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1. Pricing Options through the Trinomial Tree
Ciaran Cox
Mathematical Sciences,
School of Information Systems, Computing and Mathematics,
Brunel University
Supervisor: Jacques-´Elie Furter
2014
December 15, 2014
2. Abstract
I begin by a basic definition of option contracts and the pricing of these options through the
Black-Scholes model, which is based upon the Geometric Brownian Motion (GBM). Using this
model, one can solve for the implied volatility on an option through Newton’s iteration for find-
ing a root of a function. Binomial and trinomial lattice methods are alternate ways of pricing
options, but still assume that the stock price follows the GBM and constant volatility throughout
the option. European, American, Barrier and double barrier knockout options are priced using
the trinomial tree. Delta, Theta and Gamma Greeks are calculated through the Black-Scholes
model, with a comparison of prices on the Delta and Gamma through the trinomial tree. Using
these greeks, I move on to the delta-hedge rule and a delta tolerance applied practically to mar-
ket data from the Bloomberg terminals, with comparison of different strike prices on different
companies, concluding with a brief overview in to transaction costs.
3. Acknowledgements
A special thanks to my supervisor Jacques-´Elie Furter for his increased support throughout my
project. Also like to thank my mum, dad and my sister for inspirational motivation throughout
my university studies, along with all my friends and a special thanks to Joel Johnson for excellent
support during the project.
4. Contents
1 Introduction 6
1.1 Geometric Brownian Motion (GBM) . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Implied Volatility by the Black-Scholes Equation . . . . . . . . . . . . 10
2 Lattice models 11
2.1 Formulation of the Binomial Option Pricing Model by replicating portfolios . . 14
2.2 Binomial model to the Trinomial model . . . . . . . . . . . . . . . . . . . . . 15
3 Trinomial Tree 17
3.1 Pricing a European option through the Trinomial Tree . . . . . . . . . . . . . . 17
3.2 Pricing an American option through the Trinomial Tree . . . . . . . . . . . . . 19
3.3 Pricing a Barrier option through the Trinomial Tree . . . . . . . . . . . . . . . 20
3.4 Pricing a Double Barrier Knockout option through the Trinomial Tree . . . . . 23
4 The Greeks 25
4.1 Greeks via the Black-Scholes Model . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Delta and Gamma through the Trinomial Tree . . . . . . . . . . . . . . . . . . 29
5 Hedging Strategies 30
5.1 Delta-hedging rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.1.1 Delta-hedge rule across different companies . . . . . . . . . . . . . . . 32
5.2 Rebalancing under delta tolerance . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Delta-hedging including Transaction Costs . . . . . . . . . . . . . . . . . . . . 35
6 Conclusion 36
1
5. 7 Recommendations and Further Work 37
7.1 Implied Trinomial Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
7.2 Further Hedging Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
A my impvol.m Matlab file 39
B t treesize.m Matlab file 40
C tree barrier upcall.m Matlab file 41
D t double bar.m Matlab file 43
E g com.m Matlab file 44
F delta rebalance.m Matlab file 45
G delta rebalance tol.m Matlab file 47
H Background and Project Plan 49
Bibliography 57
Bibliography 59
2
9. Chapter 1
Introduction
An option is a type of contract that gives the holder the right to, but not obligation to buy (call),
or sell (put) an underlying asset or instrument at a specified strike price on or before a specified
date. A European option can only be exercised at the maturity date while an American option
can be exercised at any point up to and including maturity. Pricing of these options, has been
around for a while, but in 1973 Fisher Black and Myron Scholes published a paper, ’The Pricing
of Options and Corporate Liabilities’([4]). They had an idea to hedge the option by buying or
selling the underlying asset in such a way to eliminate risk. With this they derived a stochastic
partial differential equation which estimates the price of the option over time. The Black-Scholes
equation led to a boom in finance and more specifically option trading around the world ([8]).
6
10. 1.1 Geometric Brownian Motion (GBM)
A Geometric Brownian Motion is a continuous-time stochastic process where the logarithm of
the randomly varying quantity, follows a Wiener process with drift ([9]). A stochastic process St
is said to follow the GBM, if the process satisfies the following stochastic differential equation
(SDE);
dSt = µStdt +σStdWt, (1.1)
where,
• Wt is a Wiener process,
• σ is the percentage volatility,
• µ is the percentage drift,
solving equation (1.1) under It¯o’s interpretation leads to ([10]),
St = S0e(µ−σ2
2 )t+σWt
. (1.2)
The GBM assumes constant volatility when realistically in practice the volatility changes over
time, maybe stochastic ([13]). Further Extensions of the GBM are ”Stochastic Volatility models
in which the variance of a stochastic process is itself randomly distributed ([12]).” Below are
some stochastic volatility models ([13]);
• Heston Model.
• Constant Elasticity of Variance Model, CEV Model.
• Stochastic Alpha, Beta, Rho, (SABR Volatility model)
7
11. 1.2 Black-Scholes Model
This model was first published by Fisher Black and Myron Scholes in their paper ”The Pricing
of Options and Corporate Liabilities”. (1973,[4]). Pricing of European options is based upon
there should not be any opportunities for risk-free profit in the market, (no-arbitrage principle).
Black and Scholes showed that ”it is possible to create a hedged position, consisting of a long
position in the stock and a short position in the option, whose value will not depend on the price
of the stock” ([4]). The Black-Scholes model is based upon the following assumptions;
• the stock price follows a Geometric Browian Motion,
• buying and selling any amount of stock with no transaction costs incurred,
• borrowing and lending takes place at the risk-free interest rate,
• non-dividend paying stock.
Under these assumptions the dynamic hedging strategy by Black and Scholes leads to the fol-
lowing partial differential equation;
∂V
∂t
+
1
2
σ2
S2 ∂2V
∂S2
+rs
∂V
∂S
−rV = 0. (1.3)
This equation is solved with boundary conditions depending on the characteristics of the options.
For the standard European vanilla options we find explicit solutions leading to the Black-Scholes
formulas. The values of the European vanilla call and put options, respectively, are:
C(S,t) = N(d1)S−N(d2)Ke−r(T−t)
, (1.4)
P(S,t) = Ke−r(T−t)
−S+C(S,t),
= N(−d2)Ke−r(T−t)
−N(−d1)S, (1.5)
where
d1 =
1
σ
√
T −t
[ln(
S
K
)+(r +
σ2
2
)(T −t)], (1.6)
d2 =
1
σ
√
T −t
[ln(
S
K
)+(r −
σ2
2
)(T −t)],
= d1 −σ
√
T −t. (1.7)
The quantities appearing in (1.4-1.7) are
8
12. • T −t being the time left till maturity,
• S is the stock price,
• K is the strike price,
• r is the risk-free rate,
• σ is the volatility of the underlying stock,
• N(·) being the cumulative normal distribution.
