A Gold Cliquet-Pricing with the Term Structure of Volatility
May 21, 2006
This structure was recently transacted between a European bank and a commodities hedge fund in Asia . It is a very simple Cliquet option, but the pricing of this option needs to take into account the term structure of Gold's implied volatility.
The hedge fund recently sold a Gold cliquet to the bank. The payoff of the cliquet is given by:
X(1) = Price of Gold at the end of year 1 (t = 1)
X(2) = Price of gold at the end of year 2 (t = 2)
At the onset two sets of volatilities should be assumed. Assuming the volatilities to be constant and flat let us assume that all European option on gold trade at 12% volatility for t = 1 (one year options) and for two year maturity (t = 2) assume that all European options on gold trade at 10% volatility.
The cliquet becomes an ATM option at the end of year 1 or when t = 1. Therefore, at t = 1 one can use the Black-Scholes equation with the volatility equal to the future implied volatility between year 1 and year 2 (i.e. between t = 1 and t = 2). (Actually, for commodities one needs to use Black's modified formula for futures but for illustration we will stick to Black-Scholes).
Therefore, the price of the cliquet at the end of the year 1 is given by (and if the interest rates are low enough then this can be taken to be today's price):
It can be easily shown that the value of the above BS price can be reduced to:
The key issue here is the volatility, which is actually the forward volatility (some traders call it the forward-forward volatility). This is the implied volatility that exists between year one and year two. Therefore, from the spot volatilities the forward volatilities need to be estimated to create a term structure. The formula for estimating the forward volatility between year 1 and year 2 from the spot volatilities of year 1 and year 2 is
So if we assume the following parameters for gold as shown below
Year 1 (t = 1) implied volatility (at t = 1) = 12%
Year 2 (t = 2) implied volatility (at t = 2) = 10%
Then the forward volatility between year 1 and year 2 will be given by equation (2) as 7.48%. And hence the cliquet price will be given by equation (1) as 2.99% .
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