Practical Option Valuations of Futures Contracts with Negative Underlying Prices
Anatoliy Swishchuk, Ana Roldan-Contreras, Elham Soufiani, Guillermo Martinez, Mohsen Seifi, Nishant Agrawal, Yao Yao
PPRACTICAL OPTION VALUATIONS OF FUTURESCONTRACTS WITH NEGATIVE UNDERLYING PRICES
ANATOLIY SWISHCHUK, ANA ROLDAN-CONTRERAS, ELHAM SOUFIANI,GUILLERMO MARTINEZ, MOHSEN SEIFI, NISHANT AGRAWAL, AND YAO YAO
Abstract.
Here we propose two alternatives to Black 76 to value Europeanoption future contracts in which the underlying market prices can be negativeor mean reverting. The two proposed models are Ornstein-Uhlenbeck (OU)and continuous time GARCH (generalized autoregressive conditionally het-eroscedastic). We then analyse the values and compare them with Black 76, themost commonly used model, when the underlying market prices are positive. Introduction
In March 2020, the prompt month WTI futures contract settled below zerofor the first time in the contracts history. Many market participants apply theBlack 76 model or some variation when calculating the value of the options onthis futures contract as a relatively straightforward, parametric valuation method.This calculation model is hard wired into many Commodity Trading and RiskManagement Systems. Traders and risk managers rely on its straightforward andreproducible output.However, Black 76 requires positive underlying market prices. The negativeprompt month settlement price caused considerable consternation among energytraders and risk managers.More generally, OTC options are also available on basis or differential prices.These transactions are options on the difference between two published indexes suchas NYMEX Henry Hub and AECO (for natural gas) or Cushing WTI and Houston(for crude oil). As such, these instruments frequently have negative underlyingmarket prices.Our task is to propose alternative models to Black 76 to valuate option priceswhen the underlying future contracts can assume negative values.Our methodology is the following one:(1) Take data (prices), sketch their behaviour, i.e., their evolution in time;(2) If the prices are positive and not mean-reverting, then use geometric Brown-ian motion (GBM) model for their evolution and Black-76 ([5]) formula foroption valuation of futures (see also formulas (BlCall) and (BlPut) in [7])(3) If the prices are positive and mean-reverting, then use continuous-timeGARCH (or, another name, inhomogeneous GBM model) model [8] andoption pricing formula (35) from [8], Theorem 5.1;(4) If the prices are both positive and negative, but not mean-reverting, thenuse Bachelier model and his formula, [2] (see also formulas (Ba 1) and (Ba 2)in [7]); a r X i v : . [ q -f i n . M F ] S e p OPTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES (5) If the prices are both positive and negative, and mean-reverting with mean-reverting level 0 , then use Ornstein-Uhlenbeck model [10] and the formulas(OUCall 1) and (OUCall 2) from [7];(6) If the prices are both positive and negative, and mean-reverting with mean-reverting level non-zero, then use Vasicek model [11] and the formulas(VasCall 1) and (VasCall 2) from [7].We note, that the most general, to the best of our knowledge, stochasticmodels for spot prices (both arithmetic and multiplicative) in electricityand related markets were presented in [3].In this paper we show how this methodology works on data sets presented byScott Dalton (Ovintiv Services Inc.), namely, we use WTI data set and NYMEXNG data set. 2. Definitions A primary security (or securities for short) is any asset that can be tradedindependently from any other asset, such as stocks. A derivative security or (orderivatives) are legal contracts conferring financial rights or obligations upon theholder.A forward contract is an agreement to buy or sell a risky asset (such as crudeoil or natural gas) at a determined future date T, known as delivery date , at aspecified price K, known as delivery price . The price of the asset (or commodity)at time t is known as forward price and denoted by F ( t, T ) . Notice K = F (0 , T ) . Similarly, a future contract (or futures for short) involves an underlying asset,which we typically take as a forward contract, and a specified delivery date T. Afuture price set at time t with delivery date T will be denoted as f ( t, T ) . An European option is a derivative security contract that gives the holder theright, but not the obligation to buy or sell the underlying asset, for a price K fixedin advance, known as exercise or strike price, at a specified future time T e , knownas exercise or expiry date. An option contract with expiry date T e stops beingvalid after this time. The option is known as a call option if the holder has theright to buy the asset, while a put option gives the holder the right to sell theasset.Forwards and futures are legal agreements between two parties giving obligationsbetween them, in contrast, options are legal agreements giving rights to the holder.Because of this advantage intrinsic in options (the holder may trigger the contractshould it be in their favour) is that they are to be purchased. We are concernedis valuing them, specifically, we are interested in valuing European call options forfutures prices. 3. Proposed alternative models to Black 76.
