FINANCIAL RISK
MANANGEMENT
ANGELO LUISI
2023-2024
,Table of Contents
1. Interest rates .............................................................................................................. 5
1.1 Time has value ................................................................................................................ 5
1.2 Time value of money....................................................................................................... 5
1.3 Credit risk ....................................................................................................................... 5
1.4 Fair value of a financial asset .......................................................................................... 6
1.5 Typologies of rates .......................................................................................................... 6
1.6 Bond pricing .................................................................................................................... 9
1.7 Determining treasury zero rates: the bootstrap method ................................................10
1.8 Forward rates ................................................................................................................11
1.8.1 Formula .......................................................................................................................................... 11
1.8.2 Upward vs Downward Sloping Yield Curve .................................................................................... 11
1.8.3 Forward rate agreement ................................................................................................................ 11
1.8.4 FRA: example ................................................................................................................................. 11
1.9 Introduction to derivatives.............................................................................................12
1.9.1 Why are derivatives important? .................................................................................................... 12
1.9.2 How are derivatives traded? .......................................................................................................... 12
1.9.3 The OTC market ............................................................................................................................. 12
1.10 Forward Contracts .........................................................................................................13
1.10.1 Example of usage ....................................................................................................................... 13
1.10.2 Payoffs from forwards ............................................................................................................... 13
1.11 Futures ..........................................................................................................................14
1.11.1 Closing out positions.................................................................................................................. 14
1.11.2 Specification .............................................................................................................................. 15
1.11.3 Price ........................................................................................................................................... 15
1.11.4 Margins ...................................................................................................................................... 16
1.11.5 Example ..................................................................................................................................... 16
1.11.6 The clearing house ..................................................................................................................... 16
1.11.7 Cashflow when futures price … ................................................................................................. 17
1.12 Futures vs Forwards .......................................................................................................17
2. Risk measurement .................................................................................................... 18
2.1 Value-at-risk ..................................................................................................................18
2.2 VaR vs Expected Shortfall ...............................................................................................18
2.3 Expected shortfall ..........................................................................................................18
2.4 VaR and ES .....................................................................................................................19
2.4.1 Historical simulation ...................................................................................................................... 19
2.4.2 Model-building ............................................................................................................................... 19
2.4.3 Multi-day VaR................................................................................................................................. 21
2.5 Backtesting ....................................................................................................................21
3. Futures and forwards: hedging and pricing ............................................................... 22
3.1 Introduction to derivatives.............................................................................................22
3.1.1 How derivatives are traded ............................................................................................................ 22
3.1.2 The OTC market ............................................................................................................................. 22
1
, 3.1.3 Market size ..................................................................................................................................... 23
3.2 Forward contracts ..........................................................................................................23
3.2.1 Example: hedging foreign currency risk ......................................................................................... 23
3.2.2 Forward prices and spot prices ...................................................................................................... 24
3.3 Futures ..........................................................................................................................25
3.3.1 Specifications of future contracts .................................................................................................. 25
3.3.2 Price quotes ................................................................................................................................... 25
3.3.3 Price limits and position limits ....................................................................................................... 25
3.3.4 Convergence of futures to spot prices ........................................................................................... 26
3.3.5 Margins .......................................................................................................................................... 26
3.3.6 The clearing house ......................................................................................................................... 27
3.3.7 Cash flows when … ......................................................................................................................... 27
3.3.8 Futures vs forwards........................................................................................................................ 28
3.4 Hedging strategies using forwards .................................................................................28
3.4.1 Plus sides of hedging ...................................................................................................................... 28
3.4.2 Down sides of hedging ................................................................................................................... 28
3.4.3 Basis risk ......................................................................................................................................... 29
3.4.4 Long hedge for the purchase of an asset ....................................................................................... 29
3.4.5 Short hedge for the sale of an asset .............................................................................................. 29
3.4.6 Choice of contract .......................................................................................................................... 30
3.4.7 Example of cross hedging with basis risk ....................................................................................... 30
3.4.8 Optimal hedge ratio ....................................................................................................................... 30
3.4.9 Optimal number of contracts ......................................................................................................... 31
4. Futures & forward prices and SWAPS ........................................................................ 32
4.1 Futures and forwards prices ...........................................................................................32
4.1.1 Investment vs consumption asset.................................................................................................. 32
4.1.2 Short selling.................................................................................................................................... 32
4.1.3 Notation for valuing futures and forward contracts ...................................................................... 32
4.1.4 Arbitrage opportunity: forward price ............................................................................................ 33
4.1.5 The forward price ........................................................................................................................... 33
4.1.6 What if short sales are not possible? ............................................................................................. 34
4.1.7 Known income ............................................................................................................................... 34
4.1.8 Known yield.................................................................................................................................... 34
4.1.9 Valuing forward contracts .............................................................................................................. 34
