Financial Engineering – IC302
Autumn Term 2020/1
Seminar 7: Exotic Options Answers
The questions in this exercise can be answered using the Excel workbook FinEngV7.xlsm, which allows you to create and value multiple positions in vanilla and exotic options and explore how they can be combined to engineer structures with specific risk profiles.
1.
Payoff
We have seen that a European binary option can be hedged with a spread trade in which we go long one option and short another for the same expiry date but for two different strike prices, with the strike prices bracketing the strike price of the binary. A bull spread can be used to replicate (and therefore hedge) a binary call, while a bear spread can be used to replicate a binary put. Bull and bear spreads can be constructed using either calls or puts.
Payoff Bull spread with calls
KST KST K = exercise price (strike price)
ST = asset price at option expiry
For example, we could replicate a binary call option on GBP/USD with a bull spread in which we were long a vanilla GPB/USD call option with strike price 1.79 and short a vanilla GBP/USD call option with strike price 1.81. The maximum payout to this spread trade would be USD 0.02 per GBP. If the cash payoff to the binary option were USD 100,000, we would need to trade a nominal amount of USD 100,000 / USD 0.02 per GBP = GBP 5 million nominal in each of the vanilla options in order for the payoff to the bull spread to match the payoff to the binary call.
(a) Suppose that we write a European binary call option on GBP/USD with strike price 1.80 in the following market conditions:
GBP/USD spot 1.7500 USD interest rate 5.20 GBP interest rate 4.75 GBP/USD volatility 8%
The binary payoff is USD 100,000 and the time to expiry is 365 days.
According to the Excel workbook, what is the value of this option?
[Hint: Enter the market data into the Excel workbook under ‘Initial Market Data’ and ‘Current Market Data’. In the column labelled ‘Position 1’, select Contract Type ‘Call’ and Style ‘Binary’. Set ‘Underlying Amount’ to -100,000 (this amount is in GBP, and it is negative because we are short the option), ‘Initial Days to Expiry’ to 365, ‘Strike’ to 1.80, and ‘Cash Payout’ (which is in USD per pound sterling) to 1.0000.]
(b) Suppose that we decide to hedge this option by creating a bull spread that replicates it. In order to make the hedge more effective, we use a narrow spread based on vanilla call options with strikes of 1.7995 and 1.8005. What nominal amounts will we need to trade in each of the vanilla options? Do you see any potential problems with this strategy?
(c) According to the Excel workbook, what would be the cost of the call spread in (b)? How does this compare to the price of the binary call?
[Hint: The binary option should already be set up in position 1 of the Excel workbook. Use positions 2 and 3 in the workbook to set up the positions in vanilla European call options needed to construct the call spread. Make sure that you use the correct nominal amounts for the vanilla options.]
(d) We hedged the binary by using a spread with strikes that bracketed the binary strike. Suppose, however, that the binary contract specifies that it will pay only if the underlying trades at a level that is strictly above the strike. How might we set the strikes for the hedge in this case?
2. A knock-out forward is a forward contract that terminates if spot trades at or beyond a pre-specified level before expiry. In exchange for giving up the forward in this event, the client obtains a more favourable forward rate than the current market level. Knock-out forwards usually have zero upfront cost.
In order to construct a knock-out forward, we make use of put-call parity. Recall that being long a call option and short a put option on the same asset with the same expiry date and exercise price is equivalent to being long the forward:
Payoff
K ST
K = exercise price (strike price) ST = asset price at option expiry
We can therefore construct a synthetic forward by going long a European call and short a European put. Since we want the forward to terminate if spot trades at a given level, we use European barrier options that knock out at that level. The client chooses the forward rate that she wants, and we then choose the barrier level at which the structure knocks out such that the upfront cost is zero.
Suppose that current market conditions are as described below:
GBP/USD spot 1.7500 USD interest rate 5.20 GBP interest rate 4.75 GBP/USD volatility 8%
A client wishes to go long a 90-day knock out forward on GBP/USD at a forward price of 1.7500 (i.e. today’s spot) and contract amount GBP 10 million. (Notice that the maturity of the option is different than in question 1.)
(a) How does the forward price at which the client wishes to buy GBP compare with the market forward price?
[Hint: Enter the market data into the Excel workbook under ‘Initial Market Data’ and ‘Current Market Data’. In the column labelled ‘Position 1’, set ‘Contract Amount’ to 10,000,000 and ‘Initial Days to Expiry’ to 90. The market forward price is calculated automatically in the spreadsheet.]
(b) Use the spreadsheet to construct a synthetic long forward contract at the client’s chosen price by combining a long knock out call with a short knock out put. At what level must the barrier in the options be set if the structure is to have zero upfront cost? What does the client give up in exchange for the more favourable forward price?
