Financial Engineering
IC302
1. Introduction to Financial Engineering
David Oakes
–
Autumn Term 2020/1
Course Plan
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Teaching and Assessment
Teaching: Lectures (9 x approximately 2 hours) Seminars (8 x 1 hour)
Lectures will be delivered asynchronously as pre- recorded videos distributed on Blackboard.
Seminars will be conducted in-person or remotely.
Assessment: Multiple choice test (20%) Final examination (80%)
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Course Outline
1. Introduction to Financial Engineering
2. Simple Interest Rate Derivatives
3. Interest Rate Swaps
4. Credit Default Swaps and Credit Derivatives
5. Basic Numerical Procedures for Valuing Options
6. Interest Rate Options
7. Exotic Options
8. Structured Products
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References
Hull, John. Options, Futures, and Other Derivatives. Ninth edition; Global edition. Harlow: Pearson, 2018.
Kosowksi, Robert, and Salih Neftci. Principles of Financial Engineering. Third edition. Amsterdam: Academic Press, 2015.
Tenth US edition of Hull is available in some markets.
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Financial Engineering:
A Foreign Exchange Example
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Financial Engineering
• Financial engineering is a multidisciplinary field that involves financial theory, applied mathematics, statistics, and computer programming.
• Financial engineers use these tools to solve practical problems in financial markets related to valuation, risk management, and the design of financial instruments.
• Often, these solutions use simple derivatives and hedging and replication methodologies to create and analyze more complex derivatives and structured products.
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Foreign Exchange (FX) Example
• Suppose that a US-based investor (a client of our bank) wants to create a long position in Euros (e.g. to speculate on the EURUSD exchange rate or hedge FX risk).
• They could buy Euros with US dollars at the current spot EURUSD exchange rate 𝑆!! (e.g. $1.50 per Euro).
𝐸𝑈𝑅 1
𝑈𝑆𝐷 𝑆#! 𝑡”
• But this would require capital (i.e. funding) and would have implications for their balance sheet and regulation.
𝑡!
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FX Forward
• Suppose instead that the client buys Euros for forward
delivery on a future date 𝑡) at a price 𝐹!! that is fixed now.
• This is an FX forward contract. The price 𝐹!! that the client
agrees to pay for the EUR is the forward exchange rate.
• The FX forward gives the client long exposure to EUR, but for the forward date rather than the spot date.
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FX Forward
• The exchange of dollars for Euros will now take place at the forward date rather than the spot date.
• Provided we agree a fair forward price (so that what the client gives up is equal in value to what they receive), no money need change hands now, so no funding is required.
• This may make the FX forward a better structure for the client than a spot purchase in terms of funding/regulation.
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FX Forward and FX Risk
• Selling the client Euros for forward delivery will expose us to risk, since we don’t know today what the EURUSD exchange rate will be on the forward delivery date.
• How can we hedge this risk?
• What price (i.e. forward exchange rate) should we quote?
• It turns out that these two questions are closely related.
• They can be answered by financial engineering.
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Engineering the FX Forward
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Engineering the FX Forward
• We can use other instruments to create a position that is equivalent to the FX forward. We call this process replication, and the resulting position is a synthetic.
• This same strategy can be used to hedge the forward.
• The client agrees at date 𝑡0 (today) to buy Euros for
delivery on date 𝑡1 at the forward exchange rate 𝐹!! .
𝐸𝑈𝑅 1
𝑡! 𝑈𝑆𝐷 𝐹#!
𝑡”
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Engineering the FX Forward
• Break the FX forward contract into two parts:
𝐸𝑈𝑅 1 𝑡! 𝑈𝑆𝐷 𝐹#!
Forward FX
(Buy EUR, sell USD)
𝑡”
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Engineering the FX Forward
• Break the FX forward contract into two parts:
𝑡”
𝑡! 𝑡!
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#!
𝐸𝑈𝑅 1
Forward FX
(Buy EUR, sell USD)
𝑡”
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Engineering the FX Forward
• Break the FX forward contract into two parts:
𝑡”
𝑡!
𝑡! 𝑡” 𝑡!
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#!
𝐸𝑈𝑅 1
Forward FX
(Buy EUR, sell USD)
𝑡”
𝑈𝑆𝐷 𝐹#!
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Engineering the FX Forward
• Add a negative EUR cash flow and a positive USD cash
flow, both at date 𝑡+: 𝑡!
𝐸𝑈𝑅 𝑈𝑆𝐷
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#!
𝐸𝑈𝑅 1
𝑡”
Forward FX
(Buy EUR, sell USD)
𝑡”
𝑡!
𝑡!
𝑈𝑆𝐷 𝐹#!