4.1.10 Forward vs futures prices .......................................................................................................... 35
4.1.11 Stock index................................................................................................................................. 36
4.1.12 Index arbitrage........................................................................................................................... 36
4.1.13 Forwards and futures on currencies .......................................................................................... 36
4.1.14 Futures on commodities ............................................................................................................ 37
4.1.15 The cost of carry ........................................................................................................................ 38
4.2 SWAPS ...........................................................................................................................38
4.2.1 Mechanics of interest rate swaps .................................................................................................. 38
4.2.2 Overnight indexed swaps ............................................................................................................... 38
4.2.3 Determination of risk-free rates .................................................................................................... 40
4.2.4 Typical uses of interest rate swaps ................................................................................................ 40
4.2.5 Using the swap to convert a liability .............................................................................................. 41
4.2.6 Using the swap to transform an asset ........................................................................................... 41
4.2.7 The organization of trading ............................................................................................................ 41
4.2.8 The comparative-advantage argument.......................................................................................... 42
5. Options ..................................................................................................................... 44
5.1 Types of options ............................................................................................................44
5.1.1 Long call ......................................................................................................................................... 44
2
, 5.1.2 Short call ........................................................................................................................................ 44
5.1.3 Long put ......................................................................................................................................... 44
5.1.4 Short put ........................................................................................................................................ 45
5.1.5 Payoffs with options ...................................................................................................................... 45
5.2 Underlying assets ...........................................................................................................45
5.3 Specifications .................................................................................................................46
5.3.1 Expiration dates ............................................................................................................................. 46
5.3.2 Strike prices .................................................................................................................................... 46
5.3.3 Terminology ................................................................................................................................... 47
5.3.4 Intrinsic value ................................................................................................................................. 47
5.3.5 FLEX Options on equities and equity indices ................................................................................. 47
5.3.6 Dividends and stock splits .............................................................................................................. 47
5.3.7 Position limits and exercise limit.................................................................................................... 48
5.4 Trading ..........................................................................................................................48
5.4.1 Market makers ............................................................................................................................... 48
5.4.2 Offsetting orders ............................................................................................................................ 48
5.5 Margin ...........................................................................................................................48
5.6 Properties of stock options ............................................................................................49
5.6.1 Factors affecting option prices....................................................................................................... 49
5.6.2 Assumptions and notation ............................................................................................................. 49
5.6.3 American vs European options ...................................................................................................... 49
5.6.4 Upper bounds for option prices ..................................................................................................... 49
5.6.5 Lower bound for calls on Non-Dividend-Paying Stocks.................................................................. 50
5.6.6 Lower bound for puts on Non-dividend-Paying stocks .................................................................. 51
5.6.7 Impact of dividends ........................................................................................................................ 51
5.7 Put-call parity ................................................................................................................52
5.7.1 American option prices .................................................................................................................. 52
5.7.2 Effect of dividends ......................................................................................................................... 53
5.7.3 Put-call parity and dividends .......................................................................................................... 53
6. Trading strategies involving options ......................................................................... 54
6.1 Principal protected notes ...............................................................................................54
6.2 Option and underlying asset ..........................................................................................54
6.3 Spreads ..........................................................................................................................55
6.3.1 Bull spreads with calls .................................................................................................................... 55
6.3.2 Bull spreads with puts ................................................................................................................... 55
6.3.3 Bear spreads with calls................................................................................................................... 55
6.3.4 Bear spreads with Puts................................................................................................................... 55
6.3.5 Box spread...................................................................................................................................... 56
6.3.6 Butterfly spread with calls ............................................................................................................. 56
6.3.7 Butterfly spread with puts ............................................................................................................. 56
6.3.8 Calendar spreads with calls ............................................................................................................ 56
6.3.9 Calendar spreads with puts............................................................................................................ 56
6.4 Combinations.................................................................................................................57
6.4.1 Straddle .......................................................................................................................................... 57
6.4.2 Strip and straps .............................................................................................................................. 57
6.4.3 Strangles......................................................................................................................................... 57
7. Option pricing ........................................................................................................... 58
7.1 Binomial tree – no arbitrage argument ..........................................................................58
7.1.1 A generalization ............................................................................................................................. 60
3
, 7.1.2 Risk neutral valuation .................................................................................................................... 60
7.1.3 A multi-step example ..................................................................................................................... 61
7.1.4 Multi-step: put options .................................................................................................................. 62
7.1.5 Multi-step: American option .......................................................................................................... 62
7.1.6 Values for u and d .......................................................................................................................... 62
7.2 Black-Scholes-Merton Model .........................................................................................62
7.2.1 Implied volatility ............................................................................................................................ 64
7.3 Binomial trees vs BSM Model.........................................................................................64
4
, 1. Interest rates
- What are interest rates?
