U.U.D.M. Project Report 2008:19
Examensarbete i matematik, 30 hp
Handledare och examinator: Erik Ekström Oktober 2008
Department of Mathematics Uppsala University
Jump processes and the implied volatility curve
Daniel Skoog
DANIEL SKOOG
Abstract: In this paper we examine a jump diffusion model for option pric- ing to determine if the commonly observed presence of a skew in implied volatility graphs is attributable to market fear of negative jumps. We will show that non-continuous stock price dynamics including jumps precipitate positive monotonic skews in the implied volatility graph.
Date: October 2008.
1
Contents
1. Introduction 3
2. Model 4
2.1. Background 4
2.2. Assumptions 4
2.3. Model Formulation 4
2.4. Derivation of the PDE 5
3. Solving the PDE 6
3.1. Background 6
3.2. Finite Differencing of PDE 6
3.3. Stability and convergence of finite differencing 7
3.4. Boundaries 7
4. Numerical Implementation and Analysis 8
4.1. Matrix Form of the PDE 8
4.2. Jump Matrix 8
4.3. Calculation of Option Prices 9
5. Implied Volatility 11
5.1. Background 11
5.2. Calculations 11
5.3. Numerical Analysis 12
6. Conclusion 16
Appendix A. 17
A.1. PDE Derivations 17
A.2. Matrix Algebra 18
A.3. Matlab Code 19
References 21
Acknowledgements 21
1. Introduction
It is a commonly held belief that markets tend to both over-react and under-react to the arrival of good or bad news. Market confidence can drive up values of stocks, and market fear can lead to large negative dips in stock value. In early September 2008, Bloomberg mistakenly republished a 2002 article where it set out that United Airlines had sought bankruptcy protection from its creditors[1]. Investors believed it was current, and within three minutes of the republication, United Airlines stock had plummeted 75%.
Markets are volatile; they tend to fluctuate rapidly and regularly. Investor option is a major reason for these fluctuations and should be considered when price setting options.
Many of todays option pricing models are derived from the robust Black-Scholes model[3].
The Black-Scholes model uses mainly observable variables and disregards many outside variables, such as investor option. This universality allows for easy adaptation, but in practice, it is used less as a pricing model for ”vanilla” options and more as building block for more sophisticated models.
The Black-Scholes model calculates a theoretical option price based on the dynamics of an underlying derivative with a constant level of volatility. In theory, if the Black-Scholes model is correct, the volatility implied by the theoretical option price should be constant across all values. However, empirical market data has shown that the volatility implied by the market price increases as an option becomes increasingly in or out of the money.
Rama Cont and Peter Tankov posit: ”the presence of a skew (in the implied volatility graph) is attributed to the fear of large negative jumps by market participants”[4, p.10].
The purpose of this paper to examine if the skew observed in implied volatility curves is a consequence of adding jumps to a standard Black-Scholes model for pricing options.
2. Model
2.1. Background. In the absence of outside news, asset prices are known to follow a geo- metric Brownian motion, a stochastic process with constant log variance per unit time and a continuous sample path. If we allow for the possibility of non-local changes in the stock price, i.e. a sudden jump in value, asset prices can be modeled by a jump diffusion process.
Pioneered by Robert C. Merton[7], jump diffusion processes are defined as a type of sto- chastic process that has large discrete movements, or jumps, rather than small continuous movements. Like information arrivals about a stock, it is expected that these jumps will only arrive at discrete points in time. These jump arrivals are considered a Poisson driven process, occurring independently and identically distributed, and with intensity λ.
2.2. Assumptions. We will make the same assumptions as are made in the Black-Scholes model for option pricing throughout the analysis[3]:
(1) Frictionless markets: There are no transaction costs or differential taxes, and bor- rowing and short selling are allowed without restriction.
(2) The short term interest rate is known and constant in time.
(3) Stock pays no dividends or other distributions during the life of the option.
(4) The option is ’European’: It can only be exercised at the maturity date.
