U.U.D.M. Project Report 2009:19
Examensarbete i matematik, 30 hp Handledare och examinator: Johan Tysk September 2009
A Modified Binomial Lattice Monte Carlo
Method with Applications to European
Barrier Options
Hao Wu
Uppsala University
Department of Mathematics International master programme Master thesis
A modified binomial lattice monte carlo method
with applications to European barrier options
Author: Hao Wu
Abstract
Acknowledgment
Contents
Abstract iii
Acknowledgment v
1 Introduction 1
1.1 Basic theory of Options . . . 1
1.1.1 Definition of Options . . . 1
1.1.2 Fundamental of option pricing . . . 2
1.1.3 Arbitrage free principle . . . 4
1.2 Black-Scholes-Merton Framework . . . 4
1.2.1 The price process for underlying asset . . . 4
1.2.2 Brownian motion . . . 5
1.2.3 Black-Scholes formula . . . 7
1.3 Explicit solution for European options . . . 12
1.4 Summary . . . 15
2 Barrier options 17 2.1 Brief introduction . . . 17
2.2 Mathematical model for barrier options . . . 20
3 Binomial methods and Monte Carlo simulation 27
3.1 Binomial method . . . 28
3.1.1 One-period model and multi-period model for Euro-pean style options . . . 28
3.1.2 Binomial models and the Black-Scholes equation . . . . 41
3.2 Monte Carlo methods . . . 44
3.2.1 Principles . . . 44
3.2.2 Monte Carlo methods and derivatives pricing . . . 47
3.3 Summary . . . 50
4 Modified binomial Monte Carlo method 51 4.1 Method description . . . 52
4.2 Applications to simple European barrier option . . . 56
4.3 Results for European barrier options . . . 60
4.4 Summary . . . 63
5 Conclusion 65 A MATLAB Code 67 A.1 Code for calculating best fit N . . . . 67
A.2 Modified binomial based Monte Carlo for European barrier option . . . 68
List of Figures
2.1 Typical European up and out barrier call option . . . 18
3.1 possible stock trajectory . . . 29
3.2 price dynamics . . . 30
3.3 European call dynamics . . . 30
3.4 Calculating backwards through tree lattice . . . 34
3.5 price dynamics . . . 38
3.6 price dynamics for 2- step tree . . . 39
3.7 price dynamics for stock S . . . 40
3.8 calculate backwards . . . 41
4.1 binomial lattice for a 3 step tree . . . 52
4.2 binomial tree with node and edge numbers . . . 54
4.3 multiply 10000 and rounding numbers . . . 55
4.4 Generation of first path . . . 56
4.5 Finished first trial . . . 57
4.6 binomial tree structure example 1 . . . 58
Chapter 1
Introduction
1.1
Basic theory of Options
1.1.1
Definition of Options
First of all, we must answer the question:“What is an option?” A brief defi-nition from the book[1] of John Hull, is:
In finance, an option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or to sell a particular asset (the underlying asset) at a later day at an agreed price. In return for granting the option, the seller collects a payment (the premium) from the buyer.
1. Style of the option: the style or family of an option is a general term denoting the class into which the option falls.
2. Strike price: Strike price is the price at which the instrument(the un-derlying asset) will be bought or sold when the option is exercised.
3. Quantity and Class of underlying assets: for example, 100 shares of ABC company.
4. Expiration date(or maturity): It is the last day the option can be exercised.
Today, there are lots of styles for options, the majority of options are European option and American options. The European options can only be exercised at the expiration date. American options can be exercised at any time up to expiration date.
1.1.2
Fundamental of option pricing
The value of an option, as a derivative security, is determined by the value of the underlying asset. Because the underlying asset contains risk(e.g un-certainty of the future price), the value(or the price) of underlying asset is changes randomly. So does the value of the corresponding option. Now, let’s introduce some notions and formulate the option for better understanding. All these notions will be used throughout this thesis.
• The value of the option at time t is denote by Vt;
• The value of the underlying asset at time t is denote by St;
• From above we know there exist a function V = V (S, t), such that
• The volatility of the underlying asset is σ; • The interest rate, r;
• The exercise price of the underlying asset is K; • The expiry date of the option is T .
If the value of underlying asset at T is a certain value, then the payoff of the option is:
VT = (ST − K)+, (Call option) (K − ST)+. (Put option)
Then the problem of pricing the option is solved by:
V = V (S, t), (0 ≤ S < ∞, 0 ≤ t ≤ T ), such that: V (S, T ) = (ST − K)+, (Call option) (K − ST)+. (Put option)
So, when t = 0 if the price of the underlying asset is S0, the price of the
option(or say premium) p is
1.1.3
Arbitrage free principle
“There Ain’t No Such Thing As A Free Lunch”
by Robert A. Heinlein from The Moon Is a Harsh Mistress(1966)
This famous saying is a very popular phrase in the financial world. Its mean-ing is very simple:No Risk no gain or there exists no chance to make an
arbitrage. The arbitrage free principle is the very foundation of the option
pricing theory.
1.2
Black-Scholes-Merton Framework
1.2.1
The price process for underlying asset
Let’s introduce some basic property of the Brownian motion and explain why we use it to model the price process.
1.2.2
Brownian motion
A Brownian motion is used to describe the movement of molecules. A process
W follows a Brownian motion has two important properties: Property 1: The change of ∆W during a small time ∆t is:
∆W = N√∆t,
where N follow a standard normal distribution N(0, 1).
Property 2: The value of the ∆W for any two different intervals are
inde-pendent.
Thus, we have the following:
E(∆W ) = 0 , V ar(∆W ) = ∆t.
Moreover, let’s consider W (T ) − W (0), where n = T ∆t. Then: W (T ) − W (0) = n X i=1 Ni √ δt, (1.1) where Ni ∼ N(0, 1) (i = 1, 2, 3 . . . n).
From Equation (1.1), we know that W (T ) − W (0) has a normal distri-bution:
E(W (T ) − W (0)) = 0, V ar(W (T ) − W (0)) = n∆t = T. (1.2) Now we move on to the Generalized Brownian motion.
dS = a dt + b dW, (1.3)
where a, b are constants.
Using the same argument above we can calculate the value change in a time interval T is:
E[S(T ) − S(0)] = aT, V ar[S(T ) − S(0)] = b2T.
Now, it’s time to introduce the price process of a non-dividend paying stock. Hull[1] argues the generalized Brownian motion should be modified to satisfy the basic properties of the stock price. He points out that the generalized Brownian motion only has a constant return rate a and a constant variance rate. But in the stock market, the investors always require a return rate independent from the current price. (e.g A investor needs a 10% return on the stock, then he want 10% return whenever the price is $30 or $50) It is assumed that the volatility is also a proportional to the current price S. So, we have:
dS = µSdt + SdW. (1.4) Equation (1.4) is also known as geometric Brownian motion. By arranging Equation (1.4), we can see:
dS
S = µdt + σdW. (1.5)
Thus, we can get the discrete-time version of the model (1.5). For a small moment of time δt, we have:
dS S ≈ δS S ∼ N(µδt, σ √ δt). (1.6)
1.2.3
Black-Scholes formula
Now, after defining the price process of the underlying asset(Stock S ), we are able to model the price of European option for the stock S.
