# Put Accumulator Vega

Put accumulator vega describes the change in the fair value of a put accumulator due to a change in implied volatility. Put accumulator vega is the first derivative of the put accumulator with respect to a change in implied volatility. It is depicted as:

#### $V = \frac{dP}{dσ}$

where P is the fair value of the put accumulator and σ is the standard deviation of returns of the underlying, or implied volatility in this context.

### Evaluating Put Accumulator Vega

Put Accumulator Vega = R1 x Binary Put Vega(K1) + R2 x Binary Put Vega(K2)

+ R3 x Binary Put Vega(K3) + R4 x Binary Put Vega(K4)

where the right hand terms are the binary put option vega with strikes K1 < K2 < K3 < K4 respectively.

In this instance:

R1 = 40%, R2 = 30%, R3 = 20% and R4 = 10%

so that:

R1 + R2 + R3 + R4 = 1

### Put Accy Vega Over Time

With 2 days and over and asset price rising the vega has risen to be uniformly above zero. This is  because the put accumulator is now out-of-the-money. A rise in volatility provides the trader a higher probability that the asset price will fall and the put accumulator in-the-money.

Conversely, a fall in the asset price, say, down to 98.50 (the lowest strike) the trader now wants volatility to fall. The trader is in a winning position and a further rise in ‘vol’ increases the probability of the put ‘accy’ becoming out-of-the-money again.

### Put Accy Vega and Volatility

Figure 2 provides the put accumulator vega over a range of implied volatilities.

At the asset price of 98.10 the 2% volatility profile is roughly -5.0. All of the profiles are negative at this asset level. It is stating that a rise in ‘vol’ will cause a negative price movement in the put accumulator. If the reader looks at Figure 2 of binary put vega the 5% profile is a great deal lower than the higher volatilities below the strike. When the option is in-the-money a rise in volatility threatens its winning status and the option value decreases.

When 18% vol is combined with a 25-day expiry and a structure of four separate binary puts the average price, the put accumulator, is not going to be a dramatic mover. The flat blue line makes that point.

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