Table of Contents – Financial Derivatives (FIN6028)
Comparison of exchange-traded and OTC derivatives. 2
ICE vs. Eurex Exchange – Options and Futures for BT Group Plc. 2
Futures contract for BT Group. 4
Estimates of prices on Eurex for BT Group. 5
Evaluation of call and put prices. 10
Forward contract for BT Group. 15
Swap contract between BT Group and Eagle. 16
Question 1
Question 2
Futures contract for BT Group
A futures contract is described by Chance and Brooks (2016, p. 275) as ‘an agreement between two parties to exchange an asset for a fixed price at a future date’. At the beginning of a futures contract, time 0, the value is zero, as no cash has been exchanged. Futures contracts are marked to market every day, and the two parties make margin deposits.
For this model it is required to have inputs of; share price, risk-free rate, any dividend payments and a time to expiration. The share price at the time of calculation is 125.80. The selected time to expiration is 1 year, as normal contango and normal backwardation was visible towards the expiration date. Therefore, would be interesting to see the outcome from the calculations. Based on this, the risk-free rate used will be 0.11%, gained from the rate for UK government bonds expiring in 1-year time. Finally, the value for dividends is 15.40p[A1] . As the year is yet to close, no value for final dividend payment is. However, the interim dividend payment of 2020 is the same as that for 2019. Based on this, I have taken the dividend payment for the full year of 2019. This could also produce some difference in the outcome. When comparing it to that on Eurex, and this should be taken into consideration.
The following formula is used: f0 = S0(1 + r)T – DT, the price of the future is equal to the stock price compounded by the risk-free rate, minus the dividends paid at time T, where T is expiration.
f0 = 125.80 (1 + 0.11%)1 – (15.40) = 110.54
There value of the future contract[A2] expiring in 1-year time, April 2021, on Eurex is 114.995. This shows that the future contract is over valued by the exchange. This could be due to the transaction costs of the contract. It also doesn’t account[A3] for any carrying or storage costs of holding the stock. Any information from the spot market could also affect the price in the futures market. Investors will also look at the bond market interest rate to determine the interested given up by holding the asset.
Estimates of prices on Eurex for BT Group
The inputs that differ across the three models are: the up and down factors for the two and three period model, which subsequently alter the value for p, and the risk-free rate in the Black-Scholes-Merton model as its required to be continuously compounded. The values that remain constant throughout are explained in the succeeding text.
The options expire on the third Friday of the month, so choosing an option with expiration in May gives 32 days to expiration at the date of calculation. As a decimal is 0.08767, and will be used in this format for calculations.
The models will be based on options with exercise price of 120 taken from Eurex, and the stock price for BT Group Plc at the time of calculation was 125.80. The exercise price of 120 was selected as it would be interesting to see the outcomes from an exercise price that is very close to the stock price in the current global situation that is affecting the trading markets.
The basic risk-free rate to be used throughout is 0.2%; based on that for 1-month UK Treasury bills[A4] .
The standard deviation requires historical information of the share prices for BT Group Plc. Daily data for 1 year was used, providing 251 data entries. Using a scope too large will distort the standard deviation, as there’s potential for larger fluctuation in a stock’s price over a greater period of time. Daily data was used to increase accuracy. Using this data, I calculated the standard deviation to be 23[A5] .67%. Further details of the calculation steps can be found under appendix A, with visual evidence under appendix B. Values used for each model summarised under appendix C.
Two-period binomial model
The binomial model assumes that there are two possible outcomes in each period; an increase or decrease in value. These are known as the up and down factors, and converge to one as the number of periods increase. A standard formula with inputs is used to calculate these, and the down factor is dependent on the up. The component ‘n’ refers to the number of periods for the model, hence a value of 2.
u = e to the power of [0.2367 (σ) multiplied by the (square root of 0.08767(T) over 2(n))]
= 1.0508
d = 1 / 1.0508 (u) = 0.9517
Using the valued for u and d, we can calculate that of p; the binomial probability.
Value of P = (1 + r – d) / (u – d) = 0.5076, and 1 minus P = 0.4924, as there are only 2 possible outcomes, therefore their probabilities must total 1.
For each value at the end of the period the calculation is: the up factor, for that period, multiplied by P, added to the down factor for that period, multiplied by 1 minus P. All of this, divided by 1 plus the risk-free rate.
