Financial mathematics, 16/10 2014, KTHPer-Olov Åsén, Risk Modeling and Quantitative Analysis

2

Outline

Introduction to hedge funds

Valuation– Simple derivatives– Other derivatives

Risk– Why? – How?

3

Introduction

What is a hedge fund?– Absolute return– Low correlation with other markets– Allows investment in derivatives

– Speculation and/or– Hedging (reduce risk)

Apr

/02

Apr

/03

Apr

/04

Apr

/05

Apr

/06

Apr

/07

Apr

/08

Apr

/09

Apr

/10

Apr

/11

Apr

/12

Apr

/13

Apr

/140

50

100

150

200

250

300

BMS

HFRI Fund of Funds

MSCI World (Loc)

4

Introduction

What is a hedge fund?– Absolute return– Low correlation with other markets– Allows investment in derivatives

– Speculation and/or– Hedging (reduce risk)

Simplest building blocks– Equity, (stock, partial ownership in company)– Long (buy) or short (sell).

– Note: Short an equity means selling an equity you don’t own by first borrowing it.

Apr

/02

Apr

/03

Apr

/04

Apr

/05

Apr

/06

Apr

/07

Apr

/08

Apr

/09

Apr

/10

Apr

/11

Apr

/12

Apr

/13

Apr

/140

50

100

150

200

250

300

BMS

HFRI Fund of Funds

MSCI World (Loc)

5

Example: market neutral long/short equity fund

Find equity you like

Find equity you don’t like (in same sector)

Go long in first equity and equally short in second equity

Ex. buy 1000 SEK Microsoft (M), borrow=>sell 1000 SEK Apple (A)– Initial investment :1000 (M) – 1000 (A) = 0 SEK– Value in rising market:

– M => 1100, A => 1050: Value = 1100 – 1050 = 50 SEK

– Value in falling market:– M => 900, A => 850: Value =900 – 850 = 50 SEK

Positive return as long as first equity does better than second equity

Leverage

6

Derivatives

Contract which depends on some underlying quantity, e.g. an equity, a commodity etc.

7

Derivatives

Contract which depends on some underlying quantity, e.g. an equity, a commodity etc.

Equity forward– Contract stating that:

At time T>0 in the future, you buy the equity S for price K.

8

Derivatives

Contract which depends on some underlying quantity, e.g. an equity, a commodity etc.

Equity forward– Contract stating that:

At time T>0 in the future, you buy the equity S for price K.

Value today?

9

Derivatives

Contract which depends on some underlying quantity, e.g. an equity, a commodity etc.

Equity forward– Contract stating that:

At time T>0 in the future, you buy the equity S for price K.

Value today:

where = equity price today, r = risk-free interest rate

10

Derivatives

European equity call option– Contract stating that:

At time T>0 in the future, you can buy the equity S for price K if you want.– Payoff at time T:

where is the equity price at time T.

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960

10

20

30

40

50 Payoff, K=50

11

Derivatives

European equity call option– Contract stating that:

At time T>0 in the future, you can buy the equity S for price K if you want.– Payoff at time T:

where is the equity price at time T.

Value today?

0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 960

10

20

30

40

50 Payoff, K=50

12

Derivatives

Black-Scholes partial differential equation gives price of derivative:

where S is the equity price, is the standard deviation of the equity returns, and is the risk-free rate.

13

Derivatives

Black-Scholes partial differential equation gives price of derivative:

where S is the equity price, is the standard deviation of the equity returns, and is the risk-free rate.

Initial-condition for European call option:

Solution

where is the standard normal cumulative distribution function.

14

Derivatives

Much more complex derivatives exist such as:– Barrier options, digital barrier options, worst of options, one touch options,

swaps, options on swaps (swaptions), credit default swaps etc.

15

Derivatives

Much more complex derivatives exists such as:– Barrier options, digital barrier options, worst of options, one touch options,

swaps, options on swaps (swaptions), credit default swaps etc.

Value today?

16

Derivatives

Much more complex derivatives exists such as:– Barrier options, digital barrier options, worst of options, one touch options,

swaps, options on swaps (swaptions), credit default swaps etc.

Value today? Obtained by numerical simulation

– Solve Black-Scholes or other model using e.g. finite differences.– Monte Carlo for path dependent derivatives. Simulate many possible paths of

the equity and compute price for each path.

17

Risk

Market risk

Liquidity risk

Credit risk

Operational risk

18

Risk: Why?

Long-Term Capital Management:– Hegde fund founded 1994– Myron S. Scholes, Robert C. Merton (Nobel Prize 1997, BS)– Very successfull first years (20-40 % per year)– Highly leveraged

.

19

Risk: Why?

Long-Term Capital Management:– Hedge fund founded 1994– Myron S. Scholes, Robert C. Merton (Nobel Prize 1997, BS)– Very successfull first years (20-40 % per year)– Highly leveraged– 1998 Russia defaults. Over $4 billion in losses. Fund closed.

20

Risk: Why?

Metallgesellshaft AG– One of Germany’s largest industrial companies: 20 000 employees– Sold long term (5-10 year) fixed price oil & gasoline contracts– Hedged by short term future contracts– 1993, fall in oil prices– Cash drain threatened liquidity– Closed hedges at $1.3 billion loss– Hedges meant to reduce risk resulted in huge losses

21

Risk: How?

Value at Risk, VaR– Single number: ”How bad can things get?”– Given a confidence level defined as

where is the loss of the portfolio over a given time horizon.– Ex. , time horizon 1 day:

– With 95% certainty, there will not be a loss of more than in the portfolio until tomorrow.

22

Risk: VaR

Historical Simulation– Using e.g. the last 200 days, construct 200 possible tomorrows by applying the

historical returns on today.– Evalute the portfolio for each of the 200 possible tomorrows.– 95% VaR obtained from the 11th worst outcome, so that 5% of the 200

outcomes are worse.– Requires full evaluation for each of the 200 possible tomorrows. Can be

expensive.

23

Risk: VaR

Historical Simulation– Using e.g. the last 200 days, construct 200 possible tomorrows by applying the

historical returns on today.– Evalute the portfolio on each of the 200 possible tomorrows.– 95% VaR obtained from the 11th worst outcome, so that 5% of the 200

outcomes are worse.– Requires full evaluation on each of the 200 possible tomorrows. Can be

expensive.

Monte Carlo Simulation– Similar to historical, but thousands of possible tomorrows are constructed from a

model.– Even more expensive.

24

Risk: VaR

Historical Simulation– Using e.g. the last 200 days, construct 200 possible tomorrows by applying the

historical returns on today.– Evalute the portfolio on each of the 200 possible tomorrows.– 95% VaR obtained from the 11th worst outcome, so that 5% of the 200

outcomes are worse.– Requires full evaluation on each of the 200 possible tomorrows. Can be

expensive.

Monte Carlo Simulation– Similar to historical, but thousands of possible tomorrows are constructed from a

model.– Even more expensive.

Parametric models– Compute sensitivities w.r.t. risk factors and estimates VaR from this.– Typically cheaper.– Works well on linear instruments.

25

Risk: stress testing

Important complement to VaR.

Evaluate the portfolio for a number of (unfavourable) scenarios.

Scenarios may be– Historical events:

– 2008 financial crisis– 9/11 2001

– Hypothetical scenarios:– All equities up/down 20%, 50%– All interest rates up/down 1%, 5%

Ensure stability during extreme events not captured by VaR.

Questions?

per-olov.asen@brummer.se

26