9.3 Complement: barrier options in the Black-Scholes model
9.3.3 Static hedging of barrier options
-6
Regular down-and-in call
ST
B K S0
-6
Reverse up-and-in call
ST S0 K B
Figure 9.5: Types of barrier options.
9.3.2 Dynamic hedging of barrier options
We only indicate how the Black-Scholes hedging theory extends to the case of barrier options. We leave the technical details for the reader. If the barrier is hit before maturity, the barrier option value at that time is known to be either zero, or the price of the corresponding European option. Hence, it is sufficient to find the hedge before hitting the barrier TB∧ T with
TB:= inf {t ≥ 0 : St= B} .
Prices of barrier options are smooth functions of the underlying asset price in the in-region, so Itˆo’s formula may be applied up to the stopping time T ∧ TB. By following the same line of argument as in the case of plain vanilla options, it then follows that perfect replicating strategy consists in:
holding ∂f
∂s(t, St) shares of the underlying asset for t ≤ TB∧ T where f (t, St) is price of the barrier option at time t.
9.3.3 Static hedging of barrier options
In contrast with European calls and puts, the delta of barrier options is not bounded, which makes these options difficult to hedge dynamically. We conclude this section by presenting a hedging strategy for barrier options, due to P. Carr et al. [8], which uses only static positions in European products.
A barrier option is said to be regular if its pay-off function is zero at and beyond the barrier, and reverse otherwise (see Figure 9.5 for an illustration).
In the following, we will treat barrier options with arbitrary pay-off functions (not necessarily calls or puts). The price of an Up and In barrier option which pays f (ST) at date T if the barrier B has been crossed before T will be denoted by UIt(St, B, f (ST), T ), where t is the current date and St is the current stock price. In the same way, UO denotes the price of an Up and Out option and EURt(St, f (ST), T ) is the price of a European option with pay-off f (ST). These
9.3. Barrier options 141
functions satisfy the following straightforward parity relations:
UIt+ UOt= EURt
UIt(St, B, f (ST), T ) = EURt(St, f (ST), T ) if f (z) = 0 for z < B UIt(St, B, f (ST), T ) = UIt(St, B, f (ST)1ST<B, T )
+ EURt(St, f (ST)1ST≥B, T ) in general.
This means that in order to hedge an arbitrary barrier option, it is sufficient to study options of type In Regular. In addition, Up and Down options can be treated in the same manner, so we shall concentrate on Up and In regular options.
The method is based on the following symmetry relationship:
EURt(St, f (ST), T ) = EURt volatility. It is easy to check that this relation holds in the Black-Scholes model, but the method also applies to other models which possess a similar symmetry property.
Replication of regular options Let f be the pay-off function of an Up and In regular option. This means that f (z) = 0 for z ≥ B. We denote by TB the first passage time by the price process above the level B. Consider the following static hedging strategy:
• At date t, buy the European option EURt
a by-product, we obtain the pricing formula:
UIt(St, B, f (ST), T ) = EURt
The case of calls and puts Equation (9.33) shows that the price of a regular In option can be expressed via the price of the corresponding European option, for example,
However, unless γ = 1, the replication strategies will generally involve European payoffs other than calls or puts. If γ = 1 (that is, the dividend yield equals the risk-free rate), then regular In options can be statically replicated with a single call / put option. For example,
EURt St, ST B
γ
K −B2 ST
+ , T
!
= EURt St, KST B − B
+
, T
!
= K Bct
St,B2
K, T
.
The replication of reverse options will involve payoffs other than calls or puts even if γ = 1.
Chapter 10
Local volatility models and Dupire’s formula
10.1 Implied volatility
In the Black-Scholes model the only unobservable parameter is the volatility. We therefore focus on the deopendence of the Black-Scholes formula in the volatility parameter, and we denote:
CBS(σ) := sN d+(s, ˜K, σ2T ) − ˜KN d−(s, ˜K, σ2T ),
where N is the cumulative distribution function of the N (0, 1) distribution, T is the time to maturity, s is the spot price of the underlying asste, and ˜K, d± are given in (9.5).
In this section, we provide more quantitative results on the volatility calibra-tion discussed in Seccalibra-tion 9.2.5. First, observe that the model can be calibrated from a single option price because the Black-Scholes price function is strictly increasing in volatility:
lim
σ↓0CBS(σ) = (s − ˜K)+, limσ↑∞CBS(σ) = s and ∂CBS
∂σ = sN0(d+)
√ T > 0 (10.1) Then, whenever the observed market price C of the call option lies within the no-arbitrage bounds:
(s − ˜K)+< C < s there is a unique solution I(C) to the equation
CBS(σ) = C
called the implied volatility of this option. Direct calculation also shows that
∂2CBS
∂σ2 =sN0(d+)√ T σ
m2
σ2T −σ2T 4
where m = ln s
Ke−rT, (10.2) 143
is the option moneyness. Equation (10.2) shows that the function σ 7→ CBS(σ) is convex on the interval (0,
q2|m|
T −t) and concave on ( q2|m|
T −t, ∞). Then the implied volatility can be approximated by means of the Newton’s algorithm:
σ0= r2m
T and σn = σn−1+C − CBS(σn−1)
∂CBS
∂σ (σn)
which produces a monotonic sequence of positive scalars (σn)n≥0. However, in practice, when C is too close to the arbitrage bounds, the derivative ∂C∂σBS(σn) becomes too small, leading to numerical instability. In this case, it is better to use the bisection method.
In the Black-Scholes model, the implied volatility of all options on the same underlying must be the same and equal to the historical volatility (standard deviation of annualized returns) of the underlying. However, when I is computed from market-quoted option prices, one observes that
• The implied volatility is always greater than the historical volatility of the underlying.
• The impied volatilities of different options on the same underlying depend on their strikes and maturity dates.
The left graph on Fig. 10.1 shows the implied volatilities of options on the S&P 500 index as function of their strike and maturity, observed on January 23, 2006.
One can see that
• For almost all the strikes, the implied volatility is decreasing in strike (the skew phenomenon).
• For very large strikes, a slight increase of implied volatility can sometimes be observed (the smile phenomenon).
• The smile and skew are more pronounced for short maturity options; the implied volatility profile as function of strike flattens out for longer matu-rities.
The difference between implied volatility and historical volatility of the un-derlying can be explained by the fact that the cost of hedging an option in reality is actually higher than its Black-Scholes price, due, in particular to the transaction costs and the need to hedge the risk sources not captured by the Black-Scholes model (such as the volatility risk). The skew phenomenon is due to the fact that the Black-Scholes model underestimates the probability of a market crash or a large price movement in general. The traders correct this probability by increasing the implied volatilities of options far from the money.
Finally, the smile can be explained by the liquidity premiums that are higher for far from the money options. The right graph in figure 10.1 shows that the im-plied volatilities of far from the money options are almost exclusively explained by the Bid prices that have higher premiums for these options because of a lower offer.
10.2. Local volatility 145
400 600 800 1000 1200 1400 1600
0
Figure 10.1: Left: Implied volatility surface of options on the S&P500 index on January 23, 2006. Right: Implied volatilities of Bid and Ask prices.