Numerical calculation:
There are 3 possible stock prices at expiration at the end of two periods in a recombing tree. There are calculated as:
Su2 = 125.8 (1.0508)2 = 138.91 – The value increases twice.
Sud = 125.8 (1.0508) (0.9517) = 125. 81 – The value increases then decreases, or vice versa
Sd2 = 125.8 (0.9517)2 = 113.94 – The value decreases twice
A diagram of the two-period binomial model, including values, can be found under appendix D.
Three-period binomial model
For three period binomial model, its required for n to be 3, therefore altering the values for u and d:
u = e to the power of [0.2367(σ) multiplied by (the square root of 0.08767(T) over 3 (n))]
= 1.0413
d = 1 / 1.0413 (u) = 0.9603
Subsequently, calculating the new value for p:
Value of p = (1 + r – d) / (u – d) = 0.5148. 1 minus P = 0.4852
A diagram of the thr ee-period binomial model, including values, can be found under appendix E[A6] .
Black-Scholes Merton Model
Black-Scholes-Merton model assumes that the prices or return of the stock behave randomly, and an increase or decrease in the stock price cannot be predicted. They evolve in accordance to a lognormal distribution, and the return is normally distributed. It also assumes that the risk-free rate and volatility are constant. In reality, the risk-free rate is constantly changing. It assumes a frictionless market, where there are no taxes or transactions costs.
Due to the model’s assumptions on a stock’s return, the risk-free rate has to be continuously compounded. The log of the risk-free rate, 0.2%, is used for this model which is 0.001998, or 0.1998%. The other components required for BSM will maintain the same as used for the binomial model.
Entering this information into the BSM model provides a call price of 7.1016 and a put price of 1.2806. These values are very close to those provided by the binomial model. Visual evidence on the values can be found under appendix F.
Evaluation of call and put prices
Details of the comparative data for the call and put prices can be found in the appendix under appendix G and H respectively.
Evaluating the call prices first; the daily settlement price for a call with a strike price of 120, expiring in May 2020 on Eurex is 11.00, which is around 3.5 points higher than my calculation of 7.77 for two-period and 7.1016 for BSM model. However, for three-period binomial model, the gap is shortened with a call price of 9.67; still undervalued, just less so. As the number of periods increase, theoretically the value should converge to one value. Therefore, the increase of the value from two period to three period implies a convergence to the true value provided by Eurex.
The prices for the put are all very similar regardless of the model. The outcomes were; 1.46, 0.97 and 1.2806 for the two-period, three-period and BSM respectively. These are highly undervalued; however, this is shown by the increase in daily settlement price since the data was collected.
The differences in the prices could be due to the assumptions made for the models, which don’t reflect reality. In the BSM model, there is assumption of no dividends, however BT Group pays twice a year. The assumption of a frictionless market does not hold true, as these factors will be present when trading.
Hedge portfolio
A hedge portfolio is: h = (Cu – Cd) / (Su– Sd), where h is the number of shares needed to buy to every 1 call to achieve a hedge portfolio. Substituting in the values obtained previously:
(12.44 – 2.94) / (132.19 – 119.72) = 9.5 / 12.47 = 0.762.
Therefore, for every 1 call, 0.762 shares must be purchased in this scenario.
Numerical calculations:
Time 0
Write 100 calls @ £7.77 = (£777)
Buy 76 shares @ £125.80 = £9,560.80
Net investment = £8,783.80 = V0.
Time 0.5
The stock price goes up, Su. Values Su = 132.19, Cu = 12.44
Write 100 calls @12.44 = (£1,244)
Buy 76 shares @ 132.19 = £10,046.44
Net investment = £8,802.44 = Vu
Return = (Vu / V0) – 1
(8,802.44 / £8,783.80) – 1 = 0.0021 which is approximately the risk-free rate.
Adjusting the hedge ratio for the next period: hu = Cu2 – Cud / Su2 – Sdu = 18.91 – 5.81 / 138.91 – 125.81 = 1 new hedge ratio
To achieve this we need to either, buy back 24 calls or buy 24 shares, so that the ratio of calls to shares is 1:1. It is cheaper to buy back the calls, therefore:
Buy back 24 calls @ 12.44 = £298.56
This is borrowed at the risk-free rate
Summary:
76 shares @ 132.19 = £10,046.44
76 calls @ 12.44 = £945.44
Debt = £298.56
Alternatively, stock price goes down, Sd. Using values: Sd = 119.72, Cd = 2.98
Write 100 calls @ 2.98 = £298
Buy 76 shares @ 119.72 = £9,098.72
Net investment = 8,800.72 = Vd
Return = (Vd / V0) – 1 = (8,800.72 / 8,783.80) – 1 = 0.0019, approximately equals the risk-free rate.
