U.U.D.M. Project Report 2021:16

Examensarbete i matematik, 15 hpHandledare: Erik EkströmExaminator: Martin HerschendJuni 2021

Department of MathematicsUppsala University

Option pricing in the binomial model

Jasmine Vestman

Abstract

In this thesis, we will cover basic option theory and its pricing in thebinomial model. We will first look at the one-period model and thenmove on to the multiperiod model. Furthermore, I will look at the choiceof the volatility parameter in the multiperiod model. We will see thatwhen ∆t → 0, the best choice is for it to be as close to the underlyingparameter as possible. Moreover, the monotone relationship in the optionprice with respect to the risk-free rate will also be explored, where theresults will vary depending on which model we are looking at.

Contents

1 Background 21.1 Stock Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Binomial model 32.1 One-period time model . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Multiperiod time model . . . . . . . . . . . . . . . . . . . . . . . 7

3 Volatility 113.1 Exploring the choice of σ . . . . . . . . . . . . . . . . . . . . . . 11

4 Option prices and monotonicity in rate 134.1 Monotonicity in the one-period model . . . . . . . . . . . . . . . 134.2 Monotonicity in the multiperiod model . . . . . . . . . . . . . . . 14

References 16

1

1 Background

1.1 Stock Options

Stock options are one type of financial derivative which works as a contractbetween the seller and the buyer of the option. The agreement gives the optionholder the right but not obligation to buy or sell stock shares at a given periodin time, for a predetermined price. The predetermined price is called the strikeprice, and the end of the period is called the expiration date. If the optionholder decides to use the option, we say that the holder is exercising theoption. If the option holder is not exercising the option before or on theexpiration date, it is usually because the option is out of the money. The optioncan be either in the money, out of the money or at the money. If theoption is out of the money, the option holder will not make a financial gainfrom exercising the option contrary to when the option is in the money. Whenthe option is at the money, the strike price is the same as the underlying asset.There are several types of options, and below are given examples of commonoptions on the market today:

• European options: Gives the option holder the right but not the obligationto exercise the option at the expiration date.

• American options: Gives the option holder the right but not the obligationto exercise the option at any time up to the expiration date.

• Call options: Gives the option holder the right but not the obligation tobuy a stock share for the strike price.

• Put options: Gives the option holder the right but not the obligation tosell a stock share for the strike price.

In the thesis, the primary focus will be European call and put options. Furthermore,any reference or use of the word ’price’ will refer to the option’s premium. Theintroduction of this thesis will follow Tomas Bjork’s book, Arbitrage Theory inContinuous Time, chapter 2.

2

2 Binomial model

We start by defining the one-period time model, later on we will also define themultiperiod model.

2.1 One-period time model

Let our discrete-time model contain two financial assets, bonds and stock shares.Moreover, let t ∈ {0, 1} denote the running-time, R the interest rate and Bt, Stdenote the price of a bond and price of one share of stock at time t respectively.The price process of the bond is deterministic and given by:

B0 = 1,

B1 = (1 +R)B0

(1)

The stock price process however is a stochastic one. More specifically we assumethat today’s stock price S0 is known, and at time t1 the price will either go up,uS0, with probability pu or down, dS0, with probability pd = 1 − pu where0 < d < u and u, d, pu, pd are known variables. We also introduce the stochasticvariable Z:

Z =

{u, with probability pu

d, with probability pd(2)

and we can rewrite the stock price process as following:{S0 = s

S1 = s · Z(3)

S0

uS0

dS0

B0 (1 +R)B0

pu

pd

Figure 1: Price process in the one-period binomial model

3

Definition 2.1. Let p(x, y) ∈ R2 be a 2-tuple representing a portfolio p, wherex is the number of bonds in the portfolio at time t and y is the number of unitsof stocks. Then the value process of the portfolio is:

V pt = xBt + ySt ⇔

{V p0 = x+ yS0

V p1 = x(1 +R) + yS0Z(4)

Definition 2.2. A financial derivative, and in our case, a stock option is anystochastic variable, X = Φ(s). The function Φ is called the contract functionand is in the case of a call option defined below:

Φ(s) =

{s−K if s ≥ K0 if s ≤ K

(5)

where K is the strike price.

The price of stock options mainly depends on three elements, the time toexpiration, the underlying stock price, and the volatility of the stock.

