Intro

Asset Pricing Models are sophisticated theoretical frameworks employed to ascertain the value of an asset predicated on its anticipated future cash flows. This article shall introduce models pertinent to the valuation of options and examine the distinct classifications of this particular financial instrument. An option accords the bearer the privilege, though not the compulsion, to acquire or divest a designated quantity of an underlying asset, the value of which is subject to stochastic fluctuations, at a stipulated price known as the strike or exercise price, either on or prior to the option's maturation date. The financial remuneration associated with what is termed a 'real option' is contingent upon the manifestation of a predefined event within a delimited temporal duration.

In this article, a notable departure from the standard principle that an asset's value is the present value of its expected cash flows shall be explored. Specifically, the focus will be on assets that exhibit two distinct attributes: first, their value is derived from the valuation of other underlying assets, and second, their cash flows are dependent on the fulfillment of certain specified conditions. The cash flow properties of options will be detailed, the factors that influence their valuation explained, and the most effective models for estimating their worth will be briefly scrutinized.

There are two types of options:

- Call options grant the purchaser the right to buy the underlying asset at a predetermined price before the option's expiration date. Should the asset's value at expiration fall below the strike price, the option is left unexercised and becomes worthless. Conversely, if the asset's value exceeds the strike price, the holder exercises the option, acquiring the asset at the strike price. The gross profit from the option is the difference between the asset's market value and the strike price. The net profit, however, is calculated by subtracting the option's purchase cost from the gross profit. Hereafter, a payoff diagram will be provided to display the financial returns of the option at the time of its expiration.

- Put options bestow upon the buyer the right to sell the underlying asset at a set price anytime up until the option's expiration date. Should the market price of the underlying asset be higher than the strike price at expiration, the put option will go unexercised and lose its value. Conversely, if the market price is below the strike price, the put option holder will exercise the option and sell the asset at the strike price, with the gross profit being the difference between the strike price and the asset's market value. When the initial cost paid for the put option is subtracted from this amount, the result is the net profit derived from the option. The following figure summarizes the payoff structure of a put call.

The price of an option is influenced by various factors related to the underlying asset and the broader financial markets:

1. The current price of the underlying asset is foundational, as the option's value is derived from it. Any price movement of the underlying asset will directly affect the option's value.

2. The volatility of the underlying asset's price is also critical. With the option providing a right to transact at a fixed price, increased volatility usually means a higher value for the option.

3. Dividends on the underlying asset can affect its expected value. During the option's life, if the underlying asset issues dividends, it can decrease the value of a call option and increase the value of a put option.

4. The option's strike price is another determinant, where higher strike prices can lower the value of call options and increase the value of put options.

5. The time until the option expires also affects its value; longer durations until expiration typically increase the value of both calls and puts.

6. The risk-free interest rate over the option's duration can impact its pricing, as it represents the cost of capital in financial modeling.

Further concept pivotal to comprehending options are the following the terms, "at the money" (ATM), "out of the money" (OTM), and "in the money" (ITM) are fundamental concepts for determining the intrinsic value and potential exercise of an option. ATM options have a strike price that is equal to the current market price of the underlying asset. They possess no intrinsic value but hold potential based on volatility and time until expiration. OTM options, on the other hand, are characterized by a strike price that is less favorable than the prevailing market price for a call, or more favorable for a put, rendering them worthless if exercised immediately; their value is purely extrinsic, relying on potential future changes in the asset's price. Conversely, ITM options have intrinsic value: a strike price that is already favorable compared to the current market price, offering immediate exercise value for calls when the strike price is below the market price, or for puts when it is above. Traders leverage these option statuses to align with their market predictions and risk preferences, utilizing the intrinsic and time value of options to manage and strategize their investments.

Binomial Tree Pricing Method

In 1972, the seminal work of Black and Scholes, "The Pricing of Options and Corporate Liabilities," marked a milestone in financial theory, presenting a robust framework for pricing dividendprotected European Options through the concept of a "replicating portfolio." This paved the way for the evolution of Option Pricing Models. Advancing this domain, Cox, Ross, and Rubinstein [1979] introduced a more accessible binomial model for option valuation, based on a similar philosophical underpinning. The binomial model simplifies the process into a discrete-time framework where the price of the underlying asset is modeled to have only two potential outcomes in each time interval, further crystallizing the approach to option pricing.

In the prior figure S is the current stock price; the price moves up to Su with probability p and down to Sd with probability 1 − p in any time period.

