00. We can write the VWAP at time T as V W AP (T ) = T 0 S vu v dv T 0 U vdv Where S t is the price of the stock at time t and U t is the rate of trades of the stock at time t.

Example Of Real Stocks $ 40 38 36 34 32 30 28 26 24 22 Stock Price Arithmetic Average VWAP DELL 20 0 50 100 150 200 250 Day number

Example Of Real Stocks 95 90 Stock Price Arithmetic Average VWAP IBM $ 85 80 75 0 50 100 150 200 250 Day number

Example Of Real Stocks 35 34 Stock Price Arithmetic Average VWAP NAB 33 32 $ 31 30 29 28 0 50 100 150 200 250 Day number

Example Of Real Stocks 5.5 5 Stock Price Arithmetic Average VWAP TELSTRA $ 4.5 4 3.5 0 50 100 150 200 250 Day number

My Problem To price and hedge ( V T = max T 0 S vu v dv T 0 U vdv ) K, 0 (fixed strike) and

My Problem To price and hedge ( V T = max ( V T = max T 0 S vu v dv T 0 U vdv T 0 S vu v dv T 0 U vdv ) K, 0 ) S T, 0 (fixed strike) and (floating strike)

My Problem To price and hedge ( V T = max ( V T = max T 0 S vu v dv T 0 U vdv T 0 S vu v dv T 0 U vdv ) K, 0 ) S T, 0 (fixed strike) and (floating strike) with S and U being defined by the stochastic differential equations ds = rsdt + σsdw 1 (stock) du = α(µ U)dt + βudw 2 (trades per unit time), (Use several mean reverting models, add jumps later) For the moment assume correlation between W 1 and W 2 is zero, relax this assumption later once we know the problem better.

My Problem To price and hedge PDE method has 4 state variables, not realistic to solve

My Problem To price and hedge PDE method has 4 state variables, not realistic to solve Probabilistic approach requires us to evaluate an expectation for which we do not know the PDF

My Problem To price and hedge PDE method has 4 state variables, not realistic to solve Probabilistic approach requires us to evaluate an expectation for which we do not know the PDF Can solve by Monte Carlo, but slow.

An approximation Inspired by early work, on Asian options, assume that the volume weighted average price T 0 T 0 S vu v dv U vdv (1) has a log normal distribution at the final time.

An approximation Inspired by early work, on Asian options, assume that the volume weighted average price T 0 T 0 S vu v dv U vdv (1) has a log normal distribution at the final time. We know d S = µ Sdt + σ SdW has a log normal distribution,

An approximation Inspired by early work, on Asian options, assume that the volume weighted average price T 0 T 0 S vu v dv U vdv (1) has a log normal distribution at the final time. We know has a log normal distribution, d S = µ Sdt + σ SdW So all we need to do is find µ and σ which will match the expectation and variance of (1) and then we will have a pretty standard equation to solve.

An approximation Inspired by early work, on Asian options, assume that the volume weighted average price T 0 T 0 S vu v dv U vdv (1) has a log normal distribution at the final time. We know has a log normal distribution, d S = µ Sdt + σ SdW So all we need to do is find µ and σ which will match the expectation and variance of (1) and then we will have a pretty standard equation to solve. But how do we get these???

Approximations We write the VWAP as T 0 S vu v dv T 0 U vdv = Y Z

Approximations We write the VWAP as T 0 S vu v dv T 0 U vdv = Y Z Simple approximations to the expectation and variance where Y and Z are random variables. of Y Z

Approximations We write the VWAP as T 0 S vu v dv T 0 U vdv = Y Z Simple approximations to the expectation and variance where Y and Z are random variables. of Y Z E ( Y Z ) E(Y ) E(Z) Cov(Y, Z) (E(Z)) 2 + E(Y ) (E(Z)) 3 V ar(z)

Approximations We write the VWAP as T 0 S vu v dv T 0 U vdv = Y Z Simple approximations to the expectation and variance where Y and Z are random variables. of Y Z V ar E ( Y Z ( Y Z ) ) E(Y ) E(Z), Mood et al. (1974). Cov(Y, Z) (E(Z)) 2 + E(Y ) (E(Z)) 3 V ar(z) ( ) E(Y ) 2 ( V ar(y ) E(Z) (E(Y )) 2 + V ar(z) ) Cov(Y, Z) 2 (E(Z)) 2 E(Y )E(Z)

