An Application of Bayesian Option Pricing to the Soybean ... - CiteSeerX

An Application of Bayesian Option Pricing to the Soybean Market

F. Douglas Foster and Charles H. Whiteman*

December 1998

An Application of Bayesian Option Pricing to the Soybean Market Options pricing techniques have been an important part of finance for some time. Most approaches specify a particular stochastic process to represent the price dynamics of the underlying asset and then derive an explicit pricing model. While this may be acceptable for standard financial assets, it can be problematic for commodities. Many commodities have significant seasonalities and require a far more elaborate time-series specification of the price dynamics of the underlying asset. Hence, it becomes difficult at best to derive explicit pricing formulae. Further, with the additional complexity of a rich time-series specification, estimation risk becomes a genuine concern.

In this paper we suggest an alternative approach. We use numerical Bayes techniques to build a predictive density for the price of the underlying asset (for the example in this paper we need to predict the soybean cash and futures prices at the option’s expiration). Bayesian techniques allow for two very important additions. First, we can integrate out any estimation risk. Second, it allows us to incorporate properly any non-sample information that we may have. Once the predictive density has been computed, we use a procedure proposed by Stutzer (1996) to translate this density to its risk-neutral form. Once this is done, pricing European options is very straightforward.

To illustrate this approach we consider recent prices of options on soybean futures traded on The Chicago Board of Trade. We start with a simple vector autoregressive specification of the spot price return and the basis (defined as the log difference between the futures price and the spot price). We compare this procedure with traditional approaches as well as with a non-parametric procedure advocated by Stutzer (1996).

1

Building the Predictive

We are interested in pricing options on soybean futures and so we need to predict the soybean futures price on the option expiration day, FT , where F denotes the futures price and the T subscript refers to the future’s expiration day. To predict the futures price we work with the spot price and the basis using the cost of carry relation:

(1)

ln( Ft ) = ln( St ) + bt ,

where S denotes the spot price and b is the basis. Here the basis represents the percentage cost of carrying the spot commodity forward in time to the future’s expiration date (not the same as the options expiration date).

Using this structure we need to derive predictive densities for the spot price change and the basis. While there are many possible structures to use, here we rely on a two-equation vector autoregression. The variables we include in our vector autoregression are the change in the spot price, ln( St ) − ln( St −1 ) , and the basis, ln( Ft ) − ln( St ) :

(2)

yt = (ln( St ) − ln( St −1 ),ln( Ft ) − ln( St )) ′ .

The VAR can be written

(3)

yt = C + Dt + A( L) yt −1 + ν t , ν t ~ iid N (0, Σ),

where C and D are vectors of constants and A(L) is a vector of λ-degree polynomials in the lag operator. Henceforth, we shall refer to the parameters in C, D, and A(L) as µ, and

2

write θ = (µ,Σ). Let Y denote the T×n matrix with t-th row given by yt ′ , and let X denote the T×(2+nλ) matrix with t-th row given by (1,t,yt-1'). Using the independence of the vt’s and noting that the Jacobian of the transformation from v to y is unity, the sampling density of Y conditional on λ initial values, is (4)

P (Y |θ ) ∝

T

∏Σ

−

1 2

exp(vt Σ −1vt' )

t = λ +1

or

(5)

P(Y|θ) ∝ |Σ|-(T-λ-1)/2exp[-½ tr (Y-XB)'(Y-XB)Σ -1]

where tr denotes the trace operator. That is, the VAR can be seen to be a version of the standard multivariate regression model:

(6)

Y = XB + V,

where the (2+nλ)×n matrix B contains the VAR coefficients, and the rows of V are iid N(0,Σ).

For our examples we will adopt an “uninformative” prior. Although there are many interpretations that can be given to “uninformative”, we use the standard “flat” prior:

(7)

P(B, Σ) ∝ |Σ |-(n+1)/2

(see Zellner). The posterior distribution of the parameters is the product of the likelihood and the prior, or

3

(8)

P(B,Σ|Y,X) ∝ |Σ|-(T-λ+n)/2exp[-½ tr (Y-XB)'(Y-XB)Σ -1].

