The Greeks
An option price is a function of several parameters, e.g., and so on. can be used for hedging and risk management. For exmaple
Finite Difference: Resimulation
The resimulation method is typically used when the payoff function depdends on the entire price path rather than just the final value .
Resimulation method is the most general using the fact that
Because and are expected discounted payoffs based on price processes starting from and respectively, we can use
If we use independent simulations for and , this could yield a large standard error:
where the covariance is zero is paths are independent. With correlated paths, we could potentially get positive covariance.
Remarks:
- With simulated paths,
hence, to get a reasonable reliable estimator, we need to have . With , this requires
- Generating correlated paths for estimation of and is done by using the same Gaussian random variables for increments
- The estimators is biased, we can consider the bias reduction as follows
by averaging the two, we obtain
Pathwise Differentiation
The pathwise differentiation method uses the interchange of expectation and derivative to compute
In order to apply the pathwise differentiation, we need to write the payoff function explicitly in terms of .
For example, in the BS example, we have
which implies
This yields
As long as the payoff function is almost everywhere differentiable and the derivatives can be computed, pathwise differentiation yields an unbiased estimator.
Pathwise Differentiation for SDEs
The Pahtwise method is applicable when the payoff function is smooth (differentiable w.r.t ) almost everywhere.
For SDEs of the form
with could be stochastic but do not depend on , we have by Ito’s lemma
thus, we can write
This holds true for all processes that satisfy as long as the process does not depend on .
For SDEs of the form
with the fixed initial condition . The process of estimating without the knowledge of the transition densities is as follows:
- With , Euler discretization yields
- Denoting , we get
- If the payoff function is estimated by , then the pathwise derivative of the payoff is estimated as
Although pathwise differentiation requires the payoff function to be almost surely differentiable as a function of the parameter, we can apply this method for non-smooth payoff functions using conditional Monte Carlo.
Likelihood Method
The likelihood method also involves interchange of differentiation and integration, but transfers the smoothness assumption on the payoff function to the smoothness assumptions on the densities (likelihood).
The likelihood method is particularly useful when the payoff function is not differentiable, such as digital options where the indicator function creates a discontinuity.
Note that
the only term on the right hand side that depends on any of the problem parameters is and thus, for example,
where is the likelihood score.
Remarks:
- Pros: the likelihood method has no assumptions on the form of the payoff function
- Cons: we need to know the density exactly. Furthermore, because of the likelihood score, this method can sometimes incur variance inflation compared to the pathwise differentiation method
Digital Option
Digital option has a payoff function . Because
the price of digital option is
Resimulation Method
We should use the same Gaussians for generating the starting from and .
For , generate , and compute
Resimulation estimator of is
Note that is non-zero iff lies in the narrow interval
The width of the interval is
Hence, the standard MC method requires number of samples that scales much larger compared to .
The bias of the estimator is and the variance is . The MSE of the estimator is , which is minimized if .
Likelihood Method
Note that has a log-normal distribution with density
Hence
American Options
American options can be exercised at any time up to expiration. For certain options with convex payoff functions, such as call options on stocks that pay no dividends, the optimal policy is to exercise only at expiration. In most other cases, there is an optimal policy but is neither analytically available nor easily estimable via Monte Carlo.
Generally, suppose is the stoppping time at which the investor will exercise the option, then the value of the option at time is
Bermuda Option
Bermuda option can be exercised once only at a finite number of time points. If the set of allowable exercise times if , then the option reduces to the European option. If the set of allowable exercise times is where is small, then the option approximates the American option.
Denote the value of the Bermuda option with allowable exercise times at time , then as , converges to the value of the American option.
Bermuda options can be priced by dynamic programming or backward induction.
If is the payoff when the option is exercised at time with the stock price at , then
for while .
Because is known and does not require any estimation. We can use the backward induction method as below:
- Generate independent paths
- This yields the realizations of and as
- For the equation
we need to estimate . Let be the Nadaraya-Watson estimator. This yields the realizations of as
- Using repeat this to get .
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