There are two types of stock options:
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Call Option: Call option gives the buyer a right to purchase the given stock at the strike price. Thus Call option is generally bought when the buyer is bullish about the underlying security.
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Put Option: Similarly buying a put option gives you the right to sell the underlying stock at the strike price. Put option is bought when the buyer has bearish views about the underlying security.
Each option comes with an "Exercise Date". European options may only be exercised on the exercise date, whereas American options may be exercised at any time up till the exercise date.
Due to the put-call parity, it is possible to create artificial call or put options if the other is not available. Put options may also be used as a hedging instrument, against possible decline in value of the underlying stock.
While stocks with high volatility (modified duration) are high risk, options whose underlying stock have high volatility are actually better. They provide a possibility of a higher payout if the stock goes up in proportion to its volatility and the same amount of loss.
Options on Forwards
In this case, the underlying asset on which the option is written is a forward contract. A market exists in which forward contracts are traded. We do not impose the martingale property on the s.d.e. for a forward price. Rather, given the current forward price $F(t,T)$,
In order to simplify the analysis we assume that μ and σ are positive constants. The mark-to-market of a forward contract with arbitrary strike K is,
V(t,T) = B(t,T)[F(t,T) − K]
where B(t,T) = exp[ − r(T − t)] and r is the risk-free rate. An option on a forward gives the buyer of the option the right to purchase a forward contract with strike K and expiry T * at some future date T < T * . Lets price this option blindly the actuarial approach. This approach requires that the price of the option is given by taking the expectation of its payoff under the ‘true’ distribution of the forward price,
![C(t,T)=B(t,T) mathbf{E}_t left{B(T,T^*) [F(T,T^*)-K]^+ right}](http://upload.wikimedia.org/math/a/8/8/a88c4c28f08623c0c1ccb5e37f181bc7.png)
where,
![F(T,T^*)=F(t,T^*) expleft[ left( mu - frac{1}{2} sigma^2 right) (T-t) + sigma sqrt{T-t} U right]](http://upload.wikimedia.org/math/a/6/7/a67fa4f64429339f92cf38cd1ee96b05.png)
and U is a standard normal random variable. The expectation has the following simple solution,
C(t,T * ) = B(t,T * )[Fexp[μ(T − t)]N(d1) − KN(d2)]
where,
N(x) = Prob(U > x) and 
. In the same way the price of a put is given by,
P(t,T * ) = B(t,T * )[ − Fexp[μ(T − t)]N( − d1) + KN( − d2)]
In the absence of arbitareg put-call parity requires the following equation to hold,
C(t,T * ) − P(t,T * ) = V(t,T * )
This is equivalent to,
Fexp[μ(T − t)] − K = F − K
This is only possible if μ = 0. This transparent approach, first proposed by Emanuel Derman and Nassim Taleb [2], generates the arbitrage-free option price without the need for unrealistic assumptions about the viability of dynamic hedging. The only assumption we made was regarding the ‘true’ probability distribution function of the forward price. If we choose a more general approach, where F(T,T * ) has an arbitrary probability distribution function, then the value of a call option is given by,
![mathbf{E}_t left{[F(T,T^*)-K]^+ right}=
mathbf{E}_t left{F(T,T^*)right} tilde{P}(F(T,T^*)>K)-KP(F(T,T^*)>K)](http://upload.wikimedia.org/math/0/6/4/0646046e3c9e1285cad97ba7a3f64532.png)
where 
is the ‘true’ probability distribution and 
is a probability distribution with the property,
In the same way, the value of a put option is given by,
![mathbf{E}_t left{[K-F(T,T^*)]^+ right}=
-mathbf{E}_t left{F(T,T^*)right} tilde{P}(F(T,T^*)<K)+KP(F(T,T^*)<K)](http://upload.wikimedia.org/math/3/3/b/33bb8b50370e21b94eb75bb68f7cff25.png)
By put-call parity,
![B(t,T^*) [mathbf{E}_t left{F(T,T^*)right}-K]=B(t,T^*) [F(t,T)-K]](http://upload.wikimedia.org/math/0/2/6/026cab9ba2da1bf50374da6cea23cf08.png)
Therefore,
i.e. under an arbitrary ‘true’ distribution the option on the forward is priced by using the martingale property for the forward price.