The purpose of constructing a replicating portfolio is to combine risk-free borrowing or lending with the investment in the underlying asset to emulate the cash flows of the option under valuation. This process is governed by the principles of arbitrage, asserting that the option's value must be identical to the value of the replicating portfolio. In the generalized scenario provided, where the stock price can ascend to 𝑆𝑢 or descend to 𝑆𝑑 within any time frame, the replicating portfolio for a call option with a strike price 𝐾 would necessitate borrowing an amount $𝐵 and purchasing a certain quantity Δ of the underlying stock, where:

Δ = Number of units of the underlyng asset bought = 𝐶𝑢 − 𝐶𝑑 /𝑆𝑢 − 𝑆𝑑

Where, - 𝐶𝑢 = value of the call if the stock price is 𝑆𝑢. - 𝐶𝑑 = value of the call if the stock price is 𝑆𝑑.

When employing a multi-binomial valuation model, the process commences from the final time period and progresses in a reverse chronology to the present. At every interval, a replicating portfolio is constructed and its value is assessed. The culmination of this backward induction is the valuation of the option, which is articulated through the replicating portfolio. This portfolio consists of Δ units of the underlying stock (referred to as the option's delta) and an amount of risk-free borrowing or lending. The value of a Call Option is then calculated by multiplying the Current Value of the Underlying Asset by Δ, and then subtracting the borrowing amount required to replicate the option's payoff.

Following is an illustrative example of how to value an American Option, a type of contract that grants the holder the flexibility to exercise the option at any point during its term.

Consider a division of the interval from 0 to 𝑡, {𝑡0 ≔ 0, … ,𝑡𝑖 , = 𝑡 ∙ 𝑖 𝑛 , … ,𝑡𝑛 = 𝑡} where there is no opportunity for arbitrage. For European Options, the price V of the option at any given time 𝑡𝑖 is the present value of the expected payoff at maturity, discounted back to time 𝑡𝑖 using a riskneutral probability measure Q, denoted as 𝑉𝑡𝑖 = 𝑒 −𝑟∙(𝑡𝑛− 𝑡𝑖 ) ∙ 𝐸𝑡𝑖 𝑄 (𝑉𝑡𝑛 . On the other hand, American-style options require a backward recursive pricing approach. Here, V represents the option's price, S is the price of the underlying asset, and 𝜙(𝑆) is the option's payoff.

At the final time 𝑡𝑛, the price 𝑉𝑡𝑛 of an Option is given by 𝜙(𝑆𝑡𝑛 ).

At the preceding time step 𝑡𝑛−1, the holder of the option must decide whether to exercise the option, should the immediate exercise value surpass the value of retaining the option until a later time.

𝑉𝑡𝑛−1 = max( 𝜙(𝑆𝑡𝑛−1 ),𝑒 −𝑟∙(𝑡𝑛− 𝑡𝑛−1) ∙ 𝐸𝑡𝑛−1 ( 𝜙(𝑆𝑡𝑛 )))

Where,

- 𝑉𝑡𝑛−1 = max( 𝜙(𝑆𝑡𝑛−1 )) is the Value of exercising an option;

- (𝑒 −𝑟∙(𝑡𝑛− 𝑡𝑛−1) ∙ 𝐸𝑡𝑛−1 ( 𝜙(𝑆𝑡𝑛 ))) is the value of keeping the option.

𝑡𝑗 𝑡𝑛−1 𝑡𝑛

𝑉𝑛,𝑛 = 𝜙(𝑢 𝑛 ∙ 𝑆0)

𝑉𝑛−1,𝑛−1 = max(𝜙(𝑢 𝑛−1 ∙ 𝑆0 ),𝑒 −𝑟∙ 𝑡 𝑛 ∙ (𝑞 ∙ 𝑉𝑛,𝑛 ∗ (1 − 𝑞) ∙ 𝑉𝑛−1,𝑛))

𝑉𝑛−1,𝑛 = 𝜙(𝑢 𝑛−1 ∙ 𝑑 ∙ 𝑆0)

𝑉𝑖,𝑗 = max( 𝜙(𝑢 𝑖 ∙ 𝑑 𝑗−𝑖 ∙ 𝑆0),𝑒 −𝑟∙ 𝑡 𝑛 ∙ (𝑞 ∙ 𝑉𝑖+1,𝑗+1 ∗ (1 − 𝑞) ∙ 𝑉𝑖,𝑗+1))

𝑉1,𝑛 = 𝜙(𝑢 ∙ 𝑑 𝑛−1 ∙ 𝑆0)