Approximations We write the VWAP as T 0 S vu v dv T 0 U vdv = Y Z Simple approximations to the expectation and variance where Y and Z are random variables. of Y Z V ar E ( Y Z ( Y Z ) ) E(Y ) E(Z), Mood et al. (1974). Cov(Y, Z) (E(Z)) 2 + E(Y ) (E(Z)) 3 V ar(z) ( ) E(Y ) 2 ( V ar(y ) E(Z) (E(Y )) 2 + V ar(z) ) Cov(Y, Z) 2 (E(Z)) 2 E(Y )E(Z) Thats all great, but how do we get all these expectations?

Approximations We write the VWAP as T 0 S vu v dv T 0 U vdv = Y Z Simple approximations to the expectation and variance where Y and Z are random variables. of Y Z V ar E ( Y Z ( Y Z ) ) E(Y ) E(Z), Mood et al. (1974). Cov(Y, Z) (E(Z)) 2 + E(Y ) (E(Z)) 3 V ar(z) ( ) E(Y ) 2 ( V ar(y ) E(Z) (E(Y )) 2 + V ar(z) ) Cov(Y, Z) 2 (E(Z)) 2 E(Y )E(Z) Thats all great, but how do we get all these expectations? From the Ito-Doeblin formula, and lots of patience

The Doeblin in the Ito-Doeblin formula

The Doeblin in the Ito-Doeblin formula

The Doeblin in the Ito-Doeblin formula

The Doeblin in the Ito-Doeblin formula

The Doeblin in the Ito-Doeblin formula

The Doeblin in the Ito-Doeblin formula

The Doeblin in the Ito-Doeblin formula Shreve (2004)

Now back to the problem We need many expectations.

Now back to the problem We need many expectations. We can find all these expectations from properties of the Ito integral.

Demonstrate the method on E(S t ) We have ds = µsdt + σsdw

Demonstrate the method on E(S t ) We have ds = µsdt + σsdw This is really shorthand for S t S 0 = t 0 µs ν dν + t 0 σs ν dw ν

Demonstrate the method on E(S t ) We have ds = µsdt + σsdw This is really shorthand for S t S 0 = t 0 µs ν dν + t 0 σs ν dw ν Taking the expectation of this we have t E(S t S 0 ) = E( 0 µs ν dν + t 0 σs ν dw ν )

Demonstrate the method on E(S t ) We have ds = µsdt + σsdw This is really shorthand for S t S 0 = t 0 µs ν dν + t 0 σs ν dw ν Taking the expectation of this we have t E(S t S 0 ) = E( 0 µs ν dν + t 0 σs ν dw ν ) Now the expectation of an Ito integral is 0, so we have t E(S t S 0 ) = E( 0 µs ν dν)

Demonstrate the method on E(S t ) Then moving the expectation inside the integral E(S t S 0 ) = t 0 µe(s ν )dν)

Demonstrate the method on E(S t ) Then moving the expectation inside the integral E(S t S 0 ) = finally differentiating we have t 0 µe(s ν )dν) de(s t ) dt = µe(s t )

Demonstrate the method on E(S t ) Then moving the expectation inside the integral E(S t S 0 ) = finally differentiating we have t 0 µe(s ν )dν) de(s t ) dt = µe(s t ) which is simple to solve given the initial condition.

Obtaining the expectations We can do this for all the expectations which we require, it is long and tedious, but doable.

Obtaining the expectations We can do this for all the expectations which we require, it is long and tedious, but doable. Final system has 19 equations which are easy to solve in Matlab or Maple

Obtaining the expectations We can do this for all the expectations which we require, it is long and tedious, but doable. Final system has 19 equations which are easy to solve in Matlab or Maple Can now use the approximations E ( Y Z ) E(Y ) E(Z) Cov(Y, Z) (E(Z)) 2 + E(Y ) (E(Z)) 3 V ar(z) V ar ( Y Z ) ( ) E(Y ) 2 ( V ar(y ) E(Z) (E(Y )) 2 + V ar(z) ) Cov(Y, Z) 2 (E(Z)) 2 E(Y )E(Z) to find the expectation and variance at any time T for T 0 S vu v dv T 0 U vdv

