How are options priced?

How are options priced?


American option pricing based on state transitions

1. Introduction

All kinds of financial products circulating in the modern financial market are inevitably affected by the state. As an effective means for hedging risks or speculation, options have received extensive attention. American options are more and more popular because of their freedom of execution time compared to European options. The pricing and valuation of financial products is a very important part of financial product transactions. Therefore, with the continuous development of financial derivatives in this country, the pricing of American options is particularly important. Since the famous Black-Scholes model was established in 1973, option pricing under the framework of partial differential equations has attracted more and more attention in the field of refinancing mathematics [1]

In the standard, risk-neutral Black-Scholes model, the changes in the underlying asset are assumed to follow standard geometric Brownian motion, and one of the important assumptions is that the volatility of the underlying asset is constant. Nonetheless, this assumption is unrealistic in practical applications, as I can see in the article published by Cont R. (2007) [2]. In fact, the volatility of the underlying asset is stochastic, especially when measured with high frequency data [3]. To make up for this deficiency, different stochastic volatility models are proposed in the option pricing problem [4]. Among the many models, the state transition model is becoming more and more popular due to its economic significance and uncomplicated computational process. In effect, the state transition framework allows one to account for certain cyclical or cyclical patterns, such as short-term political or economic uncertainty, by assuming that the market switches from time to time between states. Therefore, by introducing the state transition index of the log-normal distribution of stock prices, the market dynamics can be better fitted. Therefore, I aim to explore the impact of state transition on the execution and pricing of American options.

(2) Research status

Regarding the pricing of options, financial scholars have been exploring it for a long time. However, it was not until the advent of the Black-Scholes-Merton model in 1973 that the pricing of options finally made groundbreaking progress. However, due to the harsh assumptions of this model, and it is only applicable to standard European options, however, with the innovation of financial technology, the types of options and their pricing models are becoming more and more complex, and it is difficult to obtain options pricing only through the BS model. exact analytical solution. Cox, Ross, Rubinstein (1979) proposed a binary tree model for option pricing, which is not only intuitive but also applicable to the pricing of European and American options. In recent years, scholars at home and abroad have made in- depth research on the Black -Scholes model on the one hand, and on the other hand, they have devoted themselves to the study of numerical solutions for American option pricing, seeking to use a more accurate model to describe the Stochastic volatility, interest rates and other parameters. Heston (1993) gave some solutions to the options problem that depend on parameters ; RUBINOV and YANG (2003) showed the convergence of the penalty problem through the penalty method, based on its good behavior and substantial performance; BATTAUZ and PRATELLI (2004) gave numerical calculations for American options with discrete dividends based on the traditional BS model methods ; ACHDOU and PIRONNEAU (2005), using variational inequality theory and the ability to use equivalence relations between complementary problems and variational inequalities to show unique solutions; RAYMOND, WONG, YEUNG (2006) using Monterey Carlo simulation method; ZHAO , DAVISON, CORLESSR (2007) and TANGMAN, GOPAUL, BHURUTH (2008) all obtained the option price through the finite difference method; HUNG used the GARCH (2011) model to predict the volatility of the underlying stock, A fuzzy system volatility prediction method is presented. I mainly give the closed analytical solution of American options based on state transition.

(3) Research methods

I first set two types of state transitions, the executable state is called the upstream state, and the non-executable state is called the downstream state. Then through the application of state transition model, geometric Brownian motion model, Poisson distribution, stop-time strategy, Ito lemma, etc., the ordinary differential equations satisfying the value of American options under two types of states are obtained, and finally the analytical solutions are obtained by solving the equations.

