This will be discussed further in the next section. We have just discussed the simplest option contract possible, a European call option.

Many generalizations are possible, such as for options on dividend paying stocks, currencies, interest rates, indices or futures, combi or exotic options, etc. Also path integral methods familiar from physics may be useful [50]. In fact, one can solve the Black—Scholes equation 4. That such a method works is hardly surprising from the similarity between the Black—Scholes and Fokker— Planck equations. For the latter, both path-integral solutions, and the reduction to quantum mechanics, are well established [37].

We will use the path integral method in Chap.

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Given values for these variables, an operator will only invest his money, e. One can therefore assume any risk preference of the agents, i. In particular, the assumption of a risk-neutral risk-free world is both possible and practical. In such a world, all assets earn the risk-free interest rate r.

The solution of the Black—Scholes found in a risk-neutral world is also valid in a risky environment our solution of the problem above takes the argument in reverse. The reason is the following: in a risky world, the growth rate of the stock price will be higher than the risk-free rate.

Risk-neutral valuation is equivalent to assuming martingale stochastic processes for the assets involved up to the risk-free rate r. In other words, if an option price was calculated according to 4. Martingales are tied to the notion of risk-neutral valuation. In principle, a riskless hedge of the option position is possible by holding a suitable quantity of the underlying asset.

The valuation therefore can be based on equivalent martingale processes, with the risk-free rate r as the drift. The basic principle for the valuation of an American option can be illustrated easily. An American option then can be exercised at any ti. For geometric Brownian motion, the probability distributions 4. The transition probability conditional 82 4. Of course, a closed solution of this problem is impossible because for every possible price Si , a decision on early exercise must be taken at each step i.

Monte Carlo simulations are an obvious choice. Random price increments are drawn from a normal distribution in the case of geometric Brownian motion to simulate the price history of the underlying, and the average over many runs is taken when ensemble properties are required. For plain vanilla options, the use of binomial trees provides an alternative. However, for exotic, path-dependent options, the discretization of the price increments is an undesirable feature.

General arguments suggest that American call options should never be exercised early in the absence of dividend payments. Dividend payments have not been considered for European options, and will not be discussed here for American options. Most of them are labelled by greek letters. We already encountered one of the Greeks, Delta, and its application in hedging, when setting up the riskless Black—Scholes portfolio in 4.

For European options described by the Black—Scholes equations 4.

### Account Options

Delta describes the dollar variation of an option when the price of the underlying changes by one dollar. This quantity is given by 84 4. Delta of a European call option described by the Black—Scholes equation as a function of the price of the underlying, for times to maturity of one, two, four and twelve months, from bottom to top at the left margin.

The dependence of the leverage on the price of the underlying is displayed in Fig. Quite generally, out-of-the money options possess a higher leverage than inthe-money options, and the leverage of a call option decreases when the price of the underlying increases. The downside risk of an option therefore always 4. Also, all other things remaining constant, the leverage of an option increases when the time to maturity decreases.

As a consequence of these two observations, speculative investments in options are advisable only when the investor holds a strong view on the price movement of the underlying, and on the time scale over which this price movement is realized. The dependences of Theta on the price of the underlying and on time to maturity is shown in Fig. Theta diverges for an at-the-money option when the time to expiration goes to zero.

Theta converges to zero for an outof-the money call, i.

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The dependence on time to maturity is more interesting and shown in Fig. When an option expires at the money, Gamma diverges. Gamma tends towards zero, on the other hand, both for options in and out of the money. Theta for European call options. For an option at the money, the dependence 4. We will come back to Vega later in Sect. The use of the Greeks in hedging option positions is discussed in Chap. We can transform 4. For a short position in a European put option, the risk-free Black—Scholes portfolio is 4. These equivalences are general and do not assume the validity of the Black—Scholes model.

Also, they are not limited to call and put options. The important message is that any option can be created synthetically by a suitable combination of a position in a riskless asset and another position in the underlying. This is a result of great practical importance.

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Whenever an investor wishes to take a position in an option which is not available in the market, he can synthetically replicate the option by taking positions in a risk-free asset and in the underlying. Many portfolio managers and risk managers use this technique to implement their trading and hedging strategies when standard options are not available.

At best, it can be estimated from historical data on the underlying — a procedure which leaves many questions unanswered. This is shown in Fig. For comparison, the Black—Scholes solution is also displayed as solid lines. Figure 4. In the absence of an accurate ab initio estimation of the volatility, a rough and pragmatic procedure consists in taking the traded prices for granted and invert the Black—Scholes equation 4.

For the series of options used in Fig.

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Apparently, there are deviations of traded option prices from a Black—Scholes equation which depend on the contract to be priced. In this representation, they turn into an implied volatility which explicitly depends on the moneyness of the options. In a purist perspective, implied volatility adds nothing new to the theory of option call price 0. Black—Scholes Theory of Option Prices implied volatility 0. Geometric Brownian motion and the Black—Scholes theory take volatility independent of the option contract to be priced.

The two solid lines mark the contract-independent volatilities used to generate the solid lines in Fig. However, it is a simple transformation of option prices and therefore is an observable on equal footing with the prices.

This is similar to physics: When temperature is measured, the basic observable most often is an electric current or voltage drop, or height of a mercury column, etc. Also, implied volatility is the standard language of derivatives traders and analysts to describe option markets. The generic shapes of implied volatilities against moneyness are shown in Fig.

Apparently, a pure smile was characteristic of the US option markets before the october crash [53]. Ever since, it has become a rather smirky structure.