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Figure 1.1 shows calculation for the fair price of BT Groups future contract. BT`s stock price at the 24th Apr 2020 was used at £115.45 for a one-year futures contract. There are two assumption for this calculation, the first of which is the risk – free rate which has been valued at 0.05[A1] which is the UK Gilt 2 Year Yield. The second assumption is the dividends which is valued at 10.78p, from the[A2] 9th of September 2019 as it is BT`s earliest final dividends paid to this date. For this calculation DT is the present value of dividend, equalling 11.32p. Giving a fair value of £109.90.
The actual price of BT Groups future contract[A3] on Eurex Exchange is £105.15. This shows that the calculated fair value of the future illustrated in figure 1.1, when compared to the actual value on the Eurex exchange is overvalued. We are willing to pay £109.90 for 1 stock of BT Group one year from now due to our calculations. However, their actual price is less than this and so we are overvalued. This could be due to the fluctuation of interest rates. This can affect the risk-free rates and either increase or decrease the future`s fair value. Also, market efficiency can also affect the price as a more efficient market allows for more accurate and frequent price changes.
BSM Model
Below, are calculations for the BSM model in pricing put and calls. BT`s stock price at the 24th Apr 2020 was used at £115.45 for a one-year call and put calculation. An exercise price of 100 as this seems to fit the model well. Standard deviation was calculated from the logarithmic returns of BT Group for the past year and is 0.3739 (37.39%), giving a variance of 0.1398. This was an assumption made as no accurate figure was present. Another assumption would be the risk – free rate at 0.51%. This was derived from the 10-year UK gilts as they are regarded as risk free returns. Giving a call price of 25.1470 and a put price of 9.1883.
The call and put prices calculated however, are different from the actual call and put prices available on the Eurex market. At a strike price of 100, the actual call price is 6.50, and the actual put price is 1.50. These differences are mainly due to the high volatility of 37.89% in this case and the long-time horizon of one year and a market crash. The call calculated of 25.1470 is overvalued in comparison to the actual price of 16.75 as it is much cheaper on the Eurex market. The same is true for the calculated put as we can sell it for our price of 9.1883 but it is valued at 0.75 on the Eurex market[A4] .
BSM calculations
standard deviation = 0.3739
variance = 0.1398
time to expiration = 1 year
risk free interest rate = 0.51%
stock price =115.45
exercise price =100
dividends yield = 0
Put = 9.1883 Actual put = 1.50
Call = 25.1470 Actual call = 6.50
C = S0N(d1) – Xe-rcTN(d2)
C = 115.45(0.7206) – 100-0.051(0.5835)
= 25.1470
d1 = ln(S0 / X) + (rc + σ2 / 2)T
σ √T
d1 = ln(115.45/100) + (0.051 + 0.37392 / 2)1
0.3739√1
= 0.5848
d2 = d1 – σ √T
d2 =0.5848 – 0.3739√1
= 0.2109
N(d1) = 0.7206
N(d2) = 0.5835
P = Xe-rcT(1 – N(d2)) – S0(1 – N(d1))
P = 100-0.051(1 – 0.2109) – 115.45(1 – 0.5848)
Calculations for the BT Groups call and put options can be demonstrated below in the two-period model. For both the call and put option, stock price was values as at 24th Apr 2020 of £115.45, exercise price is 115 and was the chosen option from the Eurex market and the expiration for these options is 90 days. Assumptions were made regarding the volatility which was calculated as the standard deviation of BT Groups returns for the past year. This was necessary to be able to determine the up and down factors which totalled to 1.14% and 0.87% [A6] respectively. The risk-free rate was derived from the UK Gilt 2 Year Yield of 0.05. This gave a call price of 14.27 and put price of 2.93. when comparing these to the actual price traded on the Eurex exchange, with a put price of 1.50 and call price of 6.50 at the exercise price of 115. Showing that the figures calculated are overvalued in comparison to the actual figure on the Eurex market. Figure 2.1 demonstrates the two-period binomial stock, call, and put path.