Black 76 model is obtained from the more general Black-Scholes model (1973).Black-Scholes is a model for the price of a stock at time t and it is given by thefollowing Stochastic Differential Equation (SDE) dS t = µS t dt + σS t dW t , where 0 ≤ t ≤ T represents time ( T is the expiry date), µ ∈ R is a number knownas the “drift”, σ > W t ) t ≥ is Wiener process (or Brownian PTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES 3 motion). In this model, S is deterministic (not random) and known in advance.Using It¯o’s formula, it can be deduced that S t = S e ( µ − σ ) t + σW t . This shows that under the assumptions of Black-Scholes, the stock price will bepositive (assuming S >
0) for all times.3.1.
Orsntein-Uhlenbeck (Vasicek) model (1930/1977).
The first alternativewe propose to Black 76 is given by the following Orsntein-Uhlenbeck SDE dS t = a ( b − S t ) dt + σdW t , (3.1)where a, σ > b ∈ R . Here a is known as the “reversion rate”, b as the meanand σ as the volatility. Again, using It¯o’s formula, it can be shown that the solutionto the OU SDE is given by S t = e − at S + b (1 − e − at ) + σe − at t (cid:90) e as dW s . (3.2) { Eq: OU SDE solution }{ Eq: OU SDE solution } This is a Gaussian random variable with mean e − at S + b (1 − e − at ) and variance σ (1 − e − at ) / a. It is readily seen it can assume negative values and as t → ∞ , thisGaussian random variable converges in distribution to a Gaussian with mean b andvariance σ / a, the rate of convergence is given by a. The value of an Europeancall option at time t with delivery date T e , rate of risk-free investment r, and strikeprice K is given according to (see formulas (VasCall 2) in [7]) C ( F, T e ) = e − r ( T e − t ) (cid:20) ξ + ( t, T e )Φ (cid:18) ξ − ( t, T e ) ζ (cid:19) + ζ Φ (cid:48) (cid:18) ξ − ( t, T e ) ζ (cid:19)(cid:21) , (3.3) { Eq: Euro Call Opt Price for OU }{ Eq: Euro Call Opt Price for OU } in which Φ is the distribution function of a standard Gaussian random variable and ξ ± ( t, T e ) = e ± aT e ( F ( t, T e ) − b ) − Kζ = σ (cid:114) − e − aT e a The future prices of this model will be modelled using (3.1).3.2.
Continuous Time GARCH model:
Some times the commodity prices ex-hibit different behavior with respect to time, which is known as Mean-Reversion.It means that, unlike stock prices that tend to change around zero, they tend toreturn to a non-zero long-term mean. Therefore for a risky asset S t which has amean reverting stochastic process, we have the following SDE: dS t = a ( b − S t ) dt + σS t dW t (3.4) { Eq: C-T GARCH }{ Eq: C-T GARCH } where W is a standard wiener process, σ > b ∈ R is the mean reversion level (the long term mean), and a > OPTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES Methodology and results
OU model.
According to (3.1), we need to calibrate the parameters a, b and σ. Using (3.2) (in which S is substituted for the future price F ) it can be seen thatobservations of the future price are in a linear relation plus normally distributederror terms. As such, least-squares linear regression can be used. In Fig. 3 wedo the calibration of the parameters for the OU model using Natural Gas futureprices provided by Ovintiv. We can see the prices are around the mean, which is anassumption of validity of the model.4.2. WTI Dataset.
