Compute risk neutral standardized moments of an asset's return distribution from volatility smile interpolation. Part of the IMOMBOX.
S = MVOL2STAT(S0,VOL,RF,Q,TAU,N)
Given the volatility smile function handle VOL = @(X) smile(X), the spot asset level S0, the annualized logarithmic risk free rate rf and dividend yield q as well as a time to maturity TAU in years, S will store the first N standardized moments of the risk neutral distribution of the underlying asset's return.
VOL is a function handle expecting an array of inputs X and returning an array of the same dimension as X. S0, RF, Q, TAU are scalars, N is a vector containing the required return powers.
Let us assume a flat volatility of 25%
vol = @(X) 0.25;
and the following parameters
S0 = 100; rf = 0.05; q = 0; tau = 0.25; N = 1:4;
The corresponding standardized moments of the risk neutral distribution is
ans = 0.0047 0.0156 -0.0000 3.0000
I.e., the return expectation, or the first standardized moment, is 0.0047. From finance 101, we know that the expected log return on an asset with constant vol is , i.e.
ans = 0.0047
which corresponds with our solution.
Let us now change the specification of the volatility smile to
vol = @(X) 0.3.*exp(-1./100.*X)+0.2; plot([0:300],vol([0:300])),ylim([0.1 0.5]),title('volatility smile');
The corresponding standardized moments are now
ans = 0.0004 0.0247 -0.3363 3.2203
Thus, our vol smile brought negative skewness and leptokurtosis to the table.