When Option Pricing Requires Numerical Methods
Summary
Whether an option has a closed-form price depends on both its payoff and the assumed dynamics of the underlying asset. Some European vanilla options have tractable formulas under certain models, while other combinations of payoff and dynamics do not. The document notes that a formula available under one model does not necessarily make that model appropriate: a payoff may depend on risks the model leaves out, such as forward skew or changing interest rates.
When an analytical solution is unavailable under the chosen dynamics, numerical approaches include finite differences, Monte Carlo simulation, and trees; analytical approximations are another possibility. The discussion offers examples and general guidance rather than pricing comparisons, historical data, or a trading strategy. It also does not explain how to choose among numerical methods or how to turn a calculated option value into a trading decision.
Key ideas
- Closed-form availability depends on the combination of payoff and underlying dynamics.
- A tractable formula under a model does not ensure that the model captures the risks driving the payoff.
- Finite differences, Monte Carlo simulation, and trees are numerical alternatives when a closed form is unavailable.
- Analytical approximations can also be used to estimate option values.
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Full text
# Which options do not have a closed form pricing formula like BS? # Which options do not have a closed form pricing formula like BS? Q1 Are there options that doesn't have a closed form pricing formula like BS? Is this the situation when we have to use Finite Difference Method? Can someone give an example? (I hope this option really exist so that I can look for its historical data from Yahoo Finance) Q2 After pricing the option, what should we do to develop trading strategies? ## Answer by LocalVolatility (score 2) https://quant.stackexchange.com/a/31669 The question about the availability of closed-form solutions can generally not be answered for types of options alone but only for the combination of a payoff function and the underlying asset dynamics. Consider for example European plain vanilla options. These have a (quasi) closed-form solution when the underlying follows an exponential Levy processes including geometric Brownian motion (GBM), in many stochastic volatility models but not under local volatility dynamics. However, you will usually find that options that cannot be priced in closed-form under GBM cannot be priced in closed-form under other (sensible) underlying asset dynamics either. Furthermore, many more exotic options that can be priced in closed-form under GBM should not be priced using this model. The reason is that the value of the payoff is heavily driven by risk-factors whose dynamics are not modelled. Examples are the forward skew sensitivity of American binary options, the stochastic interest rate exposure of equity linked bonds with an uncertain maturity, ... . Once the option cannot be priced in closed-form under the respective model dynamics, you need a numerical method such as finite differences, Monte Carlo simulation, trees, ... . ## Answer by SmallChess (score 0) https://quant.stackexchange.com/a/31666 Most of the options in finance don't have closed form solutions. That includes American and many exotic options. When you don't have a formula, you may use numeric methods such as FDM. You may also approximate the option with an analytic solution.
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