A little rewriting using the least squares estimate of B, B$ , and the sum of squares matrix,

(9)

S = (Y-X B$ )'(Y-X B$ ),

yields

(10)

P(B,Σ|Y,X) ∝ |Σ|-(T-λ+n)/2exp{-½ tr [S + (B- B$ )'X'X(B- B$ )]Σ -1}.

A little more rearrangement reveals that

(11)

P(B,Σ|Y,X) ∝ P(B|Σ,Y,X)P(Σ|Y,X),

where

(12)

P(B|Σ,Y,X) ∝ |Σ|-k/2exp{-½ tr(B- B$ )'X'X(B- B$ )Σ--1}.

is the normal distribution, (k = 2+nλ), and

(13)

P(Σ|Y,X) ∝ |Σ|-ν/2exp{-½ trSΣ -1}

is the “inverse-Wishart” distribution (ν = T-λ-k+n+1).

Sampling from the posterior distribution and the predictive are straightforward because sampling from the inverse-Wishart distribution is straightforward. In particular, to sample from the posterior distribution, simply sample from the appropriate inverse-

4

Wishart, use this drawing of Σ to condition the normal, and draw a B.1 For each drawing of B and Σ, a drawing from the predictive is calculated as follows: first make h drawings from the N(0,Σ) to generate realizations of shocks vT+1, vT+2, ...,vT+h. Then starting from the last λ sample data points, perform a dynamic simulation of the VAR using the previously drawn B and the newly drawn shocks.

For our example we use the “flat” prior and 13 lags (one quarter of a year of weekly data) in the vector auto-regression. We also include monthly dummy variables in our specification of C. Our Monte Carlo construction of the predictive density of FT uses 5000 draws.

The Risk-Neutral Density

Once we have computed the predictive density we need to risk-adjust the probabilities to form the risk-neutral or pricing density. To do this we use a procedure advocated by Stutzer (1996). This procedure uses the maximum entropy principle of information theory to transform the predictive density to its risk-neutral form. This section describes his basic approach.

Using the Monte Carlo predictive density for FT we compute a futures return factor, Ri (T − t ) , for each draw, i=1,2,…,5000:

(14)

FTi = Ft Ri (T − t ) ,

5

where t denotes the current date and T is the options expiration date. We now need to transform the Monte Carlo probabilities for each draw, π$ (i ) =

1 (an equal weighting 5000

using the number of Monte Carlo draws in building the predictive density), so that the resulting estimated risk-adjusted density, π$ * (i ) , prices the futures contract properly. That is, we require the true risk-adjusted density to satisfy the following:

(15)

∑π

*

(i ) R(T − t ) = 1 .

i

Of course, there are many choices of the 5000 component vector π * satisfying expression (15). We use an estimate, π$ * , satisfying expression (15) that is chosen to minimize the Kullback-Leibler Information Criterion (KLIC) distance between the risk-adjusted probabilities and those from the predictive density formed with our numerical Bayes procedure.2 This optimization is of the form:

(16)

π$ * = arg

FG H

IJ K

n π * (i ) * * $ min I ( π , π ) = π ( i ) ln , given: ∑ π * ( i ) > 0 ,∑ π * ( i ) =1 π$ (i ) i =1 ∀i

Because the weights π$ (i ) =

∑π

*

(i ) R(T − t ) = 1.

i

1 are equal for all 5000 Monte Carlo draws, the 5000

constrained optimization in expression (16) is identical to a constrained maximization of Shannon entropy, − ∑ π * (i ) ln(π * (i )) . Using the Lagrange multiplier method gives the ∀i

Gibb’s canonical distribution:

(17)

π$ (i ) = *

c

exp γ * Ri (T − t )

∑ expcγ ∀i

*

h

R (T − t ) i

i = 1,2,...,5000 ,

h, 6

whose Lagrange multiplier, γ * , is found by solving:

(18)

c

h

γ * = arg min ∑ exp γ R i (T − t ) − 1 . γ

i

Finally, to price a European option on a future contract, use the risk-neutral density to compute the discounted expected value at the option’s expiration. For a call option with a strike price of X we have:

(19)

C(t , T ) = ∑ i

max Ft Ri (T − t ) − X ,0 r

(T −t )

π$ * (i ) ,

where r ( T − t ) is the risk-free discount factor from the current time, t, to the option expiration date T.