𝑉0,𝑛−1 = max(𝜙(𝑑 𝑛−1 ∙ 𝑆0 ),𝑒 −𝑟∙ 𝑡 𝑛 ∙ (𝑞 ∙ 𝑉1,𝑛 ∗ (1 − 𝑞) ∙ 𝑉0,𝑛))

𝑉0,𝑛 = 𝜙(𝑑 𝑛 ∙ 𝑆0)

Boundary conditions: 𝐹𝑜𝑟 𝑖 ∈ [0, 𝑛]:

𝑆𝑖,𝑛 = 𝑢 𝑖 ∙ 𝑑 𝑛−𝑖 ∙ 𝑆0;

𝑉𝑖,𝑛 = 𝜙(𝑆𝑖,𝑛).

Backward Recursion: 𝐹𝑜𝑟 𝑗 ∈ [𝑛 − 1, 0]:

𝐹𝑜𝑟 𝑖 ∈ [0,𝑗]:

𝑆𝑖,𝑗 = 𝑢 𝑖 ∙ 𝑑 𝑗−𝑖 ∙ 𝑆0;

𝑉𝑖,𝑗 = max( 𝜙(𝑢 𝑖 ∙ 𝑑 𝑗−𝑖 ∙ 𝑆0),𝑒 −𝑟∙ 𝑡 𝑛 ∙ (𝑞 ∙ 𝑉𝑖+1,𝑗+1 ∗ (1 − 𝑞) ∙ 𝑉𝑖,𝑗+1))

Upon reaching its expiration, the value of an American option is equivalent to that of its European counterpart. The holder of an American option possesses greater privileges than that of a European option, as the American option can be exercised at any chosen moment. This added flexibility justifies that, at any given time, the price of an American option is at least as much as, if not greater than, the price of a European option.

Options Pricing Models including the Binomial Tree, which was previously discussed, the renowned Black-Scholes model, and jump process models, are tailored for valuing options that have specific exercise prices and expiration dates tied to tradable underlying assets. However, in the context of investment analysis or valuation, options on real assets rather than on financial assets are frequent. These are termed 'real options' and tend to have more complex structures. This section will explore some of the more intricate variations of these real options.

Capped & Barriers Options:

Capped Options are types of contracts that have a predetermined profit limit or cap price. To illustrate how these options function, let's consider a Call Option. A standard call option allows for unlimited profit potential for the buyer, as the price of the underlying asset could theoretically increase indefinitely, with the gains from the option increasing in tandem. However, with certain call options, there is a ceiling to the profits; the buyer is entitled to earnings only up to a certain price level. For example, imagine a call option with a strike price of 𝐾1. While the profit from a standard call option would rise as the price of the underlying asset climbs above 𝐾1, this particular capped call option would limit the payoff to the difference between 𝐾2 and 𝐾1, should the asset price reach 𝐾2. When the asset's price hits 𝐾2, the option loses any associated time value and becomes ripe for exercise. Capped Options fall under the umbrella of Barrier Options, where the payout and the duration of the option depend on whether the price of the underlying asset hits a specified threshold within a set timeframe. It stands to reason that the worth of a Capped Call will invariably be lower than an equivalent option without a cap on the payoff. To estimate the value of such an option, one can value the option twice, using both the exercise price and the cap price, and then calculate the differential between the two. Barrier Options are diverse; for instance, Knockout Options become void if the asset's price reaches a certain level. A Knockout Call Option, also known as a “Down-and-Out” Option, has a knockout level set below the strike price, while a Knockout Put Option, referred to as an “Up-and-Out” Option, has its knockout level above the exercise price.

The scholarly discourse on Barrier Option pricing was initiated by Merton [1973], with his closedform solution for pricing a down-and-out European call option subject to continuous monitoring. Subsequently, plenty of research has expanded the theoretical framework for these instruments. Notably, Kunitomo and Ikeda [1992], Geman and Yor [1996], and Kolkiewicz [1997] have offered formulas for valuing double barrier options. Broadie and Detemple [1995] focused on the 7 valuation of capped options, while Gao, Huang, and Subrahmanyam [1996] introduced quasianalytic expressions for American options with a single barrier. Despite the significant analytical advances, these models often share a common limitation: they typically presuppose that the underlying asset price adheres to a geometric Brownian motion, a premise that may not be ideal. Boyle and Tian [1997] highlight this by demonstrating substantial pricing discrepancies in barrier and lookback options when the underlying asset follows a Constant Elasticity of Variance (CEV) process rather than a lognormal distribution, emphasizing that the choice of model is particularly critical for path-dependent options. For the most part, the methods considered have been some form of binomial or trinomial tree. In 1997, Zval, Vetzal, Forsyth and Waterloo provide an implicit model to convert the Black-Scholes PDE into a forward equation,

