Stochastic Volatility

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Stochastic Volatility

Kristina Andersson

U.U.D.M. Project Report 2003:18

Examensarbete i matematik, 20 poäng

Handledare: Ola Hammarlid , Swedbank Markets

Examinator: Johan Tysk

November 2003

Department of Mathematics

Uppsala University

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Abstract

In the original Black-Scholes model, the risk is quantified by a con-stant volatility parameter. It has been proposed by many authors that the volatilities should be modeled by a stochastic process to obtain a more realistic model. The volatility that corresponds to actual mar-ket data for option prices in Black-Scholes model is called the implied volatility. This volatility is in general dependent on the strike price, in contrast to the underlying assumption of Black-Scholes model. As a function of strike it forms a curve called ”volatility smile”. To explain this smile it has been proposed to study models allowing for a volatil-ity driven by a stochastic process. In the present paper a review of stochastic volatility is presented and three stochastic volatility models are studied in some detail. We study the volatility smile of these mod-els and show that in some cases we can reproduce a smile similar to the curves occuring in reality. We also study a corrected Black-Scholes pricing formula.

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1 Introduction

A financial derivative, for example an option, is a contract defined in terms of some underlying asset which already exists on the market, such as a stock. The simplest financial derivative is a European call option. A call option gives the holder of the option the right, but not the obligation, to buy the underlying asset to a given price, the strike price, at a given time, the expiration day. European options can thus only be exercised on the expiration day. A European option can be priced by the Black-Scholes formula, see [1].

In the original Black-Scholes model, the risk is quantified by a constant volatility parameter. A natural generalization is to model the volatility by a stochastic process. In reality, the volatility process cannot be directly observed. However, through empirical studies of the stock price returns one has observed that the estimated volatility fluctuates randomly around a mean level. The process is said to be mean-reverting.

The implied volatility I is defined to be the value of the volatility pa-rameter that must go into the Black-Scholes formula (121) in the Appendix to match the observed price, Cobs,

CBS(t, S, K, T, I) = Cobs, (1)

where CBS is the Black-Scholes price. When studying a real market price

the implied volatility is not constant as assumed in Black-Scholes model, but varies. The result when plotting the implied volatility against K or the ratio of the strike price to the current price is called the smile curve or volatility smile. It is interesting to investigate if there are any smile effects in reality. In Figure 1 an example from the Swedish option market is shown. The implied volatility is plotted against the ratio of the strike price to the current price for three different underlying stocks, AstraZeneca, Skandia, and Nokia. As seen in the figure there is a distinct smile effect for all options.

2 Stochastic volatility

In a stochastic volatility model the volatility is changing randomly according to some stochastic differential equation or some discrete random processes. Our main reference for this theory is [2]. A stochastic volatility model introduces more random sources then traded assets. According to general market theory, see for instance the meta-theorem in [1], the model is not complete since the number of random sources is greater than the number of underlying traded assets. Pricing in a market with stochastic volatility is thus an incomplete market problem, which means that there does not exist a unique martingale measure, and the derivative cannot be perfectly hedged

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0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Strike Price/Current Price

Implied Volatility

Skandia AstraZeneca Nokia

Figure 1: Observed implied volatility for three different call options, the underlying stocks are AstraZeneca, Skandia, and Nokia.

with just the underlying asset and a bank account. In this chapter three different stochastic models will be introduced, Hull-White, Cox-Ingersoll-Ross, and Log Ornstein-Uhlenbeck. The Hull-White model is discrete and the other two are continuous. All models considered contain a risk-free asset with the dynamics dB = rBdt.

In a stochastic volatility model the asset price S satisfies the following stochastic differential equation

dS(t) = αS(t)dt + σ(t)S(t)dW (t) , (2) where σ(t) is the volatility process and W (t) is a geometric Brownian motion (see Appendix). When comparing the density function for the stock price with and without stochastic volatility, it can be seen that when the volatility is stochastic the density function allows for fatter tails.

2.1 General pricing

A method used to derive the price equation is to form a risk-less portfolio. The model used in this general pricing is

dS(t) = αS(t)dt + σ(t)S(t)dW (t) , (3)

σ(t) = f (Y (t)) , (4)

dY (t) = (a + bY (t))dt + cY (t)dV (t) , (5) where a and b are constants and

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The processes W and Z are uncorrelated but V and W are correlated. In the model described by equations (3)-(6) there is one traded asset S and two random sources W (t) and Z(t). In this case there is two sources of randomness instead of one as in the standard Black-Scholes case. When constructing a portfolio the derivatives cannot be perfectly hedged with just the underlying asset. Instead we also need a benchmark derivative called G. A risk-less portfolio Π is formed, contain the quantity −∆Sof the underlying

asset S, the quantity −∆G of another traded asset G (Benchmark option)

and the priced derivative P . The total value of the portfolio is

Π = P − ∆SS − ∆GG . (7)

The differential of the portfolio value is needed to construct a risk-less and arbitrage free portfolio, equation (7) gives

dΠ = dP − ∆SdS − ∆GdG , (8)

can, by using Itˆo´s formula be written as

dΠ = ∂P ∂t + 1 2 ∂2P ∂S2f (y) 2_S2₊1 2 ∂2P ∂y2c 2₊ ∂2P ∂S∂yf (y)Scρ dt +∂P ∂sdS + ∂P ∂ydY − ∆SdS − ∆G ( ∂G ∂t + 1 2 ∂2G ∂s2f (y) 2_S2₊1 2 ∂2G ∂y2c 2₊ ∂2G ∂S∂yf (y)Scρ dt +∂G ∂sdS + ∂G ∂ydY ) . (9)

Collected the dS and dY terms gives

dΠ = ∂P ∂S − ∆G ∂G ∂S − ∆S dS + ∂P ∂y − ∆G ∂G ∂y dY + ∂P ∂t + 1 2 ∂2P ∂S2f (y) 2_S2₊1 2 ∂2P ∂y2c 2₊ ∂2P ∂S∂yf (y)Scρ dt − ∆G ∂G ∂t + 1 2 ∂2_G ∂S2f (y) 2_S2₊1 2 ∂2_G ∂y2c 2₊ ∂2G ∂S∂yf (y)Scρ dt . (10)

The portfolio is made risk-less by eliminating the coefficients in front of dS and dY , ∂P ∂S − ∆G ∂G ∂S − ∆S= 0 , (11) ∂P ∂y − ∆G ∂G ∂y = 0 . (12)

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The relative parts ∆G and ∆S can be solved for, giving ∆G = ∂P ∂y ∂G ∂y −1 , (13) ∆S = ∂P ∂s − ∂G ∂s ∂P ∂y ∂G ∂y −1 . (14)

Thus, when balancing the portfolio according to (13) and (14) the risk is eliminated. The portfolio can be made arbitrage free by putting the dt-terms in (10) equal the risk-free rate r

dΠ = ∂P ∂t + 1 2 ∂2P ∂S2f (y) 2_S2₊1 2 ∂2P ∂y2c 2₊ ∂2P ∂S∂yf (y)Scρ dt + ∆G ∂G ∂t + 1 2 ∂2_G ∂S2f (y) 2_S2₊1 2 ∂2_G ∂y2c 2₊ ∂2G ∂S∂yf (y)Scρ dt = rΠdt (15) where Π = P − ∆SS − ∆GG. By inserting ∆G and ∆S from (13)and (14)

and multiplying both side by ∂G/∂y one obtains ∂P ∂t + 1 2 ∂2_P ∂S2f (y) 2_S2₊1 2 ∂2_P ∂y2c 2 + ∂ 2_P ∂S∂yf (y)Scρ − rP + rS ∂P ∂S ∂P ∂y −1 = ∂G ∂t + 1 2 ∂2G ∂S2f (y) 2_S2₊1 2 ∂2G ∂y2c 2 + ∂ 2_G ∂S∂yf (y)Scρ − rG + rS ∂G ∂S ∂G ∂y −1 . (16)

It is important to note that the left hand side in (16) does only depend on P and the right hand side does only depend on G. Both side are thus equal to some function k(t, S, y). The equation governing P can be written as

∂P ∂t + 1 2 ∂2P ∂S2f (y) 2_S2₊ 1 2 ∂2P ∂y2c 2₊ ∂2P ∂S∂yf (y)Scρ − rP + rS∂P ∂S = k(t, S, y) ∂P ∂y . (17) The terminal condition for P is the contract function h(S), i.e., P (T, S, y) = h(S(T )). The function k cannot be determined by arbitrage theory alone. However, it is completely determined in terms of the traded benchmark asset G. One can say that the market knows the function k. Without loss of generality one can write k as [2]

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where

Λ(t, S, y) = ρ α − r f (y)

+ γ(t, S, y)p_{1 − ρ}2_. ₍₁₉₎

Now, the unknown function is γ(t, S, y), which is called the market price of risk. Notice that when the processes W and B are totally correlated, i.e., ρ = 1, the model in equations (3)-(6) does only contain one random source and the market is complete. Hence, in this case the function k should be uniquely determined and not depend on the unknown market price of risk.

2.2 Stochastic volatility models

The volatility cannot be observed directly, since it is not traded. However from empirical studies of the stock price one can derived the stock price return, dS/S, and from this estimate the volatility. From these observation, the volatility seems to be low for several days, then high for a period and so on, in other words like a mean-reverting process. Figure 2 shows simulated volatility and corresponding returns for the LogOU model. In this section three different stochastic volatility models will be considered. The first is the Hull-White model, which is lognormal and not a mean-reverting model. The other two models, Cox-Ingersoll-Ross and Log Ornstein-Uhlenbeck, are mean-reverting models.

2.2.1 Hull-White

The general Hull-White model [3] is a special case of the model given by equations (3)-(6), and has the following dynamics

dS(t) = αS(t)dt + σ(t)S(t)dW (t) , (20) dY (t) = bY (t)dt + cY (t)dV (t) , (21) where the volatility is given by σ(t) = f (Y (t)), f (y) = √y, b < 0, and W is uncorrelated to V . Here a special case will be studied where the volatility can only take on two values, i.e., jump between high and low volatility. It is assumed that the volatility is uncorrelated, that is ρ = 0. This means that under the risk-neutral probability Q the volatility is uncorrelated with W (t). In this case the price of a call option can be shown to be given by, compare [2], C(t, S, y) = E(γ)hCBS(t, S, K, T, p σ2_{) | Y (t) = y}i_, ₍₂₂₎ where σ2₌ 1 T − t Z T t f (Y (x))2dx , (23)

and Y (t) is a Markov process with two states. Notice that σ2 _{is a stochastic}

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European option prices produce a smile curve for any volatility process un-correlated with the Brownian motion driving the price process. The σ2 _{is a}

Bernoulli variable under the measure Q(γ)

σ2 ₌ σ₁2 with probability p , σ2 2 with probability 1-p . (24)

Now, applying Hull-White formula (22) with (24) gives

CBS(K; I(p, K)) = pCBS(K; σ1) + (1 − p)CBS(K; σ2) , (25)

where CBS denote the standard Black-Scholes formula.

2.2.2 Cox-Ingersoll-Ross

Cox-Ingersoll-Ross model (CIR), is a mean-reverting model with the dy-namics

dS(t) = αS(t)dt + σ(t)S(t)dW (t) , (26) dY (t) = (a + bY (t))dt + cpY (t)dV (t) , (27) where a, b, and c are constants. The volatility is given by σ(t)=f (Y (t)) where f (y) = √y. The processes V and W are correlated according to dV (t) = ρdW (t) +_{p1 − ρ}2_{dZ(t), where W and Z are uncorrelated. The}

Y (t) process is mean-reverting if a > 0 and b < 0. In the mean-reverting case Y (t) tends to revert around a level −a/b with a reversion rate −b according to [5]. In the CIR model Y (t) is a non-central chi-square distribution and the expectation and variance are given by

E [Y (t)|Y (0) = y] = −a_b +y +a b e−|b|t, (28) Var [Y (t)|Y (0) = y] = ac 2 2b2 − c2 b y +a b e−|b|t+c 2 b y + a 2b e−2|b|t. (29)

Note that the limiting distribution of Y(t) is a gamma distribution with expectation −a/b and variance ac2/2b2.

2.2.3 Log Ornstein-Uhlenbeck

The last model to be introduced is the Log Ornstein-Uhlenbeck (LogOU) process which is also mean-reverting. The model dynamics is given by

dS(t) = αS(t)dt + σ(t)S(t)dW (t) , (30) dY (t) = (a + bY (t))dt + cdV (t) , (31)

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0 0.05 0.1 0.15 0.2 0.25 0.45 0.5 0.55 Time Volatility 0 0.05 0.1 0.15 0.2 0.25 −2 −1 0 1 2 Time Returns

Figure 2: Simulated volatility, ey _{path in Log OU model and corresponding}

return. The parameters are a=-70, b=-100, c=0.5,ρ = −0.2, rate=0.029, and time=3 months.

where a, b, and c are constants. The volatility is given by σ(t) = f (Y (t)) where f (y) = ey. Here V and W are correlated according to dV (t) = ρdW (t) +p1 − ρ2_{dZ(t), where W and Z are uncorrelated. Figure 2}

il-lustrates a realization of the volatility process and the corresponding stock price returns with parameters a = −70, b = −100, c = 0.5, rate=0.029, and 3 months to maturity.

Y (t) is normally distributed with expectation

E [Y (t)|Y (0) = y] = −a_b +y +a b e−|b|t, (32) and variance Var [Y (t)|Y (0) = y] = c 2 2 |b|+ 1 − e−2|b|t. (33)

Note that, when t → ∞, the limiting distribution for Y (t) is a normal distri-bution with expectation −a/b and variance c2/2 |b|. It is also interesting to compute expectation and variance for the volatility, i.e., E[σY] and Var[σY],

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can be found to be FσY(s) = P (σY ≤ s) = P (e Y _{≤ s) = P (Y ≤ log s) = F} Y(log s) , (34) giving fσY(s) = d dsFσY(s) = fY(log s) · 1 s, (35)

The density function for Y is given by

fY(y) = √1

2π˜σe

−1

2(y−˜µ)2/˜σ2, (36)

where ˜µ=E[Y ] and ˜σ2_{=Var[Y ] are given by equation (32) and (33). The}

density function for σY is then given by

fσY(s) = 1 s√2π˜σe

−1

2(log(s)−˜µ)2/˜σ2, (37)

which is the density function for a lognormal distribution. The expectation for the volatility is given by

EheY (t)i= eµ+˜ 12σ˜2, (38)

and the variance

VarheY (t)i= e2˜µe2˜σ_{− e}σ˜2. (39) We will return to these models in Chapter 4.

3 The corrected pricing formula

In this section the corrected Black-Scholes price formula is derived. The formula is general for all models with mean-reverting volatility and does not require estimation of the current volatility. The corrected Black-Scholes price is given by P = P0− (T − t) V2S2∂ 2_P 0 ∂S2 + V3S 3∂3P0 ∂S3 , (40)

where P0 is the Black-Scholes price with constant volatility ¯σ and (T − t)

is the time to maturity. The parameters V2 and V3 will be computed later

for a European call option. Formula (40) is not dependent on the choice of volatility model, but in the present text the LogOU model is assumed. Before deriving the above formula, the concept of invariant distribution is reviewed.

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3.1 Invariant distribution and important parameters

The invariant distribution of the stochastic process Y can be obtained through finding an initial distribution for the process such that the process has the same distribution at a later point in time. The invariant distribu-tion does not change in time and for Y the invariant distribudistribu-tion will be a distribution for Y (0) such that

E[Lg(Y (0))] = 0 , (41)

where L is the infinitesimal generator of the Y process. For the LogOU process it is given by

L = (a + by)_∂y∂ +1 2c

2 ∂2

∂y2 . (42)

The density function for the invariant distribution for the LogOU process is given by, according to [2]

φ(y) = _√ 1 2πν2 exp −(y − m) 2 2ν2 , (43)

where m=−a/b and ν2=−c2/2b. The parameter b is the rate of mean re-version, and the inverse of b is the typical correlation time ε = −1/b. The variance ν2 _{controls the size of the fluctuations.}

3.2 The pricing partial differential equation in terms of ε

The partial differential equation for the LogOU process can be written in terms of ε ∂Pε ∂t + 1 2f (y) 2_S2∂2Pε ∂S2 + ν2 ε ∂2Pε ∂y2 + ρν√2 √ ε Sf (y) ∂2Pε ∂S∂y + r S∂P ε ∂S − P ε + 1 ε(m − y) − ν√2 √ ε Λ(y) ! ∂Pε ∂y = 0 , (44) where m = −a/b and y is the current value of volatility level.

It is convenient to write the partial differential equation above in terms of the operators L0, L1, and L2

1 εL0+ 1 √_εL1+ L2 Pε = 0 , (45)

where the operators are defined by L0= ν2 ∂2 ∂y2 + (m − y) ∂ ∂y, L1= √ 2ρνSf (y) ∂ 2 ∂S∂y − √ 2νΛ(y) ∂ ∂y, L2= LBS(f (y)) = ∂ ∂t + 1 2f (y) 2_S2 ∂2 ∂S2 + r S ∂ ∂S− 1 , (46)

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where LBS is Black-Scholes operator.

3.3 Asymptotic solution

The solution Pε has a limit when ε goes to zero and the method used here is to expand the solution Pε _{in powers of} √_ε

Pε= P0+√εP1+ εP2+ ε√εP3+ . . . (47)

and subsequently solve for the coefficients P0, P1, P2, . . .. The terminal

con-dition for P0 is the contract function h(S), i.e., P0(T, S, y) = h(S), and the

terminal condition for Pi, i ≥ 1, is Pi(T, S, y) = 0. Inserting the expansion

(47) into equation (45) and collecting powers of ε, the result is 1 εL0P0+ 1 √ ε(L0P1+ L1P0) + (L0P2+ L1P1+ L2P0) +√_ε(L0P3+ L1P2+ L2P1) + . . . = 0 . (48)

Step by step, the terms of order 1/ε, 1/√ε,. . . will be studied. The first term in (48) is

L0P0 = 0 . (49)

The operator L0contains partial derivatives with respect to y but no

deriva-tives with respect to S. One would therefore like to find a function which is independent of y, P0 = P0(t, S) with terminal condition P0(T, S) = h(S).

The next term in (48) is

L0P1+ L1P0 = 0 . (50)

It is known from above that P0 only depends on t and S, and L1 involves

derivatives with respect to y, yielding L1P0 = 0. The equation (50) is thus

reduced to L0P1 = 0. The operator L0 involves derivatives with respect to

y and one can once again search for a function which only depends on t and S, P1= P1(t, S), with the terminal condition P1(T, S) = 0. When summing

up, the two first terms in the expression (47) will not depend on y. The next term in order is the term of order 1,

L0P2+ L1P1+ L2P0 = 0 . (51)

It is known from the discussion above that P0 and P1 only depend on y, and

that L1 and L0 only involves derivatives with respect to y. Hence, L1P1 is

equal to zero and equation (51) reduces to

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Now, P0is only dependent on t and S. When regarding S as fixed, L2P0only

depends on y, and equation (52) is a Poisson equation for P2 with respect

to L0. In order to have a solution to the Poisson equation, L2P0 must be in

the orthogonal complement of the null space of L∗0, where

L∗0 = ν2 ∂2 ∂y2 + a b + y ∂ ∂y + 1 , (53)

and ν2_=−c2_{/2b for the LogOU process. According to [7] this solvability}

condition is equivalent to saying that L0P2 has mean zero with respect to

the invariant distribution, i.e.,

E [L2P0] = hL2P0i =

Z ∞

−∞L

2P0φ(y)dy = 0 , (54)

where φ in equation (43) solves L∗0φ = 0. The solvability condition above

requires that hL2P0i = 0. Because P0 does not depend on y the solvability

condition is reduced to

LBS(¯σ)P0 = 0 , (55)

where hL2i = LBS(¯σ) and ¯σ is the effective volatility defined by

¯

σ2 = Ef (y)2_{= hf}2_{i =}

Z ∞

−∞

f (y)2φ(y)dy , (56)

which is the average with respect to the invariant distribution of Y .

In conclusion, P0 is the solution of Black-Scholes equation with terminal

condition P0(T, S) = h(S) and σ = ¯σ, where ¯σ is given above.

Equation (52) can be written as

L0P2= −L2P0. (57)

By applying the solvability condition in equation (54), L2P0 in the right

hand side of (57) can be written as

L2P0 = L2P0− hL2P0i = 1

2 f (y)

2

− ¯σ2 S2∂2P0

∂S2 . (58)

Equation (57) is now given by

L0P2 = − 1 2 f (y) 2 − ¯σ2 S2∂2P0 ∂S2 . (59)

The solution of the Poisson equation in equation (59) is given by

P2 = − 1 2L −1 0 f (y)2− ¯σ2 S2 ∂2_P 0 ∂S2 (60) = −1 2(ψ(y) + c(t, S)) S 2∂2P0 ∂S2 , (61)

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where ψ(y) is a solution of the Poisson equation

L0ψ = f (y)2− ¯σ2. (62)

Continuing to the next order, the terms of order √ε must also equal zero, giving the equation

L0P3+ L1P2+ L2P1 = 0 . (63)

This is a Poisson equation for P3 with respect to L0. Moving L1P2+L2P1

to the right hand side of equation (63) gives

L0P3 = −(L1P2+ L2P1) . (64)

Again, applying the solvability condition in equation (54) we obtain

hL1P2+ L2P1i = 0 . (65)

Here P2 is already known, so hL1P2i is moved to the right hand side to yield

hL2P1i = −hL1P2i , (66)

and now we investigate hL2P1i. We note that P1 does not depend on y

and hL2i=LBS(¯σ) giving that the left hand side of equation (66) is equal to

LBS(¯σ)P1. Focusing on the right hand side of (66), we find that

−hL1P2i = 1 2hL1(ψ(y) + c(t, S))i S 2∂2P0 ∂S2 = 1 2hL1ψ(y)i S 2∂2P0 ∂S2 , (67)

where ψ is a solution of the Poisson equation (62). The result of the discus-sion above is that the solvability condition yields the equation

LBS(¯σ)P1 =

1

2hL1ψ(y)i S

2∂2P0

∂S2 . (68)

We compute, for a general function ϕ,

hL1ψ(y)ϕ(S)i = √ 2ρνSf (y) ∂ 2 ∂S∂y − √ 2νΛ(y) ∂ ∂y (ψ(y)ϕ(S)) =√2ρνSf (y)ψ0_(y) ∂ ∂Sϕ(S) − √ 2νΛ(y)ψ0_{(y) ϕ(S)} (69) Inserting the above expression into equation (68) gives

LBS(¯σ)P1 = 1 2 √ 2ρνSf(y)ψ0_(y) ∂ ∂S− √ 2νΛ(y)ψ0_(y) S2∂ 2_P 0 ∂S2 = √ 2 2 ρνf (y)ψ(y) 0_S3∂3P0 ∂S3 + √2ρνf(y)ψ(y)0₋ √ 2 2 νΛ(y)ψ(y) 0 ! S2∂ 2_P 0 ∂S2 , (70)

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and the terminal condition is P1(T, S) = 0. Following Fouque [2], the first correction is denoted by ˜ P1(t, S) =√εP1(t, S) , (71) which is a solution of LBS(¯σ) ˜P1 = H(t, S) , (72)

and the terminal condition ˜P1(T, S) = 0. Here H(t, S) is defined by

H(t, S) = V2S2 ∂2_P 0 ∂S2 + V3S 3∂3P0 ∂S3 , (73) where V2= ν √ −2b 2ρhfψ 0_{i − hΛψ}0_i_, V3= ρν √ −2bhfψ 0_{i .} (74)

It can be shown, see [2], that equation (72) together with the terminal condition has the solution

˜

P1(t, S) = −(T − t)H(t, S) . (75)

Finally, the corrected Black-Scholes price is given by

P ' P0+ ˜P1= P0− (T − t) V2S2 ∂2_P 0 ∂S2 + V3S 3∂3P0 ∂S3 . (76)

3.4 Expression for V2 and V3

In this section the calculations of V2 and V3 in terms of the model

param-eters for the Log Ornstein-Uhlenbeck and Cox-Ingersoll-Ross model will be presented. The expression for V2 and V3 in equation (74) can equivalently

be written as, compare [2],

V2 = 1 ν√_−2b Dh −2ρF + ρ(µ − r) ˜F +p_{1 − ρ}2_Mi _f2_{− hf}2_iE , V3 = −ρ ν√_−2bF f 2_{− hf}2_i_, (77)

where F, ˜F and M are primitive functions of f, 1/f and γ. So far the specific choice of model is not important. However, to proceed further it is necessary to specify f (y).

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3.4.1 Log Ornstein-Uhlenbeck

In order to calculate the expectation values in equation (77) the invariant distribution is needed. For the LogOU model it is given by

φ(y) = _√1 2πνe

−(y−m)2_/2ν2

, (78)

where m = −a/b and ν2 = c2_{/2 |b|. As mention before, in this model} f (y) = ey_{. By using the expression for V}

2 given in equation (77) and the

explicit expression for φ and f we obtain V2=

1 c

Dh

−2ρey− ρ(µ − r)e−y+p_{1 − ρ}2_Mi _e2y_{− he}2y_iE

= 1 c

Z ∞

−∞

h

−2ρey− ρ(µ − r)e−y+p_{1 − ρ}2_Mi _e2y_{− he}2y_i φ(y)dy = −2ρ c e9ν2/2+3m_{− e}5ν2/2+3m₋ρ c(µ − r) eν2/2+m_{− e}5ν2/2+m + √2 −2b p 1 − ρ2_{γ ¯}_σ2_{ν .} (79) Similarly V3 can been calculated, starting from V3 in equation (77)

V3 = −ρ c e y _e2y − he2yi = −ρ c Z ∞ −∞ ey e2y_{− he}2y_i φ(y)dy = −ρ c e9ν2/2+3m_{− e}5ν2/2+3m. (80) 3.4.2 Cox-Ingersoll-Ross

In this model Y (t) has a non-central chi-square distribution and the invariant distribution is given by a gamma distribution

ξ(y) = y

(α−1)_e−(y/β)

Γ(α)βα , y > 0 (81)

where α = 2a/c2 _{and β = −c}2_{/2b . Using equation (77) and f (y) = √y, V} 2 can be calculated as V2 = 1 β√_−2bα −4₃ρy3/2_{+ 2ρ(µ − r)}√y +p_{1 − ρ}2_γy (y − hyi) = 1 β√_−2bα Z ∞ 0 −4₃ρy3/2_{+ 2ρ(µ − r)}√y +p_{1 − ρ}2_γy (y − αβ) ξ(y)dy = _√ 1 −2bα αβγ p 1 − ρ2₊ Γ 1 2 + α Γ (α) pβρ [(µ − r) − β(1 − 2α)] ! . (82)

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Similar calculations for V3 yield V3= −ρ β√_−2bαh √ y (y − hyi)i = −ρ β√_−2bα Z ∞ 0 √ y (y − αβ) ξ(y)dy = −ρβ 3/2 √ −2bα Γ 3₂ + α Γ (α) , (83)

3.5 Implied volatility for a European call option

In this section the implied volatility for a European call option will be calcu-lated in terms of V2and V3. The implied volatility is computed by solving the

relation between theoretical and observed price given in equation (1) with respect to the implied volatility I. The approximating price is, according to the discussion above, given by P0+ ˜P1, where P0 = CBS. Black-Scholes

formula for a call option is given in Appendix by (121), d1 and d2 are given

by (122) and (123), in this case is σ the effective volatility ¯σ. In order to apply the approximative solution (76) it is necessary to calculate the second and third derivative of P0 with respect to S. The first derivative is known

as Delta,

∂P0

∂S = N (d1) , (84)

and the second derivative is called Gamma, ∂2_P 0 ∂S2 = ed2 1/2 S ¯σp2π(T − t). (85)

Fouque, see [2] introduced a new ”Greek” called Epsilon, defined as the third derivative, ∂3P0 ∂S3 = −ed21/2 S2_σ_¯_{p2π(T − t)} 1 + d1 ¯ σ_{p(T − t)} ! . (86) Equation (73) gives H(t, S) = Se d2 1/2 ¯ σ_{p2π(T − t)} V2− V3− V3d1 ¯ σ_{p(T − t)} ! , (87)

and the expression for ˜P1(t, S) is

˜ P1(t, S) = Sed21/2 ¯ σ√2π V3 d1 ¯ σ + (V3− V2) √ T − t . (88)

Taking the corrected pricing formula as the observed price, Cobs(K, T ) =

P0(t, S) + ˜P1(t, S) in equation (1), the relation that determines the implied

volatility is thus

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Equation (89) can be solved by expanding I as I = ¯σ +√εI1 + · · · , and

inserting this in the left hand side of (89). The implied volatility is then given by I = ¯σ + ˜P1(t, S) ∂CBS ∂σ (t, S, K, T, ¯σ) −1 + O(1/α) . (90) The derivative of the Black-Scholes price with respect to σ is the Vega,

∂CBS

∂σ =

Sed21/2√T − t √

2π . (91)

When putting (88), (91), and d1 into equation (90) the implied volatility

can be written as I = ¯σ + V3 ¯ σ3 r + 3 2σ¯ 2 −V_σ_¯2 −V_σ_¯3₃ log( K S) T − t ! + O(1/α) . (92)

The implied volatility is given in terms of strike price K, asset price S, time to maturity T − t, and the parameters A1 and A2 as the following

I = A1 " log K_S T − t # + A2+ O(1/α) , (93)

where A1 and A2 are defined as

A1= − V3 ¯ σ3 , A2= ¯σ + V3 ¯ σ3 r + 3 2σ¯ 2 −V_σ_¯2 . (94)

The parameters V2 and V3 are given by

V2 = ¯σ (¯_{σ − A}2) − A1 r + 3 2¯σ 2 , V3 = −A1σ¯3. (95)

The formula for the implied volatility given in equation (93) is not valid when the time to maturity is short or if the option is far out of money. One reason is that the Y (t) process does not have sufficient time to mean-revert many times.

4 Simulation and estimation

In this chapter simulation methods, results from the simulations and pa-rameter estimations will be presented. It is important to note that the calculations are done directly in the probability measure Q. The market price of risk is thus assumed to be zero. Common for all studied models are that the implied volatility has been calculated and plotted against strike price. In all cases it has resulted in smile curves.

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0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.2 0.4 0.6 0.8 1 1.2

Implied Volatility

Figure 3: Observed implied volatility for OMX call option (with circles) and Skandia call option (with stars).

4.1 Examples from the Swedish option market

Here, two examples from the Swedish option market will be considered. The underlying assets under consideration are Skandia and OMX. The deriva-tives are call options with one month to maturity. The data for the options is shown in Table 4.1. The implied volatility was calculated with help of Matlab´s function blsimpv which solves equation (1) numerically. The in-puts to this function are the current asset price, the exercise price, rate, time to maturity, and the option value. In Figure 3 the implied volatility is plotted against the ratio of the strike price to the current price for the call options introduced above. Looking at the figure, it is easy to see that there are distinct smile effects for these options.