Lots of +ve eigenvalues, but the one to look out for is the combination of β 2 2α which appears in many places. Eigenvalues 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 0 5

Now use the log normal approximation Now we know that the expectation and variance of our underlying d S = µsdt + σsdw are E( S(t)) µt = S 0 e and V ar( S(t)) = S 2 0e 2 µt (e σ 2t 1)

Now use the log normal approximation Now we know that the expectation and variance of our underlying d S = µsdt + σsdw are E( S(t)) µt = S 0 e and V ar( S(t)) = S 2 0e 2 We can rewrite these as µt (e σ 2t 1) µ = 1 E( S(t)) log t S 0 1 σ = t log V ar( S(t)) + (E( S(t))) 2 (E( S(t))) 2

How well does it work?

How well does it work? 0.125 µ From Simulation and ODEs, ds=0.2sdt+0.5sdw 1, du=110(100 U)dt+2UdW 2 s0=110, u0=10, time of VWAP is from 0 to 0.5, 1e7 Simulations, Time split up into 1e3 intervals 0.12 0.115 µ 0.11 0.105 σ 0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time σ From Simulation and ODEs, ds=0.2sdt+0.5sdw, du=110(100 U)dt+2UdW 1 2 s0=110, u0=10, time of VWAP is from 0 to 0.5, 1e6 Simulations, Time split up into 1e3 intervals 0.35 0.34 0.33 0.32 0.31 0.3 0.29 0.28 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time Solid - results from simulations, Dashed - results from ODEs

PDF at final time for different σs σ = 0.05 0.18 0.16 PDF for the VWAP, ds=0.15sdt+0.05sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 105 110 115 120 125 VWAP

PDF at final time for different σs σ = 0.1 0.09 0.08 PDF for the VWAP, ds=0.15sdt+0.1sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 100 105 110 115 120 125 130 VWAP

PDF at final time for different σs σ = 0.15 PDF for the VWAP, ds=0.15sdt+0.15sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals 0.06 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.05 0.04 0.03 0.02 0.01 0 90 95 100 105 110 115 120 125 130 135 140 VWAP

PDF at final time for different σs σ = 0.2 PDF for the VWAP, ds=0.15sdt+0.2sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals 0.045 0.04 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 90 100 110 120 130 140 VWAP

PDF at final time for different σs σ = 0.25 PDF for the VWAP, ds=0.15sdt+0.25sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals 0.035 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.03 0.025 0.02 0.015 0.01 0.005 0 80 90 100 110 120 130 140 150 160 VWAP

PDF at final time for different σs σ = 0.3 PDF for the VWAP, ds=0.15sdt+0.3sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals 0.03 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.025 0.02 0.015 0.01 0.005 0 60 80 100 120 140 160 180 VWAP

PDF at final time for different σs σ = 0.35 PDF for the VWAP, ds=0.15sdt+0.35sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals 0.025 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.02 0.015 0.01 0.005 0 60 80 100 120 140 160 180 VWAP

PDF at final time for different σs σ = 0.4 0.02 PDF for the VWAP, ds=0.15sdt+0.4sdw 1, du=100(110 U)dt+2UdW 2 s0=110, u0=200, time of VWAP is from 0 to 0.5, 2e7 Simulations, Time split up into 1e3 intervals Log Normal Fit From Emprical Log Normal Fit From ODE Empricial 0.015 0.01 0.005 0 60 80 100 120 140 160 180 VWAP

Comments on Result Approximation is better for lower σ, which is not unexpected - this method when applied to the normal Asian option which makes an approximation to t 0 S νdν is only good for small σ.

Comments on Result Approximation is better for lower σ, which is not unexpected - this method when applied to the normal Asian option which makes an approximation to t 0 S νdν is only good for small σ. Approximation is bad for small times.

