3.1 Butterfly Spreads and the Implied Distribution

Vanilla option prices contain probability information about the market's guess at the future level of the underlying asset S. For example, suppose that Kroger Co. trades at $24 and that one-year calls struck at $24 and $25 trade at $1 and $0.60 respectively. If interest rates are zero, we may then infer that the probability of the terminal spot price S_T in one year to be above $24 must satisfy:

because the digital payoff dominates the call spread as shown in Figure 3.1.

Similarly the price of a butterfly spread with strikes $23, $24, and $25 (i.e., long one call struck at $23, short two calls struck at $24, and long one call struck at $25) will give a lower bound for as shown in Figure 3.2.

Generally, if all option prices are available along a continuum of strikes—such as option prices generated by an implied volatility surface model—we may consider butterfly spreads with strikes leveraged by 1/ϵ² and obtain in the limit as ϵ → 0 the implied distribution density of S_T:

3.1

where r is the continuous interest rate for maturity T, and c(K) denotes the price of the call struck at K.

Note that by put-call parity we have and thus put prices may alternatively be used to compute the implied distribution density.

Figure 3.3 compares the Black-Scholes lognormal density to the implied distribution density generated by an SVI model fit of S&P 500 option prices with about 2.4-year maturity. We can see that the implied density is skewed toward the right and has a fatter left tail.

The implied distribution density may be obtained directly from a smooth volatility surface by differentiating twice with respect to K (see Problem 2.4.2 (c)). The corresponding formula is:

3.2

where , F is the forward price of S for maturity T, and N′(·) is the standard normal distribution density. It is worth emphasizing that Equation (3.2) relies on partial derivatives of σ^* with respect to the dollar strike K, and that one must be careful when using an implied volatility model such as the SVI model, which is based on forward-moneyness k_F = K/F. In the latter case, we must substitute and .

The first factor, , is the Black-Scholes lognormal distribution at point K using implied volatility. Without the second factor between brackets, the integral does not sum to 1, unless the smile is flat.

The implied distribution reveals what options markets “think” in terms of the future evolution of the underlying asset price. It is a useful theoretical concept, but in practice it can be difficult to exploit this information for trading.

3.2 European Payoff Pricing and Replication

Consider an option with arbitrary European payoff f(S_T) at maturity T, and let be the implied distribution density. The corresponding option value is then and by direct substitution of Equation (3.1) we obtain the Breeden-Litzenberger (1978) formula:

The knowledge of the implied distribution thus allows us to value any European option consistently with the vanilla option market. It turns out that this value is in fact an arbitrage price, at least in theory when we can trade all vanilla options along a continuum of strikes K > 0.

To see this, assume that f is bounded and smooth (twice continuously differentiable) to perform an integration by parts and write:

because infinite-strike calls are worthless. Integrating by parts again yields:

Furthermore zero-strike calls are always worth ; additionally, by put-call parity , thus by differentiation and since zero-strike puts are worthless we have . Substituting into the previous equation we get:

This expression suggests that the option may be hedged with a portfolio:

• Long zero-coupon bonds in quantity f(0)e^–rT
• Long zero-strike calls in quantity f′(0)
• Long all vanilla calls struck at K > 0 in quantities

The definite proof of this result is provided by establishing that the option payoff perfectly matches the portfolio payoff:

which simply turns out to be the first-order Taylor expansion of f over [0, S_T] with remainder in integral form after noticing that the bounds of the integral are actually 0 and S_T.

An alternative approach mixing puts and calls is developed in Problem 3.4.2.

This is a strong fundamental result that directs how to price and hedge European payoffs, with the following limitations:

• It only applies to European payoffs. Other option payoffs that depend on the history of the underlying asset price (such as Asian or barrier options) or have an early exercise feature (such as American options) require more sophisticated valuation methods and cannot be perfectly replicated with a static portfolio of vanilla options.
• In practice, only a finite number of strikes are available. While it is possible to overhedge convex payoffs with a finite portfolio of vanillas, exact replication cannot be achieved.

3.3 Pricing Methods for European Payoffs

As stated earlier, the implied distribution makes it possible to price any European payoff f(S_T). Several numerical integration methods, such as the trapezoidal method, are then available to compute . One issue is that these methods become inefficient in large dimensions (i.e., multi-asset payoffs), which is the topic of Chapters 6 to 9.

Another approach is Monte Carlo simulation, which is easy to generalize to multiple dimensions. Using a cutoff A 0 we may approximate f₀ by:

where u₁,…, u_n are n independent simulations from a uniform distribution over [0, 1].

We can improve this approach by means of the importance sampling technique, which exploits the fact that the implied distribution is bell-shaped and somewhat similar to the Black-Scholes lognormal distribution with sensible volatility parameter σ (e.g., at-the-money implied volatility for maturity T). To do so, rewrite:

where X is lognormally distributed with density (K) and x₁,…, x_n are simulated values of X. The ratio measures how close the implied and lognormal distributions are.