4.2 Hull-White model

When calculating the implied volatility for the Hull-White model the start-ing point is the pricstart-ing formula given in equation (96) and equation (1), giving the equation

CBS(K, I(p, k)) = pCBS(K, σ1) + (1 − p)CBS(K, σ2) , (96)

where CBS denote the standard Black-Scholes formula with the only

un-known arguments p, σ1, and σ2. The equation for the implied volatility can

be written as

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Skandia Strike price Option price Implied volatility Current price:22.30 18 5.50 1.22 Time:1 month 20 3.30 0.78 Rate:0.035 22 1.55 0.53 24 0.85 0.58 26 0.30 0.53 28 0.20 0.61

OMX Strike price Option price Implied volatility

Current price:590.90 580 25.00 0.26 Time:1 month 590 18.00 0.24 Rate:0.029 600 12.80 0.23 610 8.25 0.22 620 5.25 0.21 630 3.90 0.23 640 2.00 0.21 650 1.15 0.21

Table 1: Strike price, option price, implied volatility, current asset price, rate and time to maturity for Skandia and OMX call options.

which can be solved with respect to I. Equation (97) can be solved by using some numerical method, e.g., the Newton-Raphson method or one of Matlab´s solver functions. The iterative scheme for the numerical solution using Newton-Raphson is In+1= In− F (In) V(In) , (98) where V = ∂F_∂I = ∂CBS ∂I . (99)

In the results as follows, Matlab’s solver function fzero has been used and the parameters in the model are arbitrary chosen. The next step is to plot the implied volatility for different values of the strike price, which results in a smile shaped curve. It is interesting to vary parameters in the model and investigate what happen with the smile curve.

Figure 4 shows volatility smile for different times to maturity in the Hull-White model. The shortest time is one half month, the longest is two years and the middle times are one, three and six months. A short time to maturity gives deeper smile then a long times gives. In agreement with market observations, ”options die smiling”.

Now, the volatility smile curve for four different values of p is studied. In this example σ1 = 0.9, σ2 = 0.4, rate=0.035, time=3 months, and current

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70 80 90 100 110 120 130 0.5 0.55 0.6 0.65 0.7 0.75 0.8 Strike Price Implied Volatility Short time Long time

Figure 4: Volatility smile for different times in Hull-White model using the parameters σ1= 0.9, σ2 = 0.4, rate=0.035, p = 0.3, and current price=100.

the bottom to the top 0.1, 0.5, 0.9, and 1.0. One can see that p = 1 gives constant implied volatility on the level 0.9. This is exactly the value of the parameter σ1 and p = 0.1 gives implied volatility around 0.45 which

almost is the value for the parameter σ2. The result agree with the model

in equation (24).

Back to the Swedish option market example introduced in Section 4.1, and the Skandia call option with one month to maturity. In Figure 1 the observed volatility smile for the option is shown. To calculate the observed implied volatility Matlab’s function blsimpv is used. It is interesting to see what parameters in Hull-White model which give the best fit for the observed curve. The unknown parameters are σ1, σ2, and p, and the price

formula is given by equation (96). The method used to find the unknown parameters is known as ”backing out” the parameters from the model. One uses least squares estimates to find the parameters, i.e., one solves the fol-lowing minimization problem

min

n

X

i=1

(Cobs(Ki) − Cmodel(σ1, σ2, p, Ki))2, (100)

where K is the strike price. Here Cobs(K) is the observed price which is

given from the market for different strike prices and Cmodel(σ1, σ2, p, Ki) is

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70 80 90 100 110 120 130 0.4 0.5 0.6 0.7 0.8 0.9 1 Implied Volatility Strike Price p=0.1 p=0.5 p=0.9 p=1.0 p=0.1 p=0.5 p=0.9 p=1.0 p=0.1 p=0.5 p=0.9 p=1.0

Figure 5: Volatility smile for different p values using the parameters σ1 = 0.9,

σ2 = 0.4, rate=0.035, time=3 months, and current price=100.

ˆ

σ2 = 9.2, and ˆp = 0.97. In Figure 6 the observed volatility smile is

com-pared to the model. The modeled smile is calculated using the estimated parameters. The results shows that the ”backed out” parameter are not so good estimates, perhaps depending insufficient data.

4.3 Methods for the continuous models

Two different methods have been used to calculated the implied volatility for the continuous models. In both methods the process Y has been simulated. As mention before, the volatility in CIR model is given by √y and in LogOU given by ey_{. The algorithm for this method is step by step:}

• Simulate the Y process according a to higher order schemes in [8], giving one trajectory.

• Calculate the mean value of the simulated trajectory.

We repeat the above procedure many times (in most cases 1000 times is sufficient) and then continue as follows:

• Calculate the price of the option by using Matlab’s function blkprice. The inputs in this function are the forward price of the underlying asset at time zero, strike price, rate, time to maturity and volatility. Use the simulated volatilities and calculate the mean value of the obtained prices. The mean value is taken as Cobs.

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0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

Implied Volatility

Figure 6: Implied volatility plotted against strike price/current asset price. The dotted line is observed implied volatility for Skandia given in exam-ple Section 4.1 and the solid line is the estimated implied volatility with parameters ˆσ1 = 0.33, ˆσ2 = 9.18, ˆp = 0.97, rate=0.035, and current asset

price=22.3.

To motivate above procedure:

P = e−rTEQ[h(S(T ))] = e−rTE[EQ_{[h(S(T ))|¯σ] ≈} 1 n n X i=1 EQ_{[h(S(T ))|¯σ}i] , (101) where EQ_{[h(S(T ))|¯σ}

i] is the Black-Scholes price with the volatility ¯σi.

• The function fzero in Matlab is then used to calculate the implied volatility, by solving the equation

F (I) ≡ CBS− Cobs= 0 , (102)

with respect to the implied volatility.

The second method is a Monte-Carlo simulation as in [9]. Apart from the simulated volatility one needs to simulate the stock price process S. After each simulated trajectory the contract function h(S) for the option is determined. In the case of a call option h(S(T )) = max(S(T )− Strike, 0). The price of the option at time T is then given by Cobs = e−rTE[h(S(T ))]. In

the same way as for the first method the implied volatility is found by using the function fzero in Matlab. The first method is relatively fast but can only handled the uncorrelated case. The Monte Carlo method can handle both uncorrelated and correlated cases but the simulations take long time.

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80 85 90 95 100 105 110 115 120 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.76 Strike Price Implied Volatility Short time Long time

Figure 7: Volatility smile for different times in CIR model, a=50, b=-100, c=40, rate=0.035, and current price=100.