To illustrate the effect of non-continuous stock price dynamics on option pricing, we will make a slight adjustment to the final assumption of the Black-Scholes model. We assume that the stock follows a geometric Brownian motion between any two jump points in time.
2.3. Model Formulation. The following dynamic is proposed to model the asset price, S(t):
dS(t) = (rS(t) − λγS(t))dt + σS(t)dW + γS(t)dN, (1)
where r is the rate of interest, σ is the volatility of the asset, γ is the relative jump size and λ is the jump intensity. W is a standard Wiener process describing the part of the unanticipated return due to normal price fluctuations. N is a Poisson process with rate λ describing the part of the unanticipated return due to abnormal price fluctuations.
If we divide the option pricing model into a continuous stochastic part, where a poisson event does not occur, and a discontinuous part, where a poisson event does occur, we can rewrite the asset price as follows:
S(t) = S(0)exp(r−γλ−σ22 )t+σW (t)(1 + γ)N (t) (2)
where,
dN (t) = 0, if no jump occurs at t
= 1, if a jump occurs at t
It is clear that in the case that a Poisson event does not occur, i.e. λ=0, the asset dynamics follow a standard geometric Brownian motion, similar to the Black-Scholes model.
2.4. Derivation of the PDE. Given the underlying asset dynamics of (1), we suppose that the dynamics of the option price, Φ(S), can be written as a C2 function of the stock price S, and time t, namely Φ(S) = U(S,t). This model has two sources of randomness, the Wiener process W and the Poisson process N, and one risky traded asset, S(t).
Since the number of sources of randomness are greater than the number of risky traded assets, the model is incomplete[2, p.118]. In incomplete models, there is no possibility of constructing a perfect hedge for the option price, and further, option prices cannot be differentiated from the underlying asset price by arbitrage arguments alone. A solution to this problem is to use risk-neutral modeling to model directly the risk-neutral dynamics of the asset by choosing a pricing measure Q, such that the qualitative properties of the asset price are respected.
Thus, under Q, we observe that the option price dynamics can be subdivided into two parts: a continuous stochastic function and a jump function. If a jump of size γ occurs, then the resulting change in U is U (S +γS, t)−U (S, t). Using Ito’s lemma on the continuous part, and an analogous lemma of Ito’s formula for jump diffusion processes on the jump part, we arrive at the following partial differential equation1:
∂ U
∂t + (rS − λγS)∂U
∂S +1
2σ2S2∂2U
∂S2 − rU (s, t) + λ(U (s + γs, t) − U (s, t)) = 0 Alternatively, this PDE can be written:
∂U
∂t + rS∂U
∂S +1
2σ2S2∂2U
∂S2 − rU (s, t) + λ(U (s + γs, t) − γS∂U
∂S − U (s, t)) = 0 (3)
1See appendix A.1. Also, see[7, p.8]
3. Solving the PDE
3.1. Background. To solve the partial differential equation (3), we will use an explicit finite differencing scheme to backward solve values of a European call option. With this time-stepping scheme, the approximate solution values at any given time step are explicitly derived from the known values of the preceding time step. In particular, values at the (t − ∆t) time level (price level S) are derived from values at the (t) time level and the price levels S + ∆S, S, S − ∆S and jumped level, S+γ S.
To proceed, we define a discrete grid of points within the problem domain, and replace the derivatives in the PDE (3) by finite difference approximations. To form this grid, we choose nt time steps to subdivide the time domain, and ns price steps to subdivide the price domain. The size of the time step is ∆ t = T/nt, where T is the time of maturity of the option, and the price step size is ∆ S = SM AX/ns, where SM AX is the maximum price level.
We find a numerical solution to each of the grid points by solving the system of equations created by the finite differencing of the PDE for each time step. This solution is the option value, U(i,j), where i is the price level index i, 0 ≤ i ≤ ns, and j is the time index, 0 ≤ j ≤ nt. The final vector of values generated by the finite differencing scheme will be a time 0 vector, U(S,0) which will be equal to the option value, Φ(S) at the price level S.