(4) Research content

I describe the lack of liquidity of option execution opportunities through the state transition model. More precisely, I study the execution time problem of feasible execution. The execution of American options is only flexible in one of the two modes of state transition. And the transition of the two models depends on the Poisson jump timing. The text constructs ordinary differential equations through the state transition model and geometric Brownian motion, and then solves them by derivation and gives the following. In the follow-up paper, the general applicability of the analytical formula is discussed, and the analytical formula is applied to other models, and the same analytical formula is still convergent. In the end, by setting the parameter values, I has carried on the mathematical verification to the analytical formula. Therefore, the research on the pricing of American-style options based on state transition will further deepen investors’ choice of execution points for American-style options and help domestic financial markets to price American-style options more accurately.

2. Introduction of related concepts

(1) Related to American options

Before pricing American options, this article first introduces the economics vocabulary or mathematical models involved in the article.

Option refers to a contract that will give the holder of the option the right to buy or sell an asset on or at any time before the contract expires, and the asset is called the underlying asset. Options originated in the European and American markets in the eighteenth century, in which European-style options can only be exercised on the expiration date of the contract, while American -style options can be exercised at any time before the contract expires. Options are further divided into call and put options, where a call option gives the holder the right to buy the underlying asset, and a put option gives the holder the right to sell the underlying asset. This article considers the pricing of American options. For the pricing of options, it is necessary to first assume the price change of the underlying asset.

Long side: The long side is also known as the buyer, which in this article refers to the side that holds the option. The long party is given the right to buy the underlying stock at the contract price on the expiration date of the option (European-style options) or before the option expires (American-style options).

Short side: The short side is also known as the seller, and in this article refers to the side that sells the option. On the expiration date of the option (European-style options) or before the option expires (American-style options), if the long side chooses to exercise the option, the short side conveniently needs to sell the underlying stock at the contract price.

Exotic options are a general term for options with more complex execution conditions or content. Exotic options are also divided into many types. The main types are:

1. Package contracts to appropriately portfolio spot, forward contracts, standard European call and put options of the underlying asset, resulting in new contracts.

2. Bermuda options, also known as non-standardized American options, compared to standardized American options usually traded in regular trading venues, such Bermuda options can only be issued on the expiration date of the option, that is, at certain specific time points in the validity period. Early exercise is permitted, not at any point on the option expiration date, as is the case with standard American options. And even some American options have different strike prices at different strike times.

3. Compound options, the meaning of compounding is overlapping the underlying assets on which the pricing of compound options depends are not the stocks corresponding to ordinary options, and the underlying assets of compound options are other options, that is, options of options.

4. Optional options, as the name implies, optional options will allow the holder of the option to set the option as a call or put option at a certain point in the validity period of the option, which means that the optional option holds. People can not only choose the execution time point of the option, but also gain greater benefits when the underlying stock price is much lower or higher than the option execution price. Because optional options give the option holder greater rights, the price of optional options is generally higher than the price of ordinary options.

5. Retrospective options, similarly retrospectives are retrospectives, retrospective options are a special kind of European-style options. When the option expires, the return of the option is not determined by the current price of the underlying asset and the exercise price of the option, but by the option validity period. The highest or lowest price of the underlying asset and the strike price of the option are obtained. For example, looking at a call option, the payoff of the holder of the option on the expiry date is equal to the highest price of the underlying stock during the duration of the option minus the strike price of the option or equal to zero, then it means that the price of the underlying stock has been lower than the strike price of the option.

6. Asian options, the calculation method of income of similar Asian options is also different from the calculation method of standard option income. The income of Asian options depends on the exercise price of the option and at least a period or even the entire validity period of the option. The average value of the underlying stock.

7. Barrier option, barrier means that conditions are set on the exercise opportunity of the option. Barrier options are further divided into knock-out options and knock-in options, in which “out” means out, that is, when the price of the underlying asset reaches a certain node, the option loses the right to exercise. For example, when the price of the underlying asset is higher than the set barrier price, the option will lose the right to exercise; for example, when the price of the underlying asset reaches After the corresponding node, the option is eligible to be exercised, that is, it has just begun to take effect. For example, a drop knockout option, when the price of the underlying asset, that is, the stock, fluctuates below the set barrier price, drop knockout. Options can begin to be exercised.