Two period call calculations
S = 115.45 σ = 0.374 T = 90 days
X = 115 u = 1.14%
R = 0.05 d = 0.87%
u = eσ√T/n = e σ√0.247/2 = 1.14%
d = 1/1.14 = 0.87%
U2s = 115.45 (1.14)2 = 150.04
Uds = 115.45 (1.14)(0.87) = 114.50
D2s = 115.45 (0.87)2 87.38
Cu2= Max(0,U2S, – X)
= (0, 150.04 – 115) = 35.04
Cud= Max(0,Uds – X)
= (0, 114.50 – 115) = 0
Cd2= Max(0,d2s– X)
= (0, 87.38 – 115) = 0
= (1.05-0.87) / (1.14 – 0.87) = 0.67
1 – P = 1 – 0.67 = 0.33
Cu= Pcu2 + (1-P)Cud
(1+r)
= ((0.67)35.04 + (0.33)0) / (1.05) = 22.36
Cd= PCud+ (1-P)Cd2
(1+r)
= ((0.67)0 + (0.33) 0) / (1.05) = 0
C= PCu + (1-P)Cd
(1+r)
= ((0.67)22.36 + (0.33)0) / (1.05) = 14.27
Two period put calculations
S = 115.45 σ = 0.374 T = 90 days
X = 115 u = 1.14%
R = 0.05 d = 0.87%
u = eσ√T/n = e σ√0.247/2 = 1.14%
d = 1/1.14 = 0.87%
U2s = 115.45 (1.14)2 = 150.04
Uds = 115.45 (1.14)(0.87) = 114.50
D2s = 115.45 (0.87)2 87.38
Pu2= Max(0, X – U2s)
= (0, 115 – 150.04) = 0
Pud= Max(0, X – Uds)
= (0, 115 – 114.50) = 0.5
Pd2= Max(0, X – d2s)
= (0, 115 – 87.38 = 27.62
P= (1+r-d)/(u-d)
= (1.05-0.87) / (1.14 – 0.87) = 0.67
1 – P = 1 – 0.67 = 0.33
Pu= Ppu2 + (1-P)Pud
(1+r)
= ((0.67)0 + (0.33)0.5) / (1.05) = 0.16
Pd= Ppud+ (1-P)Pd2
(1+r)
= ((0.67)0.5 + (0.33)27.62) / (1.05) = 8.99
P= Pu+ (1-P)Pd
(1+r)
= ((0.67)0.16 + (0.33)8.99) / (1.05) = 2.93
Three period binomial model
For the three-period binomial model, the same option on Eurex was used at 115 exercise prices as the two-period binomial model and all other inputs as demonstrated below. However, when calculating the binomial up and down parameter, these differed to 1.11 up and 0.90 down. The decrease is due to the extended n periods with a fixed times frame of 90 days. The same assumptions were made for the risk-free rate and volatility as the two-period model. Deriving the call price to 17.32 and put price of 5.46. when comparing these to the actual price traded on the Eurex exchange, with a put price of 1.50 and call price of 6.50, the values calculated are still overvalued. Figure 2.2 demonstrates the three-period binomial stock, call and put path.
Three period call calculations
S = 115.45 σ = 0.374 T = 90 days
X = 115 u = 1.11%
R = 0.05 d = 0.90%
u = eσ√T/n = e σ√0.247/3 = 1.11%
d = 1/1.11 = 0.90%
U3s = 115.45(1.11)3 = 157.89
U2ds= 115.45(1.11)2 (0.90) = 125.73
Ud2s= 115.45(1.11 (0.90)2 = 103.80
D3 s= 115.45(0.90)3 = 84.16
Cu3= Max(0,U3S – X)
= (0, 157.89 – 115) = 42.89
Cu2d= Max(0,U2ds – X)
= (0, 125.73 – 115= 10.73
Cud2=Max(0,Ud2s – X)
= (0, 103.80 – 115) = 0
Cd3=Max(0,D3s – x)
= (0, 84.10 – 115) = 0
P= (1+r-d)/(u-d)
= (1.05-0.90) / (1.11 – 0.90) = 0.71
1 – P = 1 – 0.71 = 0.29
Cu2= Pcu3 + (1-P)Cu2d
(1+r)
= ((0.71)42.89 = (0.29)10.73) / (1.05) = 31.97
Cud= Pcu2d + (1-P)Cud2
(1+r)
= ((0.71)10.73 + (0.29)0) / (1.05) = 7.26
(1+r)
= ((0.71)0 = (0.29)0) / (1.05) = 0
Cu= Pcu2 + (1-P)Cud
(1+r)
= ((0.71)7.26 + (0.29)0) / (1.05) = 23.62
Cd= Pcud + (1-P)Cd2
(1+r)
= ((0.71)7.26 + (0.29)0) / (1.05) = 4.91
C= Pcu + (1-P)Cd
(1+r)
= ((0.71)23.62 + (0.29)4.91) / (1.05) = 17.32
Three period put calculations
S = 115.45 σ = 0.374 T = 90 days
X = 115 u = 1.11%
R = 0.05 d = 0.90%
u = eσ√T/n = e σ√0.247/3 = 1.11%
d = 1/1.11 = 0.90%
U3s = 115.45(1.11)3 = 157.89
U2ds= 115.45(1.11)2 (0.90) = 125.73
Ud2s= 115.45(1.11 (0.90)2 = 103.80
D3 s= 115.45(0.90)3 = 84.16
Pu3= Max(0, X – U3S)
= (0, 115 – 157.89) = 0
Pu2d= Max(0, X – U2ds)
= (0, 115 – 125.73) = 0
Pud2= Max(0, X – Ud2s)
= (0, 115 – 103.80) = 11.20
Pd3= Max(0, X – d3s)
= (0, 115 – 84.16 = 30.84
P= (1+r-d)/(u-d)
= (1.05-0.90) / (1.11 – 0.90) = 0.71
1 – P = 1 – 0.71 = 0.29
Pu2= Ppu3 + (1-P)pu2d
(1+r)
=((0.71)0 + (0.29)0) / (1.05) = 0