3.2. Finite Differencing of PDE. The following equations are used to approximate the first and second order derivatives:
Backward Dif f erence : dU
dt (si, tj) = U (si, tj) − U (si, tj−1)
∆t Central Dif f erence : dU
ds(si, tj) = U (si+1, tj) − U (si−1, tj) 2∆s
Second Order Dif f erence : d2U
ds2 (si, tj) = U (si+1, tj) − 2U (si, tj) + U (si−1, tj) (∆s)2
Applying the above approximations to (3) and changing notation from U (si, tj) to Uij, we get the following finite difference approximation of our model:
Uij− Uij−1
∆t + rS(Ui+1j − Ui−1j
2∆s ) +σ2S2
2 (Ui+1j − 2Uij+ Ui−1j
(∆s)2 ) − rUij +λ(Ui+γij −γS(Ui+1j − Ui−1j )
2∆s − Uij) = 0 (4)
Solving this for Uij−1 we get:
Uij−1= Ui+1j (rSk
2h +σ2S2k
2h2 −λγSk 2h ) + Uij(1 −σ2S2k
h2 − rk − λk) + Ui−1j (−rSk
2h +σ2S2k
2h2 +λγSk 2h ) + Ui+γij (λk)
(5)
Eq.(5) gives the approximate solution value at time step j-1 in terms of values that are available from the preceding time step, j.
3.3. Stability and convergence of finite differencing. The explicit method may not be as accurate as other implicit methods which can lead to some truncation errors that can grow with time. For convergence, the local truncation error must approach zero. The truncation errors are known to be linear over the time step and quadratic in the price step.
Thus, selecting a significantly large number of time steps will ensure the consistency of the model. However, this consistency comes at the cost of increased computational time.
Further, depending on the size of the time step, the model may become unstable. To maintain the stability of the model, we choose a time step such that it conforms to the CFL condition2. The CFL condition is a necessary condition for the stability of explicit finite difference schemes, requiring that for each mesh point, the domain of dependence of the PDE must lie within the domain of dependence of the finite difference scheme.
3.4. Boundaries. Assuming the finite differenced equation (5), we need to carefully exam- ine the boundary conditions of the PDE. We will impose the Dirichlet boundary conditions, specifying the values that the option price takes at the boundaries of the problem domain.
At the time of maturity, T, the owner of a European call option has the right to buy the stock at the strike price K. If the stock price is less than the strike price, the option is worthless. If the price is above the strike price, then the value of the option is S - K:
U (S, T ) = (S − K)+= M AX(S − K, 0), 0 ≤ S ≤ SM AX (6)
Further, when the asset value is worthless, clearly the option value U is likewise 0:
U (0, t) = 0, 0 ≤ t ≤ T (7)
Finally, when the asset value is SM AX, the call option value is equal to the SM AX value minus the time discounted strike price3, K:
U (SM AX, t) = SM AX − K exp−r(T −t), 0 ≤ t ≤ T (8)
2Courant-Friedrichs-Lewy Condition. See [6, p.458]
3It would also be possible to use the Neumann Boundary condition: dUdS=1.
4. Numerical Implementation and Analysis
4.1. Matrix Form of the PDE. We will now seek to solve the system of equations (5) by solving the recursion generated by the modified explicit Euler (forward) method:
Un−1= A · Un, 1 ≤ n ≤ nt + 1, (9)
where A is formed using the coeffecients of the Ui+1j , Uij, Ui−1j and Ui+γij terms in (5), and of dimension (ns + 1, ns + 1).
If we let:
a = rSk
2h +σ2S2k
2h2 − λγSk 2h b = 1 − σ2S2k
h2 − rk − λk c = −rSk
2h + σ2S2k
2h2 +λγSk 2h d = λk
We form A as follows, where A is such that A + J = A and J is a jump matrix4:
A =
b a 0 . . . 0 c b a 0 . . 0 0 c b a 0 . 0 . . . . . . . . 0 . . 0 c b a 0 . . . 0 c b
4.2. Jump Matrix. With A formed using the coeffecients of the Ui+1j , Uij, and Ui−1j , we now form the jump matrix, J, to hold the terms corresponding to the Ui+γij . We observe that the U (S + γS, t) terms are shifted from the current price level S by a factor γS. The jump matrix is comprised of these price shifted terms.