9
13. 1.2.1 Implied Volatility by the Black-Scholes Equation
The volatility of the market is a quantity difficult to measure. One possibility is to assume that
the market satisfies the Black-Scholes model, and so the value of the traded options satisfy the
Black-Scholes formulae (1.4,1.5). With all the other quantities known except for σ, we can
set-up an equation in σ using (1.4) where C is the known traded value of the European vanilla
call option. Solving for σ
F(σ) = C −SN(d1(σ))+Ke−r(T−t)
N(d2(σ)), (1.8)
we find the Implied Volatility ([5]).
Solving (1.8) can be done by using the following Newton’s iteration for finding the root of a
function,
σi = σi−1 −
F(σi−1)
F (σi−1)
, (for i > 0), (1.9)
”In practice this iteration would be halted once |F(σi)| < ε for some user-specified tolerance
ε.”[5] A function file my impvol.m was created in Matlab to solve for the implied volatility
with inputs, Appendix A;
• S - Stock price,
• C - Call option price,
• E - Exercise/Strike price,
• r - Risk-free rate (annual),
• T - Time till maturity (in years),
• eps - convergence tolerance,
• vol0 - Initial volatility (annual).
10
14. Chapter 2
Lattice models
Another method for pricing a stock option is, a lattice model. Which divides time from now up
to expiration into N discrete time points with each point going to two possible states (Binomial)
or three possible states (Trinomial) all the way up to expiration date where the expected payoffs
are calculated. Taking these payoffs and iteratively discounting the values back to the present by
the continuously discounted risk-free interest rate, applying risk-neutral probabilities through a
series of one-step trinomial trees until back with one node being the option price.
The first lattice model was the Binomial Option Pricing Model (BOPM) By Cox, Ross and
Rubinstein (CRR)(1979,[1]). From the source node the underlying asset either goes up with
probability p or down with probability 1− p. This process repeats until you reach the expiration
date with all the possible stock outcomes. The expected payoffs are then calculated at expiration
by:
Call Option = max{ST −K,0},
Put Option = max{K −ST ,0}.
ST being the stock price at expiration (T) with K being the strike price on the option. Expected
payoffs are then discounted back through the tree by risk-neutral probabilities until you are back
at the source node and reach the option price. Figure 2.1 below shows the Binomial Tree.
It is computationally efficient to have a recombining tree over a non-recombining tree, because
if the tree recombines, there are only N +1 nodes at stage N, whereas there will be 2N nodes at
stage N on a non-recombining tree. To make the tree recombine CRR ([1]) made ud = 1. There
are three parameters in Binomial model u,d, and p, we therefore need three equations to solve
uniquely for the parameters. First equation comes from matching the expectation of return on
11
15. Figure 2.1: Non-Recombining Binomial Tree and a Recombining Binomial Tree
http://www.mathworks.co.uk/help/fininst/overview-of-interest-rate-tree-models.html
the asset in a risk-neutral world. The second from matching the variance. We get
pu+(1− p)d = er∆t
,
pu2
+(1− p)d2
−(er∆t
)
2
= σ2
∆t.
The third equation comes from making the tree recombine ([1]),
u =
1
d
.
After some rearranging and solving for the three parameters in the three equations above, results
showed[3]:
p =
er∆t −d
u−d
,
u = eσ
√
∆t
,
d = e−σ
√
∆t
,
where σ is the assets volatility and r being the risk-free interest rate. When the Binomial tree
has been created shown in Figure 2.2[17] with the expected future payoffs (leaf nodes), these
need to be continuously discounted back to earlier nodes by the risk-free interest rate, taking
into account the risk-neutral probabilities.
The formula for calculating each node,
Cn,j = e−r∆t
(pCn+1,j+1 +(1− p)Cn+1,j−1).
12
16. Figure 2.2: Multi Step Binomial Tree
Where Cn,j is the current option price for tier n with Cn+1,j+1 being the upper node and Cn+1,j−1
being the lower node at the next point in time. This process iterates through all time levels until
you are back at the source node at the present with the option price.
13
17. 2.1 Formulation of the Binomial Option Pricing Model by
replicating portfolios
Let a portfolio contain ∆ shares of the stock and an amount B invested in risk-free bonds with
a present value of ∆s+B. We want the option payoff equal to the portfolio payoff ([6]). Value
of replicating portfolio at time h with stock price Sh is ∆Sh + erhB. At the two possible states,
Su = uS and Sd = dS the replicating portfolio must satisfy ([7]):
(∆uSeδh
)+(Berh
) = Cu,
(∆dSeδh
)+(Berh
) = Cd,
with δ being the dividend yield, Cu being the upper option node and Cd being the lower option
node. Then solving for ∆ and B:
Cu −∆uSeδh
= Cd −∆dSeδh
,
∆ = eδh
(
Cu −Cd
uS−dS
), (2.1)
Berh
= Cu −∆uSeδh
, (2.2)
substituting (2.1) into (2.2) yields
Berh
= Cu −
uS(Cu −Cd)
uS−dS
,
B = e−rh
(
uCd −dCu
u−d
.
The cost of creating the option is the cash flow required to buy the shares and bonds ([7]):
∆S+B = e−rh
[
uCd −dCu
u−d
+e(r−δ)hCu −Cd
u−d
],
= e−rh
[Cu
e(r−δ)h −d
u−d
+Cd
u−e(r−δ)h
u−d
],
= e−rh
[Cu p+Cd(1− p)],
arriving at the required equation.
14
18. 2.2 Binomial model to the Trinomial model
One step in the trinomial tree can be seen as a combination of two steps of the binomial model,
either two up or down jumps or one of each. Taking equations from the binomial model and
applying two steps, Figure 2.3 shows the probabilities and the jump sizes for the Trinomial
model.
Figure 2.3: Binomial to Trinomial Formulation
Option prices were calculated using the trinomial tree with different values for the jump sizes
and probabilities depending on either CRR, RB and JR, and the equal probability tree for Google
between 23/04/2013 up to 20/12/2013 on a Call option with a strike price=1000. The volatility
used is the average of all the implied volatilities over the time of the contract at each discrete
daily time step. Plotted below in figure 2.4 are the trinomial trees calculated prices compared
to the market prices in the first row and the corresponding difference between the two on the
second row.