For WTI crude oil futures, we compare the option pricescalculated by the Black-76 model and the Vasicek model. Let C ( t, T e ) be the valuefor the European call option written on a forward F. Then the Black-76 formula forEuropean call option price is: C ( t, T e ) = e − r ( T e − t ) [ F ( t, T ) N ( d ) − KN ( d )] , (4.1)where d , := ln ( F/K ) ± σ ( T e − t ) σ √ T e − t .The European call option formula for Vasicek model is similar to equation (3)with slight differences: C ( F, T e ) = e − r ( T e − t ) (cid:20) ξ ∗ + ( t, T e )Φ (cid:18) ξ ∗− ( t, T e ) ζ (cid:19) + ζ Φ (cid:48) (cid:18) ξ ∗− ( t, T e ) ζ (cid:19)(cid:21) (4.2) { Eq: Euro Call Opt Price for OU }{ Eq: Euro Call Opt Price for OU } in which Φ is the distribution function of a standard Gaussian random variable and ξ ∗± ( t, T e ) = e ± aT e ( F ( t, T e ) − b ∗ ) − Kζ = σ (cid:114) − e − aT e a where b ∗ = b − λσ/a, λ ∈ R is a market price of risk.We use the above formula for black 76 and monte carlo simulation to get thegraphs 1 of option prices. Each graph in Figure 1 shows the option prices withdifferent strike price K. Except for the chart with the price date of 2020-04-20, whenwe have a negative future price, the others are all positive. From the graphs we cansee that option prices on futures computed by the Vasicek model are very close tothe prices calculated by the Black-76 model when future prices are positive. Whenfuture prices are negative we can employ Vasicek model again to come up with thecall option prices. Here Black 76 model would fail as it does not accepts the negativeprices. Figure 2 shows the price of the option for various strike prices where wewere also able to calculate prices for negative strike prices. These prices have beencalculated using Monte Carlo simulation.4.3. NYMEX Natural Gas Dataset.
For the Natural Gas (NG) dataset, wewill see that all the underlying future prices are positive. However, they exhibita non-zero mean-reversion process over the time (figure 3). In this case, for thecorresponding option prices, we used the Continuous-Time GARCH model (as inequation 3.4).In this methodology, first we should consider the model (3.4) under a risk-neutralprobability P ∗ . Therefore, in a risk-neutral world our model will take the followinglook (for more details see [8], sec. 5.4): dS t = a ∗ ( b ∗ − S t ) dt + σS t dW ∗ t (4.3) { Eq: C_T GARCH risk-neutral }{ Eq: C_T GARCH risk-neutral } PTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES 5
Figure 1.
Black76 vs Vasicek models for Call option prices
Figure 2.
Option prices if futures prices becomes negativewhere: a ∗ := a + λσ, b ∗ := aba + λσ and W ∗ t is defined as W ∗ t := W t + λ (cid:90) t S ( u ) du. Here, λ ∈ R is the market price of risk .For this model (4.3) we have an explicit option pricing formula for European Call OPTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES
Figure 3.
Calibrations of parameters for different initial “pricedates.” Here the x -axis is the expiry date ( T ) and the y -axis is theprice per unit. The dotted line is the mean value b. option [8]: C ∗ T = e − ( r + a ∗ ) T S (0)Φ( y + ) − e − rT K Φ( y − )+ b ∗ e − ( r + a ∗ ) T (cid:20) ( e a ∗ T − − (cid:90) y zF ∗ T ( dz )] (cid:21) where, y is the solution of: y = ln KS (0) + (cid:18) σ a ∗ (cid:19) Tσ √ T − ln (cid:18) a ∗ b ∗ S (0) (cid:19) (cid:82) T e a ∗ s e − σy √ s + σ s dsσ √ T with, y + := σ √ T − y , and y − := − y , and, F ∗ T ( dz ) is the probability distribution under the risk-neutral probability P ∗ , asin [8].4.3.1. Methodology and results.