𝜕𝑉/𝜕𝑡 ∗ = 1/ 2 𝜎^ 2𝑆^2 𝜕^2𝑉/ 𝜕𝑆^ 2 + 𝑟𝑆 𝜕𝑉/𝜕𝑆 − 𝑟𝑉

where 𝑡 ∗ = 𝑇 − 𝑡, 𝑉 denotes the value of the derivative security under consideration, 𝑆 is the price of the underlying asset, 𝜎 is its volatility, and 𝑟 is the continuously compounded risk-free interest rate. A point-distributed finite volume scheme is applied.

Therefore, in such setup the discrete version of the equation is the following,

Although a fully implicit scheme is only first-order accurate in time, the paper experiences that the Black-Scholes PDE can be solved accurately using such scheme.

Asian Options

An Asian option (or average value option) is a special type of option contract. For Asian options, the payout is determined by the average underlying asset price over a pre-set period of time. This differs from regular European or American options, where the option contract's payment depends on the price of the underlying instrument at the time of exercise. Therefore, Asian options are one of the basic forms of exotic options. There are two types of Asian options: Average Price Option (fixed strike), where the strike price is predetermined and the averaging price of the underlying asset is used for payoff calculation; and Average Strike Option (floating strike), where the averaging price of the underlying asset over the duration becomes the strike price. There are different types of asian options also based on how the average is calculated: we can have an arithmetic mean, geometric mean, or the average computed at discrete time steps. One of the benefits of Asian options is that they reduce the risk of market manipulation of the underlying asset at expiration. Another advantage of Asian options is the relative cost of Asian options compared to European or American options. Due to the averaging feature, Asian options reduce the volatility inherent in options. Therefore, Asian options are usually cheaper than 8 European or American options. This may be an advantage for companies subject to Financial Accounting Standards Board Revised Statement No. 123, which requires companies to expense employee stock options.

Different types of Asian options will be hereby presented, differing in the method of averaging, arithmetic or geometric, and the frequency of the pricing sampling, continuous or discrete. Therefore, we have:

- Continuous arithmetic average Asian call or put with,

Φ(𝑆) = ( 1 𝑇 ∫ 𝑆(𝑡)𝑑𝑡 − 𝐾) 𝑇 0 or Φ(𝑆) = (𝐾 − 1 𝑇 ∫ 𝑆(𝑡)𝑑𝑡 𝑇 0 )

Where the payoff is determined by calculating the arithmetic average of the asset's prices taken continuously over the option's life. For a call, the payoff is the difference between this average price and the strike price (if positive); for a put, it's the difference between the strike price and the average price (if positive).

- Continuous geometric average Asian call or put with,

Φ(𝑆) = (𝑒 1 𝑇 ∫ log 𝑆(𝑡)𝑑𝑡 𝑇 0 − 𝐾) or Φ(𝑆) = (𝐾 − 𝑒 1 𝑇 ∫ log 𝑆(𝑡)𝑑𝑡 𝑇 0 )

Similar to the arithmetic average option, but uses the geometric average of the continuously sampled prices. The geometric average is the nth root of the product of n prices, which typically results in a lower average than the arithmetic method due to its dampening effect on higher values.

- Discrete arithmetic average Asian call or put with,

Φ(𝑆) = ( 1 𝑚+1 ∑ 𝑆 ( 𝑖𝑇 𝑚 ) 𝑚 𝑖=0 − 𝐾) or Φ(𝑆) = (𝐾 − 1 𝑚+1 ∑ 𝑆 ( 𝑖𝑇 𝑚 ) 𝑚 𝑖=0 )

This option type calculates the average based on discrete, periodically sampled prices of the asset over the option's term. The payoff mechanism is the same as the continuous version but is based on a set number of observations, like daily or weekly closing prices.

