Volatility smile

Volatility smile

In finance, the volatility smile is the pattern in which in-the-money and out-of-the-money options are observed to have higher implied volatilities than at-the-money options. In the following, I will simulate a series of implied volatility using excel macros.

Implied volatility

Implied volatility of an option contract is the value of volatility of the underlying which is calculated from the pricing model after taking the observed market price as the input. Here the famous Black-Scholes formula will be used, and for simplicity vanilla Eur Call option is considered.

$$C(S,t) = N(d_1)S - N(d_2)Ke^{-r(T-t)}$$ with
$$d_1 = \frac{ln(\frac{S}{K})+(r+\frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$$ $$d_2 = \frac{ln(\frac{S}{K})+(r-\frac{\sigma^2}{2})(T-t)}{\sigma\sqrt{T-t}}$$ then the problem becomes how to work out sigma from the above equation by taking the option's market price as input and Newton's method is used as follows:
$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$$ which indicates that we need to calculate:
$$vega = \frac{\partial C}{\partial \sigma} = S\sqrt{T- t}N'(d1)$$ the iteration formula will be:
$$\sigma_{n+1} = \sigma_n - \frac{C(\sigma_n)}{vega(\sigma_n)}$$
a simple implementation in Excel macros:

Function impvol(MarketPrice As Double, StockPrice As Double, Strike As Double, 

                Interest As Double, Expiry As Double) As Double

Dim error As Double
Dim volatility As Double
Dim d1 As Double, d2 As Double
Dim vega As Double
Dim PI As Double
Dim dv As Double
Dim priceerror As Double

PI = 3.1415926
error = 0.001
volatility = 0.6
dv = 1

Do While Abs(dv) > error
d1 = Log(StockPrice / Strike) + (Interest + 0.5 * volatility ^ 2) * Expiry
d1 = d1 / (volatility * Sqr(Expiry))
d2 = d1 - volatility * Sqr(Expiry)
vega = StockPrice * Sqr(Expiry / 2 / PI) * Exp(-0.5 * d1 * d1)
priceerror = StockPrice * cdf(d1) - Strike * Exp(-Interest * Expiry) * cdf(d2) 

             - MarketPrice
dv = priceerror / vega
volatility = volatility - dv
Loop

impvol = volatility

End Function


Function cdf(x As Double) As Double

Dim a1, a2, a3, a4, a5 As Double
Dim d As Double
Dim dd As Double
Dim result As Double

a1 = 0.31938153
a2 = -0.356563782
a3 = 1.781477937
a4 = -1.821255978
a5 = 1.330274429

If x >= 0 Then
    d = 1 / (1 + 0.2316419 * x)
    dd = a1 * d + a2 * d ^ 2 + a3 * d ^ 3 + a4 * d ^ 4 + a5 * d ^ 5
    result = 1 - 1 / Sqr(2 * 3.1415926) * Exp(-0.5 * x ^ 2) * dd
    cdf = result
Else
    d = 1 / (1 + 0.2316419 * (-x))
    dd = a1 * d + a2 * d ^ 2 + a3 * d ^ 3 + a4 * d ^ 4 + a5 * d ^ 5
    result = 1 - 1 / Sqr(2 * 3.1415926) * Exp(-0.5 * x ^ 2) * dd
    cdf = 1 - result
End If

End Function

outcome: