Financial mathematics, 16/ , KTH Per-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)

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.

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

10 Derivatives

11 Derivatives

12 Derivatives

13 Derivatives

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: 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?

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/ –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?  26