- Discrete geometric average Asian call or put with,

Φ(𝑆) = (𝑒 1 𝑚+1 ∑ log 𝑆( 𝑖𝑇 𝑚 ) 𝑚 𝑖=0 − 𝐾) or Φ(𝑆) = (𝐾 − 𝑒 1 𝑚+1 ∑ log 𝑆( 𝑖𝑇 𝑚 ) 𝑚 𝑖=0 )

The payoff is based on the geometric average of the asset's prices, which are sampled at discrete intervals throughout the life of the option. This too will tend to have a lower average than the corresponding arithmetic average option due to the geometric averaging method

Let us represent the initial price of the arithmetic average Asian call and put options as 𝐶 𝐾,𝑎(𝑆0, 𝑇) and 𝑃 𝐾,𝑎(𝑆0, 𝑇) respectively, and the initial price of the geometric average Asian call and put options as 𝐶 𝐾,𝑔(𝑆0, 𝑇) and 𝑃 𝐾,𝑔(𝑆0, 𝑇).

The Monte Carlo method stands as the most accessible approach for valuing Asian Options, a technique pioneered by Boyle [1977]. This model, building upon the foundational Black-Scholes model, does not necessitate a deep understanding of complex mathematics. Monte Carlo simulations utilize computational algorithms based on extensive random sampling to produce numerical estimations. Their core advantage lies in their ability to model the likelihood of various outcomes in processes heavily influenced by random factors, which are challenging to predict with precision. In the realm of options pricing, this is invaluable as it allows for the simulation of numerous potential price trajectories of an underlying asset. By doing so, it generates a spectrum of possible results to accurately assess an option's value. The technique is especially advantageous for appraising financial instruments with intricate features, such as those that exhibit path dependency or are subject to fluctuating market conditions, where traditional analytical models might fall short.

Rainbow Options

A rainbow option is an options contract linked to the performances of two or more underlying assets. They can speculate on the best performer in the group or minimum performances of all the underlying assets at one time. Each underlying may be called a color so the sum of all of these factors makes up a rainbow. These are a type of exotic option that trade-the-counter (OTC). Rainbow options are similar to correlation options and basket options in that those also refer to a number of underlying securities; however, those types of options refer to a single price based on those underlying securities. Rainbow options are instead structured as calls and/or puts on the best or worst performer as it relates to the underlying assets involved. A rainbow option could derive value, for instance, from three relatively low-correlated assets like the Russell 3000 Index of U.S. stocks, the MSCI Pacific Ex-Japan Index, and the Dow-AIG Commodity Futures Index. In a call rainbow written on these three indexes, the option would pay out the difference between the strike price and the level of the index that has risen by the largest amount of the three. Even spread options are considered as rainbow options, bear in mind that this is different from an option spread strategy.

For this analysis we will focus on just one type of rainbow options: the call on max option. This gives the holder the right to purchase the maximum asset at the strike price at expiry and has the following payoff:

max(max(S1, S2,... , Sn)

Since we have more than one asset to model, we need to define our asset price dynamics which is based on independent Brownian motion processes (Ouwehand 2006)

dS/S = (r-q)dt + AdW

Where q is the dividend yield and A is the square root of the covariance matrix so that AA'=∑, A is not uniquely determined, but it would be typical to take A to be the Cholesky decomposition matrix of Σ (that is, A is lower triangular). Under such a condition, A is uniquely determined. Margrabe (1978) began by evaluating the option to exchange one asset for the other at expiry. This is justifiably one of the most famous early option pricing papers. This is conceptually like a call on the asset we are going to receive, but where the strike is itself stochastic, and is in fact the second asset. The payoff at expiry for this European option is: max(S1 − S2, 0), which can be valued as:

𝑉𝑚 = 𝑆1𝑒 −𝑞1𝜏𝑁(𝑑+) − 𝑆2𝑒 −𝑞2𝜏𝑁(𝑑−)

𝑑+ = 𝑙𝑛 𝑓1 𝑓2 ± 1 2 𝜎 2 𝜏 𝜎√𝜏

𝑓𝑖 = 𝑆𝑖𝑒 (𝑟−𝑞)𝜏

𝜎^2 = 𝜎1^2 + 𝜎2^2 − 2𝜌𝜎1𝜎2

Stulz (1982) derives the value of what are now called two asset rainbow options. First the value of the call on the minimum of the two assets is derived, by evaluating the bivariate integral. Then a min-max parity argument is invoked: having a two-asset rainbow maximum call and the corresponding two asset rainbow minimum call is just the same as having two vanilla calls on the two assets. Finally put-call parity is derived, enabling evaluation of the put on the minimum and the put on the maximum. Since max(max(S1, S2, S3)-K, 0) is equal to (max(S1, S2, S3), K)-K and then (max(S1, S2, S3, K)-K we can compute the price as following:

This is quite complex and the same is for delta; we have 3 different deltas for every underlying asset and the dual delta, which is Vmax/K, the delta for the first asset will be

And also the other 3 will have a similar form.

