Compare: Swaption § Valuation Bondsthe underlyers in this case, exhibit what is known as pull-to-par : as the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility.
On the other hand, the Black—Scholes model, which assumes constant volatility, does not reflect câștigați 2020 dolari rapid processand cannot therefore be applied here;  see Black—Scholes model Valuing bond options.
Addressing this, bond options are usually valued using the Black model or with a lattice-based short rate model such as Black-Derman-ToyHo-Lee or Hull—White. For American- and Bermudan- styled optionswhere exercise is permitted prior to maturity, only the lattice-based approach is applicable. Using the Black model, the spot price in the formula is not simply the market price of the underlying bond, rather it is the forward bond price.
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This forward price opțiuni de obligațiuni calculated by first subtracting the present value of the coupons opțiuni de obligațiuni the valuation date i. These calculations are performed using today's yield curveas opposed to the bond's YTM.
This allows us to assume that a the bond price is a random variable at a future date, but also b that the risk-free rate between now and then is constant since using the forward measure moves the discounting outside of the expectation term .
Thus the valuation takes place in a risk-neutral "forward world" where the expected future spot rate is the forward rate, and its standard deviation is the same as in the "physical world";  see Girsanov's theorem.
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The volatility used, is typically "read-off" an Implied volatility surface. The lattice-based model entails a tree of short rates - a zeroeth step - consistent with today's yield curve and short rate often caplet volatility, and where the final time step of the tree corresponds to the date of the underlying bond's maturity.
Then 2the option is valued similar to the approach for equity options : at nodes in the time-step corresponding to option maturity, value is based on moneyness ; at earlier nodes, it is the discounted expected value of the option at the up- and down-nodes in the later time step, and, depending opțiuni de obligațiuni option style and other specifications - see belowof the bond value at the node.
Note that the Hull-White tree is usually Trinomial : the logic is as described, although there are then three nodes in question at each point. See Lattice model finance Interest rate derivatives.
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Embedded options[ edit ] The term "bond option" is also used for option-like features of some bonds " embedded options ". These are an inherent part of the bond, rather than a separately traded product. These options are not mutually exclusive, so a bond may have several options embedded.
The holder of such a bond has, in effect, sold a call option to the issuer. Callable bonds cannot be called for the first few years of their life. This period is known as the lock out period.
Puttable bond : allows the holder to demand early redemption at a predetermined price at a certain time in future. The holder of such a bond has, in effect, purchased a put option on the bond.
Convertible bond : allows the holder to demand conversion of bonds into the stock of the issuer at a predetermined price at a certain time period in future. Extendible bond : allows the holder to extend the bond maturity date by opțiuni de obligațiuni number of years.
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Exchangeable bond : allows the holder to demand conversion of bonds into the stock of a different company, usually a public subsidiary of the issuer, at a opțiuni de obligațiuni price at certain time period in future.
Here, the bond is priced as a "straight bond" i.
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The option value is then added to the straight bond price if the optionality rests with the buyer of the bond; it is subtracted if the seller of the bond i. Relationship with caps and floors[ edit ] European Put options on zero coupon bonds can be seen to be equivalent to suitable caplets, i.
See for example Brigo and Mercuriowho also discuss bond options valuation with different models.