7.3 Risk Management and Hedging

Trading options is particularly risky due to the possibly high random component. Advanced strategies to reduce and manage this risk can be derived from Black-Scholes formula (6.24). To illustrate this issue we consider an example and some traditional strategies.

Example 7.1  
A bank sells a European call option to buy 100000 shares of a non dividend paying stock for 600000 EUR. The details of this option are given in Table 6.3.

Table 6.3: Data of the example
Current time $ t$ 6 weeks
Maturity $ T$ 26 weeks
Time to maturity $ \tau = T - t$ 20 weeks = 0.3846
Continuous annual interest rate $ r$ 0.05
Annualized stock volatility $ \sigma$ 0.20
Current stock price $ S_t$ 98 EUR
Exercise price $ K$ 100 EUR


Applying Black-Scholes' formula (6.24) for a non dividend paying stock,
$ b=r,$ gives a theoretical value of 480119 EUR, approximately 480000 EUR, of the above option. That is, the bank sold the option about 120000 EUR above its theoretical value. But it takes the risk to incur substantial losses.

A strategy to manage the risk due to the option would be to do nothing, i.e. to take a naked position. Should the option be exercised at maturity the bank has to buy the shares for the stock price prevailing at maturity. Assume the stock trades at $ S_T = 120$ EUR. Then an options' exercise costs the bank $ 100\;000 \cdot (S_T - K) = 2\;000\;000$ EUR, which is a multiple of what the bank received for selling the derivative. However, if the stock trades below $ K=100$ EUR the option will not be exercised and the bank books a net gain of 600000 EUR. 9248 SFEBSCopt2.xpl

In contrast to the naked position it is possible to set up a covered position by buying 100000 shares at $ 100\;000 \cdot S_t = 9\;800\;000 $ EUR at the same time the option is sold. In case $ S_T > K$ the option will be exercised and the stocks will be delivered at a price of $ 100\;000
\cdot K = 10\;000\;000$ EUR, which discounted to time $ t$ is about 9800000 EUR. Thus the bank's net gain is equal to 600000 EUR, the price at which the option is sold. If the stock price decreases to $ S_T = 80$ EUR then the option will not be exercised. However, the bank incurs a loss of 2000000 EUR due to the lower stock price, which is as above a multiple of the option price. Note that from put-call parity for European options (Theorem 2.3) it follows that the risk due to a covered short call option position is identical to the risk due to naked long put option position.

Both risk management strategies are unsatisfying because the cost varies significantly between 0 and large values. According to Black-Scholes the option costs on average around 480000 EUR, and a perfect hedge eliminates the impact of random events such that the option costs exactly this amount.

An expensive hedging strategy, i.e. a strategy to decrease the risk associated with the sale of a call, is the so-called stop-loss strategy: The bank selling the option takes an uncovered position as long as the stock price is below the exercise price, $ S_t < K,$ and sets up a covered position as soon as the call is in-the-money, $ S_t > K.$

The shares to be delivered in case of options exercise are bought as soon as the stock $ S_t$ trades above the exercise price $ K,$ and are sold as soon as $ S_t$ falls below the exercise price $ K.$

Since all stocks are sold and bought at $ K$ after time 0 and at maturity $ T$ either the stock position is zero, $ (S_t < K),$ or the stocks are sold at $ K$ to the option holder, $ (S_t > K),$ this strategy bears no costs.

Note that playing a stop-loss strategy bears a cost if $ S_0 > K,$ i.e. stocks are bought at $ S_0$ and sold at $ K:$

   costs of a stop-loss hedging strategy: $\displaystyle \, \max(S_0 - K,\ 0). $

Because these costs are smaller than the Black-Scholes price $ C(S_0,T)$ arbitrage would be possible by running a stop-loss strategy. However, this reasoning ignores some aspects: In practice, purchases and sales take place only after $ \Delta t$ time units. The larger $ \Delta t$, the greater $ \delta$ in general, and the less transaction costs have to be paid. Hull (2000) investigated in a Monte Carlo study with $ M=1000$ simulated stock price paths the stop-loss strategy's ability to reduce the risk associated with the sale of a call option. For each simulated path the costs $ \Lambda_m, m=1, ..., M,$ caused by applying the stop-loss strategy are registered and their sample variance

$\displaystyle \hat{v}^2_\Lambda = \frac{1}{M} \sum_{m=1}^{M} (\Lambda_m - \frac{1}{M} \sum_{j=1}^{M} \Lambda_j)^2$

is computed. Dividing the sample standard deviation by the call price measures the remaining risk of the stop-loss hedged short call position

