TOC
Credit v.s. Debt Valuation Adjustments
All uncollateralised OTC derivatives valuation must account for counterparty and own default, and any offset arrangements.
CVA is an adjustment for loss due to counterparty default, while DVA is an adjustment for gain due to own default.
Accounting for Financial Instruments: Credit impairment requires all companies to take both CVA and DVA into consideration. In contrast, Basel Regulations for Financial institutions requires adjustments due to CVA, but specifically exclude those related to DVA.
Credit Valuation Adjustment, CVA
CVA is an adjustment to the no-default value of derivatives arising from the possibility of a counterparty default.
CVA must be calculated on a counterparty-by-counterparty basis, not on a transaction-by-transaction basis
where is the counterparty default probability, is the PV of counterparty’s loss given default (i.e. portion that cannot be recovered), and is the maturity of the longest cash flow.
Debt Valuation Adjustment, DVA
DVA is an adjustment to a bank’s no-default value because the bank itself might default.
The banks’ DVA is the counterparty’s CVA
Like CVA, DVA must be calculated on a counterparty-by-counterparty basis
where is own default probability, is the PV of own loss given default (i.e. portion that cannot be recovered), and is the maturity of the longest cash flow.
Valuing Bilaterally Cleared Derivatives Portfolios
Value after credit adjustment is: No-default value - CVA + DVA
CVA and DVA adjustments should reflect collateral arrangements.
Using Index Option in Portfolio Insurance
Protective Put: Entering into put option to ensure that the value of a portfolio will not fall below a certain level
The strike price is chosen to give the appropriate insurance level.
To limit the downside potential, portfolio managers can use:
- Index put options, or
- Synthetic put options
Synthetic put option:
- Sell stocks, and deposit (buy bonds)
- Index futures: where is the futures maturity
- or
- If we sell index futures instead of stocks in the portfolio, then the amount of futures to sell is
- As stock value decreases, becomes more negative(abs is bigger because is bigger), more futrues contract have to be sold (or shorted)
Note that the “Protective Put” using index put option is a static hedge
- We decide what the level of protection should be
- We buy the right number of put options with the appropriate strike
- We hold the option until maturity
Portfolio insurance using put options:
means the portfolio is well diversified. The portfolio returns mirror the returns on the stock index.
Assume that the portfolio dividend yield is the same as the index dividend yield
where is capital gain
If the manager buys one index put with a strike price & maturity , then the value of the portfolio is protected against the possibility of the index falling below .
Example
Suppose that the portfolio has a and is currently worth $500,000. The value of the index is 1,000 and each index point is worth $100 in index options.
- Value underlying one option contract
- Number of puts to be bought contracts
- For a strike , the level of protection is
- Check:
- Assume that the index falls by 10% (from 1000 to 900).
- Then the equity portfolio value also drops from $500,000 to $450,000.
- And the payoff from put options equals to 5*$100*(960-900)=$30,000.
- The total value is $450,000+$30,000=$480,000
Portfolio insurance using put options:
The total return is the sum of capital gain and dividend income:
- Total return on the market:
- Total return on the portfolio:
According to the CAPM: , all returns are expected returns with no error term. CAPM is a total returns relationship that includes dividend but hedging using option protects only against capital losses exclude dividend.
Example
A trader owns a portfolio worth $500,000 and wants to buy S&P 500 puts with months to ensure that value does not drops below $450,000. The S&P 500 index stands at 1,000. The portfolio beta on the S&P 500 is 2.0. Dividend yield for both assets is 4% per annum. The risk-free rate is 12% per annum.
Assume the index rise to 1,040 after 3 months
- Capital gain:
- Dividend yield per 3 months:
- Total return per 3 months:
- Interest rate per 3 months:
- Excess return per 3 months:
From the CAPM model, the expected portfolio return is:
The portfolio’s expected capital gain is:
The expected portfolio value is:
We can list the value of index with the expected portfolio value as below
Thus a put option with a strike price equal to $960 ensures that the expected portfolio value does not drop below $450,000
- Protection against a 10% decline in the portfolio value
- Portfolio value is maintained at $0.45 million
Value underlying one option = $100*1000 = $100,000
Number of contracts = contracts
Check:
- Chose the strike at 960 as before
- If the index falls to $920, the portfolio value is expected to drop to $40,000
- Total value = $410,000 + $40,000 = $450,000
Arbitrage Using Stock Index Futures
Stock index futures
It can be viewed as an investment asset paying a dividend yield. The investment asset is the portfolio of stocks underlying the index
The cost-of-carry relationship is:
where is the average dividend yield on the portfolio represented by the index during life of contract.
Index arbitrage
When market quote , there is an arbitrage opportunity. That is, simultaneously trading in futures and many different stocks and always buy low and sell high.
- When
- an arbitrageur buys the stocks underlying the index and sells futures
- When
- an arbitrageur buys futures and sells the stocks underlying the index
October 1987 Crash & The Cascade Theory
October 1987 crash: The record one-day decline on October 19, 1987 was 22%.
Brady Commission’s Cascade Theory:
It attributes the downward cascade in stock prices to “mechanical, price-insensitive selling” by institution using:
- Dynamic portfolio insurance strategy using synthetic put with futures
- Index futures arbitrage strategies
When use synthetic put with futures to insurance portfolio:
- When the market goes up, we move away from the minimum threshold
- becomes smaller
- Buy more index futures (decrease the short position) and sell bonds (decrease the long position)
- When the market goes down, we move closer to the minimum threshold
- becomes larger
- Sell more index futures (increase the short position) and buy bonds (increase the long position)
Which implies that: buy stocks (futures) in an up market and sell stocks (futures) in a down market. Therefore, during stock market crashes, it triggers massive selling of index futures.
During Index arbitrage:
- When , buy and sell
- If index futures prices are falling, index arbitrage will lead to massive selling of stock in the spot (or cash) market
Cascade Theory:
- When cash price starts to fall, portfolio insurance scheme triggers massive shorting of index futures
- The sell side pressure on index futures creates an opprotunity for index futures arbitrage which leads to a massive selling of stocks in the cash market
- The secondary downward pressure on the stock cash market feeds back to dynamic portfolio insurance which triggers more index futures sales
- This goes on and on till both markets collapsed
Santoni’s arguments
The flip in the basis sign does not necessarily support the cascade theory:
- A large drop in the dividend growth may cause a massive price revaluation
- A drop in cash price should immediately trigger a drop in futures price
- Futures simply respond to it faster than cash
Also, there exist some irrational price and insensitive traders.
Santoni’s explanations for the crash
The difference in the trading mechanisms in the cash and the futures market caused the prices in the future market to react faster to news. The index is not a cash product (you have to trade all 500 stocks to move the index) while the index future has a real price.
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