This chapter illustrates 2 examples of SCO applications, including the flexibility evaluation of revenue guarantee and currency hedging by the pool of SCOs.
§8.1 Revenue Guarantee
This section illustrates a SCO application of BOT (Build-Operate-Transfer) project valuation. In this example, a 2-fold option (call on put) and a 3-fold SCO (call on call on put) are used to evaluate the promise value of revenue guarantee.
§8.1.1 Description for the Revenue Guarantee
Assume the government issues a BOT project of electric power plants in order to increase power supply. Assume the government and the company sign the contract at starting time T0. The project starts from 1.5 years of preparation period, which follows with 4-year construction. After the construction, there will be operation period of 30 years. If the project is constructed and operated well, there can be an expansion at the 24-year. The expansion, with scale size αa1 proportional to the original one, takes 2 years and won't extend the operation time of the project. The Figure 8.1 illustrates the schedule of the BOT project.
Preparation expansion expansion operation construction main operation
T0 T1 T2 T3 T4 T5
0 1.5 4 24 26 34 Figure 8-1: The time intervals of the example of revenue guarantee
For the government, the BOT project can increase the power supply without huge construction cost at one time. Hence the government will try its best to increase the project's investment incentives, such as the annual revenue guarantee. Revenue guarantee is the minimum revenue promised by the government. If the annual operation revenue is less than the guarantee amount, then the deficit is subsided by the government. Compared with the once huge construction cost, the payment of the revenue guarantee from the government is less and distributed over many years, and causes less burdens to the government. In addition, the guarantee can strengthens companies' incentives toward the BOT project, thus increase the plausibility of project execution.
The revenue guarantee can be regarded as the put option written by the government and owned by the company. Their Payoffs are exhibited in Figure 8.2 (a) and (b) respectively.
Payoff of the put
Guarantee Amount
Revenue from operation
Figure 8.2 (a) The payoff of the put option written by the government
Payoff of the put
Guarantee Amount
Revenue from operation
Figure 8.2 (b) The payoff of the put option owned by the company
Revenue from operation Guarantee Amount
Revenue from operation
Figure 8.2 (c) The operation payoff of the company
Revenue from operation
Total Revenue
Guarantee Amount
Figure 8.2 (d) The total payoff of by the company
The payoff from the operation is shown in Figure 8.2 (c), which means that the company hold simultaneously downside risk and upside potential. Downside risk can be eliminated by holding the revenue guarantee and thus the company can retain the upside potential (Figure 8.2(d)).
The value of the revenue guarantee, which essentially is the value of the put option, can be regarded as the expected value of the company's extra gain or the expected value of the government's payment. However, the revenue guarantee should be considered as a multi-fold SCO to coincide with preceding sequential decisions.
The following section describes details about the revenue guarantee with numerical examples.
§8.1.2 Guarantee Evaluation
Assume the government promises every year's revenue guarantee ( ) to the company, which lasts for 30 years. The expansion part also has every year's revenue guarantee ( ) for 8 years. All the construction costs are paid by the company and the revenue are belong to the company. The preparation period is 1.5 years before the 2.5-year's construction period. Assume there is no inflation. In other words, the inflation is accounting as parts of the risk-free rate. No depression rate (q=0) in this example. The construction payment occurs at its end time. The parameter setting is listed as follows.
1 2
Ka
2 3
Ka
The main construction cost (at Time T2): K1a1=50,000,000,000.
The expansion construction cost (at Time T3): K2a2 =αa1K1a1*1.05.
The each year's guarantee revenue of original construction (at time T1):
=100,000,000.
1 2
Ka
The each year's guarantee revenue of expansion construction (at time T3):
05 . 1
1*
1 1 2
3 a
a
a K
K =α
The expansion scale coefficient (comparing to the original scale): αa1=0.3.
The initial time: T0=0
The start time of main construction: T1=1.5
The end time of main construction and the start time of operation: T2=4 The start time of expansion construction: T3=24.
The end time of expansion construction and the start time of the expansion operation: T4=26.
The end time of all operation and transfer the plant to the government: T5=34.
The risk-free rate (through time): r=3.5%.
The annual volatility of the underlying revenues (through time):σ =0.5.
The estimated average revenue of each year (at time T0): S0=2,000,000,000.