→ The amount of money a borrower promises to pay to the lender
- Why do we need interest rates?
→ Time value of money
- Risk-free rate
→ Credit risk
- The risk that a default by the borrower causes the lender not to be paid
- To determine the fair value of any financial asset
1.1 Time has value
If you want to be convinced to use / invest your money you need a return
At the core, the intrest rate allows us to make a difference between the flow of money today and in
the future
1.2 Time value of money
Moving money across time
→ 10 C today are different than 10 C tomorrow, why?
- 10 C today can be spent
- 10 C tomorrow cannot (no matter how trustworthy the counterpart).
The interest rate is the price to convert future money into today’s money (and the other way around)
No matter how trustworthy the borrower is, if you lend him money, you cannot use it today
1.3 Credit risk
The counterpart is important
→ 10 C tomorrow from Bill Gates are different than 10 C tomorrow from Angelo Luisi, why?
- Bill Gates is a multimillionaire
- I am not (no matter how good this class will be)
↑ the creditworthiness of the counterpart, ↓ the credit risk
↓ the credit risk, ↓ the interest rate.
Lower bound: Risk-free rate
à Even if the borrower is very credible, there will always be an intrest rate for the time, which has a
monetary value
5
MANANGEMENT
ANGELO LUISI
2023-2024
,Table of Contents
1. Interest rates .............................................................................................................. 5
1.1 Time has value ................................................................................................................ 5
1.2 Time value of money....................................................................................................... 5
1.3 Credit risk ....................................................................................................................... 5
1.4 Fair value of a financial asset .......................................................................................... 6
1.5 Typologies of rates .......................................................................................................... 6
1.6 Bond pricing .................................................................................................................... 9
1.7 Determining treasury zero rates: the bootstrap method ................................................10
1.8 Forward rates ................................................................................................................11
1.8.1 Formula .......................................................................................................................................... 11
1.8.2 Upward vs Downward Sloping Yield Curve .................................................................................... 11
1.8.3 Forward rate agreement ................................................................................................................ 11
1.8.4 FRA: example ................................................................................................................................. 11
1.9 Introduction to derivatives.............................................................................................12
1.9.1 Why are derivatives important? .................................................................................................... 12
1.9.2 How are derivatives traded? .......................................................................................................... 12
1.9.3 The OTC market ............................................................................................................................. 12
1.10 Forward Contracts .........................................................................................................13
1.10.1 Example of usage ....................................................................................................................... 13
1.10.2 Payoffs from forwards ............................................................................................................... 13
1.11 Futures ..........................................................................................................................14
1.11.1 Closing out positions.................................................................................................................. 14
1.11.2 Specification .............................................................................................................................. 15
1.11.3 Price ........................................................................................................................................... 15
1.11.4 Margins ...................................................................................................................................... 16
1.11.5 Example ..................................................................................................................................... 16
1.11.6 The clearing house ..................................................................................................................... 16
1.11.7 Cashflow when futures price … ................................................................................................. 17
1.12 Futures vs Forwards .......................................................................................................17
2. Risk measurement .................................................................................................... 18
2.1 Value-at-risk ..................................................................................................................18
2.2 VaR vs Expected Shortfall ...............................................................................................18
2.3 Expected shortfall ..........................................................................................................18
2.4 VaR and ES .....................................................................................................................19
2.4.1 Historical simulation ...................................................................................................................... 19
2.4.2 Model-building ............................................................................................................................... 19
2.4.3 Multi-day VaR................................................................................................................................. 21
2.5 Backtesting ....................................................................................................................21
3. Futures and forwards: hedging and pricing ............................................................... 22
3.1 Introduction to derivatives.............................................................................................22
3.1.1 How derivatives are traded ............................................................................................................ 22
3.1.2 The OTC market ............................................................................................................................. 22
1
, 3.1.3 Market size ..................................................................................................................................... 23
3.2 Forward contracts ..........................................................................................................23
3.2.1 Example: hedging foreign currency risk ......................................................................................... 23
3.2.2 Forward prices and spot prices ...................................................................................................... 24
3.3 Futures ..........................................................................................................................25
3.3.1 Specifications of future contracts .................................................................................................. 25
