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A put becomes more valuable as the price of the underlying stock depreciates relative to the strike price. For example, if you have one Mar 12 Taser 10 put, you have the right to sell 100 shares of Taser at $10 until March 2012. If shares of Taser fall to $5 and you exercise the option, you can purchase 100 shares of Taser for $5 in the market and sell the shares to the option’s writer for $10 each, which means you make $500 (100 x ($10-$5)) on the put option.

Example of a Put Option on a Stock

Buying a Put

A Buyer thinks the price of a stock will decrease. He pays a premium which he will never get back, unless it is sold before it expires. The buyer has the right to sell the stock at the strike price.

Writing a Put

The writer receives a premium from the buyer. If the buyer exercises his option, the writer will buy the stock at the strike price. If the buyer does not exercise his option, the writer’s profit is the premium.
• ‘Trader A’ (Put Buyer) purchases a put contract to sell 100 shares of XYZ Corp. to ‘Trader B’ (Put Writer) for $50 per share. The current price is $55 per share, and ‘Trader A’ pays a premium of $5 per share. If the price of XYZ stock falls to $40 a share right before expiration, then ‘Trader A’ can exercise the put by buying 100 shares for $4,000 from the stock market, then selling them to ‘Trader B’ for $5,000.
Trader A’s total earnings (S) can be calculated at $500. The sale of the 100 shares of stock at a strike price of $50 to ‘Trader B’ = $5,000 (P). The purchase of 100 shares of stock at $40 = $4,000 (Q). The put option premium paid to trader B for buying the contract of 100 shares at $5 per share, excluding commissions = $500 (R). Thus S = P – (Q+R) = $5,000 – ($4,000+$500) = $500.

• If, however, the share price never drops below the strike price (in this case, $50), then ‘Trader A’ would not exercise the option (because selling a stock to ‘Trader B’ at $50 would cost ‘Trader A’ more than that to buy it). Trader A’s option would be worthless and he would have lost the whole investment, the fee (premium) for the option contract, $500 (5 per share, 100 shares per contract). Trader A’s total loss are limited to the cost of the put premium plus the sales commission to buy it.

A put option is said to have intrinsic value when the underlying instrument has a spot price (S) below the option’s strike price (K). Upon exercise, a put option is valued at K-S if it is ” in-the-money”, otherwise its value is zero. Prior to exercise, an option has time value apart from its intrinsic value. The following factors reduce the time value of a put option: shortening of the time to expire, decrease in the volatility of the underlying, and increase of interest rates. Option pricing is a central problem of financial mathematics.

Value of a Put

This examples lead to the following formal reasoning. Fix ⱷ an underlying financial instrument. Let Π be a put option for this instrument, purchased at time 0, expiring at time T Ɛ R+, with exercise (strike) price K Ɛ R; and let S: [0, T] → R be the price of the underlying instrument.

Assume the owner of the option Π, wants to make no loss, and does not want to actually possess the underlying instrument ⱷ. Then either (i) the person will purchase ⱷ at expiry, and then immediately exercise the selling option; or (ii) the person will not exercise the option (which subsequently becomes worthless). In (i), the pay-off would be − ST + K; in (ii) the pay-off would be 0. So if K -ST ≥ 0 (i) or (ii) occurs; if K − ST < 0 then (ii) occurs.

Hence the pay-off, i.e. the value of the put option at expiry, is

max{K – ST,0}

which is alternatively written (K – ST) V 0 or (K − ST) + .

A put option on a bond is a provision that allows the holder of the bond the right to force the issuer to pay back the principal on the bond. A put option gives the bond holder the ability to receive the principal of the bond whenever they want before maturity for whatever reason. If the bond holder feels that the prospects of the company are weakening, which could lower its ability to pay off its debts, they can simply force the issuer to repurchase their bond through the put provision. It also could be a situation in which interest rates have risen since the bond was initially purchased, and the bond holder feels that they can get a better return now in other investments.

Another benefit to a bond with this provision is that it removes the pricing risk bond holders face when they attempt to sell the bond into the secondary market, where they may have to sell at a discount. The provision adds an extra layer of security for bond holders – as it gives them a safe exit strategy. Because this option is favorable for bond holders, it will be sold at a premium to a comparable bond without the put provision.

Bonds with a put option are referred to as put bonds or puttable bonds. This is the opposite of a call option provision, which allows the issuer to redeem all of the outstanding bonds.