First, we list the basic assumptions.
• The price of underlying asset is following geometric Brownian motion:
dSt
St
= µdt + σdWt. (1.7)
• Let r be the risk-free interest rate (constant rate). • No dividend of underlying asset is paid.
• No transaction costs and taxes. • No chance to make an arbitrage.
Let’s consider a financial market consisting two assets: a risk-free asset B (can be seen as bank account), and a stock S. The dynamics of these two are:
• risk-free asset B is:
dB(t)
B(t) = rdt, (1.8)
• stock price S is
dS(t)
S(t) = µdt + σdWt. (1.9)
Remember the arbitrage free principle in 1.1.3, we define the ∆, ∆ -neutral and ∆-hedge:
Definition 1.2.2. In finance, ∆ is ∂V
Definition 1.2.3. A portfolio containing contingent claims (or derivatives) is called ∆-neutral if it consists of positions with offsetting positive and neg-ative deltas (exposure to changes in the value of the underlying asset), and these balance out to bring the net delta of the portfolio to zero.
Definition 1.2.4. ∆ hedging is the process of setting or keeping the delta of
a portfolio as close to zero as possible.
Assume now we have 1 unit contingent claim Π, which matures at T, and it has the form:
Π = Φ(S(T )), (1.10)
where the function Φ is the contract function. For example, a European call option is a contingent claim whose contract function is
Φ(S) = max [S − K, 0].
Now we form a portfolio Ξ which consisting two asset, buy one unit of con-tingent claim Π and sell ∆ unit S (underlying stock), we choose ∆, make in a small time period (t,t+dt), this portfolio Ξ is riskless. From Jiang’s book, see [2]:
Ξt = Π(t, S(t)) − ∆S(t). (1.11)
From the famous Itˆo formula: dΠt= ( ∂Π ∂t + 1 2σ 2S2∂2Π ∂2S + µS ∂Π ∂S)dt + σS ∂Π ∂SdWt. (1.16)
Putting Equation 1.14 into Equation 1.13, we have:
[(∂Π ∂t + 1 2σ 2S2∂2Π ∂2S + µS ∂Π ∂S)dt + σS ∂Π ∂SdWt] − ∆dSt.
We know that St follows the price process (1.9), so
[(∂Π ∂t + 1 2σ 2S2∂2Π ∂2S + µS ∂Π ∂S − ∆µS)]dt + [σS ∂Π ∂S − ∆σS]dWt.
Since riskless require the dW term to be zero, thus
∆ = ∂Π
∂S,
and finally we get:
∂Π ∂t + 1 2σ 2S2∂2Π ∂2S + rS ∂Π ∂S − rΠ = 0. (1.17)
Summing up these, we have the following result, which in fact the central results for derivative pricing.
Theorem 1.2.1. Black-Scholes equation:
Assume the market is specified by 1.8, 1.9, and we want to price a claim 1.10, then the only pricing function for Π is when Π is solution of following problem: ∂Π ∂t + 1 2σ 2S2∂2Π ∂2S + rS ∂Π ∂S − rΠ = 0, (1.18) Π(T, S) = Φ(S). (1.19)
derivatives, the mean rate of return is useless and plays no role here, in fact the risk-free rate r taken it. Well, this phenomenon is associate a measure called Q measure which is also called the Risk Neutral measure. We will talk about it later.
Now we want to solve the Equation (1.18) and (1.19). Instead just attacking our problem, we introduce the stochastic representation formula first. Consider the problem as follows:
∂F ∂t(t, x) + µ(t, x) ∂F ∂x + 1 2σ 2(t, x)∂2F ∂x2 = 0, (1.20) F (T, x) = Φ(x). (1.21)
where µ(t, x),σ(t, x),Φ(x) are scalar functions.
Moreover define the process X in the time interval (t,T) as:
dX(s) =µ(s, X(s))ds + σ(s, X(s))dWs, (1.22)
X(t) =x. (1.23)
Apply Itˆo formula to process F (s, X(s)), we have :
dF (s, X(s)) = ∂F ∂Sds + ∂F ∂Xd[X(s)] + 1 2 ∂2F ∂X2[dX(s)] 2 =[∂F ∂s + µ(s, X(s)) ∂F ∂X(s, X(s)) + 1 2σ 2(s, X(s))∂2F ∂X2(s, X(s))]ds+ σ(s, X(s))∂F ∂X(s, X(s))dWs =σ(s, X(s))∂F ∂X(s, X(s))dWs.
We see the time integral has vanished. Due to the fact that F satisfies Equation 1.20, 1.21, after we take the expected value, the stochastic term will also vanish. Using the initial value Xt = x and the boundary value
F (T, x) = Φ(x), we can have:
The equation above is known as the Feyman Kac stochastic represen-tation formula.
Well, we multiply Equation (1.18) by ert, and consider Z(s) = e−rtΠ(s, X(s)),
which gives us:
Π(t, s) = e−r(T −t)Et,s[Φ(S(T ))], (1.25)
where S satisfies:
dS(t)
S(t) =µdt + σdW, (1.26) S(t) =s. (1.27) But, this is not the right answer here, why?
Remember what we talked about Equation (1.17), it shows that µ is useless when we pricing derivatives. This equation brings us into a risk-neutral world, in which the price is irrelevant to the preference of any agent and mean rate of return µ. Thus we can not use 1.26 and 1.27 directly and find the Q-dynamics which exclude the risk from our object dynamics.
We know dynamics for stock in real world is contains risk (Equation 1.26 and Equation 1.27). If we define: ˜ Wt= Wt+ µ − r σ t,
Girsonov theorem1states that there exists a measure which ˜W
tis a Brownian
motion. Here,µ − r
σ
2is the market price for risk.
Thus by differentiating,
d ˜Wt= dWt+
µ − r σ dt.
1the Girsanov theorem tells how stochastic processes change under changes in measure.
Here we just use result without proof. For more details see Measure theory
2This is also called the Sharpe ratio, details can be found in Sharpe, W. F. (1994). The
Put it back to 1.26 and 1.27, we have
dS(t) = rS(t)dt + σS(t)d ˜Wt (1.28)
S(t) =s. (1.29)
This is the Q-dynamics of S in the risk-neutral world. Thus we have the following theorem.
Theorem 1.2.2. Risk-neutral valuation
Π(t, s) = e−r(T −t)EQ
t,s[Φ(S(T ))], (1.30)
where S satisfies:
dS(t) =rSdt + σSd ˜W (1.31)
S(t) =s. (1.32)
1.3
Explicit solution for European options
In last chapter, we derived the Black-Scholes formula for an undefined con-tingent claim Π. Now I will show you how to use Black-Scholes formula to get the explicit solution for European options.
Here I have to emphasize that our model is strictly modeled under the assumptions we listed in section 1.2.3 .
