Energy and Weather Derivatives Coursework SMM

Antonis Iaponas (070018822)

Problem 1

For Task 1 we have used Bloomberg and we have collected weekly WTI crude oil spot prices for the last 10 years starting from 09.06.00 to 29.06.11. We have as well collected the corresponding future price data of one month of future contracts for the light sweet crude, and we matched the periods with those of the spot price in order to be the same. All together we have 577 datapoints of prices both for spot and futures. Now the first five years of our data was used for the estimation of the OLS (minimum variance hedge ratio MVHR) and the remaining data was used in order to calculate and then test the effectiveness of each following hedge strategies.

One to one hedge {static} OLS estimation hedge (MV) {static} Rolling window OLS hedge {time varying} Exponentially weighted hedge {time varying}

One to one hedge: For a long position in one unit of a corresponding commodity, we have to short futures which are on the same unit of underlying. The opposite happens for a short position, where we have to buy (long) the same unit of underlying futures

Minimum Variance: By calculating log returns of both spot and future, we can calculate the MHVR γ. After we evaluate γ we can hedge the position of the commodity for the period needed. In order to hedge 1 unit of commodity γ future contracts are used.

γOLS=Cov (∆ lnS ,∆ lnF )

Var (∆ lnF ); (1.1)

S - spot price of the commodity;

F– futures price;

∆ lnX - series of log returns.

Rolling window hedge: This incorporates the exact same calculations, but we use a selected frequency in order to re estimate the hedge ratio. In this question since we are using only the first 5 years of the data and because estimation and testing samples have each, 5 year of data, as a result estimation window of one year is considered. So each year estimation window rolls for one year in the future. Therefore using the new data the hedge ratio for the following one year period is evaluated.

Exponentially weighted hedge ratio: It re estimates hedging position for every year. This can be shown by the following

γ EW=Covt (∆ lnS ,∆ lnF )

Var t(∆ lnF);

(1.2)

Cov t (∆ lnS ,∆ lnF )=λCov t−1 (∆ lnS ,∆ lnF )+(1−λ)∆ ln St ∙∆ lnF t; (1.3)

Var t (∆lnF )=λVart−1 (∆ lnF )+(1− λ)∆ lnF t2. (1.4)

We then collect the values for variances and covariance’s using the data. So all the following values were evaluated using equation (1.2) and (1.5). Since the data that we used was weekly, we had re estimated the hedge ratio for weekly basis, but by decreasing frequency of the calculations that were put in effect, this has an effect on the accuracy.

strategy hedging effectiveness is given by the following expression

HE=1−Var (∆ ln S t−γ̂ ∆ ln Ft)

Var (∆ ln S t); (1.5)

γ̂ – estimated hedge ratio.

The calculations that we performed were done using MATLAB and then we have placed our results into Coursework.xls file.

Strategy One-to-one MVRolling

Window OLSEWMH

Hedge ratios 1 0,985891

0,98589 0,985891

0,98333 0,9855

0,98042 -

1,00513 0,998394

1,0095 0,998537

Hedging effectiveness

94,51% 94,47% 94,49% 94,06%

Table 1.1

By looking at the table 1.2 we can observe that One to one hedging strategy performed the most effective hedge. Next best in effective hedging is considered to be Rolling Window OLS then Minimum Variance and last Exponentially Weighted Hedging Average ratio (EWAHR).

The following results are based on the data we have chosen. The techniques used are based on correlations methods and thus use the ratios obtained from correlations between the future returns and spot returns. Although spots and futures are usually highly correlated, they don’t tend to be stable in time. The strategies use correlations that existed between the series in the past in order to obtain the hedging ratio for the next period. If from one period to another period we have high correlation change, this can affect historic values and vary in values from the hedging period. As a result we have less hedging effectiveness when we compare the one to one hedge from the other strategies which are model based. Adding to that, rebalancing of our hedging portfolio is related with transaction cost, which we haven’t taken into consideration.

Other hedging methods that we should consider is first of all GARCH model, which estimates time changing covariance’s, another one is Markov regime switching approach proposed by (A.Alizadeh et al 2008). Also if correlation of future prices and spot prices changes a lot from one day to another then we can use co integration-based methods proposed by (Alexander 2009).

Problem 2

Now for this Task we have used monthly Henry Hub natural gas spot and future prices for the period January 2006 to May 2011 and in order to estimate 12 different maturities of futures we used 1 to 12 months maturities.

An assumption is made that the commodity spot prices follow a mean-reverting processes that can be mathematically expressed as:

dSS

=α (μ−lnS )dt+σdW (2.1)

Where:

μ is the long-run log-level to which the price reverts

α is the mean reversion rate

σ is the constant volatility of the spot price

W is the Wiener process

and S is the spot price of the commodity

All of the above parameters were given values from data collected and for this case the regression model was used.