Pud= Ppu2d + (1-P)pud2
(1+r)
= ((0.71)0 + (0.29)11.20) / (1.05) = 3.09
Pd2= Ppud2 + (1-P)pd3
(1+r)
= ((0.71)11.20 + (0.29)30.84) / (1.05) = 16.09
Pu= Ppu2 + (1-P)pud
(1+r)
= ((0.71)0 + (0.29)3.09) / (1.05) = 0.85
Pd= Ppud + (1-P)pd2
(1+r)
= ((0.71)3.09 + (0.29)16.09) / (1.05) = 17.69
P= Ppu + (1-P)pd
(1+r)
= ((0.71)0.85 + (0.29)17.69) / (1.05) = 5.46[A7]
Hedge portfolio
Figure 3 below shows that the price calculated in the two-period binomial model is fair as the risk-free returns denoted as “Rh” are approximately equal to the risk-free rate used in the calculations of 0.05. all hedge rations have been adjusted for accordingly and necessary actions have been made.
Put-call parity
The put and call prices used in estimating the risk-free bond were derived from the BSM model calculations committed[A8] previously. Call price equals 25.15 and put equals 9.19. the risk-free rate used is the UK Gilt 2 Year Yield of 0.05 and the current stock price is 115.45 with an exercise price of 100. Following the formula below, the put-call parity using these inputs equals 124.6 = 120.27, with a risk-free bond of 95.12. this differs from the actual put and call prices traded on the Eurex market of 0.25 and 9 respectively. giving a put-call parity of 115.70 = 104.12. the difference between the two calculations could be due to market efficiency and the recent market crash, affecting the demand for BT Group options in this declining economy and thus, reducing their call and put prices. Also, the risk – free rate was kept constant and is an assumption. This may not be the actual risk – free rate and so the price of risk- free bonds may differ, also causing a change in both put – call parities.
Put-call parity calculations
P = 9.19 r = 0.05
S0 = 115.45 t = 1 year
C =25.15 X = 100
P + S0 = C + Xe-Rt
= 124.64 = 120.27
Risk – free bond
100e-0.05*1 = 95.12
Actual put–call parity
P = 0.25 r = 0.05
S0 = 115.45 t = 1 year
C = 9 X = 100
P + S0 = C + Xe-Rt
= 115.70 = 104.12
Risk behaviour
Risk neutral behaviour in financial markets is a mindset where individuals are not fazed by the risk of an investment. This is fundamentally different from both risk seeking and aversion. Risk seeking focusses on the highest return for risk possible which may not come to fruition. Allowing individual to get trapped in the wrong mindset. Risk aversion differs from risk neutrality as it focusses specifically on controlling and reducing risk as much as possible and not choosing an investment with uncontrollable risks. Reducing their overall prospects for returns.
An example of risk neutral behaviour is when an investor has two companies, he can put his money towards. The first with a very high risk of 0.8 and the other with a lower risk of 0.5. As they are not fazed by risk, the individual may feel strongly towards the company with the higher risk solely based on feeling[A9] . This may lead them towards high risk returns investment such as the FTSE 100. Risk seeking can be an individual who is, for example, offered £100 up front or a 50% chance to win £200. As they are risk seekers, the 50-50 chance of doubling their money entices them and convinces them to take the risk. Allowing for equity investment to be more appropriate. whilst an example for risk aversion behaviour is when in individual with excess cash of £20,000 could invest this amount into fairly low risk of, say 0.4 risk. Instead, they find opportunities to further reduce this risk by investing it into a saving account with a fixed risk of 0.2. Government bonds and other fixed return investments would be most appropriate for these types of investors[A10] .