Using the formula we derived before, the problem for European call option in the domain [0, T ] × R+
∂F ∂t(t, S) + rS ∂F ∂S + 1 2S 2σ2(t, S)∂2F ∂S2 − rF = 0, (1.33) F (T, S) = (S − K)+. (1.34)
For simplicity, F denotes the price process for option. First, we set two new variable:
Then we transform out original problem into: ∂F ∂τ(x, τ ) − 1 2σ 2∂2F ∂2 (x, τ ) − (r − 1 2σ 2)∂F ∂x + rF = 0, (1.35) F (x, 0) = (ex− K)+. (1.36)
In order to solve the Cauchy problem above, we need to do some transfor-mation. Here u is some undefined function.
Set:
F = ueατ +βx, (1.37) It’s easy to see that the derivatives of the F are:
∂F ∂τ = e ατ +βx[∂u ∂τ + αu] ∂F ∂x = e ατ +βx[∂u ∂x + βu] ∂2F ∂x2 = e ατ +βx[∂2u ∂x2 + 2β ∂u ∂x + β 2u].
Put these derivatives into our equation, we have:
α = −r − 1 2σ2(r − σ2 2 ) 2, β = 1 2 − r σ2.
Note that equation 1.27 and 1.28 already transformed to:
∂u ∂τ − σ2 2 ∂2u ∂x2 = 0, (1.38)
u(0, x) = e−βx(ex− K)+ (initial condition). (1.39)
Since the original equation is transformed into a heat equation. According to the PDE theory, the solution of heat equation can be express by:
u(τ, x) =
Z +∞
−∞
K(x − ξ, τ )ϕ(ξ)dξ,
where the ϕ(ξ) is the initial condition and K(x − ξ, τ ) is the heat kernel or the fundamental solution:
K(x − ξ, τ ) = 1 σ√2πτe
Thus, the solution of the Equation (1.30), (1.31) can be written as: u(x, τ ) = Z +∞ −∞ 1 σ√2πτe −(x−ξ)22σ2τ [e−βξ(eξ− K)+]dξ, = Z +∞ ln K 1 σ√2πτe −(x−ξ)22σ2τ [e(1−β)ξ− Ke−βξ]dξ.
Remember our Equation (1.29),
F (x, τ ) = e−rτ −2σ21 (r− σ2 2 ) 2 τ −σ21 (r−σ22 )xu(x, τ ) = I1+ I2, where: I1 = e−rτ Z +∞ ln K 1 σ√2πτ exp{ 1 2σ2τ[(x − ξ) 2+ 2(x − ξ)(r −σ2 2 )τ + (r − σ2 2 ) 2τ2] + ξ}dξ = e−rτ Z +∞ ln K 1 σ√2πτ exp[− 1 2σ2τ[x − ξ + (r − σ2 2 )τ ] 2+ ξ]dξ. Let ψ = x−ξ +(r −σ2
2 )τ , and note that the cumulative probability function for a normal distribution N(x) is:
.
For I2, using the same method, we get:
I2 = −Ke−rτN(
x − ln K + (r −σ2
2 )τ
σ2√τ ).
Now replace the variables to our originals, the value of the European option is found to be: F (t, S) = SN(ln S K + (r + σ 2 2 )(T − t) σ√T − t )−Ke −r(T −t)N(lnKS + (r − σ 2 2)(T − t) σ√T − t ). (1.40)
1.4
Summary
Chapter 2
Barrier options
2.1
Brief introduction
In this chapter, we turn to our target option, the barrier option.
The European option, as we discussed, the payoff of a European option only depends on the price at expiration date. However, some other options are not the same, they can be seen as dependent option. In other words, path-dependent options are options whose payoffs are determined by the path of
the underlying asset’s price (e.g lookback options, barrier options and Asian
options etc).
The key characteristic of barrier options, which have various forms and types, is that this kind of option is triggered to be activated or, on the contrary, be exterminated upon reaching certain barrier levels. The option’s final payoff does not only depend on the price of the underlying asset at the maturity but also whether the option is triggered “on” or “off” before the maturity. The Figure 2.1 below illustrats a typical European up and out barrier call option.
Figure 2.1: Typical European up and out barrier call option
can take forms of either a “call” or “put” as well as European or American option. The four types are:
• Up and in.
• Up and out.
• Down and in.
• Down and out.
the barrier.
Thus, the payoff function of the barrier options can be listed below: • In options – Up and in: (ST − K)+[1 − I{St<SB,t∈[0,T ]}] (call), (K − ST)+[1 − I{St<SB,t∈[0,T ]}] (put). – Up and Out: (ST − K)+[1 − I{St>SB,t∈[0,T ]}] (call), (K − ST)+[1 − I{St>SB,t∈[0,T ]}] (put). • Out options: – Up and Out: (ST − K)+I{St<SB,t∈[0,T ]} (call), (K − ST)+I{St<SB,t∈[0,T ]} (put).
– Down and Out:
(ST − K)+I{St>SB,t∈[0,T ]} (call),
(K − ST)+I{St>SB,t∈[0,T ]} (put).
SB is the barrier level and IW(S) is an index function, where:
Iw(S) 1, S ∈ w, 0, S /∈ w.
If we call the European option as the “Vanilla option”, and it’s easy to see that :
Fvanilla(S, t) = Fup−and−out(S, t) + Fup−and−in(S, t)
The equantions above are also called In and Out parity for barrier op-tions.
2.2
Mathematical model for barrier options
In this chapter, we will model barrier options by using theoretical PDE meth-ods.
To set up the mathematical model for barrier options, we must sort all the situations according to “In”, “Out”, “Up”, “Down”, “Call”, “Put” to deter-mine the proper domain for the model. Moreover, we have to decided the boundary condition when S = SB and t = T .
According to Jiang’s book, see [2] the domain and boundary condition for different types of barrier options are as follow:
1. Domain: set the domain for model as D.
• for Up options:
D = {(S, t)| 0 6 S 6 SB, 0 6 t 6 T }.
• for Down options:
D = {(S, t)| SB 6 S < ∞, 0 6 t 6 T }.
2. When S = SB, boundary condition is
• “Out” options:
F (SB, t) = 0.
• “In” options:
3. Boundary condition at t = T : • Call Out: F (S, T ) = (S − K)+, 0 6 S 6 SB f or up SB 6 S < ∞ f or down • Call In: F (S, T ) = 0, 0 6 S 6 SB f or up SB6 S < ∞ f or down • Put Out : F (S, T ) = (K − S)+, 0 6 S 6 SB f or up SB 6 S < ∞ f or down • Put In : F (S, T ) = 0, 0 6 S 6 SB f or up SB6 S < ∞ f or down
Since options should satisfy Black-Scholes equation in its domain, theoreti-cally speaking, all the options can be priced by Black-Scholes equation upon specifying their “unique” boundary and initial value plus some restrictions if needed.