The last method can be more effective when using variance reduction, see [9].

It is also interesting to estimate which values of the parameter that give the best fit to the observed volatility smile. One should be able to use the same method as in the previous section. However we have not done this due to lack of time.

4.4 Cox-Ingersoll-Ross model

We study what happens with the volatility trajectory when the parameters a, b, and c in this model change. From Section 2.2.2, it is known that the parameter −b is the rate of mean reversion. Results of simulated paths when the parameter b takes three different values are shown in Figure 8, the chosen values of b are −10, −100, and −1000. The quotient −a/b is held constant, which results in that a takes the values 2.5, 25, and 250. The random processes are the same in the three different realizations. One can see that b with the smallest absolute value gives the highest path. High absolute value of b press down the path, giving small fluctuations around the mean −a/b.

Volatility smiles for different times for the CIR model is shown in Figure 7. The parameters are a = 50, b = −100 and c = 40. What happen with the smile when, for example, the parameter c is smaller than 40 ? We compare four different values, c = 10, c = 20, c = 30, and c = 40 which are arbitrarily chosen. The result is that a small c gives a flatter smile than a large c, see Figure 9. One reasonable explanation to this can be that the density function for the stock price has fatter tails. Fat tails means that the

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0 0.05 0.10 0.15 0.20 0.25 0.4 0.45 0.5 0.55 Volatility 0 0.05 0.10 0.15 0.20 0.25 0.2 0.3 0.4 0.5 0.6 0 0.05 0.10 0.15 0.20 0.25 0.45 0.5 0.55 Time b=−10 b=−100 b=−1000

Figure 8: Simulated volatility paths in Cox-Ingersoll-Ross model for three different b. The highest path b = −10, the middle b = −100 and the lowest b = −1000. The other parameter values are a = 2.5, a = 25, a = 250, c = 40, time=3 months, rate=0.035, and current price=100.

risk for large movements increases. The only difference when changing the parameters a and b is that the smile curve changes level.

4.5 Log Ornstein-Uhlenbeck model

Simulated volatility for different values on the parameter c, c = 1, c = 3, and c = 5 is shown in Figure 11, where a = −0.2, and b = −0.3. When c takes low values the volatility has small variations, whereas a high c values gives larger variations. This behaviour can be understood by consider the variance formula of eY, equation (33). If all parameter values are constant except c this formula gives a large variance when the value of c is high and a small variance when the value of c is low. The rate of mean reversion b is varied, b=-10,-100, and -1000 while the quotient −a/b is held constant. This means that a takes the values 10 log(0.5), 100 log(0.5), and 1000 log(0.5). The simulated volatility paths are shown in Figure 10. In Figures 8 and 10 the same random processes are used. When comparing CIR and LogOU one can see that the trajectories are almost the same for the both models. When

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80 85 90 95 100 105 110 115 120 0.66 0.67 0.68 0.69 0.7 0.71 0.72 0.73 0.74 0.75 Strike Price Implied Volatility c=10 c=20 c=30 c=40

Figure 9: Volatility smile for different c values in CIR using the parameters a = 50, b = −100, rate=0.035, time=1/2 month, and current price=100.

looking at the volatility smile for different times in Log Ornstein-Uhlenbeck model, it can be seen that a short time to maturity gives a deeper smile then a long time gives, which is illustrated in Figure 12. The results of changing the parameters a and b gives only a change in level, but different values of c gives changes in the shape. High values gives a deeper smile than low values, which is illustrated, Figure 13 shows this smiles. The explanation to this behavior is the same as in CIR model, see above.

4.6 Estimated V2 and V3

In equation (93) in Section 3.5 the formula for the implied volatility for a European call option is given. With help of historical price data, A1 and A2

in this formula can be estimated. The straight forward method proposed in [2] is to estimate A1 and A2 from observed implied volatility. This can be

done with least squares estimate. Put ui = (log(K(i)/S))/(T − t) where K

is the strike price, S is the stock price, and T − t is the time to maturity. Then A1 and A2 are given by

A1= 1 n(

Pn

i=1IiPni=1ui) −Pni=1Iiui 1

n(

Pn

i=1ui)2−Pni=1u2i

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0 0.05 0.10 0.15 0.20 0.25 0.3 0.4 0.5 0.6 0 0.05 0.10 0.15 0.20 0.25 0.4 0.45 0.5 0.55 Volatility 0 0.05 0.10 0.15 0.20 0.25 0.45 0.5 0.55 Time b=−10 b=−100 b=−1000

Figure 10: Simulated volatility paths in LogOU model for three different b. The highest path b = −10, the middle b = −100 and the lowest b = −1000. The other parameter values was a = 10 log(0.5), a = 100 log(0.5), a = 1000 log(0.5), c = 40, time=3 months, rate=0.035, and current price=100.

A2= 1 n n X i=1 Ii− A1 n X i=1 ui ! , (104)

where I is the observed implied volatility. Using data from the OMX exam-ple and the results of the estimates are ˆA1 = −0.0579 and ˆA2 = 0.2410.

Inserting these values for A1, A2, and ¯σ = 0.2362 as a mean value of

the observed implied volatility, into equation (95) gives ˆV2 = 0.00039 and

ˆ

V3 = 0.00076. The second method is to back out the parameters σ, V2,

and V3 from observed implied volatility. The same method as before can be

used. The estimated values from the first method is used as a starting guess in this method. The result of this estimation is ˆσ = 0.2361, ˆV2 = 0.0004,

and ˆV3 = 0.0008. Comparing these two methods one can see that they give

the same estimates up to the third digit. In Figure 14 the observed and es-timated implied volatility for the OMX option is shown for the last method in this section.

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0 0.05 0.10 0.15 0.20 0.25 0 0.5 1 1.5 2 2.5 Time Volatility c=5 c=1

Figure 11: Volatility path for different c, c = 1, 3, 5 in Log Ornstein-Uhlenbeck model, rate=0.035, time=3 months, a = −0.2, b = −0.3, and current price=100. 80 85 90 95 100 105 110 115 120 0.512 0.513 0.514 0.515 0.516 0.517 0.518 0.519 0.52 Strike Price Implied Volatility Long time Short time

Figure 12: Volatility smile for different times in LogOU model, time=1/2, 1, 3 months, a=-41, b=-61, c=1.2, rate=0.035, and current price=100.