By figure 2.4, we can see that the equal-probability tree is closer to the market price than the
CRR and the RB and JR tree with an average difference of 2, compared to an average of 14 for
the CRR and RB and JR. Also on the difference plots of CRR and RB and JR, we notice the
15
19. Figure 2.4: Implied Google Call Option prices through the trinomial tree compared with market
prices
closer the option reaches maturity, the difference is approximately increasing. This could be due
to the increasing volatility over the time of the contract and only calculating the prices with a
constant volatility.
16
20. Chapter 3
Trinomial Tree
3.1 Pricing a European option through the Trinomial Tree
Solving the trinomial tree with the initial stock price (S0), number of time steps up to maturity
(n), strike price on the option (K), t treesize.m computes the option price with the choice of a
equal-probability tree, RB and JR tree and finally the CRR tree, Appendix (B).
The payoffs on the trinomial tree will be the leaf nodes comprising of 2n+1 elements as shown
below;
max(S0un −K,0)
max(S0un−1 −K,0)
...
max(S0u−K,0)
max(S0mn −K,0)
max(S0d −K,0)
...
max(S0dn−1 −K,0)
max(S0dn −K,0)
,
The values of u,m and d are pre-calculated along with the probabilities depending on what
trinomial tree is being solved. To discount these option payoffs back to present, two vectors
A and B with lengths 2n + 1 and 2n − 1 respectively are created with assigning the payoffs to
vector A, then dynamically iterating back through n time steps with the calculation of each of
the nodes at each time level, while simultaneously making the vector shorter until we are back at
the present with one node being the option price. For each time level the vector B is computed
17
21. by;
Bi−1 = er·dt
(puAi−1 + pmAi + pdAi+1)
dt = T/n,
pu + pm + pd = 1,
with k representing the time level and n being the number of iterations back to the present.
Clearing vector A with length 2(n−k)+1 and setting this vector equal to vector B. The current
vector B is not needed, so is cleared and a new vector B is created two elements shorter than the
previous vector A. These elements of B are computed again one time step closer to the present
being the option price at k equal to n. Plotted below in Figure 3.1 are the option prices, against
share price with a strike price of 25, σ being 0.25, time till maturity being a year with a risk-free
interest rate of 0.005.
Figure 3.1: European Call and Put Option Prices against Share price
When the stock price is equal to strike price, Figure 3.1 shows the option price for a call and a
put option are the same, also known as ’at the money’. When S > K the call option becomes ’in
the money’ with the put option becoming ’out of the money’, therefore the price of a call option
is increasing as the share price becomes deeper in the money. Vice versa for S < K as the put
option is now ’in the money’ and a call option being ’out of the money’.
18
22. 3.2 Pricing an American option through the Trinomial Tree
American option contracts can be exercised at any point up to and including maturity, therefore
the iteration sequence back to the present needs to be modified by re-defining the payoff at every
node of the tree ([14]);
Calloptionpayof f = max(Si,j −K,0),
Putoptionpayof f = max(K −Si,j,0),
where,
Si,j = S0uNu
dNd mNm
with Nu +Nd +Nm = n, n being the time level in between the present and maturity. Each node
for each time level is computed by,
Cn,j = max(optionpayof f,e−r∆t
[puCn+1,j+1 + pmCn+1,j + pdCn+1,j−1].
To solve for these changes we modify the matlab m file t treesize.m (Appendix B) by creating
a vector C the same size as vector A after transferring the values from vector B in the iteration.
The initial payoffs are calculated the same way, and then computing the stock prices for the
corresponding point in time by the up, middle and down jumps applied to the initial stock price.
This vector’s length will change with the point of time by 2(n−k)+1, with k representing the
point in time.
max(S0un−k −K,0)
max(S0un−k−1 −K,0)
...
max(S0u−K,0)
max(S0mn−k −K,0)
max(S0d −K,0)
...
max(S0dn−k−1 −K,0)
max(S0dn−k −K,0)
, (3.1)
the elements of vector A will be,
Ai = max(Ci,Ai).
The process repeats again one step closer to the present.
19
23. 3.3 Pricing a Barrier option through the Trinomial Tree
A barrier option is a path-dependent option whose price is equal to that of a European option,
depending if the price crosses or doesn’t cross a barrier up to maturity, otherwise the payoff is
equal to zero. This is represented by an indicator variable ’I’ taking values 1 or 0 and multiplying
this variable by the payoff function for a European option. To use the trinomial tree to price such
an option, we simulate 2n+1 runs through the Geometric Brownian Motion (GBM), matching
the required leaf nodes on the trinomial tree;
Si = Si−1e∆t(µ−σ2
2 )+σ
√
∆tε(i−1)
,i ≥ 1.
∆t =
T
n
Once the runs have been simulated the maximum or minimum of each run is taken depending on
whether the option is an up version or a down version with B > S0 or B < S0 respectively, with
the indicator variables. Once the indicator variables have been configured for each simulated
Table 3.1: Indicator Variables for Barrier Options
Up and In
max0<S≤ST >B I=1
max0<S≤ST <B I=0
Up and Out
max0<S≤ST >B I=0
max0<S≤ST <B I=1
Down and In
min0<S≤ST >B I=0
min0<S≤ST <B I=1
Down and Out
min0<S≤ST >B I=1
min0<S≤ST <B I=0
run, the payoffs of each run are computed by,
callpayof fs = I·max(ST −K,0),
putpayof fs = I·max(K −ST ,0),
20
24. ST being the final value of the corresponding simulated run. The payoffs being 2n+1 in length
these are simply plugged into the initial vector A from the Trinomial tree and iteratively dis-
counted back to the source node giving the final option price. Matlab file tree barrier upcall.m
is shown in Appendix (C). Plotted below in Figure 3.2 are the option prices of the different
barrier options while varying the barrier value, with inputs,
• S0 = 50,
• n = 250,
• K = 50,
• σ = 0.25,
• T = 1
• r = 0.005,
• dB = 0.1,m = 250.