Definition 2.3. If the condition

d < 1 +R < u (6)

is satisfied the model is arbitrage free.

This means that there is an imbalance between two or more markets and thereexist an opportunity to make a risk free profit. An example of an arbitrageportfolio is a portfolio p with the value process{

V p0 = 0

P (V p1 > 0) = 1(7)

To understand this, assume that (1 + R) > u > d, thus investing in the bondwill always give higher return than an investment in the stock which gives usan arbitrage opportunity. So, if we short the stock and invest all the money inthe bond, we are sure to make a profit of either s(1 +R)− su or s(1 +R)− sddepending on the stochastic process of the stock’s value.

Definition 2.4. The financial derivative X is said to be reachable if there existsa portfolio p such that:

P (V p1 = X) = 1 (8)

then we say that p is a hedging/replicating portfolio.

This means that the assets in the replicating portfolio will have the sameproperties and, most importantly, the same cash flows as the asset/assets itis replicating. Replicating portfolios are a fundamental part of option pricingand financial mathematics as a whole. We will later in the thesis see how theyare used to minimise the risks for the option writer.

4

Proposition 2.1 (Pricing principle 1). If a financial derivative X is reachablewith a replicating portfolio p, then the only way to price an option at t = 0without creating any arbitrage opportunity is:

Π(0;X) = V p0 (9)

where Π(t;X) is the price of X at time t.

Proof. If we letΠ(0;X) > V p0 (10)

then we can sell the option for Π(0;X) at t = 0 and invest the money in thereplicating portfolio p. The remaining money [Π(0;X) − V p0 ] we can save. Attime t = 1 the option holder will receive V p1 which in turn will be balancedout by our replicating portfolio and we thus have made a risk-free profit of[Π(0;X)− V p0 ].Similarly, we prove that Π(0;X) < V p0 also leads to an arbitrage opportunity.

Proposition 2.2. Given that the binomial model is free of arbitrage, thearbitrage free price of an option X at time t = 0 is given by:

Π(0;X) =1

1 +REQ{X} =

1

1 +R· {Φ(su)qu + Φ(sd)qd} (11)

Q is called a martingale measure and is a mathematical construction withQ(Z = u) = qu & Q(Z = d) = qd and qu, qd ≥ 0. The probabilities, qu and qdare called the risk neutral probabilities and are specified so that:

S0 =1

1 +REQ{S1} (12)

The martingale probabilities are given by:

qu =(1 +R)− d

u− dand qd =

u− (1 +R)

u− d(13)

X can also be replicated using:

x =1

1 +R· uΦ(sd)− dΦ(su)

u− d

y =1

s· Φ(su)− Φ(sd)

u− d

(14)

5

Let us sum this up with an example:

Example 2.1. We are going to calculate the price of an European call option,X, with strike price K = 230EUR, stock value, S0 = 200EUR, risk-free rate,R = 0.02, u = 7

5 , d = 57 , pu = 0.55 and pd = 0.45.

This gives us the following price process of the stock:

200

280

142.8571

pu

pd

thus the contract function is defined as:

Φ(S1) =

{50 if S1 > 230

0 if S1 ≤ 230

We can now use proposition 2.2 to calculate the martingale probabilities and thearbitrage free price: {

qu = 3524{1.02− 5

7} = 107240

qd = 3524{

75 − 1.02} = 133

240

Π(0;X) =1

1.02EQ{X} =

1

1.02{50 · qu + 0 · qd} = 21 +

523

612≈ 21.8546

and the claim is reachable with a replicating portfolio with the followingcomposition using equation (14) in proposition 2.2:

x =1

1.02·− 5

7 · 502435

= −51 and y =1

200· 50

2435

=35

96,

where we take a loan from the bank of 51 EUR and invest the money in 3596 shares

of the stock. Thus the value of the portfolio at time t = 1 is the following:{V p1 = −51 · 1.02 + 35

96 · 200 · 75 = 50 if S1 = 280

V p1 = −51 · 1.02 + 3596 · 200 · 5

7 = 0 if S1 = 142.8571

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2.2 Multiperiod time model

Before moving on we will briefly discuss the multiperiod model, again usingBjork’s definitions. The multiperiod model is just an extension of the one-periodmodel, where t denotes the step on the specific interval and T is the expirationdate given in years. The interval from 0 to T is divided into N equally longtime-steps, ∆t = T

N , where at each time-step the stock price can either go up ordown just as before with probability pu and pd respectively. However, we willin this part of the thesis use Luenberger’s definitions in Investment Science foru and d given below: {

u = eR∆t+σ√

∆t

d = eR∆t−σ√

∆t(15)

where σ is the volatility given in years.The price process of the underlying assets for t = 0, ...T −∆t are the following:{

Bt+∆t = eR∆tBt,

B0 = 1(16)

{St+∆t = StZt,

S0 = s(17)

where all the Zt are I.I.D. and defined as above in the one period model.