Vd is approximately equal to Vu, confirming that the return is the same regardless of the stock movement; the risk-free rate.
Adjusting the hedge ratio for the next period: hd = Cud – Cd2 / Sud – Sd2 = 5.81 – 0 / 125.81 – 113.94 = 0.49
Here we should sell 27 shares to achieve the hedge, receiving money, and invest it bonds at the risk-free rate
Sell 27 shares @ 119.72 = £3,232.44
Summary:
100 calls @ 2.98 = £298
49 shares @ 119.72 = £5,866.28
£3,232,44 in bonds
Time 1
Stock increases a second time, Su2 = 138.91
Sell 76 shares @ 138.91 = £10,557.16
76 calls exercised @ 18.91 = £1,437.16
Repay the 298.56 loan @ r 0.2% = £299.16
Total portfolio: £10,557.16 – £1,437.16 – £299.16 = £8,820.84
(£8,820.84 / £8,802.44) – 1 = 0.0021, earning approximately the risk-free rate on the last period
Stock decreases after an increase, Sud = 125.81
Sell 76 shares @ 125.81 = £9,561.56
76 calls exercises @ 5.81 = £441.56
Repay the 298.56 @ r 0.2% = £299.16
Total portfolio: £9,561.56 – £441.56 – £299.16 = £8,820.84, equal to the value of the portfolio at Su2, therefore earning the risk-free rate on the last period again.
Stock increase after decrease, Sdu = 125.81, stock price equal to previous as it is a recombining tree
Sell 49 shares @ 125.81 = £6,164.69
100 calls exercises @ 5.81 = £581
£3,232.44 in bonds @ 0.2% = £3,238.90
Total portfolio: £6,164.69 – £581 + £3,238.90 = £8,822.59
(£8,822.59 / £8,800.72) – 1 = 0.0025, earning approximately the risk-free rate on the last period
Stock decrease a second time, Sd2 = 113.94
Sell 49 shares @ 113.94 = £5,583.06
100 calls expire out-of-the-money @ 0 = £0
£3,232.44 in bonds @ 0.2% = £3,238.90
Total portfolio: £5,583.06 – £0 + £3,238.90 = £8,821.96, approximately equal to the portfolio of Sdu, therefore earning the risk-free rate on the last period again
No matter which path taken by the stock price, it will always earn the risk-free rate of 0.2% on the previous period[A7] .
Put-Call parity
Isolating the risk-free bond in the equation gives us:
X(1 + r)-T = Pe(S0,T,X) + S0 – Ce(S0,T,X),
In words is: the risk-free bond is equal to a long position in the put, the stock and a short position in the call.
Initially calculating the value for the risk-free bond using theoretical values obtained from the BSM model.
Risk-free bond = 1.2806 + 125.80 – 7.1016 = 119.979
Now, using actual values for the call and put from Eurex Exchange.
Risk-free bond = 6.00 + 125.80 – 11.00 = 120.8
The difference between the values for the risk-free bond is very small; less than 1. Such a slight difference could be[A8] due to rounding in previous components of the calculation before entering them into the BSM model. Another reason for the variation could be simply that one is using theoretically calculated values, and the other is using actual. In order to prevent any opportunity for arbitrage, there will be transactions costs present. This could also affect the values and cause a difference.
However, with the call-put parity, it is important that it holds true, otherwise there is opportunity for arbitrage, allowing traders to take advantage of arbitrage profit.
Behaviour towards risk
Risk averse investors are less likely to take risks in their investments, and require to be compensated for the risk taken. The difference between investment and expected return is the risk premium. Most individuals can be characterised as risk averse. An example of risk-less investment is government bonds with the highest rating, AAA, as these are backed by the government issuing them. The return on investment is low, but certain.
