Premia 14
Closed Formula Methods

### 1 The Black and Scholes model

Suppose the underlying asset price (St,t 0) evolves according to the Black and Scholes model with continuous yield δ, that is

From now on, we denote by
In the following, we state the closed form solutions for the price of options whose payoff is given by f(ST ), being f a suitable function, i.e. for the quantity

### 2 Standard European Options

We have the general version of the Black-Sholes Formula [4] to price European options on stocks paying a continuos dividend yieald. In this section, we set:

and N as the cumulative normal distribution function:

### 3 European Barrier Options

Barrier options are known as knock-out if the value of the option nullifies if the underlying asset price reaches a fixed barrier before the maturity date and knock-in if it does not. We use the names up-out and up-in call/put options, or down-out and down-in call/put options, to stress if the considered barrier is an upper or lower one. Reiner-Rubinstein [3] have developed formulas for pricing standard barrier options with cash rebate R.

We set here:

and
where
and η,ϕ are suitable numbers belonging to {-1, 1} (see next sections for details).

Let

and take

and take

and take

and take

and take

and take

and take

and take

### 4 Double Barrier European Option

A double barrier option is knock-in or knock-out if the underlying price reaches or not a lower and/or an upper boundary prior to expiration. The exact value for double barrier call/put knock-out options is given by the Ikeda-Kunitomo formula [5], which allows to compute exactly the price when the boundaries suitably depend on the time variable t. More precisely, set

where the constants U,L,δ12in are such that L(s) < U(s), for every s [t,T]. The functions U(s) and L(s) play the role of upper and lower barrier respectively. δ1 and δ2 determine the curvature and the case of δ1 = 0 and δ2 = 0 corresponds to two flat boundaries.

The numerical studies suggest that in most cases it suffices to calculate the leading five terms of the series giving the price of the knock-out and knock-in double barrier call options.

Let τ stand for the first time at which the underlying asset price S reaches at least one barrier, i.e.

We define the following coefficients:

#### 4.1 Knock-Out Call Options

where F = Ueδ1θ and

#### 4.2 Knock-Out Put Options

where E = Leδ2θ and

#### 4.3 Knock-In Call Options

The double barrier knock-in call option is priced via the no-arbitrage relationship between knock-out and knock-in option:
European “IN” + European “OUT” = European Standard

#### 4.4 Knock-In Put Options

The double barrier knock-in call option is priced via the no-arbitrage relationship between knock-out and knock-in option:
European “IN” + European “OUT” = European Standard

### 5 Lookback European Options

Floating strike lookback options can be priced using Goldman-Sosin-Gatto formula [2] while fixed strike lookback options can be priced using Conze-Viswanathen formula[7].

We set, as 0 u v T,

and

#### 5.1 Fixed Lookback Call Options

Both price and delta depend on K and M0,t.

if K > M0,t then

if K M0,t then

#### 5.2 Fixed Lookback Put Options

Both price and delta depend on K and m0,t.

if K < m0,t then

if K m0,t then

### 6 Standard 2D European Options

Consider the pair of processes St = (St1,S t2) solution to

where (Wt1,t 0) and (W t2,t 0) denote two real–valued Brownian motions with istantaneous correlation ρ. The price of option with payoff f is:
Here, closed formulas due to Johnson and Stulz are presented [1],[6].

We set

and M as the cumulative bivariate normal distribution function:

where

where

#### 6.4 Best Of Option

The payoff is BT = (max(ST 1 - K 1,ST 2 - K 2)))+.

### References

[1]   H.JOHNSON. Options on the maximum ot the minimum of several assets. J.Of Finance and Quantitative Analysis, 22:227–283, 1987.

[2]   B.M.GOLDMAN H.B.SOSIN M.A.GATTO. Path dependent options: buy at low, sell at high. J. of Finance, 34:111–127, 1979.

[3]   E.REINER M.RUBINSTEIN. Breaking down the barriers. Risk, 4:28–35, 191.

[4]   F.BLACK M.SCHOLES. The pricing of Options and Corporate Liabilities. Journal of Political Economy, 81:635–654, 1973.

[5]   N.KUNIMOTO N.IKEDA. Pricing options with curved boundaries. Mathematical finance, 2:275–298, 1992.

[6]   R.STULZ. Options on the minimum or the maximum of two risky assets. J. of Finance, 10:161–185, 1992.

[7]   A.CONZE R.VISWANATHAN. Path dependent options: the case of lookback options. J. of Finance, 46:1893–1907, 1992.