Suppose the current volatility of an asset is 15% and you believe the volatility is going to go up to 18% in three months time and you want to take advantage of that increase in volatility what would you do? You can buy three month straddle option (could be at the money, in the money or out of the money) maturing in three months on a straddle that starts in three months and matures in six months.

There are a few ways in which an investor can take exposure to the volatility of a particular asset, say a stock, an FX pair or a stock index. These entail buying and selling of volatility and variance swaps, log forwards, a combination of vanilla options, etc. However, very recently in a seminal paper published in the Journal of Banking and Finance Menachem Brenner, Ernest Ou and Jin Zhang have proposed a new volatility product: the *straddle options *.

In a straddle option the underlying asset is an at the money forward (ATMF) straddle, i.e. an ATMF call and an ATMF put. The ATMF straddle is a traded asset priced well by the market and its relative value, i.e. call plus put divided by the stock (asset) is mainly affected by volatility.

How do we price a straddle option (STO)? There are two ways to do so: one is the closed form Black-Scholes type valuation which assumes a known volatility and the second is the stochastic volatility model using mean reversion. The Black-Scholes closed form solution of an STO is easy and straightforward and even come out with a fairly accurate price, if the constant vol assumption holds but is certainly not recommended. Since the idea behind a STO is to hedge volatility, which is changing and uncertain, a stochastic mean reverting model of volatility (such as the one proposed by Stein and Stein) is better suited for pricing an STO. But we will demonstrate the simpler closed form model here.

__Closed Form Valuation of Straddle Options__

At time t = 0 you buy a straddle option maturing at time on an ATMF straddle ST which matures at time ( ) with a strike price . There are two volatilities involved in the valuation of an STO. One is that exists between 0 and and the other is that exists between and . Since a straddle is a ATMF call and a put, using Brenner and Subrahmanyam's famous approximation we get the value of a straddle at t = that matures at t = as:

Where, is the value of the spot price of the asset at time t = . Thus if we are valuing a straddle option, STO, at any time then, with the following identities:

and

We get the Black-Scholes closed form value of an STO as:

STO Call =

__Numerical Example __

Let us take an example of valuing a straddle option (STO). Say, a stock is trading at $100 and the current volatility of a stock is 15%. In 6 months time we expect the volatility to be 18% and want to take a long exposure to this volatility by buying (going long) a STO call today that matures in 6 months time on an ATMF straddle that starts in 6 months time and matures in a year's time. So the maturity of the STO call is 6 months and the maturity of the underlying straddle (ST) is also 6 months but starting in 6 months from today. Thus our spot is $100, and . The 6 month forward rate on the stock is a good proxy for the value of the stock in 6 months time. Assume the risk free interest rate is 4%.

6-month forward rate of the stock =

Price of the underlying straddle (ST) =

The price of the underlying straddle helps us to decide what kind of a strike price we want so that we can be in the money or at the money or out of the money as far as the STO is concerned. Please note that the underlying straddle is always ATMF (at the money forward) but the STO can be at the money or away from the money on either side.

If we choose of strike price of $10, i.e. which is slightly in the money then the price of this straddle option is given by:

STO Call = = $0.62

Thus the value of a 6 month in the money STO call on an underlying ATMF straddle of $10.36 on the stock is $0.62.

STOs present many useful and interesting features that make them ideal candidates for trading volatility. We shall discuss some of them in a later article.

**Reference: ** :
*The above article is based on the excellent and seminal paper of Menachem Brenner, Ernest Ou and Jin Zhang as it appeared in Journal of Banking and Finance 20 (2006) 811- 821 *.

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