Figure 3.2: Barrier Option Prices with varying barrier value
Two separate barrier vectors were used in Figure 3.2, one for the up version (top row of Figure
3.2) with the first barrier value being 50, increasing by 0.1 up to 75. The second still starting at
50 but decreasing by 0.1 down to 25 (bottom row of figure 3.2). One can see that for a up and out
21
25. option the price of the option is increasing as the barrier value increases, due to less simulated
runs crossing the barrier due to the volatility remaining constant. If the volatility was to increase
proportional to the increase in the barrier value, the option price would expect to approximately
maintain the same price. For the up and in barrier option, the price is decreasing due to the same
reasons. The put options on all barrier options increase or decrease more dramatically, than the
call option of the barrier values. This could be the time value of money in the call’s favour
against the put, therefore incorporating higher charges on put options as barrier value varies.
For the down and in, as the barrier moves further away from the initial stock price the option
becomes cheaper, again due to the constant volatility. Vice versa for the down and out barrier
option.
22
26. 3.4 Pricing a Double Barrier Knockout option through the
Trinomial Tree
Extending the single barrier option to a double barrier can be beneficial for informed investors
betting on the price-movements of the security while still maintaining the same strike price on
a cheaper option[14]. A double barrier knockout option payoff is equal to that of a European
option payoff, if the maximum and the minimum of the underlying asset over the life of the
contract is between the two barriers. A slight modification on the calculation of the indicator
variables is needed to incorporate these double barriers. The payoffs are computed the same way
Table 3.2: Indicator Variables Double Barrier Knockout Option
Double Barrier Knockout Option
max<uB and min>lB I=1
else I=0
as before and then discounted back through the trinomial tree to the source node giving the price
of the option contract. Plotted below in Figure 3.3 are the option prices against the difference
between the two barriers on the knockout option with inputs,
• S0 = 50,
• n = 250,
• K = 50,
• σ = 0.2,
• T = 1,
• r = 0.005,
• dB = 0.1 up to 25 away from S0 in both directions.
Figure 3.3 shows as the difference between the two barriers is increasing from the initial stock
price, the option price is increasing. This would be expected as the maximum and minimum
of each of the simulated runs up to maturity, are not breaching either of the two barriers giving
the European option payoff. Matlab file t double bar.m computes the price of a double barrier
knockout, Appendix (D).
23
28. Chapter 4
The Greeks
The Greeks are partial derivatives with respect to the underlying parameter to see the sensitivity
of small changes in that parameter. Delta measures the rate of change of the option price with
respect to the underlying security ([15]), ∆ =
∂V
∂S
. Delta being between 0 and 1 for long position
and 0 and -1 for a short position, signifying the amount of stock to hold with respect to number
of option contracts in the portfolio. This idea is known as the Delta-Hedging rule. Delta can also
be seen as the probability of that option being ’in the money’ at maturity ([16]). Theta measures
the sensitivity of the value of option price given the passage of time, commonly divided by the
number of days in a year ([15]),θ =
∂V
∂t
. Gamma measures the rate of change in the delta with
respect to the underlying security, therefore being a second order derivative, Γ =
∂∆
∂S
=
∂2V
∂S2
([15]). Gamma is commonly used as an extension of the delta hedging rule allowing for a wider
range of movements in the underlying security, known as Delta-Gamma-Hedging rule.
25
29. 4.1 Greeks via the Black-Scholes Model
The solution of the Black-Scholes model for a call option at a point in time till maturity (t) for
an underlying security (x) given by [18](p159-160),
c(t,x) = xN(d+(T −t,x))−Ke−r(T−t)
N(d−(T −t,x)),
d±(τ,x) =
1
σ
√
τ
[log(
x
K
)+(r ±
σ2
2
)τ],
N(y) =
1
√
2π
y
−∞
e−Z2
2 dZ =
1
√
2π
∞
−y
e−Z2
2 dZ.
Taking partial derivatives of the above equation to show the value of the required Greek under
the input parameters of current stock price (x), time till expiration (τ), strike price (K), risk-free
interest rate (r) and the stocks volatility (σ).
Delta
Cx(t,x) = N(d+(T −t,x)),
Theta
Ct(t,x) = −rKe−r(T−t)
N(d−(T −t,x))−
σx
2
√
T −t
N (d+(T −t,x)),
Gamma
Cxx(t,x) = N (d+(T −t,x))
∂
∂x
d+(T −t,x),
=
1
σx
√
T −t
N (d+(T −t,x)).
Plotting the above equations in figure 4.1 for share prices between 0 up to 50 in increments of
0.1 and the time till maturity of a year in increments of 0.2. Strike price of 30 with a risk-free
interest rate of 0.005 and a σ of 0.2.
More than half of the shares are shown to be purchased when the share price crosses the strike
price, with holding all the shares with a delta of 1 when the option contract is ’deep in the
money’. Reflecting a less riskier portfolio and incurring a cheaper cost of buying shares if
the share price holds around the strike price, reflecting an uncertainty of the option maintain-
ing ’in the money’. Theta showing the option looses more value per the passage of time the
closer the option reaches maturity around the share price equalling the strike price. While loos-
ing less value as time reaches maturity with the option being ’in the money’ or ’out of the
money’. Gamma showing the rate of range of Delta being the greatest nearer maturity concen-
trated around the strike price on the option. This is seen with Delta’s biggest change when the
26
30. Figure 4.1: Delta, Theta and Gamma Greeks via the Black-Scholes model varying time till
maturity
share price crosses the strike price. Plotted in figure 4.2 are the same three Greeks but taking
a range of strike price values from 5 in increments of 5 up to 45 with all other variables kept
constant.
Figure 4.2: Delta, Theta and Gamma Greeks via the Black-Scholes model varying Strike Price
27
31. Can see that the option contract looses more value when crossing the strike price with the pas-
sage of time (Theta), however is proportional to the current share price. Also delta clearly show-
ing the greatest change when the share price goes through the strike price, shown by Gamma.
Figure 4.3 shows how the Greeks change up to maturity with all other variables kept constant,
with either the option being ’out of the money’, ’at the money’ or ’in the money’. Time till
maturity of a year, risk-free interest rate of 0.05, and a σ of 0.1 were used for figure 4.3.
Figure 4.3: Delta, Theta and Gamma Greeks via the Black-Scholes model up to maturity
Keeping all other variables constant, one can see that the delta decreases closer the option reach-
ing maturity. This is expected due to less time for the share price to vary, possibly coming ’out
of the money’. A lower probability of the option maturing ’in the money’ being represented by
the Delta. Option contracts still with a significant time till maturity loose more value over the
passage of time when they are ’out of the money’ than option contracts being ’in the money’.