On the other-hand, there are risk seeking investors. These are opposite of a risk averse investor and favour investments with high return, though with this comes the potential for a loss on investment equal to the amount invested. An example of a risky investment is in a new start-up company. This is because the company is brand new, with no historical data to prove its success, an investor could potentially lose all their investment if the start-up fails. Potential for high returns is proven by those who have invested in companies such as Google and Amazon in their early stages. The high risk in their initial investment has provided a high pay off in the end. However, at the time of investment in these companies, the return would’ve been unknown.
Risk Neutral have no attitude towards risk and it doesn’t affect their investment choices. They are focused and driven by the return on an investment, rather than the risk that occurs with it. Derivatives are considered risk neutral as they enable the process of hedging risk. Some examples of risk-neutral probabilities in derivatives are P, used previously in the binomial model[A9] for the probability of and increase or decrease in a stock’s value. Another is alpha that is present in the BSM model. Thorough derivatives, it is ensured that there is no opportunity for arbitrage in the market, which would favour those who are risk-adverse.
An analogy of risk behaviours can be found under appendix I.
Question 3
Forwards vs. Swaps
A forward contract is an agreement between two parties to exchange an asset for a fixed price at a fixed date, and is an over-the-counter traded derivative. At expiration, one party receives the asset, and pays the price of the future. The value of the asset must be at least equal to the price, otherwise the holder would sell the asset for profit. At initiation, the value is zero, as no cash has been exchanged. However, during the life of the contract, the value can increase or decrease depending on market behaviours. If you have a long forward contract, at expiration, you agree to receive the asset and pay the price of the forward contract. Therefore, your profit, or the value of the contract at expiration, is the price of the asset less the price of the forward contract. The price of the forward depends on the carrying costs of the underlying.
Similarly, with a swap, it is an agreement between two parties including an exchange. However, a swap is an exchange of a series of interest payments. There different types of swaps including currency, interest rate and equity swap. For the purpose of this explanation, we will focus on Plain Vanilla interest rate swaps. In this exchange, one party pays a fixed rate and the other will pay a floating or variable rate derived from the LIBOR rate. The value at the beginning is zero, the same as it is with a forward contract.
At initiation, the LIBOR rate is known only for the first payment which occurs on the settlement date. For example, a contract with quarterly payments, the first payment will be in 3 months’ time. The LIBOR rate for this payment will be known, and the 3-month LIBOR rate on the initiation date will be used. After the settlement period, the first payment is made on the 90th day. On this day, the 3-month LIBOR rate is then known for the payment that will occur on the 180th day, and so on for the future payments. Therefore, only the next payment value is known, and no further for the variable rate. The party receiving these payments will be the one paying the fixed rate. The fixed rate is determined by valuing the swap, and ensuring that the value of the fixed rate equal the value of the floating rate (VFXRB = VFLRB). This ensure there is no opportunity for arbitrage. On the settlement dates, only the net payments are made. If the LIBOR rate exceeds the fixed rate, the party paying the fixed rate will receive payment.
Both swaps and forward contracts are an exchange between two parties, and have a value at initiation of zero. However, during[A10] a forward contract, no money is exchanged hands, only at expiration is there an exchange. In a swap, there is an exchange of interest payment throughout the life of the contract.
In a swap, the credit risk is passed from one party to the other, with one being the bearer of all the contract risk. As the contract reaches expiration, the risk decreases. Arguably, the credit risk is higher in a forward contract as nothing is exchanged until the expiry date. Conflictingly with a swap, where there are payments made throughout the contract on agreed dates. These payments are netted, to reduce the amount of cash changing hands, thus reducing credit[A11] risk. This is called bilateral netting.
Forward contract for BT Group
The inputs for the formula include; stock price, risk-free rate, time to expiration and any dividends. The stock price of 125.80, as it has been throughout. The time to expiration is 1 year, therefore the risk-free rate is 0.11% based on UK Treasury Bills expiring in 1-year time. The dividends are assumed to be 15.4p for the year, based on those for 2019.
At the start of the contract, time 0, the value is 0, as no money has exchanged hands, this is represented as V0(0,T) = 0. The time doesn’t matter in a forward contract as the price doesn’t change; it is fixed at the expiration. Based on this and by using this formula: V0(0,T) = S0 – F(0,T)(1+r)-T = 0, we can isolate the future price, F(0,T), to get: F(0,T) = S0(1 + r)T.