3.3.2 Price quotes ................................................................................................................................... 25
3.3.3 Price limits and position limits ....................................................................................................... 25
3.3.4 Convergence of futures to spot prices ........................................................................................... 26
3.3.5 Margins .......................................................................................................................................... 26
3.3.6 The clearing house ......................................................................................................................... 27
3.3.7 Cash flows when … ......................................................................................................................... 27
3.3.8 Futures vs forwards........................................................................................................................ 28
3.4 Hedging strategies using forwards .................................................................................28
3.4.1 Plus sides of hedging ...................................................................................................................... 28
3.4.2 Down sides of hedging ................................................................................................................... 28
3.4.3 Basis risk ......................................................................................................................................... 29
3.4.4 Long hedge for the purchase of an asset ....................................................................................... 29
3.4.5 Short hedge for the sale of an asset .............................................................................................. 29
3.4.6 Choice of contract .......................................................................................................................... 30
3.4.7 Example of cross hedging with basis risk ....................................................................................... 30
3.4.8 Optimal hedge ratio ....................................................................................................................... 30
3.4.9 Optimal number of contracts ......................................................................................................... 31
4. Futures & forward prices and SWAPS ........................................................................ 32
4.1 Futures and forwards prices ...........................................................................................32
4.1.1 Investment vs consumption asset.................................................................................................. 32
4.1.2 Short selling.................................................................................................................................... 32
4.1.3 Notation for valuing futures and forward contracts ...................................................................... 32
4.1.4 Arbitrage opportunity: forward price ............................................................................................ 33
4.1.5 The forward price ........................................................................................................................... 33
4.1.6 What if short sales are not possible? ............................................................................................. 34
4.1.7 Known income ............................................................................................................................... 34
4.1.8 Known yield.................................................................................................................................... 34
4.1.9 Valuing forward contracts .............................................................................................................. 34
4.1.10 Forward vs futures prices .......................................................................................................... 35
4.1.11 Stock index................................................................................................................................. 36
4.1.12 Index arbitrage........................................................................................................................... 36
4.1.13 Forwards and futures on currencies .......................................................................................... 36
4.1.14 Futures on commodities ............................................................................................................ 37
4.1.15 The cost of carry ........................................................................................................................ 38
4.2 SWAPS ...........................................................................................................................38
4.2.1 Mechanics of interest rate swaps .................................................................................................. 38
4.2.2 Overnight indexed swaps ............................................................................................................... 38
4.2.3 Determination of risk-free rates .................................................................................................... 40
4.2.4 Typical uses of interest rate swaps ................................................................................................ 40
4.2.5 Using the swap to convert a liability .............................................................................................. 41
4.2.6 Using the swap to transform an asset ........................................................................................... 41
4.2.7 The organization of trading ............................................................................................................ 41
4.2.8 The comparative-advantage argument.......................................................................................... 42
5. Options ..................................................................................................................... 44
5.1 Types of options ............................................................................................................44
5.1.1 Long call ......................................................................................................................................... 44
2
, 5.1.2 Short call ........................................................................................................................................ 44
5.1.3 Long put ......................................................................................................................................... 44
5.1.4 Short put ........................................................................................................................................ 45
5.1.5 Payoffs with options ...................................................................................................................... 45
5.2 Underlying assets ...........................................................................................................45
5.3 Specifications .................................................................................................................46
5.3.1 Expiration dates ............................................................................................................................. 46
5.3.2 Strike prices .................................................................................................................................... 46
5.3.3 Terminology ................................................................................................................................... 47
5.3.4 Intrinsic value ................................................................................................................................. 47
