The Hidden Cost of Volatility Drag

That said, a key challenge in pricing European call options is capturing the complexity of volatility. In our previous chapter, we explored how to price a European call option with exercise price $22 and payoff function V (ST) = (ST − 22) + under certain assumptions about interest rates and stock prices.

The problem here is that many investors would prefer to value an option explicitly, rather than relying on simulations. However, generating large numbers of independent simulations can be time-consuming and costly. In this chapter, we'll examine two approaches: explicit pricing using a simple distribution (Q) and Monte Carlo methods as an alternative.

Explicit Pricing Using Simple Distribution

One approach is to generate a large number of independent random uniform variables on the interval [0, 1] using discrete uniform random numbers. We can then define stock prices ST for each simulation using these uniform numbers. By averaging the returns from all simulations, we can estimate the expected value of the option.

Let's use the following example: suppose we have 1000 simulations with random stock prices S̄ = [20, 21, ..., 24]. To estimate the price of a European call option EQV (ST) = EQ(ST − 22) + , we can calculate the average payoff over all simulations:

EQV (ST) ≈ ∑[ST - 22] \ P(S̄ = s)

where P(s) is the probability that S̄ takes on the value s.

Using this approach, we get an estimated price of V (ST) = 1,000[(23 − 22) + (24 − 22)] ≈ $8.33 for a European call option with exercise price $22 and payoff function V (ST) = (ST − 22) + .

Monte Carlo Methods

Another approach is to use Monte Carlo methods as an alternative to explicit pricing. This involves generating large numbers of independent random uniform variables on the interval [0, 1] using discrete uniform random numbers.

We can then define stock prices ST for each simulation using these uniform numbers. By averaging the returns from all simulations, we can estimate the expected value of the option.

One popular choice is to use a simple recursion formulae such as xn = g(xn−1) , where g is a carefully chosen function that mimics the behavior of an independent random variable.

Using this approach, we get an estimated price of V (ST) = ∑[ST - 22] \* P(S̄ = s)

where P(s) is the probability that S̄ takes on the value s.

The Hidden Cost of Volatility Drag

One key advantage of Monte Carlo methods over explicit pricing is their ability to handle complex options with multiple factors, such as interest rates and volatility. However, this also means that they require a large number of simulations to achieve reasonable accuracy.

In our previous chapter, we discussed the 10-year backtest for European call options, which showed that even at a relatively low volatility scenario (s = 1%), the estimated price was still relatively close to the true value ($8.33). However, as volatility increases (s > 1%), the estimated price becomes more accurate.

Portfolio/Investment Implications

When pricing European call options, it's essential to consider how changes in volatility might affect stock prices and option values. For example, if volatility increases significantly, a stock with an existing call option may see its value increase substantially.

However, this also means that investors should be cautious when buying or selling call options, as the potential for large price movements can be significant.

Actionable Conclusion

In conclusion, pricing European call options requires careful consideration of factors such as interest rates and volatility. While explicit pricing using simple distributions can provide a good starting point, Monte Carlo methods offer more accurate results but require a larger number of simulations. By understanding how volatility affects stock prices and option values, investors can make informed decisions about their portfolios.