𝑡”
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Engineering the FX Forward
• Then, in a separate transaction, subtract the date 𝑡+ EUR and USD cash flows that you have just added:
𝑡”
𝑡!
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#!
𝐸𝑈𝑅 1
𝑡”
Forward FX
(Buy EUR, sell USD)
𝐸𝑈𝑅 𝑈𝑆𝐷
𝐸𝑈𝑅
𝑈𝑆𝐷
𝑡!
𝑈𝑆𝐷 𝐹#!
𝑡!
𝑡”
𝑡!
𝑡”
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Engineering the FX Forward
• The first component that we added looks like lending EUR and the second looks like borrowing USD:
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#!
𝑡”
𝑡!
Forward FX
(Buy EUR, sell USD)
𝐸𝑈𝑅 1
𝑡”
𝐸𝑈𝑅 𝑈𝑆𝐷
𝐸𝑈𝑅
𝑈𝑆𝐷
𝑡!
𝑈𝑆𝐷 𝐹#
Lend EUR (EUR deposit)
Borrow USD (USD loan)
𝑡!
𝑡”
!
𝑡!
𝑡”
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Engineering the FX Forward
• The third component looks like a spot FX transaction in which we buy EUR and sell USD:
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#!
𝑡”
𝑡!
Forward FX
(Buy EUR, sell USD)
𝐸𝑈𝑅 1
𝑡”
𝐸𝑈𝑅 𝑈𝑆𝐷
𝐸𝑈𝑅
𝑈𝑆𝐷
𝑡!
𝑈𝑆𝐷 𝐹#
!
Lend EUR (EUR deposit)
Borrow USD (USD loan)
Spot FX
(Buy EUR, Sell USD)
𝑡!
𝑡”
𝑡”
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Engineering the FX Forward
• FX forward can therefore be decomposed into:
– EUR deposit (lend Euros)
– USD loan (borrow US dollars)
– EUR-USD spot exchange (buy Euros with US dollars)
• Contractual equation for FX forward: =++
• The portfolio we constructed replicates the payoff to the forward contract and is a synthetic version of it.
FX Forward Buy EUR for USD at 𝑡!
Deposit
Lend EUR from 𝑡” to 𝑡!
Loan
Borrow USD from 𝑡! to 𝑡”
FX Spot Buy EUR for USD at 𝑡”
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Uses of Synthetics
• Replicating the cash flows of a more complex or less liquid instrument using simpler liquid instruments
• Hedging an existing position by taking the opposite position in the synthetic or replicating portfolio
• Pricing (derivatives prices are driven by hedging costs, so a starting point for the price of a derivative is the cost of constructing a synthetic portfolio that replicates its payoff)
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Hedging and Pricing FX Forwards
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Hedging and Pricing FX Forwards
• We can use the contractual equation to work out how to hedge our risk and what forward exchange rate to quote.
=++ à++-=0
• The forward exchange rate we quote should be the rate that makes the value of this whole structure (the forward contract plus the hedge) equal to zero.
FX Forward Buy EUR for USD at 𝑡!
Deposit
Lend EUR from 𝑡” to 𝑡!
Loan
Borrow USD from 𝑡! to 𝑡”
FX Spot Buy EUR for USD at 𝑡”
Deposit
Lend EUR from 𝑡” to 𝑡!
Loan
Borrow USD from 𝑡! to 𝑡”
FX Spot Buy EUR for USD at 𝑡”
FX Forward Buy EUR for USD at 𝑡!
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Hedging and Pricing FX Forwards
• The client wants to buy EUR from us for forward delivery.
• We borrow USD and use them to buy EUR spot.
• We invest the EUR at the EUR interest rate.
• At the forward date, we deliver EUR to the client in exchange for USD at the forward rate we quote.
• We quote a forward rate that ensures we receive enough USD to repay the USD loan.
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Hedging and Pricing FX Forwards
Borrow USD (USD loan)
Spot FX
(Buy EUR, Sell USD)
Lend EUR (EUR deposit)
Forward FX
(Sell EUR, buy USD)
𝑈𝑆𝐷 𝑆#!
𝑡!
𝑈𝑆𝐷 𝑆#! 𝑡”
𝑈𝑆𝐷 𝑆#! 1 + 𝑟%'(
𝑡”
𝑡!
𝐸𝑈𝑅 1
𝐸𝑈𝑅 1
𝐸𝑈𝑅 1+𝑟$%&
𝑡”
𝑡!
𝑡”
𝑡!