$\displaystyle L = \frac{\sqrt{\hat{v}^2_\Lambda}}{C(S_0, T)} . $

Table 6.4 shows the results. A perfect hedge would reduce the risk to zero, i.e. $ L=0.$


Table 6.4: Performance of the stop-loss strategy
$ \Delta t$ (weeks) 5 4 2 1 $ \frac{1}{2} $ $ \frac{1}{4} $
$ L$ 1.02 0.93 0.82 0.77 0.76 0.76


7.3.1 Delta Hedging

In order to reduce the risk associated with option trading more complex hedging strategies than those considered so far are applied. Let us have a look at the following example. Sell a call option on a stock, and try to make the value of this portfolio for small time intervals as insensitive as possible to small changes in the price of the underlying stock. This is what is called delta hedging. Later on, we consider further Greeks (gamma, theta, vega, rho) to fine tune the hedged portfolio.

By the delta or the hedge ratio we understand the derivative of the option price with respect to the stock price. In a discrete time model we use the differential quotient of the change in the option price $ \Delta C$ with respect to a change in the stock price $ \Delta S:$

$\displaystyle \Delta = \frac{\partial C}{\partial S}$   oder $\displaystyle \quad
\Delta = \frac{\Delta C}{\Delta S}. $

The delta of other financial instruments is defined accordingly. The stock itself has the value $ S.$ Consequently it holds $ \Delta
= \partial S/\partial S = 1.$ A futures contract on a non dividend paying stock has a value of $ V = S - K \cdot e^ {- r\tau} $ (see Theorem 2.1) and thus its delta is $ \Delta =
\partial V / \partial S = 1$ as well. Stocks and future contracts can therefore be used equivalently in delta hedging strategies. If the latter are available they are preferable due to lower transaction costs.

Example 7.2  
A bank sells calls on 2000 shares of a stock for a price of $ C = 10 $ EUR/share at a stock price of $ S_0 = 100 $ EUR/share. Let the call's delta be $ \Delta = 0.4.$ To hedge the sold call options $ \Delta \cdot 2000 = 800$ shares of the stock are added to the portfolio. Small changes in the option value will be offset by corresponding changes in the value of the portfolio's stock shares. Should the stock price increase by 1 EUR, i.e. the value of the stock position in the portfolio increases by 800 EUR, the value of one call on 1 share increases by $ \Delta C = \Delta \cdot \Delta S = 0.4 $ EUR and following the value of the portfolio's short call position decreases by 800 EUR. That is, gains and losses offset because the delta of the option position is neutralized by the delta of the stock position. The portfolio has a $ \Delta = 0,$ and the bank takes a delta neutral position.

Since the delta of an option depends on the stock price and time, among others, the position is only for a short period of time delta neutral. In practice, the portfolio has to be re-balanced frequently in order to adapt to the changing environment. Strategies to manage portfolio risk which involve frequent re-balancing are known as dynamic hedging. We point out that the Black-Scholes differential equation (6.3) can be derived by means of a dynamic hedge portfolio whose position is kept continuously delta neutral. This approach is analogous to reproducing the option by a duplicating portfolio.

Example 7.3  
The price of the underlying stock rises within a week to 110 EUR. Due to the time decay and the increased stock price the option delta increased to $ \Delta = 0.5.$ In order to reobtain a delta neutral position $ (0.5 - 0.4) \cdot 2000 = 200$ shares of the stock have to be bought.

From the Black-Scholes formulae for the value of European call and put options on non dividend paying stocks it follows for the delta:
$\displaystyle \Delta = \frac{\partial C}{\partial S}$ $\displaystyle =$ $\displaystyle \Phi (y +
\sigma \sqrt{\tau})$ (7.27)
$\displaystyle \vspace{5mm}$   bzw. $\displaystyle \Delta = \frac{\partial P}{\partial
S}$ $\displaystyle =$ $\displaystyle \Phi (y + \sigma \sqrt{\tau}) - 1,$  

with $ y$ being defined in equation (6.25).

Figure 6.1 displays the delta (6.27) as a function of time and stock price. For an increasing stock price delta converges to 1, for decreasing stock prices it converges to 0. Put differently, if the option is deep in-the-money (ITM) it will be exercised at maturity with a high probability. That is the reason why the seller of such an option should be long in the underlying to cover the exercise risk. On the other hand, if the option is far out-of-the-money it will probably not be exercised, and the seller can restrict himself to holding a smaller part of the underlying.

