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 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

2 Plan of talk What an option is

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

4 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

5 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

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

7 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.

8 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...

9 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!).

10 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.

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

12 European Call, V T = max(s T K, 0) The strike price, K, is $105 Option Value $ Stock Price $

13 European Call, V T = max(s T K, 0) The strike price, K, is $ Time to expiry=1 Final Payoff Option Value $ Stock Price $

14 European Call, V T = max(s T K, 0) The strike price, K, is $ Time to expiry=0.5 Final Payoff Option Value $ Stock Price $

15 European Call, V T = max(s T K, 0) The strike price, K, is $ Time to expiry=0.3 Final Payoff Option Value $ Stock Price $

16 European Call, V T = max(s T K, 0) The strike price, K, is $ Time to expiry=0.1 Final Payoff Option Value $ Stock Price $

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

18 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.

19 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)

20 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

21 Running Average - 1 t t 0 S νdν Stock Price Running Average Price Time

22 Running Average - 1 t t 0 S νdν Stock Price Running Average Price Time

23 Running Average - 1 t t 0 S νdν Stock Price Running Average Price Time

24 Running Average - 1 t t 0 S νdν Stock Price Running Average 110 Price Time

25 Running Average - 1 t t 0 S νdν Stock Price Running Average 100 Price Time

26 Running Average - 1 t t 0 S νdν Stock Price Running Average Price Time

27 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

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

29 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 $ $ = $10.89 while a arithmetic weighted average price is $10+$100 2 = $55.00.

30 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 $ $ = $10.89 while a arithmetic weighted average price is $10+$100 2 = $ 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.

31 Example Of Real Stocks $ Stock Price Arithmetic Average VWAP DELL Day number

32 Example Of Real Stocks Stock Price Arithmetic Average VWAP IBM $ Day number

33 Example Of Real Stocks Stock Price Arithmetic Average VWAP NAB $ Day number

34 Example Of Real Stocks Stock Price Arithmetic Average VWAP TELSTRA $ Day number

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

36 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)

37 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.

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

39 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

40 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.

41 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.

42 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,

43 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.

44 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???

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

46 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

47 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)

48 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)

49 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?

50 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

51 The Doeblin in the Ito-Doeblin formula

52 The Doeblin in the Ito-Doeblin formula

53 The Doeblin in the Ito-Doeblin formula

54 The Doeblin in the Ito-Doeblin formula

55 The Doeblin in the Ito-Doeblin formula

56 The Doeblin in the Ito-Doeblin formula

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

58 Now back to the problem We need many expectations.

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

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

61 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

62 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 )

63 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)

64 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)

65 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 )

66 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.

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

68 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

69 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

70 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

71 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)

72 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

73 How well does it work?

74 How well does it work? 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 µ σ 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 Time Solid - results from simulations, Dashed - results from ODEs

75 PDF at final time for different σs σ = 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 VWAP

76 PDF at final time for different σs σ = 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 VWAP

77 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 VWAP

78 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 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial VWAP

79 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 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial VWAP

80 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 VWAP

81 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 Log Normal Fit From Emprical Log Normal Fit From ODE Empricial VWAP

82 PDF at final time for different σs σ = 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 VWAP

83 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.

84 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.

85 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 ( σ S) 2 2 V V + ( µ λ(t, S) σ S) rṽ = 0 S 2 S

86 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 ( 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 (σs)2 2 V S 2 + ρσs σ S 2 V S S ( σ S) 2 2 V V + rs S 2 S + ( µ λ(t, S) σ S) V S rv = 0

87 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 ( 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 (σs)2 2 V S 2 + ρσs σ S 2 V S S ( σ S) 2 2 V V + rs S 2 S + ( µ λ(t, S) σ S) V S rv = 0

88 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).

89 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

90 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

91 An Example 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= S 0

92 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 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...

93 Summary Have a way to price the option

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

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

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

97 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.

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

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

100 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.