Basket Option
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Four user interfaces:
- Data API.
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Equity Basket Option Definition and Valuation
FinPricing provides valuation models for the following basket products:
- Basket Forward
- Basket European Option
- Basket American Option
- Basket Asian Option
- Asian Basket Option with Cap and Floor
- Basket Binary Option
- Basket Quanto Option
- Basket Equity Swap
- Basket Barrier Option
- Basket Coupon Option/Note
- Callable Basket Option
- Basket Resetting Digital Option/Note
- Basket Ratchet Option
- Worst of Basket Ratchet Option
- Worst of Basket Barrier Option
- Two Backet Option
- Upside Fix Basket Option/Note
- Check all the valuation models
All the equity models in FinPricing take volatility skew/smile and dividend into account.
| 1. Equity Basket Option Introduction |
A basket option is a financial contract whose underlying is a weighted sum or average of different assets that have
been grouped together in a basket. The assets in an equity basket option could be equity indices or individual equities.
A basket option offers a combination of two contradictory benefits: focus on an investment style or sector, and
diversification across the spectrum of stocks in the sector.
A basket option can be used to hedge the risk exposure to or speculate the market move
on the underlying stock basket. Because it involves just one transaction, a basket option often costs less than
multiple single options. The most important feature of a basket option is its ability to
efficiently hedge risk on multiple assets at the same time. Rather than hedging each individual asset, the investor
can manage risk for the basket, or portfolio, in one transaction. The benefits of a single
transaction can be great, especially when avoiding the costs associated with hedging each and every component of the
basket or portfolio.
From valuaiton perspective, FX basket option is different from equity basket option. The major difference is the implied volatility representations and volatility models in two distinct markets. One needs different models for each product.
Buying a basket of shares is an obvious way to participate in the anticipated rapid appreciation of a sector of the
stock market, without active management. An investor bullish on a sector but wanting downside protection may favor a
call option on a basket of shares from that sector. A trader who think the market
overestimates a basket’s volatility may sell a butterfly spread on the basket. A relatively risk averse investor may
favor a basket buy or write. A trader who anticipates that the average correlation among different shares is going
to increase might buy a basket option, hedge against a change in volatility by selling
options on the component shares, and delta-hedge the remaining exposure to the
underlying shares.
| 2. Equity Basket Option Payoffs |
In a basket option, the payoff is determined by the weighted average prices of the underlying stocks in a basket.
This is different from the case of the usual European option and
American option, where the payoff of the option contract
depends on the price of the only one underlying instrument at exercise. Trading desks use this type of
option to construct the payoff structures in various Equity Linked Notes.
The payoff for a basket call option is given by
The payoff for a basket put option is given by
| 3. Equity Basket Option Valuation |
The basket option payoff function can be solved either analytically or using Monte Carlo simulation.
In this paper, we focus on the analytical solution. It assumes that the basket price can be approximated by a lognormal
distribution with moments matched to the distribution of the weighted sum of the individual stock prices. The model
includes two- and three-moment matching algorithms.
The model also can be used to price an basket option by including a period of dates in the averaging schedule.
The payoff types covered by the model include calls and puts, as well as digital calls and digital puts.
It is well known that the sum of a series of lognormal random variables is not a lognormal random variable. The weighted summation R is approximated by a shifted lognormal random variable (SLN), given by
We solve for a,b,c,d by matching central moments between R & RLN. The central moments of SLN are
After some math, we get the closed form formula for the present value of a call basket option
Practical Notes
This model assumes that the basket price can be approximated by a lognormal distribution with moments matched to the distribution of the weighted sum of the individual stock prices.
The asset value can be accurately expressed using a volatility skew model. This represents best market practice
Interest rates are assumed to be deterministic
The model can be easily extended to price an basket option by including a period of dates in the averaging schedule, i.e.,
| 4. Related Topics |