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The risk-free interest rate is 5%. XYZ stock is currently trading at $51.25 and the current market price of C X Y Z {\displaystyle C_{XYZ}} is $2.00. Using a standard Black–Scholes pricing model, the volatility implied by the market price C X Y Z {\displaystyle C_{XYZ}} is 18.7%, or:
The discrete difference equations may then be solved iteratively to calculate a price for the option. [4] The approach arises since the evolution of the option value can be modelled via a partial differential equation (PDE), as a function of (at least) time and price of underlying; see for example the Black–Scholes PDE. Once in this form, a ...
for a constant interest rate r, a positive constant > and an exponent , so that in this case σ ( S t , t ) = σ S t γ − 1 . {\displaystyle \sigma (S_{t},t)=\sigma S_{t}^{\gamma -1}.} The model is at times classified as a stochastic volatility model , although according to the definition given here, it is a local volatility model, as there ...
repeat until the discounted value at the first node in the tree equals the zero-price corresponding to the given spot interest rate for the i-th time-step. Step 2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve.
The Smith–Wilson method is a method for extrapolating forward rates. It is recommended by EIOPA to extrapolate interest rates. It was introduced in 2000 by A. Smith and T. Wilson for Bacon & Woodrow.
For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.
To determine the cheapest bond in a basket of deliverable bonds against a futures contract, implied repo rate is computed for each bond; the bond with the highest repo rate is the cheapest. It is the cheapest because it has the lowest initial value to yield a higher return provided it is delivered with the stated futures price.
Volatility smile. Volatility smiles are implied volatility patterns that arise in pricing financial options.It is a parameter (implied volatility) that is needed to be modified for the Black–Scholes formula to fit market prices.