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80 85 90 95 100 105 110 115 120 0.51 0.52 0.53 0.54 0.55 0.56 Strike Price Implied Volatility

Figure 13: Volatility smile for different c values in LogOU, solid line c = 1, dashed line c = 2, and dotted line c = 3. Using the parameters a = −41, b = −61, rate=0.035, time=1/2 month, and current price=100.

0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 0.22 0.23 0.24 0.25 0.26

Implied Volatility

Figure 14: Observed and estimated implied volatility for OMX. A1=-0.0579

and A2=0.2410. The dotted line is the observed volatility and the solid line

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5 Summary and discussion

The volatility smile which cannot be explained by the Black-Scholes models can be derived in stochastic volatility models. We have reproduced such a smile from market data using Hull-White model. The continuous models proved to be more difficult to handle. The least square minimization did not converge when using the standard approach. Further study is here of great interest.

A corrected pricing formula is also discussed. Expressions for the cor-rection parameters for the two continuous models are found. In the case of a call options these parameters are explicitly computed using market data. A further study should entail a comparison to market option prices.

6 Acknowledgements

I would like to thank my supervisor Ola Hammarlid at Swedbank Markets for support and guidance during this work. I will also thank my examiner, Johan Tysk at the Department of Mathematics at Uppsala University for discussions and good comments. Finally, I would like to thank my fianc´e Jan-Ove for all encouragement.

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A

Appendix

A.1 Basic arbitrage theory

In this section Wiener processes will be described. A a more comprehensive discussion of the basic arbitrage theory can be found in reference [1].

A stochastic process W is called a Wiener process if the following con-ditions hold:

1. W (0) = 0.

2. The process W has independent increments, i.e., if r < s ≤ t < u then W (u) − W (t) and W (s) − W (r) are independent stochastic variables. 3. For s < t the stochastic variable W (t)−W (s) has Gaussian distribution

N (0,√_{t − s)}

4. W has continuous trajectories.

The Black-Scholes model consists of two assets, one risk-less B and one stock S with following dynamics

dB(t) = rB(t)dt (105)

dS(t) = αS(t)dt + σS(t)dW (t) (106) where r, α, and σ is the rate, the drift, and the volatility, respectively. The model is complete because it contains one traded asset, S and one random sources, W (t). The price at time t for the option is given by F (t, S(t)). The differential dF can be found by using Itˆo’s lemma

dF = ∂F ∂t + αS ∂F ∂S + 1 2σ 2_S2∂2F ∂S2 dt +∂F ∂SσSdW . (107) By introducing the quantities

αf ≡ 1 F ∂F ∂t + αS ∂F ∂S + 1 2σ 2_S2∂2F ∂S2 , (108) and σf = σS F ∂F ∂S , (109)

the differential may be written as

dF = α_fF dt + σ_fF dW . (110) Form a risk-less portfolio with the underlying stock and the option. Π is value of this portfolio and dΠ is the differential. A risk-less portfolio is

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obtained when the dW term in the differential vanish. If Π is self-financing we obtain the following dynamics

dΠ = Π ns ∂S S + nf ∂F F = Π [(nsα + nfαf)dt + (nsσ + nfσf)dW ] (111)

where ns is the relative part in stock and nf is the relative part in option.

From the risk-less criterion and that ns and nf are relative parts of the

whole portfolio the following equations must hold

nsσ + nfσf = 0 , (112)

ns+ nf = 1 . (113)

The linear system in (112) has the solution

ns= σf σf− σ , (114) nf = −σ σf− σ . (115)

The portfolio is arbitrage free if

nsα + nfαf = r . (116)

Inserting ns, nf, and αf into equation (116) and multiplying by (σf − σ)F

one obtains σfF α − σ ∂F ∂t + αS ∂F ∂S + 1 2σ 2_S2∂2F ∂S2 = (σf − σ) rF . (117)

By using (109) in (117) the following partial differential equation governing the price of the option is obtained,

∂F ∂t + rS ∂F ∂S + 1 2σ 2_S2∂2F ∂S2 = rF . (118)

With the above results the Black-Scholes equation can be formulated as ∂F ∂t + rS ∂F ∂S + 1 2σ 2_S2∂2F ∂S2 − rF = 0 , (119) F (T, S) = h(S) , (120) where F (T, S) is the price of the option at time T , when the underlying asset has the price S, and h(S) is the contract function.

The Black-Scholes equation can be solved analytically and the price of a European call option with strike price K and time of maturity T is given by C(t, S(t)), where

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where d1 = log(S/K) + (r + 1/2σ2_{)(T − t)} σ√_{T − t} , (122) d2 = d1− σ √ T − t , (123) and N (z) = √1 2π Z z −∞ e−y2/2dy . (124)

Here N (z) is the cumulative distribution of a normal random variable, with mean 0 and variance 1.

The contract function for the European call option is given by

h(S) = max(S − K, 0) , (125)

where h(S) gives the value of the option at expiration. If the strike price is lower than the underlying stock price at expiration the value of h(S) is larger than zero and the profit will be S − K. The other possible situation is that the strike price is higher than the underlying stock price, in which case the value of the contract function is zero and the option is worthless. The contract function for the European put option is given by

h(S) = max(K − S, 0) (126)

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References

[1] Bj¨ork Tomas. Arbitrage theory in continuous time. Oxford university press, Oxford, 1998.

[2] Fouque Jean-Pierre, Papanicolaou George, and Sircar K. Ronnie. Deriva-tives in financial markets with stochastic volatility. Cambridge university press, Cambridge, 2000.

[3] Hull John and White Alan. The pricing of options on assets with stochas-tic volatilities. J. Finance, 42(2):281–300, 1987.

[4] Renault E. and Touzi N. Option hedging and implied volatilities in a stochastic volatility model. Math.Finance, 6(3):279–302, 1996.

[5] Hull John and White Alan. Hull-White on derivatives. Risk Publications, London, 1996.

[6] Gut Allan. An Intermediate Curse in Probability. Springer-Verlag, New York, 1995.

[7] Fouque Jean-Pierre and Papanicolaou George and Sircar K. Ronnie. Mean-reverting stochastic volatility. International Journal of theoreti-cal and applied finance, 3(1):101–142, 2000.

[8] Asmussen Søren. Stochastic simulation with a view towards stochastic processes. http://www.maphysto.dk, July, 2003.

[9] Luenberger David G. Inverstment Science. Oxford university press, New York, 1998.

Stochastic Volatility