5.3.5 FLEX Options on equities and equity indices ................................................................................. 47
5.3.6 Dividends and stock splits .............................................................................................................. 47
5.3.7 Position limits and exercise limit.................................................................................................... 48
5.4 Trading ..........................................................................................................................48
5.4.1 Market makers ............................................................................................................................... 48
5.4.2 Offsetting orders ............................................................................................................................ 48
5.5 Margin ...........................................................................................................................48
5.6 Properties of stock options ............................................................................................49
5.6.1 Factors affecting option prices....................................................................................................... 49
5.6.2 Assumptions and notation ............................................................................................................. 49
5.6.3 American vs European options ...................................................................................................... 49
5.6.4 Upper bounds for option prices ..................................................................................................... 49
5.6.5 Lower bound for calls on Non-Dividend-Paying Stocks.................................................................. 50
5.6.6 Lower bound for puts on Non-dividend-Paying stocks .................................................................. 51
5.6.7 Impact of dividends ........................................................................................................................ 51
5.7 Put-call parity ................................................................................................................52
5.7.1 American option prices .................................................................................................................. 52
5.7.2 Effect of dividends ......................................................................................................................... 53
5.7.3 Put-call parity and dividends .......................................................................................................... 53
6. Trading strategies involving options ......................................................................... 54
6.1 Principal protected notes ...............................................................................................54
6.2 Option and underlying asset ..........................................................................................54
6.3 Spreads ..........................................................................................................................55
6.3.1 Bull spreads with calls .................................................................................................................... 55
6.3.2 Bull spreads with puts ................................................................................................................... 55
6.3.3 Bear spreads with calls................................................................................................................... 55
6.3.4 Bear spreads with Puts................................................................................................................... 55
6.3.5 Box spread...................................................................................................................................... 56
6.3.6 Butterfly spread with calls ............................................................................................................. 56
6.3.7 Butterfly spread with puts ............................................................................................................. 56
6.3.8 Calendar spreads with calls ............................................................................................................ 56
6.3.9 Calendar spreads with puts............................................................................................................ 56
6.4 Combinations.................................................................................................................57
6.4.1 Straddle .......................................................................................................................................... 57
6.4.2 Strip and straps .............................................................................................................................. 57
6.4.3 Strangles......................................................................................................................................... 57
7. Option pricing ........................................................................................................... 58
7.1 Binomial tree – no arbitrage argument ..........................................................................58
7.1.1 A generalization ............................................................................................................................. 60
3
, 7.1.2 Risk neutral valuation .................................................................................................................... 60
7.1.3 A multi-step example ..................................................................................................................... 61
7.1.4 Multi-step: put options .................................................................................................................. 62
7.1.5 Multi-step: American option .......................................................................................................... 62
7.1.6 Values for u and d .......................................................................................................................... 62
7.2 Black-Scholes-Merton Model .........................................................................................62
7.2.1 Implied volatility ............................................................................................................................ 64
7.3 Binomial trees vs BSM Model.........................................................................................64
4
, 1. Interest rates
- What are interest rates?
→ The amount of money a borrower promises to pay to the lender
- Why do we need interest rates?
→ Time value of money
- Risk-free rate
→ Credit risk
- The risk that a default by the borrower causes the lender not to be paid
- To determine the fair value of any financial asset
1.1 Time has value
If you want to be convinced to use / invest your money you need a return
At the core, the intrest rate allows us to make a difference between the flow of money today and in
the future
1.2 Time value of money
Moving money across time
→ 10 C today are different than 10 C tomorrow, why?
- 10 C today can be spent
- 10 C tomorrow cannot (no matter how trustworthy the counterpart).
The interest rate is the price to convert future money into today’s money (and the other way around)
No matter how trustworthy the borrower is, if you lend him money, you cannot use it today
1.3 Credit risk
The counterpart is important
→ 10 C tomorrow from Bill Gates are different than 10 C tomorrow from Angelo Luisi, why?
- Bill Gates is a multimillionaire
- I am not (no matter how good this class will be)
↑ the creditworthiness of the counterpart, ↓ the credit risk
↓ the credit risk, ↓ the interest rate.
Lower bound: Risk-free rate
à Even if the borrower is very credible, there will always be an intrest rate for the time, which has a
monetary value
5