Data For our example we use recent prices and consider the January and March 1999 option on futures contracts traded on the Chicago Board of Trade (CBOT). Each futures contract entitles the holder of the long position to receive 5,000 bushels of soybeans; we consider both put and call claims on this contract. The deliverable grade is No. 2 Yellow at par and there can be delivery substitutions at differentials established by the CBOT. Prices for the futures are quoted in cents and quarter cents per bushel, with a tick size of

7

$0.0025 ($12.50 per contract). The contract year starts trading in September and each year there are futures that expire in September, November, January, March, May, July, and August. The last delivery day is the last business day of the delivery month. We use weekly historical data (Friday to Friday) on the cash and futures prices to construct our VAR, as well as to compute historical volatilities for the Black formula and to implement the nonparametric prediction procedure of Stutzer (1996). Logical Information Machines, Inc. generously provided the data used for these examples. Our data begin on the first non-holiday Friday in July (July 10), 1959 and goes through Friday, November 13 1998. We use the next-to-expire contract for each contract month and include dummy variables in the basis equation in the VAR when the contract year shifts.

Options on futures expire on the first Saturday following the last day of trading. The last trading day is the last Friday with at least five business days remaining in the month preceding the option month. Options prices for our examples are taken from the Wall Street Journal on Monday, November 16.

An Example

To illustrate our procedure we value calls and puts on soybean futures options for the date Friday, November 13, 1998.3 We use the January 1999 and March 1999 options contracts to contrast three different techniques: the Black model, the Stutzer model, and our blended approach of using numerical Bayes techniques to build the predictive density. The Black model is the familiar log normal predictive; the Stutzer model simply uses the empirical distribution of (T-t) period futures price growth rates together with the most recent futures datum to produce a nonparametric estimate of the predictive distribution.

8

Results for pricing the January options are presented in Tables 1 and 3. Results for the March options are presented in Tables 2 and 4. In each table we provide option prices from the three different approaches and compute the percentage deviation from the settlement price. Because option prices change with the strike price we net out the intrinsic value of the option (e.g., max(F-X,0) for a call) from both the model and settlement price before we compute the percentage error. Hence, we compare the pricing accuracy with the time values of the options. This allows us to compare relative errors across strike prices.

We use two different implementations of each pricing model. In the first we use no current information from the options market to price the contracts. This means that we use historical volatilities for the Black (1976) implementation and use only one constraint (that the futures contract be properly priced) for the Stutzer (1996) and numerical Bayes procedures. These values and percent errors (differences from settlement prices) are in Tables 1 and 2. Our second implementation uses the at-the-money option price to aid in pricing the other options. Hence, for the Black (1976) model we use the implied volatility of the at-the-money call option. For the Stutzer (1996) and numerical Bayes techniques we add an additional constraint that the at-the-money calls are correctly priced in generating the risk-adjusted predictive density. These option prices and percent deviations from the settlement prices are reported in Tables 3 and 4.

An inspection of the Tables shows adding the constraints (see Tables 3 and 4) improves the pricing accuracy considerably. Both the Stutzer model and the Bayes approach price the options at least as well as the Black model. Given the complexity of the VAR (13 lags plus dummy variables) parameter uncertainty appears to be well managed. Further, employment of informative prior information is straightforward by importance sampling—all that is required is a re-weighting of the drawings from the flat prior

9

posterior (each draw initially having weight

1 ) prior to the Stutzer change-of-measure 5000

procedure.

One caveat to our implementation is that the options on soybean futures referred to are American-style and we have provided European options prices. For the options that we consider, the additional value of the American early-exercise feature is very modest.

Conclusions

In this paper we outline a procedure for pricing derivative securities when the underlying asset has rich time-series properties. Many commodities are examples of such assets. We use some simple examples to show that relative to the standard Black (1976) model, as well as a non-parametric procedure advocated by Stutzer (1996) a procedure that makes use of numerical Bayes techniques to develop an underlying predictive density holds significant promise. That these techniques work well for complicated time series models (in our case, the model had 81 parameters) and without informative prior information is particularly encouraging, and suggests that additional efforts to tune the model and to employ non-sample information will be fruitful.

In subsequent work we plan to develop further the time-series specification of the predictive density. In addition, a more detailed examination of the pricing performance of these models is required to document performance over a longer sample, across expiration months, across puts and calls, and across strike prices.

