Fundamentals of Derivatives

Forward Contract

  • Agreement to buy something in the future at a certain price at a certain time.
  • Eliminate ambiguities by spelling out all the contingencies
  • Lock-in of a forward price
  • Eliminate price risk and price opportunity.
  • Example – farmers don't plant crops before selling it on contract.

       Two Questions:

               What will be the price of asset X in amount of time Y
               What will it cost to keep X for period of Y
                       Loan (cost of $)
                       + Loaning out (income)
                       + Storage
                       + Insurance
                       + Commission
                       + Transportation
                       + Transaction Cost
                       = Cost of Carry
               Spot + Cost of Cary = Forward Price

       Arbitrage Drives Pricing

               Can't deviate from "Spot + Cost of Cary = Forward Price."
               Observed opportunity will drive prices back to equilibrium.

       Change of Spot + Change of Cost of Carry = Change in Forward Price

       Forward prices tell nothing about expectation

               Expectation is built into the spot price.
               Forwards reflect cost of carry only.

       Why can't buyers just buy and store the asset?

               May not have the money upfront / nor tie up line of credit
               The seller of the forward may have lower cost of carry

       Stock as underlying

               IBM at $100 today.
               + Cost of money: 5%
               - Dividend from IBM: $3
               Quote on IBM in a year: $102

               Can be done with a basket of stocks as well.

       Bond as underlying

               $1000 Eurobond
               + Coupon: 8%
               - Borrowing cost: 4%
               $960 <- Since carrying the bond generates revenue.

               Can't be any other price since we could make arbitrage.

       Agriculture as underlying

               Need to over-buy to account for 'slippage'
               Slippage – buying more now to account for loss prior to delivery.
               Increases the cost of carry.

       Money as underlying

               Securing a rate on a loan ahead of time
               Seller (e.g. bank) borrows today and locks in a rate.
                       Invests until the loan
                       Calendar conventions (for different rates)
                       Compounding frequency.

               Simple Interest:
               PV = FV / (1 + r *t)
               PV = $1mm / ( 1 + [.10 x 366/365])
                       10% is the interest rate
                       366 / 365 because the convention is A/365 and it's a leap year

               Continuous Compounding:
               FV = PVe^(rt)
               FV = $908m * e^(.12(731/360))
                       12% is the interest rate
                       731/360 because the convention is A/360
               Use natural log (ln) to get down from e^(X)

               Compounding Interest:
               FV = PV(1+i)^n
               i = the annual rate divided by 2 due to semiannual compounding.
               Formula can only be used with 30/360

               Eliminate Arbitrage:
                       (Year 1 Loan Compounded) * (Year 2 Loan Compounded)
                               = (2 Year Loan Compounded)

       Forwards are Private Contracts

               Counterparty risk.
                       No third party guarantees at all.
               Private contracts – not made public.
               No way to guarantee the best price since can only get quotes by request.

Futures Contracts

Public contracts

Traded in pits on the exchange

Futures Margin

Just a security deposit (not a loan) similar to deposit on an apartment.

Daily Mark to Market to reflect the gains and losses

Eliminates credit risk

Prices in the forward and futures markets must be the same to avoid arbitrage.

Gold –

What happens on the last day of a futures contract?

The forward price is the same as the spot

Both parties pay or get paid the spot price: the gain/loss is already in the margin

Physical delivery not necessary

Eurodollar Future

Short Term Interest Rate Contract

Locking in a forward rata

Priced off of the LIBOR

Actual/360 Convention







Fundamentals of Structured Products

Convertibles and Structured Notes

Combinations of other securities to create a payoff structure or risk/reward tradeoff that’s atypical.

Used to circumvent something

  • Government regulation

  • Accounting rule

  • Tax law

  • Investment guidelines

Creating synthetic notes

Functionally the same as a derivative

Credit Linked Note rather than selling a CDS

Avoiding foreign withholding tax – for example if there’s a tax on foreign owners of a security, a local company can buy it and then they send the returns to the foreign owner.

Equity Linked Notes - Instead of owning stocks, you can invest in debt whose return is tied to return on some stocks. This can be important for insurance companies which have to balance out their risk with safe securities and stocks are riskier than bonds.

Religious Restrictions such as Islamic bonds (where interest is prohibited) – structure something so that it pays interest in effect but do not call it interest.

Retail Distribution

Transfer Asset Management Expertise

Outperform the S&P without making any stock decision by swapping multiple cash flows.

Convenient packaging of multiple long and short positions

Lower Cost

Simpler for investors to execute

Create securities based on investor’s economic outlook

Creating securities that don’t exist or are sold out

Convertible Notes - a bond that includes an option to convert the bond into a given number of shares of common stock. The value of the note can float up along with the stock into which it is convertible.

Valuating a convertible bond:

A = ( # shares that the bond converts to * share price ) + (present value of the interest payments – dividends of the stock)[until expected conversion date]

B = Present Value of all the principal payments and interest + the conversion option

Max of A AND B is the value of the Convertible Bond – we will value the security at its best price since we can treat it as either a bond or a stock.

CB Variables

When is conversion activated

- Sometimes the option is “always on”

- Can convert after some initial period.

- Conditional (e.g. if there’s change in control)

Takeover defense

  • If the price drops

  • Credit events on the bond

  • Etc.

Similarly, the conversion option can deactivate

What stock does the bond convert into?

  • Usually same as the bond issuer

  • Banks can issue bonds convertible into other firms’ stocks.

  • Convertible into some other security (senior debt)

  • A commodity

  • More than one of the above

Does the issuer have to really deliver the shares or just cash equivalence?

Does the bond include any other options (i.e. calls/puts?)

- Almost always callable.

- Hard options – no conditions attached

- Soft options – conditional on something (e.g. price, credit rating change)

Does the bond pay in cash or “pay in kind”

  • Convertible stock note – adds more bonds instead of paying interest.

Does the bond have anti-dilution clause

  • Matches the conversion to stock splits

  • Protects from stock dividends.

Do you get paid accrued interest at the time of conversion?

  • Sometimes forgo the interest when make big profit on conversion.

Relative Value

Convertible Bond Workout Period

Price of Bond – Conversion Value

------------------------------------------ = Number of years to ‘break even’

Bond Income – Stock Income

The shorter the workout period, the more attractive the bond is. Aggressive investors like WP < 1 year. Medium < 1.5 years. Conservative < 2.0 years. Otherwise, you might as well buy the stock.

Refine Workout Period


Reinvestment Income

Projected dividend changes


Why do investors like convertible bond?

Allows patience while waiting for the stock to go up

Better chances of finding a mis-priced bond due to illiquidity

Automatic Asset Allocation

You don’t have to guess whether stocks or bonds are going up.

Better risk/reward ratio

Factors that impact the CB’s curve – Bond Price vs. Stock Price

Interest rates (lowers low end as rates rise)

Coupon on bond (raises low end)

Stock Volatility – raises middle

The ‘option to convert is valuable at times of volatility.

Call option – lowers middle

Put option – raises low end

Stock dividends – lowers high end since bond owners miss out on the extra stock.

Acquire Inexpensive Options

  • Convertible = Bond + Option

  • Hedge away the bond

  • Get the value of option

  • Given the price of option we can get the implied volatility

  • The option can be held as investment or sold OTC at higher cost.

  • Set up a delta-neutral hedge by shorting the stock