Applying the selected inputs achieves: 125.80(1 + 0.11%)1 = 125.94. This is the carry arbitrage model.
BT Group pays cash dividends, assumed for the calculation[A12] to be 15.4p. We can adjust the spot price to account for dividends giving the formula:
Vt(0,T) = St – Dt,T – F(0,T)(1 + r)-(T-t)
Vt(0,T) = 125.80 – 15.4 – 125.94(1 + 0.11%)-(1-1) = 15.54.
This is just slightly higher than the estimated dividend, if we less the dividend, leaves 0.14. Which is the stock multiplied by the risk-free rate.
A forward contract helps to offset the risk by hedging. The future contract is an agreement to buy (or sell) the commodity at a later date for a set price. If an investor is worried about the potential decline in the price of their stock, they can enter into a forward contract to sell their stock for a set price in the future.
The model assumes that the risk of default is so low that it is irrelevant. However, in an exchange this is not true, as forward trading is unregulated and default risk is possible in the market. The contracts are also not standardised.
Swap contract between BT Group and Eagle
Pricing an equity swap involves calculating the fixed rate, at the start of the transaction, that is paid against the equity return.
The current LIBOR rate for 3 months is 0.403%. Assuming that the yield curve is upward sloping, the following rates for 6, 9 and 12 months were calculated: 0.42%, 0.437% and 0.453%.
Using a hypothetical notional[A13] amount initially of 1, we can calculate the fixed rate of the swap using the LIBOR rates. The fixed rate is represented by R, and is calculated using the present value of the fixed payments and the floating payments. Further details of the fixed payment calculation can be found under appendix J and K.
The assumptions of the model are: the swaps are not subject to margin requirements, not centrally cleared and not otherwise guaranteed by a third party.
The fixed payment = £1,000,000 X 0.452% X 90/360 = £1,130
Eagle receives quarterly fixed payments, and pays the return on BT Group’s stock. The return on the stock can only be calculated at the end of the swap term, as at initiation the return is not known, and this is done using the following formula: Cash flow: (notional amount) [(fixed rate) q – Return on stock over settlement period)]
Appendix
Appendix A – Standard deviation calculation steps.
Total up the adjusted close stock price, these are adjusted for dividends, to 43747.8, and dividing this by the number of data entries, 251, to calculate the mean of 174.3. Then subtracting the mean from each of the close stock prices, here is represented in the column headed ‘R-R bar’. Each of these values is then squared in column ‘R-Rbar2’. The mean of this column is then calculated in the same way as the adjusted close stock prices to give 557.9. Finally, the square root of this is take to give standard deviation of 23.67%
Appendix B – Visual evidence of standard deviation calculation
Appendix C – Summary of values used for binomial and BSM model calculations
| Two period | Three period | BSM | |
| T | 32 / 365 = 0.08767 | 0.08767 | 0.08767 |
| X | 120 | 120 | 120 |
| S0 | 125.80 | 125.80 | 125.80 |
| r | 0.2% | 0.2% | 0.2% |
| σ | 23.67% / 0.2367 | 0.2367 | 0.2367 |
| u | 1.0508 | 1.0413 | |
| d | 0.9517 | 0.9603 | |
| P | 0.4883 | 0.4908 | |
| rf | 0.001099 |
Appendix D – Two period binomial model recombining[A14] tree diagram (including values)
Appendix E – Three period binomial model recombining tree diagram (including values)
Appendix F – Black-Scholes-Merton model visual evidence
Appendix G –Details collected from Eurex for a Call with strike price 120
| Strike price | Bid price | Ask price | Diff to prev. day last | Last price | Daily settlem. price | Open interest date |
| 120 | 8.75 | 12.75 | + 25.71% | 11.00 | 11.00 | 02/28/2020 |
Appendix H – Details collected from Eurex for a Put with strike price 120
| Strike price | Bid price | Ask price | Diff to prev. day last | Last price | Daily settlem. price | Open interest date |
| 120 | 7.75 | 11.25 | +2.63% | 6 | 6 | 02/28/2020 |
Appendix I – Attitude to risk analogy
Suppose you are offered to play a game[A15] involving a dice roll. If the dice lands on an even number, you receive £4, if it lands on an odd, you receive £10. The chances of each happening are equal. Based on this, you would expect, if you played the game multiple times, that your average winnings would be £7. Those who are risk neutral would be willing to pay £7 to play. They don’t recognise the potential for loss of £3 if it lands on an even number, therefore not recognising the risk being taken. Risk neutral investors are indifferent to risk, and so they invest[A16] £7 for the potential to earn £7 return on average. Someone who is averse to risk would need to be compensated for the risk taken. They would recognise that, by paying £7 to play, there is the potential to lose £3 if an even number is rolled. This type of investor would be willing to pay less to play, say £5.50, but this depends on how risk adverse they are. The different between £5.50 and £7 is known as the risk premium, and it is to justify the risk taken. On the contrary, a risk seeker or taker, would be happy to play the game for more than £7, as they would be focusing on the potential to land on an odd number and winning £10, ignoring the potential risk of landing on an even number and only receiving £4, meaning a net loss.