10

Table 1: Option Values for January Puts and Calls (cents per bushel)

Black Model

Stutzer Model

Numerical Bayes

Option

Settlement

Description

Price

Price

Error4

Price

Error

Price

Error

Jan 525 Call

53.25

53.88

127%

54.06

162%

53.92

134%

Jan 550 Call

30.50

32.82

84

32.09

58

33.09

94

Jan 575 Call

12.75

16.89

41

14.59

18

17.25

45

Jan 600 Call

4.13

7.14

73

6.15

49

7.35

78

Jan 625 Call

1.13

2.44

117

2.92

159

2.51

123

Jan 650 Call

0.50

0.68

35

1.55

211

0.67

33

Jan 525 Put

0.63

1.31

109

1.49

138

1.35

116

Jan 550 Put

2.75

5.16

88

4.44

61

5.43

97

Jan 575 Put

10.00

14.15

41

11.86

19

14.51

45

Jan 600 Put

26.25

29.31

77

28.33

52

29.53

82

Jan 625 Put

48.25

49.54

129

50.01

176

49.61

136

Jan 650 Put

72.50

72.69

75

73.57

428

72.68

72

Median

80

11

100

88

Table 2: Option Values for March Puts and Calls (cents per bushel)

Option

Settlement

Description

Price

Price

Error5

Price

Error

Price

Error

Mar 525 Call

62.50

67.43

1973%

66.21

1484%

67.43

1970%

Mar 550 Call

41.00

48.79

208

46.53

147

48.93

211

Mar 575 Call

25.00

33.48

66

29.93

39

33.69

68

Mar 600 Call

14.00

21.73

55

18.37

31

22.14

58

Mar 625 Call

7.75

13.34

72

11.40

47

13.80

78

Mar 650 Call

4.00

7.75

94

7.22

81

8.22

105

Mar 525 Put

1.38

5.91

330

4.70

242

5.92

331

Mar 550 Put

4.50

11.98

166

9.73

116

12.13

170

Mar 575 Put

13.00

21.37

64

17.84

37

21.61

66

Mar 600 Put

27.00

34.33

51

30.99

28

34.76

54

Mar 625 Put

45.25

50.65

72

48.73

46

51.13

78

Median

Black Model

72

12

Stutzer Model

47

Numerical Bayes

78

Table 3: Constrained Option Values for January Puts and Calls (cents per bushel)

Option

Settlement

Description

Price

Price

Error6

Price

Error

Price

Error

Jan 525 Call

53.25

52.85

-81%

53.53

56%

53.00

-50%

Jan 550 Call

30.50

29.99

-19

30.90

15

30.29

-8

Jan 575 Call

12.75

12.75

0

12.75

0

12.75

0

Jan 600 Call

4.13

3.71

-10

4.59

11

3.95

-4

Jan 625 Call

1.13

0.70

-37

1.86

65

0.95

-15

Jan 650 Call

0.50

0.09

-83

0.87

73

0.17

-66

Jan 525 Put

0.63

0.27

-56

0.95

53

0.43

-32

Jan 550 Put

2.75

2.33

-15

3.25

18

2.63

-4

Jan 575 Put

10.00

10.01

0

10.01

0

10.01

0

Jan 600 Put

26.25

25.89

-9

26.77

13

26.13

-3

Jan 625 Put

48.25

47.80

-45

48.95

70

48.05

-20

Jan 650 Put

72.50

72.10

-161

72.88

151

72.18

-127

Median

Black Model

-28

13

Stutzer Model

35

Numerical Bayes

-12

Table 4: Constrained Option Values for March Puts and Calls (cents per bushel)