Fig.: Delta as a function of the stock price (right axis) and time to maturity (left axis). 9534 SFEdelta.xpl
\includegraphics[width=1.4\defpicwidth]{delta.ps}

Furthermore, the probability $ p$ that an out-of-the-money (OTM) option will be exercised and an ITM option will not be exercised at maturity is higher the longer the time to maturity. This explains why the delta for longer times to maturity becomes more flat (linear).

Table 6.5 according to Hull (2000) shows (in the same spirit as Table 6.4) the performance of the delta hedging strategy contingent on the time increments $ \Delta t$ between re-balancing trades. If $ \Delta t$ is small enough the risk associated with a sold call option can be managed quite well. In the limit $ \Delta t \rightarrow 0$ continuous re-balancing underlying the derivation of the Black-Scholes formula follows, and the risk is perfectly eliminated $ (L=0).$

Table 6.5: Performance of the delta-hedging strategy
$ \Delta t$ (weeks) 5 4 2 1 $ \frac{1}{2} $ $ \frac{1}{4} $
$ L$ 0.43 0.39 0.26 0.19 0.14 0.09


The linearity of the mathematical derivative implies for the delta $ \Delta_p$ of a portfolio consisting of $ w_1, \ldots, w_m $ contracts of $ m$ financial derivatives $ 1, \ldots, m$ with deltas $ \Delta_1, \ldots, \Delta _m:$

$\displaystyle \Delta _p = \sum^ m_{j=1} w_j \Delta _j. $

Example 7.4  
Consider a portfolio consisting of the following USD options
1.
200000 bought calls (long position) with exercise price 1.70 EUR maturing in 4 months. The delta of an option on 1 Doller is $ \Delta_1 = 0.54.$
2.
100000 written calls (short position) with exercise price 1.75 EUR maturing in 6 months and a delta of $ \Delta_2 = 0.48.$
3.
100000 written puts (short position) with exercise price $ 1.75$ EUR maturing in 3 months with $ \Delta_3 = - 0.51.$
The portfolio's delta is (increases in values of written options have a negative impact on the portfolio value):
$\displaystyle \Delta_p$ $\displaystyle =$ $\displaystyle 200\;000 \cdot \Delta _1 - 100\;000 \cdot \Delta_2 - 100\;000 \cdot \Delta_3$  
  $\displaystyle =$ $\displaystyle 111\;000$  

The portfolio can be made delta neutral by selling 111000 USD or by selling a corresponding future contract on USD (both have a delta of $ \Delta = 1$).

7.3.2 Gamma and Theta

Using the delta to hedge an option position the option price is locally approximated by a function which is linear in the stock price $ S.$ Should the time $ \Delta t$ passing by until the next portfolio re-balancing be not very short this approximation is no longer adequate (see Table 6.5). That is why a more accurate approximation, the Taylor expansion of $ C$ as a function of $ S$ and $ t,$ is considered:

$\displaystyle \Delta C = C(S + \Delta S,\ t+\Delta t) - C(S,t) = \frac{\partial...
...al ^ 2 C}{\partial S^ 2}
(\Delta S)^ 2 + {\scriptstyle \mathcal{O}}(\Delta t),
$

where (as we already saw in the demonstration of Theorem 6.1) $ \Delta S$ is of size $ \sqrt{\Delta t}$ and the terms summarized in $ {\scriptstyle \mathcal{O}}(\Delta t)$ are of size smaller than $ \Delta t.$ Neglecting all terms but the first, which is of size $ \sqrt{\Delta t},$ the approximation used in delta hedging is obtained:

$\displaystyle \Delta C \approx \Delta \cdot \Delta S . $

Taking also the terms of size $ \Delta t$ into account it follows

$\displaystyle \Delta C \approx \Delta \cdot \Delta S + \Theta \cdot \Delta t +
\frac{1}{2} \Gamma (\Delta S^) 2 , $

where $ \Theta = {\partial
C} /{\partial t }$ is the options theta and $ \Gamma =
\partial ^ 2 C/\partial S^ 2 $ is the options gamma. $ \Theta$ is also called the options time decay. For a call option on a non dividend paying stock it follows from the Black-Scholes formula (6.24):

$\displaystyle \Theta = - \frac{\sigma S}{2\sqrt{\tau}}\, \varphi (y+\sigma \sqrt{\tau}) -
rKe^ {-r\tau} \Phi (y) $

and

$\displaystyle \Gamma = \frac{1}{\sigma S\sqrt{\tau}} \, \varphi (y + \sigma \sqrt{\tau} ),$ (7.28)

where $ y$ is defined in equation (6.25).