Instrument Models

The terms for exercising the option’s right to sell it differ depending on option style. A European put option allows the holder to exercise the put option for a short period of time right before expiration, while an American put option allows exercise at any time before expiration.

The most widely-traded put options are on equities, but they are traded on many other instruments such as interest rates or commodities.

The put buyer either believes that the underlying asset’s price will fall by the exercise date or hopes to protect a long position in it. The advantage of buying a put over short selling the asset is that the option owner’s risk of loss is limited to the premium paid for it, whereas the asset short seller’s risk of loss is unlimited (its price can rise greatly, in fact, in theory it can rise infinitely, and such a rise is the short seller’s loss.) The put buyer’s prospect (risk) of gain is limited to the option’s strike price less the underlying’s spot price and the premium/fee paid for it.

The put writer believes that the underlying security’s price will rise, not fall. The writer sells the put to collect the premium. The put writer’s total potential loss is limited to the put’s strike price less the spot and premium already received. Puts can be used also to limit the writer’s portfolio risk and may be part of an option spread.

A naked put, also called an uncovered put, is a put option whose writer (the seller) does not have a position in the underlying stock or other instrument. This strategy is best used by investors who want to accumulate a position in the underlying stock, but only if the price is low enough. If the buyer fails to exercise the options, then the writer keeps the option premium as a ‘gift’ for playing the game.

If the underlying stock’s market price is below the option’s strike price when expiration arrives, the option owner (buyer) can exercise the put option, forcing the writer to buy the underlying stock at the strike price. That allows the exerciser (buyer) to profit from the difference between the stock’s market price and the option’s strike price. But if the stock’s market price is above the option’s strike price at the end of expiration day, the option expires worthless, and the owner’s loss is limited to the premium (fee) paid for it (the writer’s profit).

The seller’s potential loss on a naked put can be substantial. If the stock falls all the way to zero (bankruptcy), his loss is equal to the strike price (at which he must buy the stock to cover the option) minus the premium received. The potential upside is the premium received when selling the option: if the stock price is above the strike price at expiration, the option seller keeps the premium, and the option expires worthless. During the option’s lifetime, if the stock moves lower, the option’s premium may increase (depending on how far the stock falls and how much time passes). If it does, it becomes more costly to close the position (repurchase the put, sold earlier), resulting in a loss. If the stock price completely collapses before the put position is closed, the put writer potentially can face catastrophic loss.

Many bonds include a call feature that allows the issuer to redeem or “call” all or part of the issue before the maturity date.

The call option has value to the issuer for several reasons. In the future the borrower may wish to remove restrictions placed on him by the bond indenture. For a corporation these might be restrictions on merger, the sale of assets, or the payment of dividends. Without the call provision, the bondholders by hard bargaining might be able to utilize their monopoly position to extract a large premium from the issuer before selling back the bonds or agreeing to change these clauses.

A second source of value is that the borrower may find that he wants to decrease the amount of his borrowing before the bonds mature. Essentially the same effect as retiring his own bonds could be obtained by buying on the market similar bonds issued by someone else. However, because of the transaction costs involved in making interest payments, the cost of bonds to the issuer will always be somewhat greater than their value on the market. There will therefore be some saving to the issuer in retiring his own bonds rather than buying someone else’s.

The third and probably most significant source of value of the option to the issuer is the ability it gives him to refinance the issuer in the future if interest rates should fall. This implies three risks from the investor:
(a) The cash flow pattern becomes uncertain;
(b) The investor becomes exposed to reinvestment risk because the issuer will call the bond when interest rates drop;
(c) The capital appreciation potential of a bond will be reduced, because the price of a callable bond may not rise much above the price at which the issuer will call the bond.

Option writing for bonds is used both for risk and hedge purposes. Bond call options provide liquidity with little cash requirement. Bond options also have a finite life and can be expensive to purchase and hold until maturity.

Mechanics of Bond Option Writing

The seller of the contract must have on account securities or funds equal to the amount of the potential sale until the contract expires. The seller is also called the writer of the contract. The incentive for writing a bond call option is the premium payment reflecting the risk of loss of the bonds. The premium payment is in addition to the strike price. The strike price is the market value the bond call option must reach in order to be exercised, or called.