Take European Down-and-out call option as the example, the mathe-matical model for European Down-and-out call option is:
∂F ∂t + 1 2σ 2S2∂2F ∂S2 + (r − q)S ∂F ∂S − rF = 0 (2.1) F (S, T ) = (S − K)+ (SB 6 S 6 ∞) (2.2) F (SB.t) = 0 (0 6 t 6 T ) (2.3)
Also Jiang, see [2] gives a method to solve this PDE. First, like what we did before, make a transformation:
x = ln S SB
Thus, ∂u ∂t + 1 2σ 2∂2u ∂x2 + (r − q − σ2 2 ) ∂u ∂x − ru = 0, (x ∈ R+, 0 6 t 6 T )(2.4) u(x, T ) = (ex− K B)+, (0 < x < ∞) (2.5) u(0, t) = 0. (0 6 t 6 T ) (2.6) Here, KB = K SB
. Again, we make a transformation :
u = eαx+β(T −t)V, (2.7) and, α = − 1 σ2(r − q − σ2 2 ), β = −r − 1 2σ2(r − q − σ2 2 ) 2.
Under this transformation, V should satisfy with the following PDE in the domain {x ∈ R+, 0 6 t 6 T } . ∂V ∂t + 1 2σ 2∂2V ∂x2 = 0 (2.8) V (x, T ) = e−αx(ex− K B)+ (2.9) V (0, t) = 0 (2.10)
Here, the author uses a trick, which is to set:
ϕ(x) = e−αx(ex− K B)+, x > 0 −eαx(e−x− K B)+. x < 0
Standard calculations yield: V (x, t) = 1 σp2π(T − t) Z ∞ 0 [e−2σ2(T −t)(x−ξ)2 − e− (x+ξ)2 2σ2(T −t)]e−αξ(eξ− K B)+dξ. (2.13) Transforming back to u(x, t),gives:
u(x, t) = √ex 2π Z x−ln KB+(r−q+σ22 )(T −t) σ√T −t −∞ e−ω22 dω −√1 2πKBe −r(T −t) Z x−ln KB+(r−q−σ22 )(T −t) σ√T −t −∞ e−ω2 2 dω −e −2 σ2(r−q)x √ 2π Z −x−ln KB−(r−q+σ22 ) σ√T −t −∞ e−ω2 2 dω +√KB 2πe −r(T −t)−σ22 (r−q−σ22 )x Z −x−ln KB+(r−q−σ22 )(T −t) σ√T −t −∞ e−ω22 dω. (2.14)
Thus, transforming to our F (S, t), we get:
F (S, t) = SN(d1) − Ke−r(T −t)N(d2) − SB( S SB )−σ22 (r−q)N(d3) − Ke−r(T −t)( S SB )1−σ22 (r−q)N(d4), (2.15) here, d1 = ln S K + (r − q + σ2 2 )(T − t) σ√T − t , d2 = d1 − σ √ T − t, d3 = lnSB2 SK + (r − q +σ 2 2 )(T − t) σ√T − t , d4 = d3 − σ √ T − t.
Note that Equation (2.15) can be rewritten as:
.
Thus the Fdown−and−in call option is:
Fdown−and−in(S, t) = ( S SB )−σ22 (r−q)+1Fvanilla(S 2 B S , t).
Moreover, like European option, there exists a Put-Call parity for barrier options.
Theorem 2.2.1. For a Down and knock out option, there exist the put-call parity: pdown−and−out(S, t) + SN( ˆd1) =cdown−and−out(S, t) + Ke−r(T −t)N( ˆd2) + µ S SB ¶−2 σ2(r−q)+1 [SB2 S N( ˆd3) − Ke −r(T −t)N( ˆd 4)], (2.16) where: ˆ d1 = ln S SB + (r − q + σ2 2 )(T − t) σ√T − t , ˆ d2 = ˆd1− σ √ T − t, ˆ d3 = lnSB S + (r − q + σ2 2 )(T − t) σ√T − t , ˆ d4 = ˆd3− σ √ T − t.
With all these, we can get all pricing formulas for other barrier option.
2.3
Summary
Chapter 3
Binomial methods and Monte
Carlo simulation
With the development of global financial markets, more and more financial instruments are invented to satisfy the need of risk management. People need some very complex derivative securities to control risk and making profit. Then pricing those derivatives is become a very important issue. But only a few derivatives have explicit solutions, thus we need to use some numerical procedure to solve the problem numerically.
Among many numerical methods, there are three commonly-used methods, see [4]:
• Lattice methods, which include binomial and trinomial models, are based on a discrete-time framework, assume that price of the under-lying asset “jumps” to a finite setof values (each one with a certain probability) in a small time period.
can calculate the payoff for each simulated path of underlying asset and then these payoffs are averaged and discounted to today.
• Finite difference methods, which approximate the solutions to differen-tial equations using finite difference equations to approximate deriva-tives.
In this paper, we focus on the first two methods, and using a combined method to price European barrier options.
3.1
Binomial method
The binomial method was first introduced in the context of option prices in the paper Options pricing: A simplified approach(1979) by Cox, Ross and Rubinstein.
3.1.1
One-period model and multi-period model for
European style options
We start with the one-period model and later we move to multi-period model.
Model description
Assume we have a market that only have two asset. One is a bond B (risk-free) and the other is a stock with price process S. We have two time points, one is time zero and the other is time T . Assume that p + q = 1 and u > d. The two process of B and S are follows.
The bond process is given by:
and,
B0 = 1.
Here r is a constant interest rate.
The stock process is stochastic and given by:
S0 = s, ST = S2 = uS0 with probability p S1 = dS0 with probability q .
We can use Figure 3.1 to illustrate this.
T S0 S2 S1 0 StockPrice Time
Figure 3.1: possible stock trajectory
It’s important to point out that at time 0, the parameters we already known are: s, r, σ(the standard deviation of S), time length T .
Now, we can illustrate the price dynamics by the Figure 3.2.
Delta hedge
S0 S2 = uS0 S1 = dS0 p q B0 BT
Figure 3.2: price dynamics
Definition 3.1.1. A Contingent claim is any stochastic variable of the form VT = Ψ(ST), where Ψ is the payoff function.
Using simple European call option as the example, thus:
Ψ(ST) = (ST − K)+,
and dynamics shows in Figure 3.3.
V
0V
T2= (S
2− K
)
+V
T1= (S
1− K
)
+Figure 3.3: European call dynamics
Our main problem is to find out a “fair” value for the contingent claim Ψ, and if such value exist for all.
Definition 3.1.2. For a given contingent claim V , if there exists a portfolio
Γ such that ΓT = VT with probability 1, then Γ is called a replicating portfolio.
Since there is no difference between holding a claim or portfolio, the price of the claim should be as same as the value of the portfolio:
V0 = Γ0. (3.1)
Moreover, we have the following theorem.
Theorem 3.1.1. Uniqueness: Suppose that a claim V can be replicated by the portfolio Γ. Thus there exists arbitrage possibility if and only if the price at time 0 of the claim V is other than Γ0 .
Proof. Suppose the price of V is cheaper then Γ0:
V0 < Γ0.
then at time 0, we short one unit of Γ and buy Γ0
V0
unit claim V . At time T, we get Γ0
V0
ΓT and pay ΓT, thus we get (
Γ0
V0
− 1)ΓT from nothing.