However gradually gets reversed closer the option reaches maturity ending with ’in the money’
options loosing greater value than ’out of the money’ options. Gamma, again showing the rate
of change of Delta with respect to the underlying asset, has the greatest change with ’out of the
money’ options linked to a decreased probability (Delta) of the option maturing ’in the money’
the closer the option reaches maturity.
28
32. 4.2 Delta and Gamma through the Trinomial Tree
Creating a function and denoting my trinomial tree by f(S,n,K,σ,T,r), Delta and Gamma
Greek’s are calculated through the tree by[14](p8-9),
∆ =
f(S+dS,n,K,σ,T,r)− f(S,n,K,σ,T,r)
dS
,
Γ =
f(S+dS,n,K,σ,T,r)−2 f(S,n,K,σ,T,r)+ f(S−dS,n,K,σ,T,r)
dS2
,
dS = Sσ
√
T,
• dS is chosen such that the amount is proportional to the volatility and the current share
price, while taking into consideration time till maturity.
Plotted in Figure 4.4 are the Delta and Gamma values against share price, via the trinomial tree
and the Black-Scholes model. From the figure we can clearly see the trinomial tree approxi-
mately follows the same values as the Black-Scholes model. This would be expected as both
methods are built on the assumption the stock price follows the Geometric Brownian Motion
(GBM), with assumed constant volatility up to maturity.
Figure 4.4: Delta and Gamma Greeks via the Trinomial tree and the Black-Scholes model
Done using matlab file g com.m, Appendix (E).
29
33. Chapter 5
Hedging Strategies
Investors like to diversify their risk against stock movements by going short on European call
options, while at the same time being long on the underlying asset. Or long on European put
option, while going short on the underlying asset. The amount of stock held is equal to the
Delta of the option multiplied by the number of option contracts purchased in the portfolio,
along with the multiple of lot size, (number of share’s the option contract gives right to buy/sell
at strike price at maturity). A portfolio with this characteristic is known to be delta-neutral,
the share price will vary leading up to maturity and in turn the Delta will change value. To
keep the portfolio delta-neutral, the underlying asset needs to be bought or sold appropriately
on the change of the delta leading up to maturity. Rebalancing the portfolio keeps the portfolio
more risk averse to small changes in the stock price. Ideally the number of rebalances would be
continuous, called self-financing portfolio but in practice is impossible due to transaction costs.
A high Gamma showing a high rate of change of Delta, indicates the portfolio becomes more
riskier the longer the time interval becomes between the portfolio rebalancing. The stock price
moving from S to S indicates the option price to move from C to C , however moves to C , the
difference between C −C is the hedging error[19](p361). Fixing this error will allow for larger
price jumps in the share price, making the portfolio less riskier than just the delta-hedging rule,
extending on to the delta-gamma-hedging rule.
30
34. 5.1 Delta-hedging rule
To begin the delta-hedging rule, an initial cost is incurred of setting up the portfolios positions.
Going long on the shares with a short European call option (lot size being a 100 shares), therefore
borrowing the initial cost minus the cost of the option contracts,
C0 = ∆0N100S0,
B0 = C0 −N f0.
C and B being the cost an amount borrowed respectively, N the number of option contracts
and f being the price of the call option with ∆ being calculated via the trinomial tree. At each
rebalancing point (i) the cumulative cost and borrowed money being ([21]),
Ci = Ci−1e
r
252 +N100(∆i −∆i−1)Si,
Bi = N100∆iSi −N fi.
At maturity the option can be exercised if ST ≥K giving the replication cost,
repcost = CT −N100K,
leading on to the gain after taking in to account the initial price of the option contracts,
netgain = f0Ne
rx
252 −repcost,
x being number days between initial purchase of option contract and maturity.
Call Option price data and stock price data was taken from the Bloomberg Terminals for Mi-
crosoft (MSFT) between 19/08/2013 up to 20/12/2013 with a strike price of 34, the delta-hedge
rule was applied to the data while rebalancing every day. The volatility used for the calculations
was the average of all the Implied Volatilities leading up to maturity. 10 option contracts were
purchased with a lot size of a 100 and a risk-free interest rate of 0.005, plotted in Figure 5.1 is
the cumulative cost, delta, stock price and the amount of money needed to borrow up to maturity
on the contract. Matlab file delta rebalance.m was used for calculations, Appendix (F).
The option matured in the money with a final stock price of 36.8 making the European call op-
tion ’in the money’, therefore exercisable with a strike price of 34. The cumulative cost of the
hedge was 34011 resulting in a replication cost of 11.0826. Giving final net value of -2.77. Such
a small loss in size, in comparison to the cost showing the delta-hedge rule eliminates more risk,
but in turn giving a lower return.
31
35. Figure 5.1: Delta-hedge rule, rebalancing every day
5.1.1 Delta-hedge rule across different companies
The same process was run again on Microsft (MSFT), Google (GOOG) and Apple (AAPL) each
with a variety of three strike prices, 10 option contracts with a risk-free interest rate of 0.005.
Following table shows key results along with net loss/gain.
Table 5.1: Delta-hedging rule comparison of companies
MSFT AAPL
ST 36.8 549.02
K 34 35 36 450 500 550
CT 34,027 34,723 29,446 513,770 525,280 245,330
Rep Cost 26.7219 -277.419 6,553.5 63,774 2528.1 -257,030
Net Value -18.4073 285.5298 6557.2 -63,085 -24,924 257,200
Net Value/CT -0.054 0.822 22.27 -12.28 -4.74 104.83
Table 5.1 showing the delta-hedge rule eliminating a lot of potential loss when ST hasn’t crossed
the strike price, while taking a nice return on options maturing ’in the money’. Seen with MSFT
with strike price of 36 and GOOG with a strike price of 1100, taking returns of 22.27% and
32
36. GOOG
ST 1100.6
K 900 1000 1100
CT 922,310 998,240 329,040
Rep cost 22,310 -1759.2 -770,960
Net value -21,994 1836.3 770,980
Net value/CT -2.38 0.184 234.311
234.311% respectively. Compared with loss return of 0.054% and 2.38% for MSFT and GOOG
respectively for the lower strike prices. Signifying option contracts with higher strike price in
the future become cheaper, reflecting a lower probability that the contract will mature ’in the
money’. This is shown by a small difference between the replication cost and net value for the
higher strike prices.