Figure 6.2 displays the gamma given by equation (6.28) as a function of stock price and time to maturity. Most sensitive to movements in stock prices are at-the-money options with a short time to maturity. Consequently, to hedge such options the portfolio has to be rebalanced frequently.

Fig.: Gamma as a function of stock price (right axis) and time to maturity (left axis). 9747 SFEgamma.xpl
\includegraphics[width=1.4\defpicwidth]{gamma.ps}

Assuming a delta neutral portfolio gamma hedging consists of buying or selling further derivatives to achieve a gamma neutral portfolio, i.e.  $ \Gamma
= 0,$ and thereby making the portfolio value even more insensitive to changes in the stock price. Note that on the one hand neither stocks nor future contracts can be used for gamma hedging strategies since both have a constant $ \Delta$ and thus a zero gamma $ \Gamma=0.$ On the other hand, however, those instruments can be used to make a gamma neutral portfolio delta neutral without affecting the portfolio's gamma neutrality. Consider an option position with a gamma of $ \Gamma$. Using $ w$ contracts of an option traded on a stock exchange with a gamma of $ \Gamma_B,$ the portfolio's gamma is $ \Gamma +
w \Gamma_B.$ By setting $ w=- \Gamma / \Gamma_B$ the resulting gamma for the portfolio is 0.

Example 7.5  
Let a portfolio of USD options and US-Dollars be delta neutral with a gamma of $ \Gamma = - $150000. On the exchange trades a USD-call with $ \Delta_B = 0.52 $ and $ \Gamma_B = 1.20.$ By adding $ - \Gamma / \Gamma _B = 125\;000$ contracts of this option the portfolio becomes gamma neutral. Unfortunately, its delta will be $ 125\;000 \cdot \Delta _B = 65\;000.$ The delta neutrality can be achieved by selling 65000 USD without changing the gamma.

Contrary to the evolution of the stock price the expiry of time is deterministic, and time does not involve any risk increasing randomness. If both $ \Delta$ and $ \Gamma$ are 0 then the option value changes (approximately risk free) at a rate $ \Theta = \Delta
C/\Delta t.$ The parameter $ \Theta$ is for most options negative, i.e. the option value decreases as the maturity date approaches.

From Black-Scholes's formula (6.24) it follows for a delta neutral portfolio consisting of stock options

$\displaystyle rV = \Theta + \frac{1}{2} \sigma ^ 2 S^ 2 \Gamma, $

with $ V$ denoting the portfolio value. $ \Delta$ and $ \Gamma$ depend on each other in a straightforward way. Consequently, $ \Delta$ can be used instead of $ \Gamma$ to gamma hedge a delta neutral portfolio.

7.3.3 Rho and Vega

Black-Scholes' approach proceeds from the assumption of a constant volatility $ \sigma.$ The appearance of smiles indicates that this assumption does not hold in practice. Therefore, it can be useful to make the portfolio value insensitive to changes in volatility. By doing this, the vega of a portfolio (in literature sometimes also called lambda or kappa) is used, which is for a call option defined by $ {\cal V} =
\frac{\partial C}{\partial \sigma}.$

For stocks and future contracts it holds $ {\cal V} = 0.$ Thus, in order to set up a vega hedge one has to make use of traded options. Since a vega neutral portfolio is not necessarily delta neutral two distinct options have to be involved to achieve simultaneously $ {\cal V} = 0$ and $ \Gamma=0.$

From Black-Scholes' formula (6.24) and the variable $ y$ defined in equation (6.25) it follows that the vega of a call option on a non dividend paying stock is given by:

$\displaystyle {\cal V} = S\sqrt{\tau} \varphi (y + \sigma \sqrt{\tau}).$ (7.29)

Since the Black-Scholes formula was derived under the assumption of a constant volatility it is actually not justified to compute the derivative of (6.24) with respect to $ \sigma.$ However, the above formula for $ {\cal V}$ is quite similar to an equation for $ {\cal V}$ following from a more general stochastic volatility model. For that reason, equation (6.29) can be used as an approximation to the real vega.

Figure 6.3 displays the vega given by equation (6.29) as a function of stock price and time to maturity. At-the-money options with a long time to maturity are most sensitive to changes in volatility.