Pricing the options This is the easy part. We can easily obtain PDEs which describe the option price from standard techniques Fixed strike(bc max( S T K, 0)) V t + 1 2 ( σ S) 2 2 V V + ( µ λ(t, S) σ S) rṽ = 0 S 2 S

Pricing the options This is the easy part. We can easily obtain PDEs which describe the option price from standard techniques Fixed strike(bc max( S T K, 0)) V t + 1 2 ( σ S) 2 2 V V + ( µ λ(t, S) σ S) rṽ = 0 S 2 S Floating strike(bc max(s T S T, 0)) V t + 1 2 (σs)2 2 V S 2 + ρσs σ S 2 V S S + 1 2 ( σ S) 2 2 V V + rs S 2 S + ( µ λ(t, S) σ S) V S rv = 0

We could also price American options as well as exotic products without too much more work? Ie Barrier, Lookback... Pricing the options This is the easy part. We can easily obtain PDEs which describe the option price from standard techniques Fixed strike(bc max( S T K, 0)) V t + 1 2 ( σ S) 2 2 V V + ( µ λ(t, S) σ S) rṽ = 0 S 2 S Floating strike(bc max(s T S T, 0)) V t + 1 2 (σs)2 2 V S 2 + ρσs σ S 2 V S S + 1 2 ( σ S) 2 2 V V + rs S 2 S + ( µ λ(t, S) σ S) V S rv = 0

Solutions In the case that the market price of risk is constant, the fixed strike has the analytic solution where V fixed (0) = e (r µ+ d 1 = d 2 + σ T and σλ)t S(0)Φ(d 1 ) Ke rt Φ(d 2 ) d 2 = log(s(0)/k) + ( µ σλ 1 2 σ2 )T σ T where Φ( ) is the cumulative normal distribution function, Benth (2004).

Solutions In the case that the market price of risk is constant, the fixed strike has the analytic solution where V fixed (0) = e (r µ+ d 1 = d 2 + σ T and σλ)t S(0)Φ(d 1 ) Ke rt Φ(d 2 ) d 2 = log(s(0)/k) + ( µ σλ 1 2 σ2 )T σ T where Φ( ) is the cumulative normal distribution function, Benth (2004). Otherwise we must use a numeric technique such as finite differences, Monte Carlo, FFT, etc

An Example Method demonstrated on the system ds = 0.1Sdt + 0.5SdW 1 du = 100(110 U)dt + 2UdW 2 U 0 = 200, K = 100, r = 10% and time from 0 to 0.5

An Example 60 50 40 Fixed Strike VWAP Price, K=100, r=0.1, ds=0.15sdt+0.5sdw 1, du=100(110 U)dt+2UdW 2 u0=200, time of VWAP is from 0 to 0.5 Bounds λ= 0 λ= 2 λ= 2 BS price Asian Call λ= 2 30 Price 20 BS 10 Asian Call 0 λ=2 10 60 70 80 90 100 110 120 S 0

Share Purchase Plans V T = ( S T2 D T1 T 0 T1 T 0 S v U v dv U v dv ) +,T 1 T 0 typically 3-10 days, T 2 T 1 typically 10-30 days, D a discount factor usually 70%-90% We can value this using the method just described. Raises capital easily, no prospectus Aimed at small investors, max $5000 IAG, Suncorp, AMP We can immediately now say how much it is worth to participate in a share purchase plan(actually what the companies are giving away for free!!) I am not suggesting you do this, but since they have given you this payoff...

Summary Have a way to price the option

Summary Have a way to price the option Can price exotics

Summary Have a way to price the option Can price exotics FAST

Summary Have a way to price the option Can price exotics FAST Can use as a control variate in Monte Carlo

Summary Have a way to price the option Can price exotics FAST Can use as a control variate in Monte Carlo Can tell you how much companies are giving to you when they offer shares at a VWAP to you in a share purchase plan.

Future Work Find a region where this approximation is good is some sense. Take more moments? Find a practical hedge More Monte Carlo

Thanks MASCOS for financial assistance. Dr Chandler for comments and suggestions. Thanks for Josh for helping with the tex.

References Benth, F. E. (2004), Option Theory with Stochastic Analysis An Introduction to Mathematical Finance, Springer. Mood, A. M., Graybill, F. A. & Boes, D. C. (1974), Introduction To The Theory Of Statistics, Third Edition, McGraw-Hill. Shreve, S. (2004), Stochastic Calculus for Finance II Continuous-Time Models, Springer.