Discuss the management of market risk
Develop and recommend strategies for the reduction of market risk using financial derivatives
Demonstrate an awareness of the practical issues related to dynamic hedging using market instruments
Exam Note:
Need to be able to recommend strategies for dealing with market risk
(e.g. how to apply appropriate hedging strategies)
Market risk in financial institutions is typically linked with credit risk and op risk (e.g. both may worsen when markets are subdued or in turmoil)
Market risk management strategies include:
Diversification:
Holding a range of assets so as to limit losses within a portfolio investment strategy
Ensuring it is appropriate for the liabilities of the org
Hedging:
Use derivatives to manage risk
Diversity: definition and measurement
Can be measured in:
Absolute term
(e.g. breadth, balance of assets in each class, economic sector or geography)
Relative term
(i.e. compared to the diversity of a suitable benchmark)
Use factor analysis (e.g. SVD from Module 23) to determine how broad a range of economic and financial variables influence the market value of the portfolio
Key activities of market risk management:
Set and monitor policies
Set and monitor limits
(for overall and each asset class, individual security and counterparty)
Reporting
Capital management
Implementing risk-portfolio strategies
(e.g. matching, hedging)
Purpose of documented market risk policies is to ensure:
All market risks are identified, measured, monitored, controlled and regularly reported
Policies should reflect the complexity of businesses and be tailored to the activities of the company
Scope of policies :
Roles and responsibilities:
Person responsible for developing, implementing, monitoring and reviewing policies
Delegation of authority and limits:
Who has permission to execute market risk positions and to what extent
front office) and settlement (back office) functions should be segregatedRisk measurement and reporting:
How risks are measured and reported (esp. critical issues e.g. limit violation)
Valuation and back-testing:
How positions are valued (esp. when no market price exist)
Hedging policy:
What risk to hedge, the products, limits and strategies for hedging and how the effectiveness is measured
Liquidity policy:
How to measure liquidity and what the contingency plan is during distress
Exception management:
How to handle and exceptions
Be aware that no policies or risk measuring will protect fully against rogue traders or incompetence (but that’s consider op-risk)
| Basic | Standard | Best | |
|---|---|---|---|
| Market risk factors | Evaluates earning impact of interest rate and FX changes, perform gap analysis |
Central management of interest rate and FX risk through internal transfer pricing; Considers internal hedges |
Seeks competitive advantage (e.g. exploiting mispriced securities and better intelligence) |
| Modeling | Spreadsheet | Robust capabilities (incl. sensitivity analysis and simulation) |
Sophisticated tools (incl. hot-spot analysis, best hedge analysis, best replicating portfolio and implied view) |
| Market risk function | Policy, analysis and reporting | Actively manages the b/s to optimize return given risk constraints | Corporate control and profit centre: seeks to maximize profits within risk limits |
Even the best practice companies with the most sophisticated tools need to apply common sense judgement and integrity to their management of risks
Gain or loss from a derivative depends on changes in the market price of underlying assets or index (e.g. interst rate, FX rate, commodity, equity, etc)
Derivative redistribute risk from those who wish to hedge to those who are prepared to accept the risk in return for the possibility of a large reward (i.e. speculators)
4 main types of derivatives:
Options:
Right but not the obligation (for 1 party) to exercise the contract in return for payment of a premium
Forwards:
Obligation (for both parties) to complete a transaction on a future data at a known prize
Futures:
Standardized forward contract traded on an exchange
Swaps:
Obligation (for both parties) to exchange a series of cash flows
Futures (and some options) are exchange-traded contracts
Characteristics:
Standardized assets and indices and only for specific delivery dates as determined by the exchange
Trading is done through the exchange based on market prices
(Which may be subject to a minimum price movement or ticks)
Deals are settled through a clearing-house
Clearing-house takes on the counterparty risk as they act as both the buyer and seller of the contract
Reduced counterparty risk for the clearing house
By the pooling of many contracts
Collateral were required from the trading parties
Highly liquid market
Prices being minimally affected by large transaction
Comparatively low transaction cost
Characteristics of OTC markets:
Trading is done at the convenience of the parties
No money changes hands until the delivery date
Price is negotiated between the parties
(one that take on the risk of counterparty defaulting)
Very flexible (underlying and delivery data)
Documentation is usually based on standard terms and conditions, such as those published by the International Swaps and Derivatives Associations (ISDA)
Types of OTC contracts
Forwards and swaps
Some other types of options as well but regulators are aiming to increase transparency by moving more derivatives trading onto exchanges
Main factors that influences the price of an OTC contract
Spot price of the underlying
Time to delivery
Expectations of interest rates over this period
Expected income yield on the underlying over this period
Residual counterparty risk (allowing for effect of any collateral)
There is a high degree of counterparty risk with OTC contracts
Collateral is typical
Greater residaul counterparty risk and the more bespoke the contract
\(hookrightarrow\) Higher the price (due to admin cost, default risk and difficulty in hedging)
Cost of carry
(incl. opportunity cost of providing collateral; cost of storage if a commodity)
May be held by one or both parties to provide protection from the costs of any potential default (cash deposit or other securities)
Exchange-traded contracts
Mark-to-market process
Cash is deposited by the counterparty into a margin account at the start of an exchange traded contract (initial margin)
Determined based on the size and anticipate volatility of the contract
Clearing house periodically (e.g. daily) considers whether to add or remove amounts from the margin account based on the intervening movement in the price of the underlying
The additional or deduction reflect the respective P&L position of the counterparty (marking-to-market process)
If the margin account drops below a specific level (maintenance margin), the counterparty much top-up the account to the starting level by adding to the margin account (variation margin)
Exchanges may accept securities (rather than cash) as collateral
Amount of collateral required may be reduced to account for the degree of diversification in the counterparty’s total set of positions with that exchange
OTC contracts
Requirements for collateralization are typically specified in a credit support annex (CSA) such as that published by the ISDA
Types of security that can act as collateral
When collateral must be calculated and transferred
Minimum transfer amount, below which no transfer is required
Advantages
It maybe cheaper and easier to deal in a derivative than the underlying assets
Very flexible and exposure can be changes quickly without the need to deal in the underlying asset
(e.g. investment allocation can be changed quickly while still holding the original assets)
Caveat
Hedging strategies can be ineffective and may even result in losses
By hedging a risk we also might eliminate potential gains
Costly process, involving transaction cost, spreads, premiums and management time and effort
Requires experienced staff
Potential risk exposed from derivatives
Credit
Settlement
Aggregation
Operational
Liquidity
Legal
Reputational
Concentration
Basis
For future contracts:
\(\underbrace{\text{Basis}}_{B_t} = \underbrace{\text{Price of an asset}}_{S_t} - \underbrace{\text{Price of the future}}_{F_t}\)
Basis risk: Risk that the basis changes over time
Arise in practice due to:
Hedger is uncertain as to the exact date when the asset will be bought or sold
Hedger requires the futures contract to be closed out well before its expiration date
Hedger requires the future position to be rolled over at, or prior to, expiration
(Because the futures contract is shorter than the desired period of the hedge)
Differences in income, benefits and/or costs between the futures contract and the underlying asset are not known precisely in advance
Forward is bespoke to reflect the exact risk being hedged but futures are standardized hence may be difficult to match the risk exactly
Cross hedging risk (component of basis risk):
Price risk resulting from the asset whose price is to be hedged is not exactly the same as the asset underlying the futures contract