Regression model ∆ ln S t=α0+α1 ln S t−1+ϵ t; (2.2)

ϵ tapproximates ¿ ∈(0 , σ2).

For the regression model the coefficients are approximated by OLS however the parameters are computed with

α̂ 0=αμ Δtα̂ 1=−αdt }=¿α=

− α̂1Δt

,μ=− α̂0α̂1

;

(2.3)

With the help of Matlab software (Script Question2.m) all the estimations are computed and then uploaded into the Microsoft excel (Coursework.xls) file, the results can be manipulated and the following results are given:

α=1.8064; (2.5)

μ=1.6602; (2.6)

σ=0.1833; (2.7)

dSS

=1.8064 (1.6602−lnS )dt+0.1833dW . (2.8)

As it has been discussed mathematical expression (2.1) is based on an assumption process for the spot prices therefore it can be used to conclude to another mathematical equation for the forward price:

F (t ,T n )=exp(e−α (Tn−1 )ln St+ (1−e−α (Tn−t ) )(μ− σ 2

2α )+ σ2

4 α(1−e

−2α (T n−t ) )) (2.9)

Now by substituting the parameters previously computed by MATLAB (α=1.8064and

σ=0.1833) into the equation (2.9) the forward curve that corresponds to the mean-reverting process for the spot price can be graphically drawn.

The next step is to approximate volatilities of futures prices and once the prices are estimated the exponential function must be fitted in their structure. The mathematical model for the spot dynamics can be expressed as:

σ F ( t , Tn )=σ e−α (T n−t )

.

(2.10)

We approximate α > 0 by minimizing the sum of squared residuals and using the coefficient σ=0.1833 from equation (2.7) and the historical volatilities.

Therefore:

∑i

ϵ i2=∑

i(σ F ( t ,T i)

−σ e−α (T i−t ))→min

(2.11)

Using the excel solver we have:

αF=0.11412(2.12)

At this point it is immediately noticed that αF<α parameter obtained differs from the one in (2.5). One possible reason is that the futures prices and volatilities using the model are derived both from spot price dynamics. Now the forward curve in figure 2.1 in general prices derivatives and more precisely the swap contracts. The fixed for floating swap price can be computed using the expression:

SP (t )= 1m∑k=1

m

P (t , sk) (F (t , sk )−K ) (2.13)

Where:

K is the fixed price

P(t,sk) is the discount factor for the period from t to sk

and F(t,sk) is forward price at time t for with the delivery date sk

Now when we fix the price at the origin to be zero we get:

K=∑k=1

m

P (t , sk) F ( t , sk )

∑k=1

m

P (t , sk )

(2.14)

Then by means of the forward curve and by substituting the parameters in the expression (2.14) the value of K can be calculated assuming an annual risk free rate of 10%:

Therefore:

K̂=4.973 (2.15)

The value of K based on the observed forward curve:

K=4.976 (2.16)

It is observed that there is a slight variation in the K values and the reason is obviously because of the difference in the forward and observes curves. The Figure 2.1 curve shows the mean-reversion on spot prices. The long-run expectations on the price are the ones who give rise to the forward. The only information that is taken into account at present is the current spot price. Provided that the final observed price is higher from long-run mean. The model 2.1 assumes that the price will steadily converge to this mean.

The forward prices in the models because they are heavily depended on the spot prices, they experience a price decrease. Therefore we can see a decrease in the swap price in order to compensate the floating rate which is related with the forward price. On the other hand the current observed forward curve shows the fact that market participant expectations about spot

prices are in a shorter viewpoint. So by looking at the observed graph, Henry Hub price is expected to go up, thus increasing the swap price as well.

Problem 3

In the Task 3 we are going to find an intrinsic value by means of forward curve, this intrinsic value is the storage contract, model based and with chosen parameters. In Figure 3.1 we can see the forward curve from period July 2011 to June 2012.

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03 N

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2011

23 D

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2011

11 F

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2012

01 A

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4.6

4.7

4.8

4.9

5

5.1

5.2

5.3FWD curve (Natural Gas)

Pric

e

Figure 3.1 Forward curve for Henry Hub natural gas

Using the current forward curve we are going to maximize cash flows from usage of storage facility.

