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In financial mathematics, the Ho-Lee model is a short-rate model widely used in the pricing of bond options, swaptions and other interest rate derivatives, and in modeling future interest rates. [1]: 381 It was developed in 1986 by Thomas Ho [2] and Sang Bin Lee. [3] Under this model, the short rate follows a normal process:
For both, the option strike price is the specified futures price at which the futures is traded if the option is exercised. Futures are often used since they are delta one instruments. Calls and options on futures may be priced similarly to those on traded assets by using an extension of the Black-Scholes formula, namely the Black model. For ...
As above, the PDE is expressed in a discretized form, using finite differences, and the evolution in the option price is then modelled using a lattice with corresponding dimensions: time runs from 0 to maturity; and price runs from 0 to a "high" value, such that the option is deeply in or out of the money. The option is then valued as follows: [5]
Applying the Black-Scholes formula with these values as the appropriate inputs, e.g. initial asset value S 1 (0)/S 2 (0), interest rate q 2, volatility σ, etc., gives us the price of the option under numeraire pricing. Since the resulting option price is in units of S 2, multiplying through by S 2 (0) will undo our change of numeraire, and ...
The Black formula is similar to the Black–Scholes formula for valuing stock options except that the spot price of the underlying is replaced by a discounted futures price F. Suppose there is constant risk-free interest rate r and the futures price F(t) of a particular underlying is log-normal with constant volatility σ.
Here the price of the option is its discounted expected value; see risk neutrality and rational pricing. The technique applied then, is (1) to generate a large number of possible, but random , price paths for the underlying (or underlyings) via simulation , and (2) to then calculate the associated exercise value (i.e. "payoff") of the option ...
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based ) model of the varying price over time of the underlying financial instrument, addressing cases where the closed-form Black–Scholes formula is wanting.
Martingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. options , futures , interest rate derivatives , credit derivatives , etc.