For our model, we will assume that the jumps occur with intensity λ = 1. We will test several cases for the relative size of the jump, γ, ranging from 10 to 50 percent5 of the current price level, S. This means that when a jump occurs, the jump will be to a value γ percent lower than the current price level, S. If this point is between two price levels, then the jump matrix will use a linear combination of the two nearest price levels, weighting the smallest level greater than (1 + γ)S and the greatest level smaller than (1 + γ)S with combined weight equal to 1.
4See Appendix A.2 for more detail on this matrix construction.
520 percent is default relative jump size
The weights are formulated as follows:
US−n: n − n
US−n: n − n = 1 − (US−n)
where n = -γS, n is the smaller integer greater than n, and n is the largest integer less than n.
For example, if we are at the 6th price step, the jump would be to a point (6 x 0.2 =) 1.2 levels below the current level. Then, we will weight the point S6−2=4with (1.2 - 1 =) 0.2 and the point S6−1=5 with (2 - 1.2 =) 0.8.
Forming the Jump matrix, J, with dimension (ns + 1, ns + 1) and gamma level 0.20:
J =
1 0 . . . 0
0.2 0.8 0 . . . 0
0 0.4 0.6 0 . . . 0
0 0 0.6 0.4 0 . . . . 0
0 0 0 0.8 0.2 0 . . . 0
0 0 0 0 1 0 . . . 0
0 0 0 0 0.2 0.8 0 . . 0
0 0 0 0 0 0.4 0.6 0 . 0
. . . .
∗ d
where d = λk. Adding this Jump matrix to A gives us the A matrix use in the calculation Un−1 = A·Un.
4.3. Calculation of Option Prices. To find the value of the option prices, we numerically solve the system of equations, (9), using Matlab6. The settings used are as follows:
Variable Settings
r σ SM AX K ∆s ∆t T γ λ
0.05 0.4 50 12.5 1 0.002 1 -0.2 1
From the above, we can observe that the time domain has 501 steps [0:500], and the price domain has 51 steps [0:50]. The strike price, K, is defined as SM AX/4. A condition of using an explicit finite differencing scheme is that ∆s and ∆t fulfill the CFL condition. In practice, the choosing (∆s)∆t2 < 0.02 yields the most consistent results.
From the Figure 1 and Figure 2, we can observe that both graphs are in accordance with standard models for option pricing, and we see no irregularities. Further testing with γ values between [0.1:0.5] yield similar results. It can be noted that the option price values increase as the relative jump size increases, which is in accordance with general theory of properties of option prices in models with jumps7.
6See A.3 for Matlab code 7See Ekstrom/Tysk[5]
0
10
20
30
40
50
0 0.2 0.4 0.6 0.8 1 0 10 20 30 40
Price Plot of Option Values
Time
Option Value
Figure 1. Plot of option price values, time, and asset price level
0 10 K 20 30 40 50
0 5 10 15 20 25 30 35 40
Stock Price
Option Value
Plot of Option price and (S!K)+
(S!K)+ Option Price
Figure 2. Plot of option price values vs. time, and speculative option value vs. time
5. Implied Volatility
5.1. Background. Above, we solved the PDE (3) by finite differencing to give the the- oretical option prices based on the underlying derivative S(t). To price the option, we assumed that the volatility was constant, σ = 0.40 over the life of our option. However, in real market data, the volatility is not constant and varies over time. Consequently, if our goal is to price our option consistently with respect to other derivatives that are already priced by the market, we should use the market expectation of the volatility over the life of the option, known as the implied volatility.
The implied volatility of an option contract is the volatility implied by the market price of the option derived from an option pricing model. Our pricing model assumes a constant volatility, which implies that the implied volatility surface should be flat. However, it has been frequently observed that the implied volatility surface is rarely flat, instead looking more like a smile or skew. Cont and Tankov list three empirically observed properties of implied volatility surfaces[4, p.9]:
(1) Smiles and skews: for equity and foreign exchange options, implied volatilites dis- play a strong dependence with respect to the strike price: this dependence may be decreasing (”skew”), or more U-shaped (”smile”).
