BFF2701 Chap.8 Risk, VaR and Arbitrage
Risk, VaR and Arbitrage
Risk analysis distinguishes realised historical variation from forward-looking option-implied volatility. Historical volatility is estimated from past returns and depends on sampling frequency and window length. Implied volatility is backed out from option prices under a pricing model, so it reflects market prices and model assumptions rather than a directly observed future path.
Value at risk gives a loss threshold for a specified confidence level and horizon under a chosen distribution or empirical method. A one-day 95% VaR of $100,000 does not mean the worst loss is $100,000; it means the model places 5% of days beyond that threshold. Scaling across horizons requires assumptions about independence and stable volatility.
Arbitrage strategies seek relative mispricing: classic arbitrage aims for offsetting cash flows, pairs trading uses a relationship between two securities, and statistical arbitrage applies systematic signals across many positions. Real trades face funding costs, short-sale constraints, model error, execution risk and the possibility that mispricing widens before it converges.
A good recommendation states the trigger, hedge, exit rule and loss control rather than calling a pattern risk-free.
What this chapter covers
- 01
Historical volatility
- 02
Implied volatility
- 03
Value at risk interpretation
- 04
Confidence levels and horizons
- 05
Classic and relative-value arbitrage
- 06
Pairs and statistical arbitrage
- 07
Funding, model and convergence limits
Worked example · free
Calculate and interpret parametric VaR
- +1Daily dollar volatility = $2,000,000 times 0.014 = $28,000.
- +2One-day 95% VaR = 1.645 times $28,000 = $46,060.
- +1Ten-day scaling factor = square root of 10 = 3.1623.
- +2Ten-day VaR = $46,060 times 3.1623 = $145,658 after rounding.
- +1Interpretation: under the model, losses exceed about $46,060 on 5% of one-day periods and exceed about $145,658 on 5% of ten-day periods.
- +1VaR does not cap the loss beyond the threshold. Horizon scaling assumes stable independent returns and can fail during stressed markets.
Key terms
- Historical volatility
- An estimate of return variability based on observed past data.
- Implied volatility
- The volatility input consistent with an observed option price under a model.
- Value at risk
- A modelled loss threshold for a stated probability and horizon.
- Pairs trading
- A relative-value strategy using the relationship between two securities.
- Convergence risk
- The risk that a price relationship fails to return before losses or constraints force exit.
- Short-sale constraint
- A restriction or cost that limits selling a borrowed security.
Risk, VaR and Arbitrage FAQ
Is implied volatility a forecast?
It is forward-looking market information extracted through a model, but it is not a guaranteed forecast of realised volatility and depends on the selected option-pricing assumptions.
Does 95% VaR describe the worst 5% of losses?
It identifies the threshold that separates the model's central 95% from the tail. It does not describe how severe losses inside that tail may become in practice.
Can I multiply one-day VaR by ten?
Not under square-root-of-time scaling. The conventional factor is the square root of ten, and even that requires restrictive assumptions about independent returns and stable volatility.
Why is pairs trading not risk-free?
The historical relationship can break, borrowing can become costly, execution can be asymmetric and funding pressure can force an early exit before the expected convergence occurs.
What controls belong in an arbitrage recommendation?
State the mispricing signal, both position legs, hedge logic, entry trigger, expected convergence event and exit rule. Then stress borrowing cost, short availability, model error and adverse widening. This structure distinguishes a risk-managed relative-value trade from a claim that similar historical prices guarantee convergence.
Exam move
For any risk number, write amount, confidence, horizon and method before interpreting it. Practise translating VaR into exceedance language without calling it a maximum loss. Compare historical and implied volatility by data source and assumption. For arbitrage questions, draw both legs, show why they hedge, and list the event that closes the trade.
Finish with a concrete limit such as funding, short availability, model breakdown or non-convergence. Build a VaR audit in a fixed order: portfolio value, periodic volatility, tail coefficient, horizon scaling, numeric threshold and verbal exceedance statement. Keep percentages as decimals in multiplication, round only after the final step and attach currency.
Reverse the calculation once to check whether dividing the reported VaR by the coefficient recovers the expected dollar volatility. Compare a short and long historical window to see how responsiveness trades off against estimation noise. Place implied volatility in a separate column because it comes from option prices and a model, not directly from realised returns.
For an arbitrage case, write entry conditions and both position legs before discussing profit. Then stress the relationship: widen the spread, raise borrowing fees, remove short availability or delay convergence. This turns a diagram of relative mispricing into a risk-managed strategy.
Never call pairs or statistical arbitrage risk-free merely because market exposure appears hedged; basis, model, execution and funding risks remain. End with a stop or exit rule tied to the original thesis.
Working through Risk, VaR and Arbitrage in BFF2701? Sia is AskSia’s AI Finance tutor — ask any BFF2701 Risk, VaR and Arbitrage question and get a clear, step-by-step explanation grounded in how BFF2701 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