This is a arbitrage.
The same holds in the case :
V0 > Γ0.
Thus we get what we need.
Now, let’s construct our portfolio Γ. Our portfolio has the form: Γ = τ S + βB.
At time T, we have:
Γ1T = τ S1+ βB1 = τ dS0+ βαB0 = VT2, (3.2)
Since u > d, then we solve the system of equations above and a simple calculate shows that:
β = 1 α · dV2 T − uVT1 (d − u)B0 , (3.4) τ = VT2− VT1 (u − d)S0 . (3.5)
Now, it’s easy to calculate the price for the claim V :
V0 = βB0+ τ S0 = 1 α · µ α − d u − dV 2 T + u − α u − dV 1 T ¶ . (3.6)
Assume d < α < u, and define the probability measure Q: p = α − d u − d, (3.7) q = u − α u − d. (3.8)
It’s easy to observe that 0 < p, q < 1 and p + q = 1. Denoting expectation w.r.t this measure by EQ, we have:
V0 =
1
αE
Q(V
T). (3.9)
It is worthwhile to talk more about properties of this Q measure. Take underlying asset S as example, consider EQ(ST
BT ): EQ(ST BT ) = 1 αB0 (pS2+ qS1) = 1 αB0 (α − d u − duS0+ u − α u − ddS0) = S0 B0 .
just the same as the meaning of Risk-neutral world. In a Risk-neutral world, investors requires no extra premium for risk, thus the expected return for all risky assets are equal to the return for risk-free asset.
Now, only two parameters u, d are still unknown. Remember that the process we developed for stock S, and we know the variance σ2 for the stock
S, then we see that the variance of the percentage change of stock S in a short period T is σ2T , then by a simple calculation:
σ2T = E(S2) − [E(S)]2.
It follows that:
pu2+ (1 − p)d2− [pu + (1 − p)d]2 = σ2T. (3.10)
The second condition comes from the fact that in this risk-neutral world the expected value for S after time T is:
erTS = puS + (1 − p)dS.
or simply,
erT = pu + (1 − p)d. (3.11) The third condition used by Cox,et al[5] is:
u = 1
d. (3.12)
Use these three condition, we get:
u = eσ√T, (3.13)
d = e−σ√T. (3.14)
Now, we can just use eqn 3.9 to calculate V0. Using the binomial lattice, we
V
0V
T2V
T1p
q
= 1 α(pV 2 T + qVT1)Figure 3.4: Calculating backwards through tree lattice
to root.
This procedure can be illustrated by Figure 3.4 below. With this result, we have the following definition.
Definition 3.1.3. Let V is some risky asset, and B is a bank account, we denote EQ(Vt
Bt
) as the discounted price of the risky asset V .
Since B is risk free asset, so BT = αB0. Then we can rewrite Equation
(3.7) as: V0 B0 = EQ(VT BT ). (3.15)
This equation tells us the following theorem:
Theorem 3.1.2. Under the Q measure, the discounted price of option V is the expectation of the discounted price of V at expiration date(or say the discounted payoff at expiration date).
For our European call option:
V0 B0 = EQ((ST − K) + BT ).
Proposition 3.1.1. Suppose a market only consist of risk-less asset B and risky asset S, the necessary and sufficient condition for d < α < u is that arbitrage free.
Although the meta-theorem already tells us the market with one risk-free asset B and one risky S is arbitrage risk-free. But this time, we will prove this by using another method. From Bjork’s book, see [7], we have the following proof.
Proof. First prove that if market is arbitrage free, then d < α < u holds. if α > u, construct the portfolio:
Ψ = −S + S0
B0
B.
Then we calculate the value of Ψ at time 0 and T Ψ0 = −S0+ S0 B0 B0 = 0, (3.16) ΨT = −ST + S0 B0 BT. (3.17)
Since at time T, ST has two values:
Ψ2 T = −uS0+ S0 B0 αB0 = (α − u)S0 > 0, Ψ1 T = −dS0+ S0 B0 αB0 = (α − d)S0 > 0.
Thus, we have that ΨT > 0.
Moreover,
Prob{ ΨT > 0} > 0 = Prob{ΨT = Ψ1T} > 0.
It shows that if α > u it would be arbitrage chance for portfolio Ψ.
Using the same method we can prove that if α < d, it would be also a arbitrage chance for Ψ.
other words: If Ψ0 = τ S0+ βB0 = 0, and ΨT = τ ST + βBT > 0, (3.18) Then ΨT must be ΨT = τ ST + βBT = 0.
In fact, since d < α < u holds, we can use the Q-measure that we defined before. Consider: EQ(Ψ T) = α − d u − d(τ uS0+ βαB0) + u − α u − d(τ dS0+ βαB0) = α(τ S0+ βB0) = αΨ0 = 0. (3.19)
Since we have Equation (3.18): Ψ2 T > 0, Ψ1T > 0. Then, Ψ2 T = Ψ1T = 0. Thus we have: Prob{ ΨT > 0} = 0.
Now we get what we need.
To sum up, we have the following theorem:
Multi-period model
It may seem odd that the price of stock S can only be either of two values at
T in our model, while real stock can be a lot of values. Thus, we introduce the
multi-period model, which is a more realistic model for stock S by spitting the time period into many smaller periods “ ∆T ”. For example, if we use a 20-step model, then we have 21 × 22
2 − 1 = 230 elementary outcomes for S. In fact, we usually use multi-period model in practice rather than one-period model.
Like one-period model, we discretized our the price process of our underlying
S and risk-free B first. As before we also assume ud = 1. Then, B0 = 1, Bt+1 = αBt (t = 0, 1, 2...N − 1), and, S0 = S0, St+1 = uSt(with probability p) dSt(with probability q) .
This can be illustrated as follows:(note that we only draw first 3 steps) When we dealing with the one period model, we have created a portfolio to replicate our claim. It’s easy when we only have one period, but how can we create the replicate portfolio when there is more than one period?
First, we define the definition.
Definition 3.1.4. A portfolio strategy is a stochastic process, which: {gt = (τt, βt); t = 0, 1...T },
then the corresponding value process is:
S0 uS0 dS0 u2S0 u3S0 d3S0 d2S0 udS0 u2dS0 ud2S0
Figure 3.5: price dynamics
Here, α is also the growth rate for each step, and for simplicity we can erase the B0, since B0 is 1.
Definition 3.1.5. A portfolio strategy is regarded as self-financing if follow-ing condition holds for all t, where t = 0, 1, 2...T − 1
βtα + τtSt = βt+1α + τt+1St+1.
A self-financing portfolio is the portfolio without any exogenous infusion or withdraw of money. Since a self-financing portfolio can rebalanced itself without cost, thus we can replicate any claim if the market is complete like what we had done before. Also, same as one-period model, we need to find condition that made market arbitrage free. Actually, the condition that fits multi-period model is just as same as the one-period models. It’s easy to see that a multi-period model can be considered to be a summary of many one-period model. Thus the condition that fits for all of the one-one-period model is also fits the multi-period model:
Moreover, the Q measure and other parameters are defined and calculated just as before: p = α − d u − d , q = u − α u − d α = erδt, u = eσ√δt, d = 1 u. Here δt = T
N as the length of time step. Then we can get the root value by
calculating backwards from the end values.