Derivatives pricing is risk neutral as Instead, these individuals when assessing an investment do not use calculation but uses the assumption of no arbitrage rather than risk aversion to drive the pricing process. Thus, controlling risk is not a key[A11] factor in this process, eliminating risk seeking and aversion behaviour when calculating derivative prices.
Forward and swap contracts
A forward’s contract is a flexible contract between two parties to buy or sell assets on a set price on a set date, one taking the long position and the other a short. After the contract is draw up and both parties are in agreement of the terms, there will be one settlement date where the contract ends. Before this, the price of the forward contract will stay fixed and not be affected by any fluctuations. However, at settlement date the parties to the forward contracts are obliged to buy or sell the underlying securities.
swap contracts are agreements between two parties to exchange amounts of cash flows for a determined period. The most common is a vanilla interest rate swap. How this works is, for example, company A agrees to enter to a 10-year swap. Company A pays Company B 15% yearly of £10 million. Company B in return will pay company A one year of LIBOR rate plus 5% annual on the £10 million. This agreement will be developed because both groups believe they will gain in the end.
Like forward contracts, swaps do not involve any upfront cash. Swaps also have a notional value unlike forward[A12] contracts, but forward contracts are bilateral contracts and can be specifically tailored to meet individual needs. There are credit risks associated with forward contracts as assets may not be delivered on the date set. Due to the gains or losses on the contract cannot be locked in prior to the agreed settlement date, the size of the contract’s value serves as a measure for the credit risk involved. Credit risk associated with swaps are that the other company may default in the future. The risk increases the[A13] longer the contract is intact as more and more money is accumulated where a possible default on one end will increase.
BT Group forward contract
S0 = 91.62
T = 180/360 = 0.5 = 180 days
r = 0.69%
F = S0( 1 + r )T
F = 91.62(1 + 0.69)0.5 = 119.11[A14]
For this hypothetical BT Group Forward contract above, the inputs used are a stock price of 91.63, risk free rate of 0.69 as this and a time frame of 180 days. These inputs were chosen as the comfortable illustrate BT Groups past forward contract and reflect their current performance. The first step in calculating this forward calculation is to evaluate what “T” is. As demonstrated above, once this is determined, all inputs are available to calculate formula “F”. This will derive the final figure for this hypothetical forward contract of £119[A15] .11.
This forward contract could help BT hedge their currency risk as if they were to buy the underlying asset at £119.11 and the currency appreciates, BT Group will experience a gain. Also, for the time before the contract is settled, BT Group will lock in a price of £119.11 for whatever the assets is and so appreciation in the foreign[A16] currency fluctuation on this asset will not affect BT. The same can be said to hedge their interest rate risk as it is currently 0.69% which is locked in for the time of the contract. Any fluctuations in the interest rate before the settlement of the contact upon the underlying asset will not affect BT.
Eagle Interest Rate Swap
To calculate this hypothetical swap contract with Eagle, an assumption of the Libor rate had to be made 0.07913%. This was derived from Libor, 1-week figure as of 05-13-2020. This is an accurate assumption as it would show what Eagle would receive at this moment in time. The first step to calculating Eagle`s fixed payments is to determine the first section of the formula in the calculations below, (1/q). Since there will be quarterly payments, q = 90/30, and so 1/q will be 360/90. Then, to calculate the rate that will be used in each term, the Libor rate will be multiplied by the term as shown in figure 4. This will give us input B0 in the formula and input tn will simply be the term divided by 1 year. Once this has been calculated for each term, all the inputs are now available to calculate R (fixed rate). In this case, R is 0.11 and is the fixed rate Eagle will receive in every period from £1000000. The swaps fixed payment will then be the £1000000 multiplied by R and q totalling £27500. The interest rate swap has allowed Eagle to guarantee himself a £27500 pay-out. If LIBOR is decreases, the corresponding party will owe Eagle under the swap. However, if LIBOR is higher, Eagle will owe the corresponding party the difference in money.