Options on the Product of Two Asset Prices
The growth of the financial sector has resulted in products which are covered under the broad term "exotic derivatives". These derivatives are often written on indices which are derived from traded prices but which themselves are not traded. Depending on investor preferences an index can be a function of more than one asset prices and can be determined from the value of these asset prices from a single or a series of observations. Exotic derivatives can either be priced using analytic methods or numerical techniques. The framework used to price all exotic derivatives is based on the Black-Scholes option pricing theory, in which dynamic hedging is used to obtain an arbitrage-free equation for the option price. Although we can always obtain a p.d.e. for all exotic derivatives, an analytic solution cannot always be obtained. However, there exists a large range of exotics where an analytic solution is possible. An option on the product of two asset prices has an analytic solution.
Given two traded assets, an index can be created where the value of the index at some time t is defined as,
where t = 0 is the time at which the index is created and S(0) = 1. An option can be written on this index with payoff at expiry T,
C(T) = max[S(T) − 1,0]
Since the option is only a function of P1, P2 and t, given the s.d.e.s for the prices of the two assets,
(where 
) Itô’s lemma can be applied the price of the option to give,
![dC=left[ frac{partial C}{partial t}
+ m_1 P_1(t)frac{partial C}{partial P_1}
+ m_2 P_2(t)frac{partial C}{partial P_2}
+ frac{1}{2} sigma_1^2 P_1(t)^2 frac{partial^2 C}{partial P_1^2}
+ frac{1}{2} sigma_2^2 P_2(t)^2 frac{partial^2 C}{partial P_2^2}
+ sigma_1 sigma_2 rho P_1(t) P_2(t) frac{partial^2 C}{partial
P_1partial P_2}right] dt
+ sigma_1 frac{partial C}{partial P_1} P_1(t) dW_t^1+ sigma_2
frac{partial C}{partial P_2}P_2(t) dW_t^2](http://upload.wikimedia.org/math/6/7/1/6715f1228c61bfb2cec266d1f668d67a.png)
A portfolio consisting of $1 of the option, 
of asset 1 and 
of asset 2 must therefore have an s.d.e. given by,
![dleft( C - frac{partial C}{partial P_1} P_1(t) - frac{partial
C}{partial P_2} P_2(t)right)=left[ frac{partial C}{partial t}
+ frac{1}{2} sigma_1^2 P_1(t)^2 frac{partial^2 C}{partial P_1^2}
+ frac{1}{2} sigma_2^2 P_2(t)^2 frac{partial^2 C}{partial P_2^2}
+ sigma_1 sigma_2 rho P_1(t) P_2(t) frac{partial^2 C}{partial
P_1partial P_2}right] dt](http://upload.wikimedia.org/math/2/a/5/2a5edb0c54dec7c8dfc65d60011ab920.png)
Since this portfolio has no sources of risk, in the absence of arbitrage it must have an instantaneous return equal to the risk-free rate r. Therefore the last equation gives rise to the following p.d.e.:
From the payoff function of this option we can deduce that the pricing equation can be transformed into a two-dimensional one with variables t and P = P1P2. Note that,
Therefore the p.d.e. can be simplified to,
where,
m = 2r + σ1σ2ρ
and,
and boundary condition C(T) = max[P(T) / P(0) − 1]. This p.d.e. is the Black-Scholes p.d.e. for a call option and can be solved to give,
C(0) = exp[(m − r)T]N(h1) − exp[ − rT]N(h2)
where,
The same result can be obtained by starting with the risk-neutral processes for the two assets,
Using Itô’s lemma, the process for the product of the two prices is,
and the pricing equation derived using the p.d.e. follows.
Advanced Structures
Theoretically, the price of an option (or option premium)consists of two elements: the Intrinsic value and time value of an option. Therefore, Premium=Intrinsic value+Time value.
The price of an option consists of five things: Strike Price, Price of the underlying asset, Time to maturity, Risk-Free interest rate and Volatility. Since the first four can be read from the markets, the only unknown factor in the price of the option is volatility.