Assumptions for storage facility:

Maximum Capacity, Qmax 20 000 000 MBTU

Maximum Withdrawal rate, Wmax 300 000 MBTU/day

Maximum Injection rate, Imax 1 000 000 MBTU/day

Injection cost, ci 0.02 $/MBTUWithdrawal cost, cw 0.01 $/MBTUr 0.01

Table 3.1 Characteristics of the storage facility

Spreads discounting is done in this way:

ΔF ij=(F i−cw )∙ e−r t i−(F j+ci) ∙ e−r t j; (3.1)

where Fi – current forward price for time ti, at which gas is sold;Fj – current forward price for time tj, at which gas is bought;cw – cost of withdrawal of 1 MBTU;ci – cost of injection of 1 MBTU;r – risk free rate.

Using Table 3.1 we get the following spreads shown in Table 3.2. Then we form a table (3.3) with injections and withdrawals which are arbitrary. Then the following spreads, withdrawals and injections which are cash flow related, are selected and calculated using the following

C F ij=V ij ∙max (ΔF ij ,0). (3.2)

The max (ΔF ij ,0) states that if F ij (spread which is discounted) is positive then we make a transaction ij. This is a common sense assumption that takes place only if we have a profitable transaction.

Each cash flow which corresponds to the parameters and the forward curve can be seen in Table 3.4. Here total cash flows are given as the summation of all other cash flows.

TCF=∑i∑j>i

C F ij. (3.3)

Now by maximizing the total cash flows we can compute the intrinsic value of the storage facility but it has to be subject to the amount of constrains we have thus we get:

TCF→max, (3.4)Subject to:

{V ij≥0 ,

I i=∑j

V ij ≤ Imax ,

W j=∑i

V ij≤Wmax ,

Qi≤Qmax .

(3.5)

where Imax – maximum injection per day;Wmax – maximum withdrawal per day;Qmax – maximum storage capacity.

The following calculations were done in Excel in the file Coursework.xls. The storage facility value is given as:

StorageFacilityValue=$7 144 680,82

(3.6)The prices that we consider are from the current forward curve, thus the

StorageFacilityValue is just a static value, but the forward curve varies through time, thus making the chosen positions incorrect. Therefore an extrinsic value exists on the storage facility which shows how it reacts with price ups and downs. We can make this possible by re-adjusting the position in order to follow the movements of the forward curve. Thus we need to optimize each period and conditioning on the movements of the forward curve. Thus if forward curve moves in un-ideal way, we then keep the previous position. The storage facility extrinsic value is obtained using the American option pricing method (Thompson et al, 2008).

Problem 4

This time we have daily spot prices for Henry Hub Natural Gas commodity from 30.05.06 to 27.06.11, resulting in a sample of 1268 observations.

Generic GBM process is characterised by equation

dSS

=μdt+σdW (4.1)

with S being the spot price, μ and σ characterising constant drift and volatility respectively, and W being Wiener process. To estimate the parameters we fit a simple OLS regression:

Δ ln S t=μ+ϵ t ; with ϵ t ∈(0 , σ2) and receive the implied coefficients from it.

With MRJD, GBM is enhanced by addition of mean reversion facto α , long-run mean price μ

and an average jump size λ characterised by a Poisson process q:

dSS

=α (μ−lnS )dt+σdW + λdq

As proposed, we use recursive filtering to estimate λ, jump

frequency φ and jump series volatility γ, by consequently eliminating the jumps (outliers being further than 3σ from current

subsample μ), finding new subsample σ and repeating the process until no outliers are left. Arranging the eliminated terms into a series, we receive jump observations and find the implied parameters from with basic statistical techniques.

The resulting parameters estimates are given to the left.

Having estimated the processes’ parameters, we are able to use discretized versions of processes to simulate price paths of the commodities predicted by GBM and MRJD:

St=S0 e(μ−1

2σ 2)t+ϵ tσ √t for GBM and

St=S0e(α (μ−ln St)−

12σ 2)t+σ ϵ1 , t√ t+ (λ+ γ ϵ2 , t) (ut<ϕt )for MJRD, whereϵ t is a plain random and ut is a uniformly

distributed random variable between 0 and 1.

Further, we merely plug the simulated price series into for Asian call and put options price equations:

C ( t ,T )=e−r (T−t )( 1N ∑j=1

N

max(∑i

S j ,i−K ,0)); P(t ,T )=e−r (T−t )( 1N ∑j=1

N

max (K−∑i

S j ,i ,0)) And take into account the in-the-moneyness spread.