Fig.: Vega as a function of stock price (right axis) and time to maturity (left axis). 9903 SFEvega.xpl
\includegraphics[width=1.4\defpicwidth]{vega.ps}

Finally, the call option's risk associated with movements in interest rates can be reduced by using rho to hedge the position:

$\displaystyle \rho = \frac{\partial C}{\partial r}. $

For a call on a non dividend paying stock it follows from equation (6.24)

$\displaystyle \rho = K\ \tau\ e^ {-r\tau} \Phi (y) . $

When hedging currency options domestic as well as foreign interest rates have to be taken into account. Consequently, rho hedging strategies need to consider two distinct values $ \rho_1$ and $ \rho_2.$

7.3.4 Historical and Implied Volatility

A property of the Black-Scholes formulae (6.22), (6.24) is that all option parameters are known except the volatility parameter $ \sigma.$ In practical applications $ \sigma$ is estimated from available stock price observations or from prices of similar products traded on an exchange.

Historical volatility is an estimator for $ \sigma$ based on the variability of the underlying stock in the past. Let $ S_0, \ldots, S_n$ be the stock prices at times $ 0, \Delta t, 2 \Delta t, \ldots, n
\Delta t.$ If the stock price $ S_t$ is modelled as Brownian motion, the logarithmic relative increments

$\displaystyle R_t = \ln \frac{S_t}{S_{t-1}} \, ,\ \, \, t = 1, \ldots, n $

are independent and identical normally distributed random variables. $ R_t$ is the increment $ Y_t - Y_{t-1}$ of the logarithmic stock price $ Y_t = \ln S_t$ which as we saw in Section 5.4 is in a small time interval of length $ \Delta t$ a Wiener process with variance $ \sigma^{2}.$ Consequently the variance of $ R_t$ is given by

$\displaystyle v = \mathop{\text{\rm Var}}(R_t) = \sigma^ 2 \cdot \Delta t .$

A good estimator for $ \mathop{\text{\rm Var}}(R_t)$ is the sample variance

$\displaystyle \hat{v} = \frac{1}{n-1} \, \sum^ n_{t=1} (R_t - \bar{R}_n)^ 2 $

with $ \bar{R}_n = \frac{1}{n} \, \sum^ n_{t=1} \, R_t $ being the sample average. $ \, \hat{v} $ is unbiased, i.e.  $ \mathop{\text{\rm\sf E}}[\hat{v}] =
v,\ $ and the random variable

$\displaystyle (n - 1) \, \frac{\hat{v}}{v} $

is $ \chi^ 2_{n-1}$ distributed (chi-square distribution with $ n-1$ degrees of freedom). In particular this implies that the mean squared relative estimation error of $ \hat{v}$ is given by

$\displaystyle \mathop{\text{\rm\sf E}}\left( \frac{\hat{v}-v}{v}\right)^ 2 = \f...
...,
\mathop{\text{\rm Var}}\left((n-1) \frac{\hat{v}}{v}\right) = \frac{2}{n-1} .$

Since it holds $ v = \sigma^ 2 \Delta t$ an estimator for the volatility $ \sigma$ based on historical stock prices is

$\displaystyle \hat{\sigma} = \sqrt{\hat{v}/\Delta t}. $

By means of a Taylor expansion of the square root function and by means of the known quantities $ \mathop{\text{\rm\sf E}}[ \hat{v}]$ and $ \mathop{\text{\rm Var}}({\hat{v}}/{v})$ it follows that $ \hat{\sigma}$ is unbiased neglecting terms of size $ {n}^{-1}:$

$\displaystyle \mathop{\text{\rm\sf E}}\hat{\sigma} = \sigma + {\mathcal{O}}\left( \frac{1}{n}\right)\ , $

and that the mean squared relative estimation error of $ \hat{\sigma}$ is given by

$\displaystyle \mathop{\text{\rm\sf E}}\left( \frac{\hat{\sigma}-\sigma}{\sigma}...
... 2 =
\frac{1}{2(n-1)} + {\scriptstyle \mathcal{O}}\left( \frac{1}{n}\right)\ , $