Swap calculations
R = (1/q) ((1-B0(tn) / (Σ B0(ti))
Libor rate (L0) = 0.07913[A17] %
Q = 90/360
1/Q = 360/90
| Figure 4 | |||
| Term | Rate | Discounted Bond Price | |
| 90 | L0(90) = 7.12% | B0(90) = 1 / (1 +0.0712(90/360)) = 3.73 | |
| 180 | L0(180) = 14.24% | B0(180) = 1 / (1 +0.1424(180/360)) = 1.75 | |
| 270 | L0(270) = 21[A18] .37% | B0(270) = 1 / (1 +0.2137(270/360)) = 1.10 | |
| 360 | L0(360) = 28.47 | B0(360) = 1 / (1 +0.2847(360/360)) = 0.79 | |
R = (360 / 90) ((1-0.76) / (3.73 + 1.75 + 1.10 + 0.79)) = 0.11
Swaps fixed payments = £1000000 (0.11) (90/360) = £27500
Appendix
| Figure 4 | |||
| Term | Rate | Discounted Bond Price | |
| 90 | L0(90) = 7.12% | B0(90) = 1 / (1 +0.0712(90/360)) = 3.73 | |
| 180 | L0(180) = 14.24% | B0(180) = 1 / (1 +0.1424(180/360)) = 1.75 | |
| 270 | L0(270) = 21.37% | B0(270) = 1 / (1 +0.2137(270/360)) = 1.10 | |
| 360 | L0(360) = 28.47 | B0(360) = 1 / (1 +0.2847(360/360)) = 0.79 | |
References
Investopedia. 2020. Over-the-Counter Derivative. [ONLINE] Available at: https://www.investopedia.com/ask/answers/052815/what-overthecounter-derivative.asp. [Accessed 29 april 2020].
Investopedia. 2020. Exchange Traded Derivative Definition. [ONLINE] Available at: https://www.investopedia.com/terms/e/exchange-traded-derivative.asp. [Accessed 14 May 2020].
Eurex Exchange – BT Group. 2020. Eurex Exchange – BT Group. [ONLINE] Available at: https://www.eurexclearing.com/exchange-en/products/equ/opt/BT-Group-48848. [Accessed 1 May 2020].
Eurex Exchange – BT Group. 2020. Eurex Exchange – BT Group. [ONLINE] Available at: https://www.eurexclearing.com/exchange-en/products/equ/fut/BT-Group-947960. [Accessed 10 May 2020]
BT Group PLC | ICE. 2020. BT Group PLC | ICE. [ONLINE] Available at: https://www.theice.com/products/38716793/BT-Group-PLC. [Accessed 14 May 2020].
Clearing Risk Management for Futures, Options & CDS | ICE Clear Europe. 2020. Clearing Risk Management for Futures, Options & CDS | ICE Clear Europe. [ONLINE] Available at: https://www.theice.com/clear-europe/risk-management. [Accessed 14 May 2020].
Eurex Marketing. 2020. Eurex Clearing – Default Management Process. [ONLINE] Available at: https://www.eurexclearing.com/clearing-en/risk-management/default-management-process. [Accessed 14 May 2020].
Harrison, M. and D. Kreps. “Martingales and Multiperiod Securities Markets.” Journal of Economic Theory, 20 (July, 1979)
Finance Train. 2020. Credit Risk and Forward Contracts – Finance Train. [ONLINE] Available at: https://financetrain.com/credit-risk-and-forward-contracts/. [Accessed 14 May 2020].
Mozumdar, Abon, Corporate hedging and speculative incentives: Implications for swap market default risk, Journal of Financial and Quantitative Analysis, 36 (2001)
Chance, D. and Brooks, R., 2016. An introduction to derivatives and risk management. 10th ed. Australia: Cengage Learning.
[A1]This may be too high given current rates are less than 1%.
[A2]It may not be plausible to assume this higher dividend into the future given COVID and economic outlook.
[A3]Good work.
[A4]Good work
[A5]Good work.
[A6]Not in percentage but in times i.e. 1.14 of current price and 0.87 of current price.
[A7]Good work.
[A8]These should have come from your earlier calculations and you did not need to calculate these a new as the inputs assumed here are different and not relate to the actual options on Eurex.
[A9]How does this then differ from risk seeking should have been clearly outlined.
[A10]Risk attitudes should have been more objectively and clearly described with more explicit examples.
[A11]Risk neutral pricing should have been explained in line with arguments of risk free arbitrage and hedge portfolio.
[A12]So can be swaps.
Both differences and similarities of swaps and forwards should have been adequately covered.
[A13]This is the case for forwards only and not swaps.
[A14]Typo error as rate is 0.69%(.0069) so the price is not correct.
[A15]Price is in pence and not in £s.
[A16]No interim dividends have been assumed. This should have been discussed too.
[A17]!!
[A18]Actual LIBOR rates for £s should have been used from the link provided in the coursework. The rates used here are too high given the prevailing LIBOR rates.
Calculations however, are correct.