The results of this are in 4 tables below:

GBM Call Options          Moneyness/T 1M 2M 3M 4M 5M 6M

90% 0,420995430,4168968

30,4139258

40,4120709

30,4063470

60,4042271

9

95% 0,210357230,2080228

90,2068817

40,2070996

90,2038870

90,2041094

5

ATM 0,011999850,0236239

80,0318556

50,0388949

20,0426281

70,0474565

9

105% 0 1,6142E-060,0001513

10,0006161

70,0016130

50,0024669

4110% 0 0 0 0 1,5596E-06 2,5941E-05

MRJD Call Options          

Moneyness/T 1M 2M 3M 4M 5M 6M90% 0,5335098 0,7460761 0,9174926 1,0699252 1,2029969 1,3134463

GBM parameters

mean 0,00029678

std 0,04381888

MRJD parameters

alpha 1,40721772

mean 1,66422574

std 0,03815041

lambda 0,02602158

phi 7,69534333

gamma 0,16009174

6 6 5 6 4 8

95%0,3343075

80,5614325

4 0,73541030,8897040

8 1,02419931,1349745

1

ATM0,1482817

90,3913268

70,5670495

70,7228696

1 0,8576931 0,9681799

105%0,0713395

40,2590701

30,4198282

50,5735803

30,7073628

40,8150612

1

110%0,0435142

30,1888349

60,3168435

80,4509460

80,5755669

20,6777412

7

GBM Put Options          Moneyness/T 1M 2M 3M 4M 5M 6M

90% 0 0 0 0 03,7251E-

06

95% 02,3096E-

068,4945E-

050,000425

20,001218

370,0018605

8

ATM0,012280

820,024479

650,032187

890,037616

870,043637

790,0471823

3

105%0,210919

180,209733

540,207612

60,204734

570,206301

01 0,2041673

110%0,421557

380,418608

180,414590

330,409514

850,408367

860,4037009

1

MRJD Put Options          Moneyness/T 1M 2M 3M 4M 5M 6M

90%0,007604

030,026433

250,031094

560,034293

760,036498

020,0347873

6

95%0,019039

950,050665

880,056141

250,059469

020,061378

720,0582900

9

ATM0,043652

370,089436

460,094909

56 0,0980310,098550

860,0934700

9

105%0,177348

320,166055

980,154817

280,154138

160,151898

950,1423260

1

110%0,360161

220,304697

070,258961

650,236900

360,223781

360,2069806

9

For illustrative comparative purposes, we have made a simple table showing when MRJD simulated option price is higher than that of GBM version (green if MRJD price is greater by more than 0.5, yellow if it is greater by a number between 0 and 0.5 and red otherwise):

We can see that Call option MRJD-simulated prices is always greater than predicted by estimated by GBM – which reflects general empirical notions. We can see that the smallest difference occurs for short-term maturities coupled with in-the-moneyness and difference grows as maturities become longer and we move from in-the-money options to out-of money cases. This can be accredited to each process specifics, namely presence of jumps and mean-reversion components in MRJD: taking into account that long-run price mean for MRJD is very high and much higher than the most recent spot, it is very much expected that price will grow due to mean reversion. Additionally, jumps allow for spiking crosses of in-the-moneyness ‘barrier’, while relatively continuous GBM price path does not. The same reasoning, partly reversed, can be used to explain the situation with put option price difference – explosive growth patterns of MRJD process predicted higher price levels in general and effect of this on put options is, essentially, directly opposite.

For the last part of the question, we use the following equations to calculate implied volatilities using Black-Scholes pricing formula and construct volatility surfaces:

δ call ( x ,C ,S ,K , r ,T )=SN (d1 )−e (−rT ) KN (d2 )−C ;

δ put ( x ,P ,S , K ,r ,T )=e−rT KN (−d2 )−SN (−d1 )−P;

where d1=

lnSK

+(r+ x2

2 )Tx √T

; d2=d1−x√T ;

What we see here for call options is a very peculiar volatility surface, with the whole volatility range being negative across all strikes and maturities, and exhibiting a sharp drop in volatility for short-term low strike-price contracts. The reasoning behind this could be best addressed to using Black-Scholes European Call pricing model for Asian options (explaining effectively zero volatility

levels by essentially lower prices of Asian options) and, again, particularly spiked behaviour of MRJD process when it comes to low strikes – with call options, similar to intuition in previous part, this results in sharp reduction in volatility and could be explained by the nature of the process (synthetic substitution of process parameters could make the volatility surface smoother). Volatility spike at a higher strike and lower term follows the same intuition of MRJD process sensitive response to price path breaking the barrier.

Put option volatility surface, on the other hands, features a fairly expected shape in that case, which follows more obvious specifics of MRJD process in general – high volatility for long maturities and high strike prices is explained by greater time frame allowed for a jump in the process to occur – which essentially increases the variance of option price. On the other hand, we could say that relatively low volatility at short-term maturities could be accredited to distinctive mean-reversion component in the MRJD process.