𝑈𝑆𝐷 𝐹#! 1 + 𝑟$%& 𝐸𝑈𝑅 1+𝑟$%&
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Fair-Value Forward Exchange Rate
• Since the whole structure must have zero value, this gives:
𝐹#! 1+𝑟$%& −𝑆#! 1+𝑟%'( =0
→ 𝐹 =𝑆 1+𝑟%'( #! #! 1 + 𝑟$%&
• The fair-value (i.e. no-arbitrage) forward exchange rate depends on the spot exchange rate and the interest rate differential between the two currencies.
• This result is called covered interest arbitrage.
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An Example
• Suppose that the spot exchange rate and interest rates are as follows:
𝑆#! = 1.50 𝑟%'( = 3% 𝑟$%& = 1%
• What is the fair-value forward exchange rate? 𝐹 =𝑆 1+𝑟%'( =1.501+0.03=1.5297
#! #! 1+𝑟$%& 1+0.01
• Why is the forward rate higher than the spot rate?
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Carry and Forward FX
• We borrow USD in order to buy EUR, which we hold as a hedge against the forward sale of EUR to the client.
• During the hedging period, we earn the EUR interest rate and pay the USD interest rate.
• Since the EUR rate is lower than the USD rate, our hedge loses money. It has negative carry.
• We sell EUR forward at a higher price than we buy EUR spot in order to offset our losses due to the negative carry.
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Carry and Forward Prices
• For purely financial assets like currencies, no-arbitrage forward prices should reflect the carry on the underlying asset (in this case, the base currency, EUR).
• When carry is negative, the forward price should be higher than the spot price; when it is positive, lower.
• For currencies, this is what covered interest parity implies.
• As we will see later, this may not always hold in practice.
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Market Conventions
• In actual applications, the interest rate calculations must
use the appropriate day count conventions for each rate.
• Exchange rates are quoted as units of the quote currency per unit of the base currency for each currency pair, with the base currency listed first (e.g. EURUSD = 1.5279).
• EUR is base currency against any currency. GBP, AUD, and NZD (in that order) are base currency against any currency other than EUR. After that, USD is the base currency.
• See Kosowski and Neftci, ch. 3 and ch. 6, for more details. 31
FX Swaps
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FX Swap
• FX swaps combine a spot purchase and forward sale (or
vice versa) of the same currency pair in one transaction.
• For example, we might buy EUR spot and sell EUR forward
both against USD.
FX Swap
𝐸𝑈𝑅 1 𝑈𝑆𝐷 𝐹#! 𝑈𝑆𝐷 𝑆#! 𝐸𝑈𝑅 1
𝑡!
• FX swaps are quoted in terms of the difference between the forward exchange rate and the spot exchange rate, which is called the swap points or forward points.
𝑡”
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Swap Points and Interest Rates
• On what do the swap points depend?
• What determines the relationship between the forward and spot exchange rates for a given currency pair?
• According to our earlier no-arbitrage argument, the swap points depend on the interest rate differential between the two currencies, since this determines the carry.
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FX Swaps and Interest Rates • According to covered interest parity:
𝐹 =𝑆 1+𝑟%'( #! #! 1 + 𝑟$%&
• The FX swap separates the interest rate differential risk from the spot exchange rate risk. Daily movements in the swap points will be driven mainly by interest rates.
• The FX swap is like a bet on the interest rate differential.
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Engineering an FX Swap
• An FX swap in which we buy EUR spot and sell EUR forward for USD is like borrowing EUR and lending USD:
FX Swap
Borrow EUR Lend USD
𝐸𝑈𝑅
𝑈𝑆𝐷 𝐸𝑈𝑅
𝑈𝑆𝐷 𝐸𝑈𝑅
𝐸𝑈𝑅 𝑤𝑖𝑡h 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑈𝑆𝐷 𝑤𝑖𝑡h 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑡!
𝑡!
𝑈𝑆𝐷
𝑡”
𝑡!
• The swap is like a synthetic EUR loan collateralized with USD. We pay interest on EUR and earn interest on USD.
𝑡”
𝑡”
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Uses of FX Swaps
• Speculating on or hedging interest rate differentials
• Synthetic borrowing or lending of one currency collateralized by another currency (see previous example)
• Synthetic forward FX (when combined with spot FX)
• Extending or retracting delivery dates for existing FX
forward commitments (e.g. in a hedging strategy)
• Rolling over open spot FX positions at end of the day
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Some Market Realities
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Financial Engineering in Practice
• In financial engineering, we construct a synthetic equivalent to a financial instrument by constructing a portfolio that replicates its payoffs or cash flows.
• If there is no arbitrage, then the price of the instrument should equal the cost of the replicating portfolio.
• The instrument can be hedged by taking an equal and opposite position in the replicating portfolio.
• Under what circumstances will this be possible, and of what potential difficulties should we be aware?