(2) Related to the construction of the state transition model

Geometric Brownian motion, also known as exponential Brownian motion, is a random process based on continuous time, and the change in the logarithm of a random variable is consistent with Brownian motion. The random process is a set of random variables, and the change of this set of random variables depends on a certain parameter, usually the parameter is time. I get random variables by quantifying random phenomena. In I, it is assumed that the price of the underlying asset, that is, the stock, obeys the geometric Brownian motion, which is also a commonly used description of the price fluctuation of stocks in finance.

Poisson distribution is commonly used in statistics or probability theory, and it is also a very important discrete distribution. It is used to represent the number of events in X in a certain space or time, for example, within a certain time, farm the number of eggs laid by chickens, and in I, the Poisson distribution is used to describe the number of occurrences of state transition nodes within a certain period.

The stop time, also known as the stop time, means that during the duration of the time, when the expected result occurs, this moment can be regarded as the stop time, of course, the stop time is random, so it is a random time. moment. In I, the study is about American options. According to the above knowledge, American options can be exercised at any time before the option expires. When the holder of the option thinks that the time is ripe, the option will be exercised, otherwise it will wait until the option expires. expiry date. Therefore, it is very important for the pricing of American options to study a “random moment”. It is obvious that American options have additional value compared to European options because American options give the holder the ability to execute in advance. And this “random moment” can be very good by stopping time. Set τ as the execution date of the option, it is obvious that at the current time t≥τ , the domain flow It should be known, and the execution of the option can be known when τ≤t means that the option has been exercised, and when τ≤t means that the option has been exercised, not executed in advance. You only need to see which conditions have been met to know whether the option is executed or not. The domain flow refers to the information flow within a period, and this information flow generally contains all available information, that is, if the domain flow It is known at the current time , all information from zero to time can be known .

The Ito process is regarded as a general Wiener process. In the Ito process, Brownian motion is regarded as a general random disturbance, which makes Brownian motion in its most general sense. Among them, the Wiener process is a well-known and important process, also known as the Brownian motion process, with the following characteristics:
  1. All the messages needed to make predictions about the future are the current values of the process.

2. The probability distribution of the Wiener process changing at any time is equivalent to the probability distribution changing at any other time the increments of the process are independent of each other.

3. The variance of the process value change has a linear relationship with the length of time, and the process is subject to a normal distribution based on a limited time.

Markov process, similar to the above-mentioned process, Markov process is also a kind of random process, its important property is no aftereffect or no memory, also known as Markov property, that is to say, if “at this moment” Generalized to the “now” in the above stop time, in the case of knowing the current information domain, the changes that will occur in the “future” have nothing to do with the changes in the “history”.

3. Construction and solution of state transition model

(1) State transition model assumptions

   First consider an investor who owns an American call option. In the first state, which is the good state, the investor can exercise the option at any time, which is to buy the stock, and in the other, which is the bad state, the investor will Options are not allowed to be exercised. A bad state transforms into a good state at a Poisson jump with an arrival rate λ (>0), while a good state transforms into a bad state at a Poisson jump with an arrival rate η (>0) . When the option is exercised, the cost of buying the stock is I and it will generate a return of X(t), that is, the stock price when the option is exercised is X(t), and its price follows the geometric Brownian motion

where B(t) represents the standard Brownian motion, which is defined in the domain probability space (Ω, F, Fu, Р), and μ, σ(>0), x(>0) are all constants. For the tractability of the model, I assumes that B(t) depends on the Poisson process. r greater than 0 means the discount rate, for convergence I assumes r>μ.