Bermudan options

A Bermuda option is a type of exotic options contract that can only be exercised on predetermined dates—often on one day each month. A spin on American-style options, which permit holders to exercise early at any time, Bermudian options allow investors to buy or sell a security or underlying asset at a preset price on a set of specific dates as well as the option`s expiration date, these dates can be monthly, quarterly, or at other intervals, depending on the terms of the contract. The ability to exercise an option early is a benefit to the holder, and this feature adds value to the contract. The premium (price) on a Bermuda option will often be higher than a European option with the same terms, and lower than an American option due to its limitations on early exercise. The name reflects that the Bermudan option can be seen as something between an American-style option, just as Bermuda itself is located between the US and Europe.

Introducing this concept with the American style options we need to decide whether to exercise it or to keep it for every exercise point. The determination of the exercise value at every time-step is intuitive: for American Put options it’s max (K-S, 0), while for American Call Options it’s max (SK, 0). The computation of the continuation value, instead, represents the key issue for the determination of the option value, and here it comes to the LSMC. In this section we will focus on the Least Square Monte Carlo (LSMC) algorithm, which is a technique for valuing early-exercise options, first introduced by Jacques Carriere in 1996 and further developed by Longstaff and Schwartz in 2001.

It is based on the iteration of a two-step procedure: The first step consists in generating, under adequate price dynamics, a group of sample paths. Successively, as we said before, the interval of possible exercise times, which should be continuous for all the life of the option as by definition of American options, will be instead approximated with a discrete set of time points. Then, all the discounted future payoffs realized (by exercising the option) will be regressed on functions of the state which can variate at each of the time steps. In this way we can obtain a complete estimation of the strategy of optimal early exercise. This implies that the option should be exercised as soon as both it’s in-the-money (having payoff higher than 0) and it has an estimated value of conditional expectation of continuation lower than the value it would have when exercised immediately. The option’s estimated value results to be, then, the estimated expected payoff discounted at time 0.

We will show this method by first generating 10 paths for a stock that has spot price equal to 100 and strike price for the option equal to 100, volatility equal to 20% and drift equal to . We adopt the GBM process for the stock price.

To better understand this method we focus on just two steps in which the buyer of a call option could exercise his Bermudan option. This is what we get. To identify the optimal strategy we must look at time 2 and see where the spot price S is higher than the strike price K (1,2,5,7,9,10)

Now we will get the values for time t=1. Here the option holder needs to decide whether to exercise the option or to keep it until the next time spot. That is if the path is in-the-money, otherwise it becomes not relevant to choose, since exercising would be useless as the option’s payoff is zero at that time. We define S the stock price at time 1 when in the money and Y the discounted value of the payoff at time 2 if the option is not exercised.

Regressing the discounted cash flow Y to a function of the current stock price S and to a constant, it is possible to estimate the conditional expected value of the continuation to time 2, in this way we can see when the price at time 1 is favorable to exercise the option. In our case when 𝑠̃(1) ≥ 13 3,98 the payoff from immediate exercise is higher than the conditional expectation of continuing, in this case exercising the option at t=1 seems the best strategy. And the average of all the discounted values should correspond to the price of the option, which equals 12.49, provided that in this example we considered a Bermudan option that can be exercised once every year. Another simulation we run with 200 samples for a hypothetical Bermudan option that lasts one year and can be exercised every month, given a spot price of 100 and a strike price of 100, at 10% volatility, should be priced around $6. This is just a simplified example, since larger simulations can be run in other programming languages. We also want to try and simplify the mechanism, since computing the conditional expectations of the values given a positive premium is not easy and straightforward we want to try and determine if it is convenient to exercise the option basing on a moving average process, which compare all the values obtained by exercising the option when in the money to the average of all the premiums of in the money option at t=ti. This data is always referred to our simulation of 200 paths, with 10% volatility and stock price and strike price equal to 100. This is what we get by plotting the average premiums at different time steps.

And this is the distribution of premiums when the payoff of exercising the option is positive at t=1.

When the payoff is higher than the mean of the positive expected payoffs, we should exercise the option. To compute the price, we just take the mean of the discounted expected payoff in the simulation in order to have an expected value of 0, using this method we end up with a $5.50 price for this Bermudan option. Running more simulations could surely make this approach more precise.

We must say that many other methods could be used to price Bermudan options, such as the finite differences method or neural network regression

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