Option

Settlement

Description

Price

Price

% Error7

Price

% Error

Price

Mar 525 Call

62.50

63.13

250%

63.91

565%

63.48

390%

Mar 550 Call

41.00

42.03

27

42.83

49

42.29

34

Mar 575 Call

25.00

25.00

0

25.00

0

25.00

0

Mar 600 Call

14.00

13.08

-7

13.39

-4

13.71

-2

Mar 625 Call

7.75

5.97

-23

7.13

-8

7.03

-9

Mar 650 Call

4.00

2.37

-41

3.79

-5

3.39

-15

Mar 525 Put

1.38

1.60

17

2.39

74

1.96

42

Mar 550 Put

4.50

5.21

16

6.01

34

5.48

22

Mar 575 Put

13.00

12.89

-1

12.89

-1

12.90

-1

Mar 600 Put

27.00

25.68

-9

25.99

-7

26.31

-5

Mar 625 Put

45.25

43.27

-26

44.44

-11

44.34

-12

Median

Black Model

-1

14

Stutzer Model

-1

Numerical Bayes % Error

-1

References

Barone-Adesi, G., and R.E. Whaley. “Efficient Analytic Approximation of American Option Values.” The Journal of Finance 42 (June 1987):301-320.

Black, F. “The Pricing of Commodity Contracts.” Journal of Financial Economics 3 (March 1976):167-179.

Chicago Board of Trade. Contract Specifications, Chicago, Illinois, 1994.

Gerber, H., and E. Shiu. “Option Pricing by Esscher Transforms.” Transactions of the Society of Actuaries 46 (1994), 99-140.

Hobson, A. Concepts in Statistical Mechanics. Gordon and Breach: New York, New York, 1971.

Hull, J.C. Introduction to Futures and Options Markets (second edition). Prentice-Hall Inc.: Englewood Cliffs, New Jersey, 1995.

Stoll, H.R., and R.W. Whaley. Futures and Options: Theory and Applications. SouthWestern Publishing Co.: Cincinnati, Ohio, 1993.

Stutzer, M. “A Bayesian Approach to Diagnosis of Asset Pricing Models.” Journal of Econometrics 68, no. 2 (August 1995):367-397.

Stutzer, M. “A Simple Nonparametric Approach to Derivative Security Valuation.” The Journal of Finance 51, no. 4 (December 1996):1633-1652.

15

Stutzer, M., and M. Chowdhury. "A Simple Nonparametric Approach to Bond Futures Option Pricing." Journal of Fixed Income, March 1999.

Zellner, A. An Introduction to Bayesian Inference in Econometrics. Krieger Publishing Company: Malabar, Florida, 1971 (reprint edition, 1987).

16

Endnotes

*

F. Douglas Foster is Professor of Finance, and Charles H. Whiteman is Professor of

Economics, both at The University of Iowa. The authors thank Logical Information Machines for their extensive data support. Discussions with Tom Smith and Michael Stutzer improved the paper substantially. 1

Our sampler may remind the reader of the recently developed Gibbs Sampler. The

distinction is that our procedure samples directly from the posterior (by sampling from the marginal distribution of Σ and then the distribution of B conditioned on Σ.) The Gibbs sampler is useful in cases in which direct sampling from the joint posterior is not possible, but the full set of conditional distributions is available (e.g., the conditional posteriors P(Σ|B,Y,X) and P(B|Σ,Y,X) are both known.) Starting from, say, an initial Σ0 the Gibbs sampler produces a sequence B1 ~ P(B|Σ0,Y,X), Σ1 ~ P(Σ|B1,Y,X), B2 ~ P(B|Σ1,Y,X), and so on. This sequence can be shown to be the output of a Markov chain which converges in distribution to P(B, Σ|Y,X). In our case, there are no convergence issues since we are sampling directly from the posterior P(B, Σ|Y,X) = P(B|Σ,Y,X) P(Σ|Y,X). 2

See Hobson and Stutzer (1995) for a more detailed review of the KLIC. For its use in

deriving well-known parametric option pricing models, see Gerber and Shiu who dub the transformation from actual to risk-neutral probabilities the “Esscher Transform”. 3

For pricing bond futures options see Stutzer and Chowdhury (1999).

17

4

For each option and technique (both price and % error) we report prices computed

without reference to current information in the options market. Percent errors are computed net of the intrinsic value of the option:

% error =

Predicted - Actual *100 . Actual − max[ F − x ,0]

5

See note 4.

6

For each option and technique, we use information from the current options market (for

the Black model we use an implied volatility, and the Stutzer and the numerical Bayes techniques we add the additional constraint that the closest-to-the-money option be correctly priced. This is in addition to the constraint that the futures contract be correctly priced.) Percentage errors are calculated as in note 4. 7

See note 6.

18