To better understand of this, we give a simple example on how to calculate the price for a European call option.
Example:
Consider a 6 month European call option. The stock price start at $50, strike price is $50, the volatility is 40% and risk-free interest rate is 10% per annum. For simplicity, we only construct a 2-step model.
Now we calculate the parameters:
S
0uS
0u
2S
0S
0dS
0d
2S
0T = 0.5, δt = 0.25 , α = erδt = 1.0253, u = eσ√δt= 1.2214 , d = 0.8187, p = α − d u − d = 0.2067 0.4026 = 0.5134, q = 0.4866.
With all these parameters we can build the binomial tree for our case. Note the lower number at each node is the payoff for that node. For example,
50 61.065 74.579 50 40.935 33.513 A B C D E F 24.579 0 0
Figure 3.7: price dynamics for stock S
the payoff for node D is 24.579.
Now we calculate backwards. At node B, the value of option is:
(pVD + qVE)e−rδt= (0.5134 × 24.579 + 0.4866 × 0)e−0.1ב0.25 = 12.3072,
and for node C, it’s easy to see that:
(pVE + qVF)e−rδt= (0.5134 × 0 + 0.4866 × 0)e−0.1ב0.25 = 0.
Then the price for option at time 0 is:
(pVB+ qVC)e−rδt = (0.5134 × 12.3072 + 0.4866 × 0)e−0.1ב0.25 = 6.1625.
50 61.065 74.579 50 40.935 33.513 A B C D E F 24.579 0 0 12.3072 0 6.1625
Figure 3.8: calculate backwards
Well, it’s just a rough result, and if we set the steps to 200 or even more, we finally get the result which very close to its theoretical value 6.7902.
3.1.2
Binomial models and the Black-Scholes equation
In the first chapter, I made a short introduction on Black-Scholes framework. This is the fundamental framework of the option pricing theory nowadays. Is the binomial method is consistent with the Black-Scholes-Merton framework? The answer is yes.
Here we follow the book by Jiang, see[2], we can proof this answer. First of all, we split the time period [0, T ] into N equal time intervals:
tn = n∆t (∆t =
T N).
As before, we build a binomial model. We set t = tn, thus at t = tn+1, Sn+1
In a risk-neutral world, we know that dynamics for stock S is:
dS
S = rdt + σdW, (3.20) S(tn) = Sn. (3.21)
Using Itˆo formula,
d ln S = (r − σ2
2 )dt + σdW.
Integrate it from tn to tn+1, and consider the initial condition at tn, we have:
lnSn+1
Sn
= (r −σ
2
2 )∆t + σ∆W, ∆W = Wtn+1− Wtn.
We calculate its expectation and variance:
E(lnSn+1 Sn ) = (r −σ2 2 )∆t, V ar(lnSn+1 Sn ) = σ2∆t.
Since under the risk-neutral measure, we define the probability p and q as:
p = α − d u − d, q = u − α u − d. (3.22) Thus, we have: E(lnSn+1 Sn ) = p ln u + q ln d, and V ar(lnSn+1 Sn ) = p(ln u)2+ q(ln d)2− (p ln u + q ln d)2.
Remember that we have another condition from Cox,et al and the definition of α:
ud = 1 (3.23)
Thus, we have the following 2 equations:
(r − σ2
2 )∆t = p ln u + q ln d, (3.25)
σ2∆t = p(ln u)2+ q(ln d)2− (p ln u + q ln d)2. (3.26)
We use the condition (3.22), (3.23), and rearrange the Equation (3.25), (3.26), then we get:
(2p − 1) ln u = (r − 1 2σ
2)∆t, (3.27)
4p(1 − p)(ln u)2 = σ2∆t. (3.28)
In order to solve Equation (3.27), (3.28), we assume:
u = eη√∆t,
where η is an unknown function. Put it into Equation (3.27), (3.28):
η(2p − 1) = (r − σ 2 2 ) √ ∆t, (3.29) 4p(1 − p)η2∆t = σ2∆t. (3.30)
From these two equations, we solve p :
p = 1 2 + r − σ2 2 2η √ ∆t, (3.31)
and put the value of p into Equation (3.30):
η2− σ2 = (r −1
2σ
2)∆t, (3.32)
which can be rewritten as :
If we ignore the infinitesimal of higher order part O(∆t), then we get following two equations: p = 1 2+ r − σ2 2 2σ √ ∆t, (3.33) q = 1 − p = 1 2 − r − σ2 2 2σ √ ∆t. (3.34) Moreover, we have: u = eσ√∆t, d = e−σ√∆t.
It is just the same results as what we get by using the binomial model.
3.2
Monte Carlo methods
This section includes two parts. The first part will develop principle of the Monte Carlo methods. It will begin with a general description and then present some examples. In the second part, will show how to expand the Monte Carlo simulation to price derivatives.
3.2.1
Principles
Monte Carlo methods are a class of stochastic techniques which means they are based on the use of random numbers and probability statistics to examine the problem. These methods may be visualized as black box in which we enter a stream of pseudo-random numbers, then an estimate of a quantity of interest can be obtained by analyzing the output. To explaining this in a formal way, we will summarize the typical pattern of the Monte Carlo methods[6]:
• Generate random numbers as our inputs from the domain by using a particular probability distribution.
• Perform a deterministic computation using the inputs.
• Aggregate the results of the individual computations into final results Since we need use computer to handle all numerical methods, thus it is important to point out that we must speak of pseudorandom numbers instead of random numbers since nothing is a real random on a computer. Although these sequences of numbers are deterministic, we can still use them and, in fact, they works very well.
When we are pricing derivatives, we always encounter the problem that need computing integrals.
Consider the problem of
I =
Z 1
0
f (x)dx. (3.35) We can represent I as an expectation E[f (γ)], withγ uniformly distributed between 0 and 1. Then if we picking points γ1, γ2, ... independently and
uniformly from [0, 1]. Thus we can evaluate I by averaging n of these random function f : ˆ In= 1 n n X i=1 f (γi). (3.36)
If f is integrable over period [0, 1], then by using law of large numbers, ˆ
In→ I with probability 1 as n → ∞.
Example: Let’s consider a trivial case:
I =
Z 1
0
exdx.
1. We define the domain for input. It is easy to know that this domain is [0, 1] for our case.
2. We use computer algorithm to generate uniformly distribute numbers. For example, we can use MATLAB rand function.
3. Calculate exi according to each random number x
i
4. Calculate the sample mean, 1 n n X i=1 exi.
Covert the above steps into computer algorithm: for i = 1, ...n Generate xi Set f (xi) = exp(xi) Set meanf (xi) = 1 n n X i=1 exi print meanf (xi)
Using MATLAB, we can simply type: mean(exp(rand(1,n))) in the command window, where n is the number of trials. For different n, we can get following results:
Results of Monte Carlo method number of trials result
100 1.6967 1000 1.7216 10000 1.7186 100000 1.7189 Note that Z 1 0
exdx = e − 1 ≈ 1.7183. It is apparently that the estimate tends
3.2.2
Monte Carlo methods and derivatives pricing
What do these methods have to do with our problem of derivatives pricing? Remember, the underlying asset’s evolution follows a stochastic process, and since the Monte Carlo methods are a class of stochastic techniques, then there must exist some methods to simulate the price process of the underlying assets.