33
37. 5.2 Rebalancing under delta tolerance
Instead of rebalancing every data point (daily), modifying the delta rebalance.m (Appendix F),
with an additional input for delta tolerance. Only rebalancing if the absolute value of the change
between the delta of the previous rebalance, and the current delta is greater than the delta toler-
ance. If the tolerance is not met, leave the holding of shares the same. Doing this will reduce
the amount of transaction costs incorporated over the life of the option, however may not give
a higher gain due to the increased volatility closer to the option reaching maturity. More fre-
quent rebalancing would be required to hedge more of the investors risk, this could be done by
the delta tolerance decreasing closer the option reaches maturity. Better still make the decrease
proportional to the change of the implied volatility over time. Plotted below in Figure 5.2 is the
net gain of the hedging rule against delta tolerance being constant throughout the option. Gain
was calculated via Matlab file delta rebalance tol.m, Appendix (G).
One can see the net gain diminishing, as the tolerance increases. Showing that a portfolio with
Figure 5.2: Delta-hedging rule while varying Delta tolerance
more frequent rebalancing is more ideal, however transaction costs were not incorporated in
Figure 5.2.
34
38. 5.3 Delta-hedging including Transaction Costs
Buying and selling stocks on the market to rebalance the portfolio incurs transaction costs.
Either a fixed charge per share, a percentage of shares bought or sold, or just a flat fee regardless
of the number of shares ([20]). At each rebalancing point, additional charges are included in the
cumulative cost of the delta-hedge,
Ci = Ci−1e
r
252 +N100(∆i −∆i−1)Si + pSiN100(∆i −∆i−1),
where (p) is the percentage charge of the transaction, only being applied to number of shares
purchased keeping the portfolio delta-neutral. More additional costs occur in practice, the dif-
ference between buying and selling from the broker, known as the bid-ask spread. Stamp duty,
tax and other over night financing costs occur with the holding of your securities. More so-
phisticated pricing techniques of these options are required to give a more accurate and realistic
option price, with additional extension on to allowing the volatility to vary up to maturity on the
option contract.
35
39. Chapter 6
Conclusion
The equal probability trinomial tree being the most accurate against market prices, even with
assumed constant volatility with an average difference of 2 between the trinomial tree option
price and the market prices. Using the equal probability tree throughout for further calculations
due to the increased accuracy compared to CRR and JR. Using this trinomial tree for the calcu-
lation of the delta and gamma of an option, allows us to rebalance an option contract with the
underlying asset to minimize the risk to the market. This is known as the delta-hedge rule where
rebalancing is carried out on the portfolio to eliminate risk, however continuous rebalancing is
infeasible due to additional transaction costs. Further extension of the delta-hedging rule would
be the delta-gamma hedging rule, rebalancing the holding of the traded option with respect to
the delta on the underlying asset. Therefore a delta-gamma hedging rule allows for a larger
price movement in the underlying asset between rebalancing points. Extending this again by a
delta-gamma-vega hedging rule, incorporating an additional option in the portfolio taking ad-
vantage of the volatility between rebalancing points. Additionally transaction cost are incurred
from the broker, bid-ask spread. Buying and selling of the underlying asset are not of the same
value. So two share price vectors will need to be included in the calculations, one for buying the
underlying asset and one for selling. Doing this will include the brokers transaction cost as well
as adding the fixed charge percentage on the rebalancing transaction.
36
40. Chapter 7
Recommendations and Further Work
7.1 Implied Trinomial Tree
The trinomial tree assumes constant volatility throughout, an extension of this being the implied
trinomial tree. Where the implied volatilities are computed through the market prices, and the
volatility smile is interpolated across the tree varying the size of the jumps and time between the
jumps. Figure 7.1 shows a constant trinomial tree and an implied trinomial tree.
Figure 7.1: Trinomial tree and an Implied trinomial tree ([22])
37
41. 7.2 Further Hedging Techniques
Delta-Gamma hedging rule rebalances the holding of option contracts between the delta-rebalance
points. ”What is required is a position in an instrument such as an option that is not linearly de-
pendent on the underlying asset” ([19],p363). Letting Γ being the Gamma of a delta-neutral
portfolio and Γτ be the Gamma of a traded option, then the the overall Gamma of the portfolio
with wτ holding of the option contract being ([19],p363),
wτΓτ +Γ (7.1)
holding − Γ
Γτ
of the option contract will in turn make the portfolio gamma-neutral, but the port-
folio may not be delta-neutral anymore, so a rebalancing of the underlying asset is needed.
Vega is an another partial derivative of the Black-Scholes equation with respect to volatility,
ν = ∂V
∂σ ([15]). Having a holding of − ν
ντ
in a traded option will make the portfolio Vega neu-
tral, a portfolio cant be gamma and Vega neutral unless another traded option is bought into the
portfolio ([19],p365). Solving simultaneously the amount of options to hold for the Gamma and
Vega making the respective partial derivative equal to zero on the portfolio. Correspondingly
buying or selling the underlying asset to maintain delta neutrality, in turn made the portfolio
delta-gamma-vega neutral.
38
52. Appendix H
Background and Project Plan
Ciaran Cox (1115773)
Jacques Furter
Pricing options with trinomial trees
Aims and Objectives
• To understand the pricing of options and implement algorithms in Matlab programming.
• To compute prices of barrier options by trinomial trees and compare the price with the
Black-Scholes equation from Mathematical Finance (MA3667) module assignment.
• To understand the concepts of an Implied Trinomial Tree (ITT).
• Use trinomial trees to calculate the option greeks.
Project Plan
I will begin my project with a brief history of option pricing and some of the key breakthroughs
in mathematical finance, along with the definition of an option contract along with its features
and properties. Then I will talk about the binomial option pricing model by Cox,Ross and Ru-
binstein (CRR)(1979) and its variant by Rendleman-Barter (RB) and Jarrow-Rudd (JR)(1979).
Then extending the idea of a binomial model to a trinomial model by Boyle (1986) and imple-
ment algorithms in Matlab to formulate a trinomial tree and calculate the price of barrier options.
I will also mention the trinomial tree with a diffusion parameter by Kamrad and Ritchken (1991).
Black-Scholes equation by Black and Scholes (1973) will be covered and compared with trino-
mial trees. Also comparing the price a barrier option valued by the trinomial tree along with
49
53. the price of the same option calculated by the Black-Scholes equation from the Mathematical
Finance (MA3667) module. The family of option greeks will be covered and a few of them
calculated through the trinomial tree.