(2) Flattening of the smile: the dependence of the implied volatility with respect to the strike price decreases with maturity; the smile/ skew flattens out as maturity increases.
(3) Floating smiles: if expressed in terms of relative strikes, implied volatility patterns vary less in time than when expressed as a function of the strike price.
Cont and Tankov suggest that these properties are attributable to the market participants fear of large negative jumps. To test this claim, we will find the volatilities implied by the theoretical option prices and use them to examine if the addition of jumps to the geometric Brownian motion results in a skew in the theoretical implied volatility graph.
5.2. Calculations. To find the implied volatility of the option, we can derive market price data from a benchmark option with the same time to maturity and written on the same underlying derivative. If C(S,t) is the theoretical value of an option in the no jumps case, it can be written in terms of the following Black-Scholes parameters:
C(S, t) = S · F (d1) − K exp−r(T −t)·F (d2) where
d1 = ln(KS) + (r + σ22)(T − t) σ√
T − t d2 = d1− σ√
T − t
and F is the cumulative distribution function of the standard normal distribution N(0,1).
Further, the value of an option can be written as a function of σ and other inputs, C = U (σ, ·). To see the effect of σ on the option price, we plot the option price as a function of volatility:
0 2 4 6 8 10 12K 14 16 18
0 1 2 3 4 5 6 7 8
Stock Price
Option Value
Option Price as a function of Volatility
Large ó
Small ó
Figure 3. Option Price as a function of σ
Figure 3 indicates that the option price is monotonically increasing with respect to σ. In fact, this can be shown analytically by differentiating C with respect to the volatility σ, i.e. V = ∂C∂σ = S · F (d1)√
T − t − Ke−r(T −t)· F (d2)√
T − t > 0.
It is clear from the above that there can at most one value for σ that corresponds to a particular value of the option price C. Consequently, provided the function is continuously differentiable, there exists an inverse function u(·) = U−1(·), such that σ = u(C, ·). Anal- ogously, there must exist a function that solves for the volatility implied by the market price C, σ. In this case, to solve for σ, we can set the theoretical option price equal to the market price and rearrange to get:
U (σ, ·) − C(σ, ·) = 0 (10)
5.3. Numerical Analysis. To find the roots of (10), we will implement Newton’s method in Matlab. The roots correspond to the values for the implied volatilities, σi, where 0 ≤ i ≤ ns corresponding to the price levels Si8. We will then plot these values as a function of the asset price, S, and analyze the characteristics of the resulting implied volatility graphs.
Recall that our model can be seperated into two cases: a Poisson event occurs, and a Poisson event does not occur. If we consider the case that a Poisson event does not occur, or λ = 0, the volatility used to price options in this model is constant (σ = 0.4) and accordingly, the graph of the implied volatility should be constant as well.
8See implied volatility section of matlab code, A.3
Figure 4 shows the result of the implied volatility calculation based on the λ = 0 case:
10 15 20 25 30 35 40
0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44
Stock Price
Implied Volatility
Plot of Implied Volatility vs. Stock Price: Gamma Level 0
Figure 4. Plot of Implied Volatility vs. Stock Price, Black-Scholes Model
This is consistent with what we would expect to see since it has a relatively constant im- plied volatility. The implied volatility graph shows a constant volatility approximately = 0.4 (= σ). It is reasonable to expect that some truncation error occurred in the finite differencing leading the implied volatility graph being slightly above 0.40 at price 40.