We know the price process of the underlying asset S under the object measure can be written as:
dS = µSdt + σSdW. (3.37) Is this the right price process that we are about to simulate? Answer is no. Review the Risk-neutral pricing we talked before, the price for an option is the discounted expectation of the option’s risk-neutral payoff . Thus we need rewrite Equation (3.37) under the Q measure:
dS = rSdt + σSdW , (3.38) or equivalently:
d ln S = (r − 1
2σ
2)dt + σdW . (3.39)
How to simulate this? We divide the life time of the option T into n small subintervals of length∆ T = T
n . Then over each interval [t, t + ∆t] , we
Here Z is a standard normal random variable.
This result comes from the fact that for a Wiener process W ,
W (t + ∆t) − W (t) ∼ Normal(0, ∆t).
Thus, for each interval, we can use an independent draw from the normal distribution, and repeating this for n steps, we can get a estimate value of
S(T ). Moreover, the law of large number tells us if n → ∞, the approximate
distribution of S(T ) will draw closer to the exact distribution.
Now we use a simple example to illustrate the general method for valu-ating derivatives.
Example:
Consider a standard European call option. Strike price is K at time T, the current price is S(0), standard deviation is γ, and interest rate is r. Review the Theorem 1.2.2 we talked before, then:
C(0, S(0)) = E(e−rT(S(T ) − K+)), (3.42)
where S has the dynamics:
dS(t) =rSdt + σSdW . (3.43)
S(0) =S(0) (3.44) Assume we can produce a sequence of Z1, Z2, Z3...Zn independent normal
random variables, then we can estimate E(erT(S(T )−K+)) by using following
algorithm: for i=1,2....n
Generate a normal distributed Zi sequence.
Set cCn=
1
n(C1+ C2+ ... + Cn)
Print cCn
Let C denote the theoretical value by Black-Scholes formula for our call option.
Since
E( bCn) = C = E[e−rT(S(T ) − K)+],
then the bCn is an unbiased estimator.
Moreover, the estimator is also consistent, meaning that when n → ∞:
c
Cn→ C with probability 1 ¤
In this section, we shows how to price a derivative by using Monte Carlo methods. Although there are many Monte Carlo methods to valuate the price for options, all of these methods are always based on the same ideology. Note: Number of Trials
When you read here, maybe you will ask: If i want to get a result that satisfies my accuracy requirement, how many trials i need to do? If n independent trials are carried out in the standard Monte Carlo simulation we described above, then we can calculate the mean and standard deviation for our estimate value. Denote standard deviation by η and mean by µ. From Hull’s book, see [1], the standard deviation of the estimate for the Monte Carlo method described above is:
η √
n.
Thus the 95% coefficient interval of the estimate is:
µ − 1.96η√
n < cCn < µ +
1.96η
This shows that the accuracy is associate with the square root of number of trials. If we want to double the accuracy we must quadruple the number of trials.
3.3
Summary
Chapter 4
Modified binomial Monte Carlo
method
following section, I will describe the detail of this method and apply it to price an European barrier option.
4.1
Method description
Now, I am going to show how to implement this method step by step. First, we build a binomial lattice. Like before, we denote the underlying asset S, the initial price of S is S0, the lifespan for the derivative which is
associated with S is T , interest rate is r and standard deviation of S is σ. Then we can illustrate the binomial lattice, which shows below.
Figure 4.1: binomial lattice for a 3 step tree
Note that in the figure above, there are two numbers above each node. So pick any node, the upper number is the price and the lower number is the probability which this node can be reached.
can be called “node numbers” . By using these node numbers, the probability of path traversing between each of two connected nodes can be determined. For simplicity, we call these probabilities “edge numbers”. The edge number for two particular connected nodes is the node number of the start node, multiplied by the node number of the end node. This procedure can be illustrated in Figure 4.2. Note that the number on the path is the path number and we always set edge number for initial node (S0) is 1 for the need
of programming.
Now our tree lattice has numbers which are associated with every node and edge, thus we can begin our path generation process. But edge numbers are always infinite decimals, we need to normalize them first by multiplying by a large number, for example 10000. It is worth while to point out, this number is just the number of trials in our exploration process later. To make this easier to understand, I will use a example to illustrate this number normalization process.
Example: Assume that we have a 3 step binomial tree. Initial value of stock S is 10 , u=2 , d=0.5, and p=0.5136. The tree structure can be illustrated in Figure 4.2.:
Multiply 10000 to edge numbers and node numbers then round them, so we get the a new tree, which can be illustrated in Figure 4.3
Thus we finish our number normalization process. ¤ Now, we begin our path generation process. Let’s starting this from the initial node S0. First we generate a random number between 0 and 1 by using
a random number generator. Second, we compare this random number with the ratio of up edge number
Figure 4.2: binomial tree with node and edge numbers
the edge number and initial node number along this path minus 1 respec-tively. Fourth, we set the end note of last path as the new initial node, and do the same thing all over and over again until our initial node is the end node of the tree. When we reach the end node, we only subtract the node number by one. Thus we get one possible trajectory of our underlying asset and finish the first trial. According to the type and payoff function of the option, we can calculate the payoff at the end node. Then calculate present value of this payoff , we know the price of option for our first trajectory. Then we repeat the same process until all the node number and edge number reach 0. I am going to use our last example to illustrate this path exploration process.
Example: Continue our last example. Since our tree has 3 steps, we need generate 3 random number at a time. For example, the first sequence is {0.2312, 0.5723, 0.3641}.
Since 0.2312 < 5136
Figure 4.3: multiply 10000 and rounding numbers
node number and up edge number are minus 1 respectively. After the first path, the new initial node is B. Since 0.5723 > 2638
5136, we choose a down path. Same as first path, both new initial node number and down path number are minus 1. Well after generate the third path, we get to the end node. S we finish our first trail. The tree structure after first trail is illustrated in Figure 4.5.
Now we get first possible trajectory for our underlying asset, repeat the path generate process until all node number and edge number are 0. Well, then we finish the whole path generate process. ¤ After the path generation process, I am going to talk about price part. Apparently, for each of the trajectory we have generated, we can calculate the price for our desirable option. For example, If we want to price a European call option, we simply calculate the deducted payoff of the option:
ϕi = e−rT(SiT − K)+.
Figure 4.4: Generation of first path
of stock at end node.
We calculate the average value of all n deducted payoffs.
E(ϕ) = n P i=1 ϕT i n .
This E(ϕ) is the Risk-neutral price for the our European call option.
4.2
Applications to simple European barrier
option
In this section, we will apply our modified method to price simple European barrier option.