Trading on an Exchange

For a stated par, or maturity value, the bond call writer initiates a contract through his broker on a nationally recognized exchange. The exchange provides a uniform bond option contract detailing the responsibilities of both the bond call option buyer and seller, including the length of the call and what bonds constitute good value for delivery. Rarely are bonds ever delivered. Instead contracts are offset by cash payments at maturity. The contract also denotes an exercise price. The fee received, the premium, is determined by market conditions. This includes the current level of interest rates, the price volatility of the bonds and the number of business days the bond option seller is at risk.

Leverage

Rather than incur the price volatility and cash payment for bonds, traders often use bond call options. Traders know from the beginning of the trade the absolute amount of loss they may incur while freeing up cash for other trading purposes. The cost of a bond option varies depending on the premium charged, but a one month bond call option may cost as little as 2 percent of the face value of the bonds. This implies that a 1 percent increase in bond values on a $1 million call would increase bond option values by nearly $10,000.

Hedging

Bond options are popular as hedging vehicles to reduce risk. Traders can buy options and use them to hedge, or offset, an equal amount of futures contracts, bonds and other options, such as bond put options. Bond call options are also used like stock options in a variety of mathematical strategies. By buying and selling different bond call and bond put options, income can be derived from the time to maturity, the difference in coupon or even bond prices. Bond call options, with their liquidity, and known maximum loss, provide traders many opportunities to employ profitable, leveraged, strategic portfolio outcomes.

Ai = number of accrued days for the ith, or last, quasi-coupon period within odd period counting forward from last interest date before redemption.
DCi = number of days counted in the ith, or last, quasi-coupon period as delimited by the length of the actual coupon period.
NC = number of quasi-coupon periods that fit in odd period; if this number contains a fraction it will be raised to the next whole number.
NLi = normal length in days of the ith, or last, quasi-coupon period within odd coupon period.

Bonds often are referred to as being short-, medium- or long-term. Generally, a bond that matures in one to three years is referred to as a short-term bond. Medium or intermediate-term bonds generally are those that mature in four to 10 years, and long-term bonds are those with maturities greater than 10 years. Whatever the duration of a bond, the borrower fulfills its debt obligation when the bond reaches its maturity date, and the final interest payment and the original sum you loaned (the principal) are paid to you.

Not all bonds reach maturity, even if you want them to. Callable bonds are common: they allow the issuer to retire a bond before it matures. Call provisions are outlined in the bond’s prospectus (or offering statement or circular) and the indenture – both are documents that explain a bond’s terms and conditions. While firms are not formally required to document all call provision terms on the customer’s confirmation statement, many do so.

Bond maturities usually range from one day up to 30 years or even more. But this maturity date must be seen as the last future date (except if the borrower is in default) on which the investor will receive the principal amount from to the issuer. Depending on redemption features, the real reimbursement date can be very different (much shorter). These redemption features usually give the right to the investors and/or the issuer to advance the maturity date of the bond:

Call

This is a provision that allows or require the issuer to repay the bond before the maturity date. The issuer will ‘call’ his bond if the interest rate index is lower than when he issued the bond. On the investor point of view, it means that the bond will be prepaid if the bond brings him too much interest compared to the current market conditions. If you purchase a bond with a ‘call option’, you have to pay less (get a premium) than without call because if the bond is prepaid, you will reinvest the money at a lower rate.

Put

The put is a provision that gives to right to the investors to require from the issuer to redeem the bond before the maturity date. Investors usually exercise this option when the current market rates are higher so that he can reinvest his money at a higher rate. The put feature is a protection for the investor against an increase of the interest rate on the market and, consequently, should pay more (pay a premium) for a bond with a put than without.

Conversion

Some corporate bonds, known as convertible bonds, contain an option to convert the bond into common stock instead of receiving a cash payment. Convertible bonds contain provisions on how and when the option to convert can be exercised. Convertibles offer a lower coupon rate beacuse they have the stability of a bond while offering the the potential upside of a stock.

Partial Prepayment (Paydown)

This kind of feature is usually seen with mortgage-backed securities. Without entering into details, a mortgage-backed security is anything else but the securitisation of a pool of mortgage loans. In mortgage loans, you have a regular (often monthly or quarterly) payment of principal and also the ability for the borrower the prepay the loan before maturity. Mortgage-backed securities prepay the principal to the investors in parallel the underlying mortgage loans. That’s the reason why mortgage-backed securities are traded on the basis of their ‘average life’.

Mortgage-backed bonds prices are more volatile than fixed rate bonds because the speed of redemption increases when the interests go down (when you have to reinvest at a lower rate) but decreases when interests go up. An increase in interest rates will increase the average life (the real maturity date) of your investment.