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4.1.3 Application, time value and implied volatility

[ SLIDE call(SMI) - implied volatility - smile - time value || VIDEO modem - LAN - DSL ]

We are ready now to use the VMARKET applet and compare the fair value obtained from a model with the market price of products that are sold by financial institutions. For example, take the European call on the Swiss Market Index (SMI) expiring on Dec 19, 2002 with a strike at 7000. Nine months before the expiry, the underlying index was trading at 6610 with a market volatility around 18% (follow the links to obtain the current market data). Under reasonable assumptions in Switzerland of a 2% risk-free rate and a 2% average dividend yield for the shares that constitute the SMI index, the VMARKET applet below calculates the fair price for this option according to the Black-Scholes model.

VMARKET applet:  press Start/Stop to run the simulation until it stops exactly nine months before the option expires. For an approximative solution, you may simply click inside the plot area to measure the payoff V(S) around the coordinate 6610. For a complete printout of the numerical solution, switch from Double-click below in the applet to Print data to console, set TimeStep=0. and press Step 1; the number output can be now read from the Java-console (with Netscape open Communicator->Tools->Java console) where x[] is the price of the underlying, f0[] is the intrinsic value in grey and f[] the solution in black. Don't forget to switch back to Double-click below avoiding to overflow the Java console...

After interpolation, the fair price (CHF 251.6) is encouragingly close to the offer from Crédit Suisse First Boston CSFB (500 x 0.51 = CHF 255.0). Considering the crude approximations we made for the input parameters and the limitations of the Black-Scholes model, this agreement may however well be fortuitous: in fact, there is no guarantee that market prices coincide with any model at all-the offer and demand from traders on the option markets need not to be rational!

You may well ask now why people use financial modeling... It turns out that the predicted values are nevertheless often in the right range. Modeling is particularly useful to estimate what should be a fair value when there are not a sufficient number of buyers / sellers to make a market, for example when an option is offered for the first time or when an exotic option is tailored by a financial institution (the market maker) to meet the specific needs of only two clients. Simple products such as the vanilla call above have more than 100 million options listed on the Swiss exchange: this is enough to set a price only by offer and demand. Instead of calculating the option price as a function of the volatility, the Black-Scholes model is then often used as a market standard to calculate an implied volatility, i.e. the volatility that has to be used in the model to reproduce the price from the market.

Virtual market experiments: application in a real market
  1. Keeping the same Volatility=0.18, calculate the fair value of a call with a StrikePrice=7500; compare with the market price (500 x 0.22 = CHF 110).
  2. Reduce the Volatility until the calculated value matches the market price; compare with the 17% implied volatility calculated by CSFB.
  3. From the two values above, estimate the Vega measuring the sensitivity of this option to changes in the volatility. You can use a finite difference approximation to calculate the derivative from Vega=(V(0.18)-V(0.17))/(0.18-0.17).

Keeping the same expiry date, the implied volatility can be measured for different strike prices $ \sigma(K)$ ; this curve is traditionally called the smile, but has a shape that really depends on the market conditions and can equally well be a frown (exercise 4.04). Although there is no guarantee to make a profit from the so-called volatility trading, some investors buy options with a low- and short options with a high implied volatility: their bet is that market forces will eventually move the prices of options so as to make implied volatilities comparable in the future.

To complete this analysis dealing with the European option payoff dynamics, simply note that the difference between the present and the intrinsic (or final) value of an option is traditionally called time value. The simulation using the VMARKET applet below shows that the time value is usually negative for a put option in-the-money and is sometimes strictly positive for a call option.

VMARKET applet:  press Start/Stop to simulate the value of a European put V(S,t) before the option expires; observe the relative position of the present value (in black) and the intrinsic or final value (in grey).

Virtual market experiments: time value
  1. Try to find a combination of financial parameters that leads to a negative time value for the call option and a positive time value for the put option.
  2. Can you produce a positive time value for both the call and the put option?

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