Basis reduces to zero as expiry approaches
If there is no cross-hedging risk and no uncertainty as to the associated cashflows (income, benefits or costs)
Example
Asset is currently held but must be sold in 1 year at the market price
A 2 years future is available on the same underlying asset
No associated cashflows other than the payment of margin
Hedging strategy:
\(t=0\):
Sell the future contract
\(t=1\):
Sell the asset and close out the hedge (or by buying the same futures contract)
\(F_0 - F_1 + S_1 = F_0 + B_1\)
Exposed to basis risk as the total cash flow is a function of the uncertain basis at \(t=1\)
Here we are ignoring the TVM…
Difference between the {spot price of the underlying} and the {corresponding futures contract price} is largely driven by differences in the amounts and timing of the associated expected cashflows
Fair price (\(F_0\)) of a future on an asset (currently priced at a spot rate of \(S_0\)) while ignoring transaction costs:
\(F_0 = S_0 e^{rT}\)
\(r\): Risk free rate
\(T\): Time to expiry of the contract
If underlying asset provides an income (\(I\)):
(Not received from holding the future contract)
\(F_0 = (S_0 - I)e^{rT}\)
If the income is in the form of a continuous yield \(q\):
\(F_o = S_0 e^{(r-q)T}\)
Cost of carry can be considered as negative income and dealt with similarly
Normal backwardation:
Future price is below the expected value of the future spot price
May occur due to the market expecting the income on the asset to outweigh any costs
(\(\therefore\) a preference to hold the asset)
Or due to high demand for short position in the future
(e.g from the asset holder protecting the value of their assets)
Contango:
Future price exceeds the expected value of the future spot price
May arise due to demand for long positions in the future
(e.g. storage costs of the underlying commodity are particularly high)
Number of contracts required to hedge the market risk of a portfolio:
\(= \dfrac{\text{Value of the portfolio}}{\text{Value of one contract}}\)
Basis risk adjustments
Need to adjust for differences in the relative volatility of the portfolio and the future using the optimal hedge ratio
Optimal hedge ratio (\(h\)):
\(h = \rho \dfrac{\sigma_S}{\sigma_F}\)
Ratio of the {size of the position taken in futures contracts} to the {size of the exposure}
\(\sigma_S\): s.d. of \(\Delta S\)
The change in spot prices over the life of the hedge
\(\sigma_F\): s.d. of \(\Delta F\)
The change in future prices over the life of the hedge
\(\rho\): Correlation coefficient of the two
Example (Cross hedging oil futures)
Airlines are exposed to the risk of changes in aviation fuel prices
No aviation fuel futures
Hedge with heating oil futures as they move in almost the same way as crude oil
(Need to allow for the costs and capacity constraints in the oil refining process)
Currency hedging:
Currency forward:
Agree to buy/sell foreign currency at an agreed rate at a future date
Forward price:
Based on current (spot) rate of exchange adjusted for the different in interest rates between two currencies
Currency swap:
Effectively a series of currency forwards
Currency future:
Similar to currency forward but via exchange
Currency option:
Company buys the right to buy/sell foreign currency at an agreed date for an agreed price
Allows to lock into a particular FX rate but can benefit from favorable FX rate changes
Currency risk:
Does not provide any additional systematic return
For overseas bonds:
Currency exposures are typically hedged
For overseas equities:
The situation is more complex (e.g. due to the need to establish an exposures), so hedging is either approximate or not attempted
Purchasing power parity of exchange rate:
In the long term, FX change in line with the difference in the inflation rates of the 2 relevant economies
(Thus mitigating the need to hedge currency risk)
Cashflow management techniques can be used to manage currency risk as well:
Netting:
Use revenue in a particular currency to meet any amounts owing in the same currency (residual amount may still need to be hedged)
Leading and lagging:
Attempt to bring forward (lead) or delay (lag) foreign currency cashflows in order to exploit expected movements in FX rates
Will go through the key points on the Greeks
Exam note:
delta, gamma and vega of a portfolio and relevant derivatives where there are one or two underlying indices or asset pricesConsider a portfolio (\(p\)):
Value: \(V\)