In this approach, to avoid the huge computations re-gards to the explicit formula, we used Least Square Regression method for calibratingthe parameters by following the methodology in [9], F i +1 = τ F i + µ + sd ( e ) , PTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES 7 to have the following equations: F x = n (cid:88) i =1 F i − , F y = n (cid:88) i =1 F i ,F xx = n (cid:88) i =1 F i − , F yy = n (cid:88) i =1 F i ,F xy = n (cid:88) i =1 F i − F i and then the following relationships can be considered: τ = nF xy − F x F y nF xx − F x ,µ = F y − τ F x n ,sd ( e ) = (cid:115) nF yy − F y − τ ( nF xy − F x F y ) n ( n − . For our purpose, we used the Euler approximation to simulate the future prices inorder to approximate the corresponding European Call option prices. F i +1 = F i exp a ∗ δ + b ∗ (1 − exp − a ∗ δ ) + σF i (cid:114) − exp − a ∗ δ a ∗ N , (4.4)Here, δ > F i prices are the exact discrete solution ofequation (3.4). Hence, we can find the following relations between the parameters: a = − ln τδ , b = µ − τ , σ = sd ( e ) (cid:115) − τδ (1 − τ )Finally, for the risk neutral parameters, the following adjustment has been applied: a ∗ = a + λσ, b ∗ = aba + λσ According to our dataset, there was not any access to the market option prices toestimate the market price of risk. Therefore, the following formula has been takeninto account: λ := dFF − rσ where, dFF is a returns on futures prices, r is the interest rates, and σ is the impliedvolatilities.The future prices were simulated 20 times (an exercise of this is shown in figure 4),and the average of them is applied in the payoff function. Then, the discount of theaverage of payoffs considered as the requested call option prices with continuous-timeGARCH model approach (results can be seen through figure (5) and (6)).Here, the risk-neutral parameters a ∗ and b ∗ has been estimated as 1 . . OPTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES
Figure 4.
The evolution of simulated future price with respect to time
Figure 5.
In this picture, we can see the evolution of the calculatedoption prices, according to the Continuous-Time GARCH model,with respect to their related strike prices is depicted5.
Conclusion
In this project, we worked with some useful alternative models which are helpfulfor valuation of options on future contracts. In spite of Black 76 is the most commonlyused model for valuating option future contracts in industry, it is necessary to havealternative models for the valuation when the prices’ behaviour differs from theprices describe by the same model. For WTI future option prices, O-U and Vasicekmodel has shown to have similar prices as Black 76 when future prices are positiveand have a valuation when negative prices, which is useful when irregular eventshappen. Also, for Natural Gas future option prices, continuous time GARCH alsodisplay comparable values as Black 76, further it allows us to calibrate a mean
PTION VALUATIONS OF FUTURES CONTRACTS WITH NEGATIVE PRICES 9
Figure 6.
In this table, the accuracy of our model comparing tothe known Black-76 model has been exhibited for the first 10 strikepricesreversion parameter to describe in a better way future option prices and theirbehaviour.As a recommendation, it would be useful for industry to keep a track on thistwo models to know how to react in unusual situations and double check their ownvaluation prices. This models have shown to be simple to understand, clear tocalculate and comparable with what the industry uses. With respect to the dataand results, in table 1 are some suggestions for the valuation according to the data’snature:
Future Prices Mean-Reversion Level Model
Positive none GBM modelPositive b Continuous-time GARCHNegative and Positive 0 OU modelNegative and Positive b Vasicek modelNegative and Positive none Bachelier model
Table 1.
Recommended model according to sign of prices andmean reversion behaviour.
Acknowledgement
We thank Scott Dalton (Ovintiv Services Inc.) for his time, for providing thedatasets, and his willingness to share his industry expertise. We also thank PIMScommittee and Professor Kristine Bauer for well-organized workshop.
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EnergyEconomics 30 (2008) . Dept. of Mathematics & Statistics, University of Calgary,, Calgary, Alberta, CanadaT2N 1N4
E-mail address : [email protected] Dept. of Mathematics & Statistics, University of Calgary,, Calgary, Alberta, CanadaT2N 1N4
E-mail address : [email protected] Department of Mathematics and Statistics, University of Regina, SK
E-mail address : [email protected] E-mail address : [email protected] Department of Statistics and Actuarial Science (SAS) Mathematics 3 (M3) Universityof Waterloo
E-mail address : [email protected] Dept of Mathematical and Statistical Sciences, University of Alberta at Edmonton,Edmonton, Canada, T6G 2G1
E-mail address : [email protected] Dept. of Mathematics & Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
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