Figure 4.5: Finished first trial Parameters Value S0 50 K 50 B 75 T 1.25 year volatility 40% risk-free rate 10% annum
Here we will build a 5-step tree. Thus, all parameters for building the tree are: T = 1.25 , δt = 0.25 α = erδt= 1.0253 , u = eσ√δt = 1.2214 d = 0.8187, p = α − d u − d = 0.2067 0.4026 = 0.5134 q = 0.4866.
Figure 4.6: binomial tree structure example 1
We see the barrier does not lies on a layer of nodes. This cause the a shift of the barrier to a nearby layer of nodes, say effective barrier(dotted line in the figure above). Here, our binomial tree lattice takes the effective barrier as “true barrier” , because the barrier condition apply on those laid on the effective barrier instead of our real barrier. The shift of barrier causes “option specification error” - the tree lattice is too coarse to present the risk-neutral stock price process.
Then we need to eliminate this error. Boyle and Lau pointed out a method of improving binomial lattice valuation in this case, see[3]. They try to make a constraint on binomial tree in such way that lattice has a layer of nodes as close as possible to barrier.
this can be expressed as:
S0um < B < Soum+1,
where m denote the m th step.
Let N be the largest integer such that:
S0um > B ⇒S0emσ √ δt > B ⇒ ln(emσ√NT) > ln(B S0 ) ⇒N 6 m2σ2T ln (B S0) 2.
This value of N produces a layer of nodes just on the barrier. For other type of barrier options, we can use the similar way to calculate this value
After adjusted the layer of nodes, we can directly use our method to price European barrier option.
Figure 4.7: binomial tree stucture example 2
4.3
Results for European barrier options
In this section we will compare results between two methods. These two methods are the standard Monte Carlo method and our modified binomial Monte Carlo method. We use same pseudo random number generator for both methods.
Here we use European barrier option to compare our modified binomial Monte Carlo method with standard Monte Carlo method.
Table 4.3.1: Option parameters
Case number Type S0 K T B σ r
1 Down and out call 60 40 1 30 40% 10% 2 Down and out put 50 50 1 30 40% 10% 3 Up and out call 50 60 1 80 40% 10% 4 Up and out put 50 50 1 70 80% 10% 5 Down and in call 50 50 1 40 80% 10% 6 Down and in put 50 60 1 45 40% 10% 7 Up and in call 50 50 1 70 40% 10% 8 Up and in put 50 60 1 70 40% 10%
Table 4.3.2: Results of 2 methods
Modified binomial Monte Carlo Standard Monte Carlo
Case number N Trial Results Steps Trial Results Theoretical value
Table 4.3.3: CPU time consumption by two methods
Case number Modified binomial Monte Carlo Standard Monte Carlo
1 185.2824 s 433.6122 s 2 126.0080 s 413.5788 s 3 208.1351 s 1283.8211 s 4 121450.1068 s 1710.4219 s 5 212870.3456 s 453.9863 s 6 101320.8091 s 566.1255 s 7 192.1398 s 411.3113 s 8 328.1578 s 597.4219 s Platform:AMD T urionT M RM70 2×2GHz
4GB memory under MAT LABr environment
From Table 4.3.2, It can be seen that our modified Monte Carlo method can make a more accurate result than standard Monte Carlo method. But standard Monte Carlo method can improve its performance by using more steps for each iteration and number of trials. Case 4 shows that when the volatility is high, the first integer N for our method will be extremely large. The same situation occurs when the barrier closes to the initial price. I have to say, this is a major deficiency to our method. A larger N means: larger sequence of pseudo-random numbers need to be generated, more edge and nodes will be evaluated. All these cause the sharp drop of efficiency. Moreover, when N is extremely large, the corresponding number of trials will also be very large. Due to there are 2n possible paths for a n-step binomial
our modified method is faster than standard Monte Carlo method. However, for those have high volatility or initial price close to barrier, the CPU have to spend extremely much more of time than standard Monte Carlo method. But if we take a closer look to our method, even for those time consuming cases, the average time spend for each step is still less than standard Monte Carlo. So we can conclude that our method is better than the standard Monte Carlo method in a lot of cases, but not efficiently when dealing with high volatility cases or those barrier conditions are close to initial price.
Moreover, if we allow m to be larger integer, then we have larger N. With larger N, we have better result. Table 4.3.4 shows the results for case 1.
Table 4.3.4: Results by changing optimal steps N N Trials Results 33 1 million 24.6939 193 100 million 24.6926 341 100 million 24.6930 523 100 million 24.6920 Theoretical Value 24.6929
These results obtained are very accurate and therefore it identified the accuracy and adaptability of my method in pricing these European barrier options.
4.4
Summary
Chapter 5
Conclusion
In the Chapter 1, we give a brief introduction, which including the concept of options, the arbitrage-free principle, the behavior of the risky asset, and the Black-Scholes-Merton framework. At the end of Chapter 1, we show how to use this framework to model European call options and obtain explicit pricing formulas.
In the Chapter 2, we introduce the basics of barrier options and described the model of simple barrier option. We also discussed the existence of explicit solutions for pricing PDEs. For most situations, we can not obtain explicit solutions.
generating a sequence of pseudo random number, we simulated the possible trajectory for our risky asset.
In Chapter 4, we present a new numerical method. Inspired by the fact that binomial lattice contains the information of the price process for our underlying asset, we can price the option by sampling the already known binomial lattice rather than simulating some unknown trajectory for under-lying asset which is done when using the standard Monte Carlo method. This method is quite universal. In this paper, we only apply this method to European barrier options, and we find out our method works well and give good approximations.
Appendix A
MATLAB Code
A.1
Code for calculating best fit N
%initial parameters s = 50 ;
end end N
A.2
Modified binomial based Monte Carlo for
European barrier option
% Binominal tree Monte-Carlo simulatoin
% This M code generates a binominal with given paramters and then travel
% the tree with decision made by random numbers. % Simulation
Parameters price =60;% initial price K = 40;% strike price
N = 20; %tree depth T = 5/12; % time
barrier =30;%barrier value sigma = 0.4;% volatility rate = 0.1; % interest rate a = exp(rate*(T/N));
u =exp(sigma*sqrt(T/N)); % for price to upper path d = 1/u; %for price to lower path
p up = (a-d)/(u-d) ; % transition possibility p down = 1 - p up;
node amount = (1+N)*N/2; % number of nodes in the N order binomial tree
%pre-allocate the nodes
% binominal = struct(’up’,{},’down’,{},’p’,{},’price’,{});
binominal = struct(’up’, {}, ’down’, {},’p up’,{},’p down’,{},’price’,
{});
b index = 1; % pointer to the last occupied element in the binominal array
% initialize the root node
binominal(1).p up = mf; %initial possibility binominal(1).p down= 0;
binominal(1).price = price; %initial price
% start construct the binominal tree for depth = 1:N
start = depth*(depth - 1)/2 + 1; %start position for this iteration in the tree array
for j = start:(start +depth - 1)
transition p = (binominal(j).p up + binominal(j).p down) * p down; b index = b index + 1;
down node = b index;
binominal(j).down = down node;