For the revenue guarantee of the main construction at time T1, it can be regarded as the guarantee revenue plus 30 1-fold forward-start put options (
∑
) writtenby the government to the company. These forward-start options can be regarded as European options because their dividend rates are zero. The company should pay the construction cost ( ) as the "option premium". It should be noted that the main construction cost is shared as the strike price of both the main guarantee and the expansion one. The strike price of the option ( ) is the revenue guarantee ( ).
=
At time T0, the revenue guarantee of the main construction can be considered as 2-fold compound options (call on put)
∑
, whose strike prices are the proportional construction cost=
Similarly, the revenue guarantee of the expansion construction can be regarded as 3-fold SCOs (call on call on put). At time T3, the SCOs can be regarded as 8 1-fold forward-start put options ( ) written by the government to the company. The company pays the main construction cost and the expansion cost as the "option premium". The strike price of the option ( ) is the revenue guarantee amount of the expansion ( ). At time T1, the revenue guarantee of expansion can be considered as 2-fold SCOs (call on put), whose strike price is the proportional cost of expansion construction . It means that the company should pay the expansion cost in order to gain the revenue guarantee. Thus the payoff of the individual option is
∑
= SCOs (call on call on put), whose strike price is proportional cost of main construction1 main construction in order to gain the expansion right. The payoff of the individual
option is ~(max 0, ) )
(
1 2 1
1 2 2
1 2 2
) ( , 2 0
2 ) ( ,
3 ⎥
⎦
⎢ ⎤
⎣
⎡
× +
− Ψ
= Ψ
a a a
a a a
a a u a
u
E K
T τ τ α
α τ
τ .
At time T0, the main revenue guarantee is 4.437 billion worth, which is evaluated by 30 2-fold options (call on put). Thus the company is expected to gain 4.437 billion from the government to eliminate the downside risk of main operation. The company can get at least 6 billion in 30 years. Similarly, the revenue guarantee of expansion is 0.287 billion worth, which is evaluated by 8 3-fold SCOs (call on call on put). In other words, the company is expected to gain 0.287 billion from the government according to the revenue guarantee of expansion construction. There the company can get at least 1.6 billion in last 8 years due to expansion.
The sensitivity analysis is listed as follows. Table 8.1 represents the guarantee amount sensitivity. The annual guarantee revenue of the expansion construction is associated with that of main construction. The 30-year guarantee revenue is the summation of 30 years' annual guarantee. Guarantee worth (30-year) is the value of the revenue guarantee, which is evaluated by 2-fold compound options. It is found that guarantee worth decreases while the guarantee revenue increase. It means that the raise of guarantee amount can increase the guaranteed revenue hugely and thus results in the subtle reduce of the guarantee worth. In other words, the increase of the certainty part (guarantee amount) will diminish the uncertainty part (guarantee worth).
Similarly, the opposite direction of the guarantee amount and guarantee worth also appears in the expansion construction.
Table 8.1: The Guarantee Sensitivity of the Guarantee Example (Unit: 10^9 NT)
Annual Guarantee
Revenue 0.100 0.200 0.300 0.500
30-year Guarantee
Revenue 3.000 6.000 9.000 15.000
Main Construction
Guarantee
Worth(30yr) 6.096 4.436 3.162 1.451
Annual Guarantee
Revenue 0.0315 0.063 0.094
5 0.158
8-year Guarantee
Revenue 0.252 0.504 0.756 1.260
Expansion Construction
Guarantee
Worth(8yr) 0.884 0.794 0.708 0.544
Table 8.2 represents the sensitivity analysis of volatility. The volatility of the annual revenue is assumed constant through time. The table shows that the guarantee worth, which is evaluated as option summation, increases with the volatility. The results correspond with general intuition.
Table 8.2: The Volatility Sensitivity of the Guarantee Example (Unit: 10^9 NT)
Volatility 0.300 0.400 0.500 0.600
Guarantee Worth (30yr) of
Main Construction 0.932 2.467 4.436 6.631
Guarantee Worth (8yr) of
Expansion Construction 0.001 0.137 0.794 1.517 The sensitivity of estimated annual revenue is tabulated in Table 8.3. The guarantee worth decreases while the estimated annual revenue increases, which is consistent with put's behavior. In other words, the increase of the estimated annual revenue will also increase the certainty of high revenue, thus causes reduce of the uncertainty (guarantee worth).