Appendix J – Visual calculation of the fixed rate for a swap
Appendix K – Explanation of fixed rate swap calculation in Appendix J
The LIBOR rate for the first period was found to be 0.403%, a value for the last period was selected, assuming the yield curve is upward sloping. The values inbetween were calculated using interpolation formula. These have been transposed into the table under LIBOR rate, represented as a decimal. At the beginning of the swap the value is 0, therefore VS = VFLRB – VFXRB; this knowledge is used to calculate the value for R, by setting VS to 0. The formula used to calculate the values for discount bond price is presented beneath the table. The formula used to calculate R can be seen in the formula bar at the top: 1 over q, q being the number of days per period dividend by 360, multiplied by 1 minus the final bond price dividend by the sum of all the bond prices. This give a final fixed rate value of 0.4522%. This is used in the formula for fixed interest payments as seen, using the notional amount of £1,000,000.
References
Chance, D.M. and Brooks, R., 2015. Introduction to derivatives and risk management. Cengage Learning.
Eurex Exchange, April 2018. Contract Specifications for Futures Contracts and Options Contracts at Eurex Deutschland
Eurex Exchange, February 2020. Single Stock Futures at Eurex Exchange
Eurex Exchange, February 2020. The International Derivatives Exchange: Equity Options at Eurex Exchange
ICE Futures Europe, Feb 27, 2020. ICE Futures Europe: BT Group PLC Contract Specifications. accessed on 27th February 2020
https://www.eurexclearing.com/exchange-en/products/equ/opt/BT-Group-48848 accessed 13th April 2020
https://www.global-rates.com/interest-rates/libor/british-pound-sterling/british-pound-sterling.aspx accessed 7th May 2020
https://www.londonstockexchange.com/exchange/prices-and-markets/stocks/summary/company-summary/GB0030913577GBGBXSET1.html accessed 13th April 2020
https://www.marketwatch.com/investing/bond/tmbmkgb-01y?countrycode=bx accessed 1st May 2020
https://markets.ft.com/data/bonds accessed 13th April 2020
https://uk.finance.yahoo.com/quote/BT-A.L/history?period1=1555113600&period2=1586736000&interval=1d&filter=history&frequency=1d accessed on 13th April 2020 (excel sheet extracted of historical stock prices)
https://uk.investing.com/rates-bonds/uk-1-month-bond-yield accessed13th April 2020
[A1]Given the COVID and economic conditions, a dividend of this amount may not be a plausible assumption over the year.
[A2]Good work as you have rightly matched the maturity of the actual contract on Eurex.
[A4]Use of short term interest rates should have been justified.
[A5]Good work justifying your assumption.
[A6]As P3 < P2, what factors could have explained this should have been discussed.
[A7]Goo work
[A8]Bond prices from put-call parity with actual and estimated prices calculated and compared as required.
[A9]Risk attitudes explained with examples. Good work.
However, the logic underlying the use of risk neutral pricing of derivatives should have been objectively explained with reference to the concept of hedged portfolio.
[A10]The similarities and differences between both should have been clearly identified and described as required.
[A11]Credit risk is minimal at start of the swap and highest in the middle of the swap tenure.
[A12]Dividends should have been adjusted in the pricing of the forward and not in the valuation. Also the valuation of forward ignores the time value of money for the dividends.
[A13]Great work. Calculations are correct and explained well.
[A14]P3< P2, should not be the case as it breaches the boundary conditions.
[A15]Great work providing the actual option contracts that have been evaluated.
[A16]Good work.
[A17]Great work. Calculations are objective and correct.