     In I, V(x) and U(x) are used to denote the situation of the payoff in the bad state and the good state, respectively:

(2) Model solutions

     I believes that the investor’s optimal strategy is the threshold strategy, which can be expressed as:

(In good condition)

Then the value equations V(x) and U(x) will satisfy ordinary differential equations.




where λ (U(x)-V(x)) and λ(x-I-V(x)) reflect the transition of V(x) to U(x) and x-I when a bad state turns into a good state , and η(V (x)-U(x)) means that U(x) turns into V(x) when a good state becomes a bad state .

The boundary conditions are

And U(x) and V(x) are continuously differentiable when x =x *.

Among them, the first boundary condition means that in a bad state, when the stock price when the option is exercised tends to zero, the option will not be exercised, and the total return V(x) is equal to 0. The second boundary condition is that the exercise price needs to be paid to exercise the option, so the total return V(x) is less than the stock price when the option is exercised, so when the stock price is close to infinity, V(x)/x also converges. The third boundary condition is that in a good state, when the stock price when the option is exercised tends to zero, the option will not be exercised, and the total return U(x) is equal to 0. The fourth boundary condition means that, according to the threshold strategy set in I, in a good state, once x>x *, the option should be executed, and U(x) is the gain obtained.

By solving the above three ordinary differential equations, I has the following proposition, which defines that for y > 0



For the convenience of calculation, I defines β=β 0 and γ=γ 0.

Fourth, the state transition model solution



where the investment threshold X * and the coefficients A 1, A 2 , A 3 are





(1) Proof

I defines f(x)=U(x)λ/η+V(x) and g(x)=U(x)-V(x). Through (1) and (3), I can get


For ordinary differential equations (1 2) and boundary conditions: the general solution obtained by is expressed as , where are the uncertain coefficients. For ordinary differential equations (13) and boundary conditions: it is derived from , expressed as , where is the uncertainty coefficient. So, for the pass order , I can get the expressions (6), (7). For ordinary differential equation (2) and boundary conditions , the general solution is expressed as (6), where is the uncertainty coefficient.

The following paper will derive the sum by continuously differentiating the sum at the time . By continuously differentiating now in I, we get,

then there is



By continuously differentiating now in I, we get,



Then we get


Having heard substituting (14) and (15) into (18), I gets


Through (19) I can solve (8) . Through (14), (15) and (16), I can solve (9), (10) and (11) , and respectively . This concludes the proof.

(2) Conclusion

In good condition, the holder of an American call option will exercise the option at time t such that x (t) ≥ x *. The upper side in Equation (7) represents the waiting value when the option value is greater than the profit from execution, for example: U(x)>xI . The bottom of equation (7) is equal to the profit after execution. The upper part of Equation (6) represents the waiting value when the option value is greater than the profit from execution, and the lower part of Equation (6) represents the waiting value when the option value is lower than the profit from execution. In the bad state, when x (t) ≥ x * , the option holder wants to exercise but must wait for the good state to come. The first and second terms of the second case in Eq. (6) represent the expected execution benefit when the good state comes. The last term, A3xγλ represents the value of the option to change the strategy when x (t) falls below x * .

It is worth noting that in Solution 1, both the value formula and the execution threshold are derived from closed form. This article can easily extend this closed analytic, turning it into a best execution timing and yield solution, although the article omits the details. Models with closed analytical expressions dominate the literature on ordinary differential equations because closed analytical expressions allow us to ignore parameter values to study relative static and economic effects. Therefore, the closed analytic in Solution 1 has advantages over the non-closed analytic, even though the non-closed interpretive is more general and allows options to be exercised in any state.

Based on the closed solution in Solution 1, I can show the following proposition.