Simulating X is straightforward through the identity where is a standard normal, for which there are many efficient pseudo-random generators. For completeness we provide the Matlab code to price the “capped quadratic” payoff according to the implied distribution shown in Figure 3.3.

function price = ImpDistMC(n)
  %Note: this algorithm assumes zero rates and dividends
  epsilon = 0.0001;
  T = 2.41; %maturity
  function val = Payoff(S)
    val = min(1, S∧2);
   end
  function vol = SVI(k)
    a = 0.02;
    b = 0.05;
    rho = -1;
    m = 0.3;
    s = 0.1;
    k = log(k);
    vol = sqrt( a + b*( rho*(k-m) + sqrt( (k-m)∧2 + s∧2 ) ) );
  end
  function density = ImpDist(k)
    vol = SVI(k);
    volPrime = (SVI(k+epsilon)-vol)/epsilon;                  %finite difference
    volDblPrime = (SVI(k+epsilon)-2*vol+SVI(k-epsilon))/epsilon∧2;                  %finite diff.
    d1 = (-log(k)+0.5*vol∧2*T)/(vol*sqrt(T));
    d2 = d1 - vol*sqrt(T);
    density = normpdf(d2)/(k*vol*sqrt(T)) * (1 + …             2*d1*k*volPrime*sqrt(T) + …
               d1*d2*(k*volPrime*sqrt(T))∧2 + …             (k*volDblPrime*sqrt(T))*(k*vol*sqrt(T)) );
  end
    %%%%%%%%%%%%%%%%%%%%%%%%%%
    %%% MAIN FUNCTION BODY %%%
    %%%%%%%%%%%%%%%%%%%%%%%%%%
  atm_vol = SVI(1);
  price = 0;
  for i=1:n
    x = exp(atm_vol*randn*sqrt(T) - 0.5*atm_vol∧2*T);
                %lognormal simulation
    price = (i-1)/i*price + Payoff(x)*ImpDist(x) …
          / lognpdf(x,-0.5*atm_vol∧2*T,atm_vol*sqrt(T)) / i;
  end
end

3.4 Greeks

Because the implied distribution is derived from vanilla option prices in the first place, it produces the same delta as the input volatility smile. Consequently, if the implied volatility smile incorporates a sticky-delta rule, the delta obtained by repricing calls using the implied distribution will be higher than the Black-Scholes delta.

The implied distribution also makes it possible to calculate smile-consistent Greeks for any European payoff. In practice this is often done using finite differences, but one must be careful with the numerical precision of the method used for pricing. In particular the Monte Carlo method has error of order where n is the number of simulations, and will generate a different price on each call unless the pseudo-random generator is systematically initialized at the same seed.

It is worth emphasizing that the implied distribution depends on the variables or parameters used for the Greeks. For example, if the payoff f(S_T) is independent from the initial spot price S₀ the delta may be written which involves the derivative of the implied distribution with respect to the spot price.

References

1. Breeden, Douglas T., and Robert H. Litzenberger. 1978. “Prices of State-Contingent Claims Implicit in Option Prices.” Journal of Business 51 (4): 621–651.
2. Demeterfi, Kresimir, Emanuel Derman, Michael Kamal, and Joseph Zhou. 1999. “More than You Ever Wanted to Know about Volatility Swaps.” Goldman Sachs Quantitative Strategies Research Notes, March.

Problems

3.2 Perfect Hedging with Puts and Calls

Show that for any smooth (twice continuously differentiable) payoff f(S_T):

where F is the forward price and α and β are constants to be identified. What is the corresponding option price f₀?

3.4 Conditional Pricing

On August 24, 2012, Kroger Co.'s stock traded at $21.795, and options maturing in 511 days had the following implied volatilities:

Strike (%Spot)	30%	50%	70%	90%	100%	110%	130%	150%	200%
Impl. Vol. (%)	49.58	36.59	30.17	25.43	24.23	22.97	21.40	20.86	22.89

The forward price was $21.366, and the continuous interest rate was 0.81%.

1. Calibrate the SVI model parameters to this data. Answer: a = 0, b = 0.1272, ρ = – 0.7249, m = –0.1569, s = 0.5388 using initial condition a = 0.04, b = 0.4, ρ = –0.4, m = 0.05, s = 0.1, and bounds a > 0, 0 < b < 2.024, –1 < ρ < 1, s > 0.
2. Produce the graph of the corresponding implied distribution and compute the price of the “capped quadratic” option with payoff using the numerical method of your choice. Answer: approximately $0.797.
3. The following graph (Figure 3.6) shows the history of Kroger Co.'s stock price since 1980. Based on this graph, you reckon that the stock price will remain above $14 within the next three years.
   1. Compute the probability that S_T > 14 using the implied distribution.
   2. Compute the value of the “capped quadratic” option conditional upon {S_T > 14}. Is this an arbitrage price?