Definition 2.5. A portfolio strategy is a stochastic process

{pt(xt, yt) : t = 1, ..., T} (18)

such that pt is a function of St−1. The value process corresponding to theportfolio p is defined by:

V pt = xteR∆t + ytSt (19)

where xt is the amount invested in bonds and yt is the number shares of stocksinvested at t− 1 respectively and kept in the portfolio until time t.

Definition 2.6. A portfolio strategy p is self-financing if the portfolio pt(xt, yt)equals the purchase value of the newly bought portfolio pt+1(xt+1, yt+1), i.e. if:

xteR∆t + ytSt = xt+1 + yt+1St (20)

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Definition 2.7. An arbitrage possibility is a self-financing portfolio p with theproperties:

V p0 = 0,

P (V pT ≥ 0) = 1,

P (V pT > 0) > 0

However as we can see that in our model with our newly defined u and d thatthey are separated by eR∆t which means that the model itself will not allow forany arbitrage opportunities and just as in the one-period model, we will use theQ measurement, defined below:

Definition 2.8. The martingale measure is defined such that the followingrelationship holds:

s =1

eR∆tEQ{St+1|St = s} (21)

Proposition 2.3 (Binomial algorithm). Consider a T-claim X = Φ(ST ). Thenthis claim can be replicated using a self-financing portfolio. If Vt(k) denotes thevalue of the portfolio at node (t, k), then Vt(k) can be computed recursively bythe scheme: {

Vt(k) = 1eR∆t {quVt+1(k + 1) + qdVt+1(k)},

VT (k) = Φ(sukdN−k)(22)

where k denotes the number of up-moves that have been made and the martingaleprobabilities are given by: {

qu = 1

eσ√

∆t+1

qd = 1

e−σ√

∆t+1

(23)

The hedging portfolio is thus calculated by:{xt(k) = 1

eR∆t

uVt(k)−dVt(k+1)u−d ,

yt(k) = 1St−1

∗ Vt(k+1)−Vt(k)u−d

(24)

In particular, the arbitrage free price of the claim at t=0 is given by V0. Fromthe algorithm above it is also clear that we can obtain a risk neutral valuationformula.

Proposition 2.4. If a claim X is reachable(i.e. there exists a self-financingportfolio p such that P (V pT = X) = 1) then the only arbitrage-free price processfor X is:

Π(t;X) = V pt , t = 0, ..., T (25)

and the arbitrage free price at t = 0 of the claim X is given by:

Π(0;X) =1

(eR∆t)NEQ{X} =

1

(eR∆t)N

N∑k=0

(N

k

)qkuq

N−kd Φ(sukdN−k) (26)

8

We will in this example calculate the price process of a European call option inthe multiperiod model and we will see that there are more than one way to getthe option price at t = 0.

Example 2.2. Let s = 300, σ = 0.30, R = 0.15, T = 3 months, pu = 0.55,pd = 0.45, K = 305 we then have that ∆t = 3

12·3 and{u = e0.15· 1

12 +0.30√

112

d = e0.15· 112−0.30

√112

qu = 1

e0.30√

112 +1

qd = 1

e−0.30√

112 +1

To find the price of the option at time t = 0 we need to work backwards in thetree since the price of the option at t = T is given by Π(T ;X) = V pT and therefordeterministic. We can then look at the nodes on t = 2

12 as starting prices forthe stock in the one-period binomial model and use proposition 3.2 to get theoption prices.