V. Applicability of the state transition model

(1) Proposition

 V(x), U(x) and x* monotonically increase (decrease) in λ (η). When η→0 , U(x) and x * converge in the standard European option pricing model [ 1] (denoted by Vi (x) and xi ) When η → ∞ , V (x), U( x) and x* both converge in the study of optimal stopping problems that depend on Poisson jumping moments (denoted by Vj(x) , Uj(x) and xj ). When λ→0, U(x) and x* converge in the American option model that studies the failure to execute when the Poisson jump moment first reaches the arrival rate η. As λ→∞, U(x) and x * converge with Vi (x) and xi. U(x) and x * converge to the NPV model as λ→0 and η→ ∞ . (represented by Vn (x) and xn )

(2) Proof

The standard European option pricing model in the first case is the Black-Scholes model [1] proposed in 1973 , whose authors believe that if options are priced correctly in the market, then it is impossible to establish options by establishing options A portfolio of long and short positions in its underlying stocks to ensure profits, as options are priced based on risk-neutral conditions and follow the no-arbitrage principle. To facilitate pricing, consider giving the option to buy a single stock, also known as a call option. For call options, when the price of the stock is higher, it means greater option value. When the stock price is much higher than the strike price, it is almost certain that the option will be exercised. At this moment, the value of the current option can be obtained by subtracting the price of the zero-coupon bond whose face value is equal to the strike value of the option and the expiration time is the same as that of the option from the price of the stock. Likewise, if the value of the stock is significantly lower than the strike price of the option, it is almost certain that the option will not be exercised, so the value of the option is approximately zero. When the expiration time of the option is very far away from the current time, the price of the bond that pays an amount equal to the strike price of the option at the expiration time will be very low, and the value of the option will be nearly equal to the value of the stock at that time. Conversely, if the option’s expiration time is very close to the current one, then the option’s value will be approximately equal to the stock’s price minus the option’s strike price, or zero if the stock’s price falls below the option’s strike price. Typically, when assuming no change in the price of the stock, the value of the option will decrease as the expiration date approaches. The above is a simple estimate of the option value. To obtain the pricing formula for options that depend on the underlying stock, the author makes a series of ideal assumptions about the stock market:

1. Know the short-term interest rate and the short-term interest rate is constant over the maturity time.

2. The stock price obeys a random walk based on continuous time and variance equal to the square of the stock price, so the distribution of the stock price in any limited interval obeys the log-normal distribution. The variance of stock returns is constant.

3. During the effective period of the option, no dividend or other income has been distributed.

4. This option contract can only be exercised on the expiry date, which means it is a European option.

5. There are no transaction costs when buying or selling stocks and options in the market.

6. The market allows for a short-term interest rate to borrow a portion of the security’s price to buy or hold it.

7. When the seller does not have the corresponding securities, it can also directly accept the purchase price of the securities by the buyer and agree to pay the option buyer an amount equal to the present value of the securities on the exercise date of the option. This also means there is no penalty for short selling.

Based on these assumptions, the value of the option will depend only on the value and timing of the underlying stock and some known variables. The authors were thus able to build a hedging scheme that included a long position in a stock and a short position in an option, and the value of the option would depend not on the stock but on time and an already constant value. By solving the differential equation, the author finally obtained the most classic and most famous Black – Scholes formula.

According to the definition of the problem, V(x) and U (x) monotonically increase (decrease) in λ(η). The monotonicity of x * follows the monotonicity of U (x) and x * =min{x≥0|U(x)=x-I}.

Through (8), I have


where (20) equals the strike threshold in the standard American option pricing model. Substituting (20) into (10), I obtain . Therefore, since x< xi and (9), I obtain


where (21) equals the price of the American option in the standard pricing model.

Paul Dupuis, Hui Wang (2002), the authors consider a class of optimal stopping-time problems, in which the stopping ability depends on an exogenous Poisson signal process, that is, stopping only at Poisson jumps [ 14]. Even though the time variables in these problems are discrete in nature, the variational inequalities are obtained by considering the continuous time structure. The function values in I exhibit different properties across optimal execution boundaries, based on whether stopping at t=0 is allowed. In fact, the function in the text is only allowed to stop passing the boundary value c 0 at t = 0, and c 2 in the other cases, of course both cases are the same as the value c 1 when the smoothing condition necessary to apply the formula There is a contradiction. The authors also discuss equivalent stochastic control formulations based on these stop-time problems. Finally, the author obtains the asymptotic property of the function value and the optimal execution boundary as the density of the Poisson distribution reaches the maximum value, which is infinite, and these same problems converge asymptotically to the classical continuous-time optimal stopping time problem.