I will then extend from the Black-Scholes equation and trinomial tree which assumes constant
volatility to the Implied Trinomial Tree by Derman, Kani, and Chriss (1996). Relevant option
price data will also need to be collected either via the Bloomberg terminals or Datastream termi-
nals for the calculation of the implied volatility at different time points throughout the implied
trinomial tree, not all option price data will be available therefore interpolation will be required
to match the volatilities to the volatility smile.
Gant Chart showing the project layout is in Appendix 1.
Background
Introduction
An option is a type of contract that gives the holder the right to, but not obligation to buy (call)
or sell (put) an underlying asset or instrument at a specified strike price on or before a specified
date.
• European option can only be exercised at the expiration date.
• American option can be exercised at any time between the purchase date and the expiration
date.
• Bermuda option can only be exercised at certain times leading up to expiration date, which
are discussed in the option contract.
Pricing of these options has been around for a while, but in 1973, Fisher Black and Myron
Scholes published a paper, ’The Pricing of Options and Corporate Liabilities’.[1] They had an
idea to hedge the option by buying or seeling the underlying asset in such a way to eliminate
risk. With this they derived a stochastic partial differential equation which estimates the price of
the option over time.The Black-Scholes equation let to a boom in finance and more specifically
option trading around the world.[8]
Overiew
50
54. In the following I cover a brief overview of where I’m taking my project and what areas of
option pricing I’ll be covering. Firstly looking at the binomial option pricing model which was
the first of its kind by Cox, Ross and Rubinstein (CRR)(1979)[1], followed by the formulation
of the model by replicating portfolios. With an extension of the model to 3 states (trinomial)
first introduced by Boyle (1986)[5]. Concluding on Implied Trinomial Trees which is a further
extension by allowing for changing volatility over the time period of the asset by matching the
implied volatilities with the volatility smile.[10]
Binomial Option Pricing Model
Another method for a pricing a stock option is a lattice model, which divides time from now up
to expiration into N discrete time periods with each point going to 2 possible states (Binomial)
or 3 possible states (Trinomial) all the way up to expiration date where the expected payoff’s are
calculated. Then discounting yourself back through the tree until you reach the option price at
the source node.
The first lattice model was the Binomial Option Pricing Model (BOPM) By Cox, Ross and
Rubinstein (CRR)(1979)[1]. From the source node the underlying asset either goes up with
probability p or down with probability 1− p. This process repeats until you reach the expiration
date with all the possible stock outcomes. The expected payoffs are then calculated at expiration
by:
CallOption = max{ST −K,0},
PutOption = max{K −ST ,0}.
ST being the stock price at expiration (T) with K being the strike price on the option. Expected
payoffs are then discounted back through the tree by risk-neutral probabilities until your back at
the source node and reached the option price. Figure 1 below shows the Binomial Tree.
Figure 1 shows a non-recombining tree and a recombining tree. It is computationally efficient to
have a recombining tree over a non-recombining tree, because if the tree recombines, there are
only N + 1 nodes at stage N, whereas there will be 2N nodes at stage N on a non-recombining
tree. To make the tree recombine CRR[1] made ud = 1, making an up jump followed by a down
jump equal to your original position. There is 3 parameters in Binomial model u,d, and p, we
therefore need 3 equations to solve for the parameters. First equation comes from matching
the expectation of return on the asset in a risk-neutral world. The Second from matching the
51
55. Figure H.1: Non-Recombining Binomial Tree(Left) and a Recombining Binomial Tree(Right)
http://www.mathworks.co.uk/help/fininst/overview-of-interest-rate-tree-models.html
variance.
pu+(1− p)d = er∆t
,
pu2
+(1− p)d2
−(er∆t
)
2
= σ2
∆t.
The third equation comes from making the tree recombine,(CRR)(1979)[1]
u =
1
d
.
After some rearranging and solving for the 3 parameters in the 3 equations above, results
showed[3]:
p =
er∆t −d
u−d
,
u = eσ
√
∆t
,
d = e−σ
√
∆t
,
where σ is the assets volatility and r being the risk-free interest rate. When the Binomial tree
has been created (Figure 2) with the expected future payoffs (leaf nodes), these need to be
continuously discounted back to earlier nodes by the risk-free interest rate, taking into account
the risk-neutral probabilities. The formula is
Cn,j = e−r∆t
(pCn+1,j+1 +(1− p)Cn+1,j−1).
Where Cn,j is the current option price for tier n with Cn+1,j+1 being the upper node and Cn+1,j−1
being the lower node at the next point in time. This process repeats until your back at the source
node with the option price.
52
56. Figure H.2: Multi-Step Binomial Tree
http://investexcel.net/binomial-option-pricing-excel/
Formulation of the Binomial Option Pricing Model by replicating portfolios
Let a portfolio contain ∆ shares of the stock and an ammount B invested in risk-free bonds with
a present value of ∆s+B. We want the option payoff = portfolio payoff.[6] Value of replicating
portfolio at time h with stock price Sh is ∆Sh + erhB. At Sh = uS and Sh = dS the replicating
portfolio must satisfy[7]:
(∆uSeδh
)+(Berh
) = Cu,
(∆dSeδh
)+(Berh
) = Cd,
with δ being the dividend yield, Cu being the upper option node and Cd being the lower option
node. Then solving for ∆ and B:
Cu −∆uSeδh
= Cd −∆dSeδh
,
∆ = eδh
(
Cu −Cd
uS−dS
), (H.1)
Berh
= Cu −∆uSeδh
, (H.2)
substituting eq(1) into eq(2) yields:
Berh
= Cu −
uS(Cu −Cd)
uS−dS
,
B = e−rh
(
uCd −dCu
u−d
.
53
57. The cost of creating the option is the cash flow required to buy the shares and bonds[7]:
∆S+B = e−rh
[
uCd −dCu
u−d
+e(r−δ)hCu −Cd
u−d
],
= e−rh
[Cu
e(r−δ)h −d
u−d
+Cd
u−e(r−δ)h
u−d
],
= e−rh
[Cu p+Cd(1− p)].
Trinomial Model
The Trinomial model is an extension of the Binomial model. but taking an additional path at
each node of the stock price staying the same. This was first introduced by Boyle (1986) [5].
The foundations of the model are similar in the fact that the first two moments are matched
but with the first two moments of the Geometric Brownian Motion (GBM)[14], which behaves
similar to stock price movements.