300

22.37

331.25

38.92

278.57

7.73

365.76

64.55

307.59

16.36

258.68

0

403.87

98.87

339.64

34.64

285.63

0

240.20

0

t0 1 2 3

Figure 2: The price process of a option and its underlying assets given over 3months

9

Repeating the process for the nodes on t = 112 we can calculate the option price

at t = 0 with the resulting tree graph given in Figure 2 where the numbers inthe rectangles represent the stock price and the cursive numbers below the optionprice.Using proposition 2.4 we get the same results:

Π(0;X) =1

(e0.15/12)3

3∑k=0

qkuq3−kd Φ(sukd3−k)

=1

(e0.15/12)3

((3

0

)q3dΦ(sd3) +

(3

1

)quq

2dΦ(sud2) +

(3

2

)q2uqdΦ(su2d) +

(3

3

)q3uΦ(su3)

)= 0.9632(0 + 0 + 12.4046 + 10.8227) ≈ 22.37

10

3 Volatility

Volatility is an essential component in options pricing. According toInvestopedia(see [3] and [4]), volatility is a statistical measure of the stock’sfluctuation around the mean over a given period of time, given in percentage.Higher volatility is often associated with a higher risk for the option writer. Ifthe volatility is high, the option premium will go up in price, and lower volatilitywill affect the premium conversely. Mathematically speaking, the generalisedvolatility for time T is the standard deviation of the stock:

σ =

√V ar

{log

(STS0

)}/T (27)

3.1 Exploring the choice of σ

For a given volatility σ over a time period [0, T ] the volatility will be constantin each step t, and because of the independence between the stock values thevolatility over a given time period can be written as the sum of the variances

V ar

{log

(S∆t

S0

)+ ...+ log

(ST

ST−∆t

)}(28)

Thus one only have to look at the first step to examine σ.

σ2 = E

{log

(S∆t

S0

)2}− E

{log

(S∆t

S0

)}2

(29)

Since the stock price S0 is deterministic we can set it to an arbitrary constantand let the R = 0 for computational simplicity. If we let S0 = 1 we get thefollowing:

σ2 = E{log(S1)2

}− E

{log(S1)

}2

= p · log(us)2 + (1− p) · log(ds)2 −[p · log(us) + (1− p) · log(ds)

]2= p[log(us)2 − log(us)2p

]+ (1− p)

[log(ds)2 − log(ds)2(1− p)

]− 2log(us)p · log(ds)(1− p)= log(us)2p

[1− p] + log(ds)2(1− p)

[1− (1− p)

]− 2log(us)p · log(ds)(1− p)

= p(1− p)[log(us)2 + log(ds)2 − 2log(us)log(ds)

]=

= p(1− p)[log(us)− log(ds)]2 = p(1− p)[log

(us

ds

)]2

=

= p(1− p)[log

(u

d

)]2

=⇒ σ =√p− p2 · log

(u

d

)/√

∆t

(30)

Instead of p we will use the martingale probabilities which can be obtained bysolving

uq + d(1− q) = eR∆t (31)

11

since R = 0 the expression will equal to one and we get that

q =1− e−σ

√∆t

eσ√

∆t − e−σ√

∆t=

1

eσ√

∆t + 1(32)

substituting σ with A and plugging it into equation (30) we get:

σ =

√1

eA√

∆t + 1−(

1

eA√

∆t + 1

)2

log

(e2A√

∆t

)/√

∆t

=

√1

eA√

∆t + 1−(

1

eA√

∆t + 1

)2

· 2A

σ2

4A2=

1

eA√

∆t + 1

(1− 1

eA√

∆t + 1

)(33)

Let x = eA√

∆t + 1

σ2

4A2=

1

x

(1− 1

x

)=x− 1

x2

x2 − 4A2

σ2x+

4A2

σ2= 0(

x− 2

σA

)2

⇒ x =2

σA

eA√

∆t + 1 =2

σA

σ =2

eA√

∆t + 1A

(34)

Thus when ∆t→ 0, σ tends to A so the only way to price the option correctlyis to let A be as close to σ as possible in this particular model so when ∆t goesto zero the two quantities will be arbitrary close.

12

4 Option prices and monotonicity in rate

We will in this section explore the monotonicity of the option price in regard tothe rate, R, and what conditions needs to be put on Φ(s) to accomplish this.We will first look at this in the one-period model and later move on to themultiperiod model.

