Table 2.3: Parameter estimates for joint density of Act and S on 31 July 1998

Figure 2.10 Joint density for price and volatility, 30 September 1998.

Table 2.4: Parameter estimates for joint density of and AS on 30 September 1998

the index and the changes in implied volatility, as reflected in the estimate of (3, and the mean daily change in the index is only -4.6.

Contrast this with the density shown in Figure 2.10, which is given by the parameter estimates of Table 2.4. On 30 September 1998 the market had been

through an extremely jumpy period and the probability of large downward moves in the index, accompanied by large upward moves in the ATM volatility, was much higher than it was on 31 July. This is evident from the estimate of beta having risen (in absolute terms) to -0.0359. The mean daily change in index price was -10.6.

This section has illustrated a simple method for examining the correlations of price changes and ATM volatility changes. Knowledge of this correlation provides information about the current market regime, and therefore the appropriate sticky model to use for calibrating binomial trees for option pricing and hedging. It shows how to construct a joint density for prices and ATM volatilities that can be used to compute the expected loss in probabilistic scenario analysis (§9.6).

However, it has told us nothing about the relationship between fixed strike volatilities and changes in the underlying price. Since positions are likely to move ITM and/or OTM during the risk horizon, we need to know what scenarios are most probable for the whole volatility skew. More than this, in the next section we show how it is extremely important to have an accurate estimate of the volatility sensitivity to price, dcs/dS, for options of all strikes. The simple model that has been presented in this section will not provide these estimates and we shall have to return to this problem later in the book, in §6.3, after some more sophisticated tools have been developed.

2.3.3 Implications for Delta Hedging

In this section we write an option price asf = f (S, a), ignoring the dependence on strike, maturity, discount rates and so on for ease of notation. The delta of an option is its price sensitivity to changes in the underlying price S:

A(S,n) = df(S,n)/dS. (2.8)

Standard delta hedging strategies are to match each unit of the option with A units of the underlying. Therefore if one is short x units in the option one goes long Ax units of the underlying, and this hedged portfolio has value V = AxS - xf(S, cr). The delta hedged portfolio is called delta neutral because the portfolio value will remain unchanged for small changes in the underlying: dV/dS = 0.

The amount of the underlying required to maintain delta neutrality will usually change every time the underlying price changes. That is, the option delta will depend on the underlying price level unless the option gamma is zero. The gamma of an option measures the sensitivity of delta to changes in the underlying price:

It is extremely important to have an accurate estimate of the volatility sensitivity to price, /dS, for options of all strikes

T(S, cr) = d2f(S, a)/dS2 = dA(S, a)/dS.

(2.9)

Gamma is usually greatest for ATM options, and typically it will increase as the option approaches expiry. Therefore to maintain delta neutrality the portfolio will have to be delta hedged on a dynamic basis, and very frequently as the option approaches expiry. This rebalancing can be very costly so it is obviously important to get the delta right. What is the right delta to use?

If one assumes volatility is constant as in the Black-Scholes model, the Black-Scholes delta is simply

ABS(5, a) = df/dS.

From the Black-Scholes formula (2.1), ABs(, °") = ( ) and the delta is easily calculated as the value of the normal distribution (-) at x, the moneyness of the option.25

However, the discussion in this section has focused on the fact that the smile surface changes each time the underlying moves. We should not be assuming that volatility is constant, and instead we shall suppose that that the volatility surface a(S, t) is deterministic. In this case applying the chain rule to compute the option delta gives:

A(5, cr) = df/dS + [df/dn][du/dS] = ABS + vega[<3cr/<35], (2.10)

where

vega = df(S, )/ . (2.11)

The option vega is the sensitivity of the option price to changes in implied volatility. ATM options can have very large vegas; in fact the option gamma and the option vega are often proportional.26 When volatility is not constant and an option has a non-zero vega, the Black-Scholes delta will not be very accurate. From (2.10) an extra term which contains the vega and the volatility price sensitivity /dS will have to be included.

When volatility is not constant and an option has a non-zero vega, the Black-Scholes delta will not be very accurate

Option traders often approximate the term /dS by / . That is, they evaluate the slope of the current skew or smile and use that in place of the volatility price sensitivity to calculate the option delta by (2.10). Note that the sticky models given in equations (2.3) will determine different values for /dS according to the market regimes. In range-bounded markets one should take /dS - O, but in trending markets da/dS = b and in jumpy markets /dS = - b. These models do not, however, provide a measure for this coefficient b. In any case, how does one know which model to apply?

-5dC/dS = (. ) + S$(x)dx/dS - Ke-"fy(y)dy/dS, where = x - ojt. Since dy/dS = dx/dS, this may be written OC/dS = ( ) + (dx/dS)[S<ii(x) - -" ( )], where ( ) = ( ). A little algebra shows that the term in the square brackets is zero.

26 For a simple example to illustrate this, suppose the pay-off function is (5, - Sq)2, where the variance V(S,) is gSqI, such as would be the case under geometric Brownian motion starting from an initial price S = S0. Then the value f(S, a) = E((S, - S0)2) = V(S,) = a2S20t. Therefore, gamma = 2a2t and vega = 2aSJjf. That is, vega = (So/cOgamma.