E[S(ti+1)|S(ti)] = er∆t
S(ti),
Var[S(ti+1)|S(ti)] = ∆tS(ti)2
σ2
,
ud = 1.
The last constraint is needed to make the tree recombine, solving for the above equations
yields[14]:
u = eσ
√
2∆t
,
v = e−σ
√
2∆t
,
with the transition probabilities being:
pu = (
e
r∆t
2 −e
−σ ∆t
2
e
σ ∆t
2 −e
−σ ∆t
2
)2
,
pd = (
e
σ ∆t
2 −e
r∆t
2
e
σ ∆t
2 −e
−σ ∆t
2
)2
,
pm = 1− pu − pd.
The same discounting method is used from the Binomial model just extended to the trinomial
model:
Cn,j = e−r∆t
(puCn+1,j+1 + pmCn+1,j + pdCn+1,j−1).
54
58. Figure H.3: Trinomial Tree Example
http:
//www.24-something.com/2011/03/07/how-to-create-trinomial-option-pricing-trees-with-excel-applescripts/
This process keeps repeating until back at the source node just like the Binomial Model. Figure
3 below shows an example of a trinomial tree with the blue being the stocks price with the option
price beneath.
Implied Trinomial Trees (ITT)
Implied trees allows for changing volatility between nodes by extracting an implied evolution
for the stock prices in equilibrium from market prices of liquid standard options on the underly-
ing stock.[2] Making implied trees an extension to the Black-Scholes equation which assumes
volatility is constant. A couple of new concepts are needed for the calculation of ITT, Arrow-
Debreu prices and the volatility smile.
Arrow-Debreu prices are the sum of the product of the risklessly discounted transition proba-
bilities over all paths starting in the root of the tree and leading to node (n,i), with n being the
nth time level and i being the highest node on that level. [10]
The Volatility Smile is the plot of implied volatility against varying strike prices as shown in
figure 4 below. ‘In the money’ meaning the option is worth something, ‘at the money’ being the
option is at the strike price and ‘out of the money’ meaning the option is worthless.
There is also a reverse skew and forward skew also known as the volatility smirk. In the reverse
skew pattern the implied volatility’s are higher at lower strike prices than the implied volatility at
higher strike prices. More frequently appears for longer term equity options and index options.
[11] In the forward skew pattern, the implied volatility for lower strike prices are lower than the
implied volatility at higher strike prices, commonly seen for options in the commodities market.
55
59. Figure H.4: Volatility Smile
http://www.investopedia.com/terms/v/volatilitysmile.asp
[11]
The Implied trinomial tree desires the following properties to model the underlying price cor-
rectly [10].
1. Reproduces correctly the volatility smile.
2. Is risk neutral.
3. Uses transition probabilities from the interval (0,1).
The study of implied trinomial trees is currently a work in progress.
56
60. Bibliography
[1] Black, Fischer; Myron Scholes (1973). The Pricing of Options and Corporate Liabilities.
Journal of Political Economy 81 (3): 637654. doi:10.1086/260062. [1] (Black and Scholes’
original paper.)
[2] MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial Models Shape
Markets. Cambridge, MA: MIT Press. ISBN 0-262-13460-8.
[3] John C. Cox, Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified
Approach. Journal of Financial Economics 7: 229-263.
[4] http://www.goddardconsulting.ca/option-pricing-binomial-index.html
[5] P. Boyle, Option Valuation Using a Three-Jump Process, International Options Journal 3,
7-12 (1986).
[6] Professor P.A.Spindt Binomial Option Pricing
[7] https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=
3&ved=0CDgQFjAC&url=http%3A%2F%2Fwww2.fiu.edu%2F~dupoyetb%2FAdvanced_
Risk_Mgt%2Flectures%2Fweek%25201.ppt&ei=IPWDUu-gMMXIhAev8oCQDg&usg=
AFQjCNEOr7-1rmnXwdsiVze2JeAMszww2A&bvm=bv.56343320,d.ZG4&cad=rja
[8] P. Clifford, O. Zaboronski. Pricing Options Using Trinomial Trees (2008)
[9] E. Derman, I. Kani, N.Chriss. Implied Trinomial Trees of the Volatility Smile (1996)
[10] P.Cizek, K.Komorad Implied Trinomial Trees SFB 649 Discussion Paper (2005-007)
[11] http://www.theoptionsguide.com/volatility-smile.aspx
57
61. Appendix 1: Gant Chart for Pricing Options with Trinomial Trees Major Project
58
62. Bibliography
[1] John C. Cox, Stephen A. Ross, and Mark Rubinstein. 1979. Option Pricing: A Simplified
Approach. Journal of Financial Economics 7: 229-263.
[2] E. Derman, I. Kani, N.Chriss. Implied Trinomial Trees of the Volatility Smile (1996)
[3] http://www.goddardconsulting.ca/option-pricing-binomial-index.html
[4] F.Black and M.Scholes. The Pricing of Options and Corporate Liabilities The Journal of
Political Economy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654
[5] P.Date and S.Virmani MA3667:Mathematical and Computational Finance Assignment
[6] Professor P.A.Spindt Binomial Option Pricing
[7] https://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=
3&ved=0CDgQFjAC&url=http%3A%2F%2Fwww2.fiu.edu%2F~dupoyetb%2FAdvanced_
Risk_Mgt%2Flectures%2Fweek%25201.ppt&ei=IPWDUu-gMMXIhAev8oCQDg&usg=
AFQjCNEOr7-1rmnXwdsiVze2JeAMszww2A&bvm=bv.56343320,d.ZG4&cad=rja
[8] MacKenzie, Donald (2006). An Engine, Not a Camera: How Financial Models Shape
Markets. Cambridge, MA: MIT Press. ISBN 0-262-13460-8.
[9] Ross, Sheldon.M (2007). ”10.3.2”. Introduction to Probability Models
[10] http://en.wikipedia.org/wiki/Geometric_Brownian_motion
[11] Wilmott, Paul (2006). ”16.4”. Paul Wilmott on Quantitative Finance (2 ed.).
[12] Gatheral, J, (2006). The volatility surface: a practitioners guide, Wiley.
[13] http://en.wikipedia.org/wiki/Stochastic_volatility
59
63. [14] P. Clifford, O. Zaboronski. Pricing Options Using Trinomial Trees (2008)
[15] Haug, Espen Gaardner (2007). The Complete Guide to Option Pricing Formulas. McGraw-
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