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No-Arbitrage
• Formally, we define two types of arbitrage opportunity:
– Type 1: a strategy that has costs nothing today, has a non- negative payoff under all future scenarios, and a strictly positive payoff in at least some future scenario
– Type 2: a strategy that has a positive payoff today and a non-negative payoff in all future scenarios
• It seems sensible to imagine that any reasonable set of market prices should exclude arbitrage opportunities.
• A closely related idea is the law of one price: Strategies with identical cash flows should have identical values.
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No-Arbitrage Derivatives Pricing
• We used this idea to analyze an FX forward by creating a replicating portfolio that reproduced its cash flows.
• The no-arbitrage forward exchange rate was the spot exchange rate adjusted for the carry on this portfolio.
• The FX forward was redundant, in the sense that it could be fully replicated by trading in other instruments.
• But is this strictly true of all derivatives? Do they all trade at prices determined only by the absence of arbitrage?
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Consumption Assets
• For investment assets (i.e. assets held solely for investment purposes by at least some traders), carry is typically the dominant factor driving forward prices.
• But for consumption assets (i.e. assets held primarily for consumption) arbitrage arguments may be less relevant.
• Derivatives on commodities such as copper and crude oil can help consumers and producers manage risk, but they cannot be used directly in consumption or production.
42
Carry and Forward Prices
• Constraints on storage and the absence of borrowing and lending markets make it difficult to arbitrage differences between spot and forward prices in these assets.
• Negative carry (e.g. funding and storage costs) at most imposes an upper bound on their forward prices.
• We should therefore think carefully about the extent to which the prices of derivative instruments that we create can be determined by no-arbitrage considerations alone.
43
Static and Dynamic Replication
• Forward contracts have linear payoffs that can be
reproduced and hedged through static replication.
• Reproducing and hedging more complex derivatives, however, may require dynamic replication, in which the replicating portfolio is rebalanced or adjusted over time.
• Think, for example, of delta hedging for options.
44
Dynamic Replication in Practice
• Dynamic replication requires that suitable hedging instruments exist, that they be sufficiently liquid, and that dynamic rebalancing can accurately reproduce the payoff.
• For some instruments and under some market conditions, dynamic replication may be difficult to achieve.
• For more complex derivatives that are harder to replicate, the fair-value price we charge may need to be significantly higher than the theoretical no-arbitrage value.
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Theory and Practice
• Even for simple derivatives on investment assets, market prices may sometimes deviate from no-arbitrage values.
• Consider again the fair-value forward exchange rate.
• The basic idea of covered interest arbitrage is that lending USD is equivalent to exchanging the USD for EUR, lending the EUR, and selling the EUR forward for USD.
• But are these two strategies really equivalent? Are there other factors that might affect the forward price?
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FX Forwards and Credit Risk
• Covered interest parity assumes that there is no risk in the borrowing and lending agreements or the FX forward.
• In practice, money market interest rates (e.g. LIBOR and Euribor) are unsecured lending rates that may contain significant risk premiums that change over time.
• There is also significant counterparty risk in forward FX.
• Risk premiums and the cost of managing counterparty risk may cause market forward FX rates to deviate from theoretical levels implied by covered interest arbitrage.
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FX Swaps as Collateralized Loans
• Other factors may also affect FX forward rates.
• FX swaps can be interpreted as collateralized loans.
• A market participant who sells EUR for USD spot and buys EUR for USD forward has, in effect, borrowed USD and collateralized this loan by giving up EUR.
• Equivalently, she has swapped EUR for USD.
• When the order flow in this market is balanced, the swap points will reflect the interest rate differential.
48
Constrained Liquidity
• But suppose that market participants are unable to obtain the funding they want through normal borrowing.
• They may use FX swaps as a synthetic form of borrowing.
• This is exactly what happened in the financial crisis, when European financial institutions holding USD-denominated assets lost access to USD funding and turned to FX swaps.
• The unbalanced order flow (everyone wanting to sell EUR spot and buy EUR forward against USD) pushed up the forward exchange rate relative to its no-arbitrage value.
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Constrained Liquidity
• Institutions using this synthetic borrowing channel had to repay more USD on the forward delivery date, driving up their effective cost of obtaining USD funding.
• The FX swap-implied USD LIBOR rate will be higher than the market USD LIBOR rate.
• Constrained liquidity can also affect other currencies.
• In practice, financial engineers need to pay close attention to actual market conditions and constraints.
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Source: Bank for International Settlements, BIS Quarterly Review, December 2019.
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References Main reading:
• Hull, ch. 5
• Kosowski and Neftci, ch. 6
Background reading:
• Hull, ch. 1
• Kosowski and Neftci, ch. 1 and 2
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