But options traders need to know more than just the price: The way traders make money is just the same way that shop-keepers do — by selling options to other people for a little bit more than they buy them for. Once they sell an option, they have some money but they also have some risk, since if the price of the underlying moves in the wrong direction, they stand to lose a large amount of money.

In the simplest case, a trader might be able to buy a matching option on the market for less than she sold the original option to her client for. Another possibility is to try and create a hedged portfolio consisting of several options and the underlying stock as well, so as to minimise the net risk of the portfolio.

Clearly, if the stock price goes up too high, she will lose more money than she received for selling the option.

One possibility might be for her to buy the stock for S t. Since she will have to pay a maximum of S T — K, but would be able to sell the stock for S T , she would cover her position in the case that the stock price goes very high and actually guarantee a profit in this case.

But she has over-hedged her position — in the case that the stock falls in price, she will lose S t — S T on the stock.

This is shown in the graph below. For a call option this will be positive and for a put it will be negative, and the magnitude of both will be between 0 when far out-of-the-money and 1 when far in-the-money. This exposure can also be hedged, but now she will need to do it by trading options in the stock as the stock price itself is independent of volatility. These sensitivities of the derivative price are called the Greeks, as they tend to be represented with various greek letters.

Some examples are delta variation with spot , vega variation with vol , theta variation with time ; and second-order greeks like gamma sensitivity of delta to spot , vanna sensitivity of delta to vol, or equivalently sensitivity of vega to spot , and volga sensitivity of vega to vol. For vanilla options in the BS model, there are simple expressions for the greeks see, for example, the Wikipedia page. Since banks will have large portfolios and want to calculate their total exposure fairly frequently, pricing procedures will typically need to be fairly fast so that these risk calculations can be done in a reasonable amount of time, which usually rules out Monte Carlo as a technique here.

These hedges will only work for small changes in the underlying price for example, delta itself changes with the underlying price according to the second-order greek, gamma. What this means is that the trader will need to re-hedge from time to time, which will cost her some money to do — one of the main challenges for a trader is to balance the need to hedge her portfolio with the associated costs of doing so.

Your email address will not be published. Skip to content Quantopia Piecewise Quantitative Finance. Home Community Forum Tutoring Blog Index Monte Carlo Pricers.

The payoff at expiry of the three portfolios shown in the text.

volatility - Intuitively speaking, why do at the money options have no volga/convexity? - Quantitative Finance Stack Exchange

The overhedged option is the reverse — now the trader loses money if the stock falls too far below the strike price — this is called a covered call, the payoff is the same as the payoff for an uncovered put option. Near-the-money, the difference between the option price and its payoff at expiry is greatest as the implicit insurance provided by the option is most useful.

Vanna–Volga pricing - Wikipedia

The delta of this option is the local gradient of the call price with spot. This is the amount of the underlying that would be required to delta-hedge the portfolio, so that its value is unaffected by small changes in the spot price. This graph shows the instantaneous change in PnL due to changes in spot for the three portfolios discussed above. Both the covered and the uncovered calls have some delta — a change in the spot price will have a direct effect in the value of the portfolio.

By contrast, the value of the delta hedged portfolio is insensitive to the value of spot for small moves. Unfortunately, it is short gamma, so large moves in either direction will reduce portfolio value, so the trader must be careful to re-hedge frequently.

As all of these portfolios are short an option, they are long theta — that is, the value of the portfolio will INCREASE with time if other factors remain constant, as the time value of the option is decaying towards expiry.

Vega is always positive — so increased vol will always increase the price.

Increasing spot tends to increase the value of a call, while it decreases the value of a put, but by a progressively smaller amount as spot increases. The Gamma, Vanna and Volga of long vanilla options these are the same for a call and a put.

Gamma is always positive for long options — this means that price is a convex function of spot. Vanna and volga tell us the sensitivity of other greeks to volatility, and are useful in hedging portfolios if vol is changing rapidly.

Leave a Reply Cancel reply Your email address will not be published. Importance of the Vol Smile. The Discount Curve, ZCBs and the Time Value of Money. Blocks correlation credit default swaps credit risk default design patterns digital options discount curve european options excel Exotic options expectation factory pattern forwards FRAs futures game theory greeks heat equation intrinsic value jointly normal Monte Carlo nash equilibrium normal distribution Path-dependent options Pricers Put-call parity quantile functions rates replication risk neutral valuation stochastic rates swaps time value Vasicek model XLW zcbs.

Quantopia Proudly powered by WordPress.