4.1 Monotonicity in the one-period model

We will start by exploring this for a call option, and than look at a more generalcase. For a call option, Φ(ds), will always equal zero, we can thus rewrite theequation and get:

Π(0;X) =1

1 +R

{(1 +R)− d

u− dΦ(us) +

u− (1 +R)

u− dΦ(ds)

}= (35)

1

1 +R

{(1 +R)− d

u− dΦ(us)

}(36)

taking the derivative of the function with respect to R gives us the followingexpression,

∂

∂RΠ(0;X) =

Φ(us)[(1 +R)(u− d)]− (u− d)[Φ(us)(1 +R− d)]

(1 +R)2(u− d)2= (37)

Φ(us)(1 +R)− Φ(us)(1 +R− d)

(1 +R)2(u− d)=

Φ(us)d

(1 +R)2(u− d)(38)

and since both Φ(us), u and d will greater than zero, and d < u we have thatΠ(0;X) will be a monotone non-decreasing function in R ∈ R, without anyadditional conditions to Φ(us).Now looking at a more general case, we get:

∂

∂RΠ(0;X) =

1

u− d

{1 +R−R(1 +R)2

Φ(us)− 1− d(1 +R)2

Φ(us)−

1 +R−R(1 +R)2

Φ(ds)− u− 1

(1 +R)2Φ(ds)

}=

Φ(us)d− Φ(ds)u

(1 +R)2(u− d)

The denominator will always be greater than zero because of u > d. So forΠ(0;X) to be a monotone increasing function of R we need Φ(us)d ≥ Φ(ds)u.Rewriting the expression we get:

13

Φ(us)d ≥ Φ(ds)u

⇔Φ(us)

u≥ Φ(ds)

d⇔

Φ(us)

us≥ Φ(ds)

ds

(39)

Now, for a generalised variable x ∈ R>0, we know that if Φ(x)x is a monotone

increasing function this will imply that the condition above is satisfied. To seethis, take for example an European call option, taking the derivative of the

expression proves that, Φ(x)x , indeed is a monotone increasing function in R.

Φ(x)

x=x−Kx

⇒ d

dx

Φ(x)

x=K

x2for ∀x 6= 0 (40)

4.2 Monotonicity in the multiperiod model

Now looking the multiperiod model with T = 2 and N = 2,

S

uS

dS

u2S

d2S

udS

t0 T

2T

Figure 3: Price process in the two-period binomial modelWe get the following expressions for u and d:{

u = erT2 +σ√

T2

d = erT2 −σ√

T2

14

and we have q such that:

uq + d(1− q) = erT2 =√ud

q =

√ud− du− d

=

√ud − 1ud − 1

=eσ√

T2 − 1

e2σ√

T2 − 1

(41)

The option price at t = 0 is given by:

Π(0;X) =1

e2R

(q2Φ(u2s) + 2q(1− q)Φ(uds) + (1− q)2Φ(d2s)

)(42)

Differentiation with respect to R gives:

∂Π

∂R= 2

1

e2Rq2

(− Φ(u2s) + su2Φ′(u2s)

)+ 4

1

e2Rq(1− q)

(− Φ(uds) + sudΦ′(uds)

)+ 2

1

e2R(1− q)2

(− Φ(d2s) + sd2Φ′(d2s)

)(43)

Thus ifxΦ′(x) ≥ Φ(x) ∀x ∈ R (44)

then∂Π

∂R≥ 0 (45)

and we have shown if equation (44) holds the option price will be monotoneincreasing with respect to the risk-free rate R, in the multiperiod binomialmodel for N = 2. This can also readily be extended into N-periods as well andwe can end this thesis by formulating a proposition:

Proposition 4.1. Consider an option X in the N-period binomial model, thenthe option price will be monotone increasing in the risk-free rate R if thefollowing condition is satisfied:

xΦ′(x) ≥ Φ(x) ∀x ∈ R (46)

15

References

[1] Tomas Bjork, Arbitrage Theory in Continuous Time, Oxford UniversityPress; 2nd edition (May 6, 2004), Chapter 2.

[2] David G. Luenberger, Investment Science, Oxford University Press;(NewYork, 1998), Chapter 11-12.

[3] Justin Kuepper: Volatility,https://www.investopedia.com/terms/v/volatility.asp,(Apr 16, 2021)

[4] J.B. Maverick: How Does Implied Volatility Impact Options Pricing?,https://www.investopedia.com/terms/v/volatility.asp, (Jan 5, 2021)

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