Depends non-linearly on a single underlying \(x\) and it’s volatility \(\sigma\)
Liquid traded derivatives available:
The greeks of an asset price or portfolio value:
\(Delta_p\): Partial derivative w.r.t. \(x\)
\(Gamma_p\): 2nd partial derivative w.r.t. \(x\)
\(Vega_p\): Partial derivative w.r.t. \(\sigma\)
Portfolio is delta, gamma, vega hedged if we buy \(u_1,..,u_n\) unites of \(D_1,...D_n\) such that:
\(Delta_p + u_1 Delta_1 + \dots + u_n Delta_n = 0\)
\(Gamma_p + u_1 Gamma_1 + \dots + u_n Gamma_n = 0\)
\(Vega_p + u_1 Vega_1 + \dots + u_n Vega_n = 0\)
Typically this can be achieved with only \(n=3\) derivatives
But there might be liquidity or regulatory constraints that make other derivatives useful
Greeks
Delta (\(\Delta\)) of a portfolio is the rate of {change in its value \(V\)} relative to {changes in the price of the underlying \(S\)}
\(\Delta = \dfrac{\partial V}{\partial S}\)
A portfolio containing options is Delta hedged (neutral) when it consist of positions with offsetting (+) and (-) deltas and the net delta of the portfolio is 0
Gamma (\(\Gamma\)) is the rate of {change of \(\Delta\)} relative to {change in the price of the underlying \(S\)}
\(\Gamma = \dfrac{\partial^2 V}{\partial S^2} = \dfrac{\partial \Delta}{\partial S}\)
Measures the curvature (convexity) of the relationship between the derivative price and the price of the underlying asset
Vega (\(\nu\)) is the rate of {change in its value \(V\)} relative to {change in the assumed level of volatility of the underlying (\(\sigma\))}
\(\nu = \dfrac{\partial V}{\partial \sigma}\)
As the price of derivative depends directly on the assumed volatility of the price of the underlying, if the volatility of the price of the underlying changes, then so must the price of the derivative
Example: Delta and Vega neutral
Option portfolio:
\(\Delta_p = 2,000\)
\(\nu_p = 6,000\)
Options and assets available for hedge:
Underlying stock (\(\Delta = 1\), \(\nu = 0\))
Traded option (\(\Delta = 0.5\), \(\nu = 10\))
Steps:
For vega neutral: sell \(\dfrac{60,000}{10}\) contracts of the traded option
Now \(\Delta = 2,000 - 0.5 \times 6,000 = -1,000\) so we need 1,000 shares of the underlying stock
More formally, given:
\((\Delta_p;\Delta_1;\Delta_2) = (2,000; 1; 0.5)\)
\((\nu_p; \nu_1; \nu_2) = (60,000; 0; 10)\)
Need:
\(0 = \Delta_p + n_1 \Delta_1 + n_2 \Delta_2\)
\(0 = \nu_p + n_1 \nu_1 + n_2 \nu_2\)
Plug in the above and solved we get the same answer
More often a portfolio is linked to the value of many underlying indices or asset prices (\(m\))
Portfolio value depends upon the underlying index values, their volatilities and their correlations
This requires potentially hedging:
\(m\) delta
\(V\) depends on \(x_1,...,x_m\) and there are \(m\) deltas (\(\dfrac{\partial V}{\partial x_i}\))
\(\dfrac{1}{2} m(m+1)\) gammas
Gamma is the 2nd derivatives and so differentiating w.r.t. to any pair of variables we get \(\dfrac{\partial V}{\partial x_i \partial x_j}\)
\(\dfrac{1}{2} m(m+1)\) vegas
Vega is the derivatives w.r.t. the entries in the \(m \times m\) covariance matrix for the \(m\) variables
(e.g. \(\dfrac{\partial V}{\partial C_{1,2}}\) where \(C_{1,2}\) is the covariance between variables 1 and 2)
To achieve neutrality, number of derivatives require is over \(m^2\)
Impractical to maintain gamma and vega neutral:
Requires a large number of derivatives
Individual traders tend to have responsibility for trading in all derivatives linked to a single underlying quantity only
They will be given limits on how far they can deviate from the greek neutrality
The overall portfolio manager then has responsibility for managing the total Greeks
Delta can be easily neutralized daily at low cost using future contracts
Dynamic Hedging: Day-to-day hedging activity undertaken by writers of options
Linear hedging instruments
E.g. futures and forwards
Can easily be hedged by finding offsetting transactions (e.g. just finding long/short position)
Options
More difficult to hedge
Few customers are prepared to write options given the substantial downside risk in a short call or put
Dynamic hedging8 is used to manage the risk from writing options***:
Re balance the option portfolio using forwards, futures and asset holding to remain delta neutral (on a daily basis)
Trader is exposed to risk and can make losses between rebalancing points (also incur cost at each rebalance)
Gamma risk
Large gamma
\(\hookrightarrow\) substantial rebalancing will be required to maintain the delta neutrality
Gamma reflects exposure to the risks associated with:
Jumps in prices
Risk of hedging at discrete time points rather than continuously
Higher the gamma \(\Rightarrow\) the greater of the above risks \(\Rightarrow\) higher transaction costs associated with rebalancing
\(\therefore\) low gamma is preferable
Vega risk
How sensitive the portfolio is to changes in the volatility of the underlying index of asset price
Higher vega the greater the risk associated with an incorrectly specified volatility
Managing Gamma and Vega
Due to lack of traded derivatives or poor liquidity, it is difficult to achieve gamma and vega neutral in practice
Typically managed using limits that will limit the volume of options that a trader can write as well as for the whole institution
Considerations to those outlined above for option writers apply to institutions as well
Component of market risk arises when assets and/or liabilities are sensitive to changes in interest rates
Types of interest rate risk:
Direct effect on the size of company’s cashflows
(e.g. rise in interest rates increase the cost of a floating rate loan)
Affect the value of future cashflows
(e.g. change in interest rates altering the PV of future payments due on an annuity portfolio)
Direct exposure can be hedged using forward rate agreement (FRA) or can be limited by using interest rate caps and floors
OTC, FRA commits 2 parties to exchange some interest rate dependent payments at a future data (or over a future period)
Payments are calculated by:
Applying 2 different pre-agreed forms of interest rates (e.g. one fixed and one floating)…
…to the same specified monetary amount (the principal or nominal)…
…over the agreed period
Generally respective payments are netted off
In their example the FRA is like swaps
OTC, provide insurance against the rate of interest on an underlying floating rate note rising above a certain level (cap rate)
Interest rate floor: provides a payoff when the interest rate on an underlying floating rate note falls below a certain rate
Common example:
Home mortgages have an arrangement where there is a clause that caps the borrower’s interest payments if the bank’s standard variable mortgage rate goes above a specific level
Floor provide insurance against a fall in floating rate, which might be useful for those receiving the floating rate (e.g. a saver)
Common techniques:
Assets and liabilities with exactly matching cashflow (size and timing)
Key factors:
Nature of the liabilities
(e.g. whether the cashflows are fixed or linked to prices or interest rates)
Term
Currency
Practical challenges
Might not have suitable assets
(e.g. No suitable term or cross-hedging risk may be introduced)
Future cashflows may not be known
Expected future cashflows may change frequently
\(\hookrightarrow\) Costly to alter the portfolio
Complete matching is rarely possible in practice
In practice it is usually approximate matching
Can view pure matching as the benchmark position
Also might not be desirable to eliminate all risks
2 means by which interest rate risk might be mitigated
Interest rate swap:
Exchanges of payments based on a fixed interest rate for those based on a floating interest rate
Swaption:
Gives the buyer the right to enter into a swap in return for paying a premium
(Can eliminate the downside interest rate risk while retaining the upside)
Similar purpose to matching:
Reduce the risk of failing to meet the liabilities as they fall due, arising from a change in investment conditions (particularly a change in interest rates)
Use when pure matching is no possible
(e.g. investment income is initially greater than the net liability outgo, then the liabilities cannot be matched but can be immunized)
Conditions for immunization
PV of liability cashflow = PV of asset cashflow
Discounted mean term of the asset cashflows = the liabilities’ (matching duration)
Convexity of the asset cashflows > liabilities’
Limitations (inaddition to matching)
Only ensure the PV of assets \(\nless\) PV liability
Only protects against changes in interest rate
Only works for small changes in interest rates
Only works for parallel shifts in the yield curve
Requires *8regular rebalancing** of the assets
May be acceptable to hedge cashflows at key reference points in the future
Process of using interest rate simulation to construct a set of possible future values for the liabilities at such key reference points (e.g. 5, 10, 15 year etc)
Similarly, a set of future cashflows from some series of swaps or bonds is created
swap cashflows and the liability cashflows