[Hint: Set up the knock-out option positions in Position 1 and Position 2 of the spreadsheet; the option ‘Style’ is ‘KO 1 barrier’. Don’t forget to set the ‘Cash payout/rebate’ on both options to zero. Link the barrier level on the put to the barrier level on the call (you can do this by writing =C24 in the cell for the put barrier, assuming that C24 is the cell that contains the call barrier). Make an initial guess for the barrier level and then use ‘Goal Seek’
to find the barrier that makes the upfront cost of the structure zero. Make sure that your initial guess for the barrier is on the correct side of the strike.]
(c) Suppose that the client wanted a lower forward price. What effect would this have on the barrier level at which the forward knocks out, assuming once against that the upfront cost is to be zero?
(d) Would an increase in exchange rate volatility have any effect on the required barrier level for the knock-out forward? How does this compare with the effect of volatility on an ordinary forward?
3. A bonus forward is a forward contract that reset to the client’s disadvantage by a pre-determined amount if the spot rate reaches a specified trigger level during the life of the contract. In exchange for accepting this reset, the client obtains an improved forward rate if the trigger level is not touched. Bonus forwards are also known as range reset forwards.
Suppose that current market conditions are as described below:
GBP/USD spot 1.7500 USD interest rate 5.20 GBP interest rate 4.75 GBP/USD volatility 8%
A client wishes to go long a 365-day bonus forward on GBP/USD at a best forward price of 1.7300 that will reset to a higher level if GBP/USD touches an agreed trigger level of 1.7800. The contract amount is GBP 10 million. (Notice that the maturity of the option is different than in question 2.)
One way of modelling this structure is to think of it as a combination of two synthetic forwards. The first of these forwards has forward price (i.e. strike price) 1.7300 and knocks out at 1.7800. The second knocks in at 1.7800 and has a forward price equal to the level required for the structure to have zero upfront cost. Each synthetic forward is constructed as a combination of a barrier call and a barrier put, as described above in the question 2. The bonus forward can also be modelled is as an ordinary forward at forward price 1.7300, for which the client pays by selling us an American binary call with strike price 1.7800 and cash rebate equal to the level required for the structure to have zero upfront cost.
(a) What must the trigger level be if the bonus forward described above is to have zero upfront cost?
[Hint: Model the bonus forward as a combination of two synthetic forwards, as described in question 2. Since each synthetic forward is a combination of a long position in a barrier call and a short position in a barrier put, you will need to use the first four columns of the spreadsheet. The first pair of
options has strike 1.7300 and knocks out at 1.7800; the second pair of options knocks in at 1.7800. Don’t forget to choose style ‘KI 1 Barrier’ for this second pair of options. Set the strikes on the call and the put in the second pair equal to each other and use ‘Goal Seek’ to find the reset level (i.e. strike) that makes the structure have zero cost. Don’t forget to set the initial days to expiry to 365.]
(b) What are the advantages or disadvantages of a bonus forward to the client relative to a knock-out forward structure?
4. A forward plus is an option that turns into a forward contract if a specified barrier level is hit. The structure is usually designed to have zero upfront cost, so that the client gets a ‘free’ option, for which she pays by accepting that it will convert to a forward position in certain conditions. Naturally, the conditions that trigger conversion are conditions in which the resulting forward position would have negative value to the client.
One way in which this structure can be modelled is as a combination of a long position in an ordinary European option with a short position in a knock-in barrier option of the opposite type with the same strike. Suppose, for example, that the client wishes to buy GBP in exchange for USD for a given forward date. The forward plus structure combines a long position in a European GBP/USD call with a short position in a European GBP/USD put that knocks in at an agreed barrier level. Both options have the same exercise price. So long as the barrier is not hit, the client is long a GBP/USD call. If the barrier is hit, however, put-call parity implies that she will be long a GBP/USD forward contract with forward price equal to the common strike price of the two options.
Suppose that current market conditions are as described below:
GBP/USD spot 1.7500 USD interest rate 5.20 GBP interest rate 4.75 GBP/USD volatility 8%
A client wishes to go long a 365-day forward plus on GBP/USD.
(a) Suppose that the barrier level at which the structure converts to a forward is set at 1.7000. What strike price on the two options would ensure that the entire structure has zero upfront cost?
[Hint: Use the spreadsheet to analyse the forward plus as a combination of a long European call and a short European knock-in put. Set the barrier on the knock-in put at 1.7000 and set the strike price on the knock-in put equal to the strike price on the European call. Then use ‘Goal Seek’ to find the strike price that gives the structure zero upfront cost.]
(b) In the forward plus that you have just constructed, what is the best price at which the client may be able to sell GBP on the forward delivery date? What is the worst price? Under what conditions would she pay each of these prices? How do they compare with the market forward price?
(c) Suppose that the trigger level for the forward plus were set at 1.65 rather than 1.7000. What effect would this have on the strike price required to make the structure have zero upfront cost? What explanation can you offer for this result?