Table 8.3: The Sensitivity of Estimated Annual Revenue of the Guarantee Example (Unit: 10^9 NT)
Estimated Annual Revenue S0 0.500 1.000 2.000 3.000 Guarantee Worth (30yr) of
Main Construction 22.581 12.972 4.436 1.691 Guarantee Worth (8yr) of
Expansion Construction 1.772 1.321 0.794 0.496 Table 8.4 exhibits the interest rate sensitivity. The hike of interest rate results in the guarantee worth decrease because the discounting of high interest rate will diminish the guarantee's value.
Table 8.4: The Interest Rate Sensitivity of the Guarantee Example (Unit: 10^9 NT)
Interest rate r 2.5% 3.0% 3.5% 4.0%
Guarantee Worth (30yr) of
Main Construction 4.815 4.623 4.436 4.256 Guarantee Worth (8yr) of
Expansion Construction 2.207 1.440 0.794 0.309
§8.2 Currency Hedging
Assume an American company participates in a project auction and may have to buy Japanese products sequentially in the future. The company wants to hedge the appreciation risk of Japanese Yen. It can take a pool of SCOs, instead of a strip of futures or a stack of futures. The pool including a 1-fold European put, a 2-fold compound option (call on put) and a 3-fold SCO (call on call on put). Compared with the strip/stack of futures, the SCO pool is a better risk management instrument because the downside risk is well protected.
Options in the pool are with the final strike price 110. The 2-fold and 3-fold option should pay the fold payment (5 Yen) when enter the next fold.
The parameters of this example are set as follows.
The current exchange rate= 123.8 (Yen/USD).
The final strike price =110.
Payment for each fold =5.
The domestic (US) risk-free interest rate: r=5%
The foreign (Japanese) risk-free interest rate: q=rf=1%
The annual volatility of the exchange rate: σ =0.4.
The time interval for each fold: 0.5 yr.
The 1-fold put option is priced as 6.51 (Yen), while the 2-fold (call on put) and the 3-fold (call on call on put) are valued as 6.69 and 5.86, respectively. The following tables show the sensitivity analysis of this example.
Table 8.5 represents the exchange rate sensitivity. It is found that the value of the SCO pool decrease while the current exchange rate rises. The result corresponds with the behavior of put option. The volatility sensitivity is tabulated in Table 8.6. The table reflects the intuition that higher volatility causes higher option prices.
Table 8.5 The Exchange Rate Sensitivity of the Currency Example (Unit:Yen)
S 115.00 123.80 130.00
1-fold 9.22 6.51 5.05
2-fold 9.04 6.69 5.38
3-fold 7.85 5.86 4.73
Table 8.7 and 8.8 exhibit the sensitivity of domestic and foreign interest rate, respectively. When the domestic (US) interest rate hikes, the US dollar becomes more strengthen and results in the exchange rate decrease. Nevertheless, the foreign (Japanese) interest rate raising will cause the exchange rate increase. The result can be
explained according to Interest Rate Parity (IRP).
Table 8.6 The Volatility Sensitivity of the Currency Example (Unit:Yen)
Volatility 0.3 0.4 0.5
1-fold 3.77 6.51 9.42
2-fold 3.32 6.69 10.47
3-fold 2.38 5.86 10.03
Table 8.7 The Domestic (US) Interest Rate Sensitivity of the Currency Example (Unit:Yen)
r 4% 5% 6%
1-fold 6.71 6.51 6.32
2-fold 7.08 6.69 6.64
3-fold 6.37 5.86 5.37
Table 8.8 The Foreign (Japanese) Interest Rate Sensitivity of the Currency Example (Unit:Yen)
rf (q) 0.5% 1% 2%
1-fold 6.43 6.51 6.68
2-fold 6.55 6.69 6.99
3-fold 5.67 5.86 6.23
The sensitivity of final strike and fold payment are shown in Table 8.9 and Table 8.10, respectively. The increase of final strike will result the value of the put-style SCO pool. The fold payment can be regarded as another premium payment. Table 8.10 prevails the fact that SCOs can support decision postponement, which is one of SCOs' advantages. The higher SCO premium payment at current time can enjoy less fold payment in the future.
Table 8.9 The Strike Sensitivity of the Currency Example (Unit:Yen)
final Strike 100 110 130
1-fold 3.54 6.51 13.16
2-fold 3.72 6.69 13.05
3-fold 3.10 5.86 11.79
Table 8.10 The Fold Payment Sensitivity of the Currency Example (Unit:Yen)
payment 1 5 10
2-fold 9.51 6.69 4.30
3-fold 11.32 5.86 2.12