again neglecting terms of size smaller than $ {n}^{-1}.$ Thanks to this relationship the reliability of the estimator $ \hat{\sigma}$ can be evaluated. Sample parameter selection:
a)
As data daily settlement prices $ S_0, \ldots, S_n$ are often used. Since $ \sigma$ is in general expressed as an annualized volatility $ \Delta t$ corresponds to one day on a yearly basis. Working with calender day count convention $ \Delta t= \frac{1}{365}.$ Unfortunately, for weekends and holidays no data is available. The following empirical argument favors to ignore weekends and holidays: If the stock dynamics behaved on Saturdays and Sundays as it does on trading days even if the dynamics were not observed then standard deviation of the change in the stock price from Friday to Monday would three times as large as the standard deviation between two trading days, say Monday to Tuesday. This follows from the fact that for the Wiener process $ Y_t = \ln S_t$ the standard deviation of the increment $ Y_{t+\delta} - Y_t$ is $ \sigma \cdot \delta.$ Empirical studies of stock markets show, however, that both standard deviations are proportional with a constant of around 1 but in any case significantly smaller than 3. Put in other words, the volatility decreases on weekend days. A conclusion is that trading increases volatility, and that the stock variability is not solely driven by external economic influences. Estimating volatility should therefore be done by considering exclusively trading days. Usually a year is supposed to have 252 trading days, i.e.  $ \Delta t= \frac{1}{252}.$

Concerning monthly data, $ \Delta t= \frac{1}{12}$ is applied. In Section 3.3 we have calculated an annual volatility of $ 19\%$ based on the monthly DAX data.
10082 SFEsumm.xpl

b)
Theoretically, the larger $ n$ the more reliable $ \hat{\sigma}.$ However, empirically the volatility is not constant over longer time periods. That is to say that stock prices from the recent past contain more information about the current $ \sigma$ as do stock prices from long ago. As a compromise closing prices of the last 90 days respectively 180 days are used. Some authors advise to use historical data of a period which has the same length as the period in the future in which the estimated volatility will be applied. In other words, if you want to compute the value of a call expiring in 9 months you should use closing prices of the past 9 months.

The implied volatility of an option is computed from its market price observed on an exchange and not from the prices of the underlying as it is case for the historical volatility. Consider a European call on a non dividend paying stock $ (d = 0,\ b = r),$ which has a quoted market price of $ C_B$, then its implied volatility $ \sigma_I$ is given by solving

    $\displaystyle S\ \Phi (y + \sigma_I \sqrt{\tau}) - e^ {-r\tau} K\ \Phi (y) = C_B$  
       
  with  $\displaystyle \qquad \displaystyle y = \frac{1}{\sigma_I \sqrt{\tau}} \, \{ \ln
\frac{S}{K} + ( r - \frac{1}{2}\sigma^2_I ) \tau \} .$  

$ \sigma_I$ is the value of the volatility which if substituted into the Black-Scholes formula (6.24) would give a price equal to the observed market price $ C_B. \, \,
\sigma_I$ is implicitly defined as a solution of the above equation, and has to be computed numerically due to the fact that the Black-Scholes formula cannot be inverted.

The implied volatility can be used to get an idea of the market view of the stock volatility. It is possible to construct an estimator using implied volatilities of options on the same stock but which are different in time to maturity $ \tau$ and exercise price $ K.$ A weighting scheme takes the option price dependence on the volatility into account.
10086 SFEVolSurfPlot.xpl

Example 7.6  
Consider two traded options on the same underlying. One is at-the-money (ATM) and the other is deep ITM with volatilities of $ \sigma_{I1} = 0.25$ respectively $ \sigma_{I2} = 0.21.$ At-the-money the dependence of option price and volatility is particular strong. That is, the price of the first option contains more information about the stock volatility and $ \sigma_{I1}$ can be considered a more reliable volatility estimate. Thus the estimator combining both implied volatilities should attribute a higher weight to $ \sigma_{I1},$ as for example

$\displaystyle \tilde{\sigma} = 0.8 \cdot \sigma_{I1} + 0.2 \cdot \sigma_{I2} .$

Some authors suggest to set $ \tilde{\sigma} = \sigma_{Im}$ with $ \sigma_{Im}$ being the volatility of the option which is most sensitive to changes in $ \sigma,$ i.e. the option with the highest vega $ \partial C/\partial \sigma$ in absolute terms.

In order to apply the concept of risk neutrality (see Cox and Ross (1976)) the probability measure has to be transformed such that the price process under this new measure is a martingale. By doing this the absence of arbitrage opportunities is guaranteed. In incomplete markets, however, a multitude of such transformations exist (see Harrison and Kreps (1979)). In contrast to complete markets the trader cannot build up a self-financing portfolio reproducing the options payoff at maturity when the market is incomplete. Therefore hedging is no more riskless, and option prices depend on risk preferences. In this context we want to point out that the lack of a perfect hedge is of great importance in practice.