Next, assume η→∞.

Note that I can get from (4) that

Then through (8)-(11), I can get

                      (2 2)

                                                (23 )

                                                    (24 )

                                    (2 5)

Among them, the formula (2 2) is equal to the threshold of the optimal stopping time at the Poisson jump time . Through (2 2) -(2 5 ), I obtains and also satisfy the value equation of V j (x) , U j (x) .

Now, suppose , by (8), I can get


where (26) is equal to the threshold X k in the American option model that cannot be executed when the Poisson jump moment reaches the arrival rate η for the first time. By substituting (2 6) into (9), I obtain . For x < x k, I can get


where (2 7) equals the corresponding option value.

Note that based on (4) and (5), I have

Substitute η→∞ into equations (2 6) and (2 7 ) to get

This result satisfies the threshold and value function in the net present value model. So, the proof is complete.

(3) The economic effect of the proposition

Proposition 2 shows that the model in I can be nested in the previous model. With higher λ or lower η , the holder of the American call option will be in good shape for longer, so the value of deferring will increase. That is, the value of the option and the investment threshold increase with higher λ or lower η . For λ→∞ or η→0 , the solution will converge to the standard American option pricing model.

With a high η , the good state will end quickly, so the value of deferring execution will decrease. As η→∞, with good conditions quickly turning bad, holders of American call options must decide whether to invest because bad conditions are coming. The solution will then converge to the constraint problem of the optimal stopping time at the Poisson jump time.

With a low λ , it is difficult to transform a bad state into a good state. For λ→0 , exercising the option will never get paid when the option holder is facing a bad state. Thus, the solution will converge to the American option model that cannot be executed when the Poisson jump moment first reaches the arrival rate η . In particular, when λ→0 and η→∞ , a good state will immediately become a bad state and will not change back. Therefore, the holder of an American option whose solution converges to the NPV model needs to decide whether to exercise it now or never.

6. Numerical verification

Finally, I do some empirical research on numerical examples. In I, the baseline parameters are set as r =0.07, μ=0.05, σ=0.2, I=1, x=1, λ=1 and η=1 . Therefore, according to the model, I can obtain V (x)=0.5205, U(x)=0.5205, x * =4.522 , and after substituting into the standard American option pricing model, Vi (x)=0.522, x i = 4.766 , and obtain V j ( x)=U j (x)=0.5194,x j =4.044 In the time-limited model based on the received Poisson jump, the American option that cannot be executed when the Poisson jump time reaches the arrival rate η for the first time is introduced In the model, V k (x)=0.0603,x k =1.179 , and finally V n (x)=0,x n =1 in the cash flow model. Note that U(x)-V(x) is close to 2*10 -2 when x=1 , so the verification is complete. Figure 1 shows the value of U (x) and x * as a function of λ and η . The description of V (x) is omitted in I , because V (x) and U(x) are almost indistinguishable and difficult to find in many cases. From the picture in I, it can be found that the value of the option and the threshold for exercising the option will be accompanied by It increases with the increase of λ or increases with the decrease of η , thus verifying the above-mentioned viewpoint.

figure 1 and changes based on and


Limit solutions do not appear in the literature on state transition models because non-closed models prevent limits from being calculated analytically. In particular, there is no known state-transition-based American option pricing model related to Paul Dupuis, Hui Wang (2002) . And I reveal the solution in Paul Dupuis, Hui Wang (2002) for the first time through the state transition model and obtains the limit solution. In fact, the closed analytical solution in I is a generalization of Paul Dupuis, Hui Wang (2002) .

Hao Chen

How are options priced?


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Hao Chen

How are options priced?