Next, we examine the implied volatility derived from the jump diffusion model, where λ = 1, γ = −0.20 for the Dirichlet boundary condition (8) and the Neumann boundary condition, ∂U∂S = 1:
5 10 K 15 20 25 30 35 40
0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.48 0.485
Stock Price
Implied Volatility
Plot of Implied Volatility vs. Stock Price: Gamma Level !0.20
Neumann Dirichlet
Figure 5. Plot of Implied Volatility vs. Stock Price, Jump Diffusion Model Figure 5 indicates that adding jumps produces a positive monotonic skew in the implied volatility graph. For the Dirichlet boundary, the change in implied volatility is approx- imately 0.025, which is a significant displacement. Near the upper end of the implied volatility graph, the slope of the implied volatility curve is nearly flat, suggesting some irregularities near the upper boundary of the model. This can be attributed to round-off errors and truncation errors generated in the finite differencing scheme near the boundary of the problem domain.
With the Neumann boundary, the approximate change in the implied volatility 0.04, is slightly higher than the Dirichlet case 0.025. We can see that the upper tail of the implied volatility curve curves upward, as opposed to the Dirichlet boundary, which curved down- ward. This again, can be attributed to round-off errors and truncation errors generated in the finite differencing scheme near the boundary of the problem domain. Due to the inclusion of jumps, both graphs show a significant positive change in the implied volatility.
This is a confirmation of the assumption that jumps precipitate positive monotonic change in the implied volatility curve.
To verify the results of the γ = −0.20 case, we test values of gamma from 0 to 50 percent:
10 12K 14 16 18 20 22 24 26 28 30
0.4 0.5 0.6 0.7 0.8 0.9 1
Stock Price
Implied Volatility
Plot of Implied Volatility vs. Stock Price over Gamma levels [0.0:0.5]
0.5 0.4 0.3 0.2 0.1 0.0
Figure 6. Gamma Levels [0.0:0.5]
We can observe that as γ increases, the change in the implied volatility over the asset price levels becomes greater. In fact, we can observe that the approximate change in implied volatility is:
Approximate change in implied volatility γ 0.0 -0.1 -0.2 -0.3 -0.4 -0.5
∆σ 0.002 0.004 0.023 0.057 0.092 0.120
This further confirms that the addition of jumps produces a positive monotonic skew in the implied volatility graph.
6. Conclusion
The objective of this paper was to analyze the claim that the skew in the implied volatility curve is attributable to fear of large negative jumps by investors. To accomplish this we derived a partial differential equation describing a jump diffusion model for option pricing for an underlying asset with jumps. The solutions of this PDE were approximated using an explicit finite difference scheme and numerically solved using an Euler forward method in Matlab, the result of which was a vector of theoretical option prices. These option prices were then used to solve for the volatilities implied by the market prices, and the implied volatilities were plotted as a function of stock price to determine if there was any monotonic skew created by the introduction of jumps.
Jumps in the market have long been theorized to cause the implied volatility of an asset to increase as it is increasingly in or out of the money. We have numerically shown that the presence of a skew in the implied volatility curve can be attributable to the market fear of negative jumps. Depending on the relative size of a jump, non-continuous stock price dynamics have been shown to precipitate a positive increasing monotonic skew in the theoretical implied volatility curve.
Appendix A
A.1. PDE Derivations. In order to derive the PDE, we assume that Ito’s formula for jump diffusions will give a correct solution. For a detailed derivation of Ito’s formula for jump diffusion models, see [4, p.274-5]. Then, to test this assumption, let:
U (S, 0) = e−rTE[(ST − K)+] Assume V(S,t) solves Ito’s formula for Jump Diffusions:
Vt+ (rS − γλS)VS+1
2σ2S2VSS− rV + [V (s + γs, t) − V (s, t)] = 0 (11)
V (S, T ) = (S − K)+ (12)
Now, we let Yt= V (St, t)e−rt.
By using Ito’s formula with jumps, we have:
dY = (∂V
∂t + (rS − γλS)∂V
∂S +1
2σ2S2∂2V
∂S2)e−r(T −t)dt + e−r(T −t)σS∂V
∂SdW + e−r(T −t)γS∂V
∂SdN Since Y solves (11), the dt term is 0. Thus,
dY = 0dt + e−r(T −t)σS∂V
∂SdW + e−r(T −t)γS∂V
∂SdN (13)
Y has no drift term, so Y is a martingale. Then,
V (S, 0) = Y0 = E[YT] = e−rTE[V (ST, T )] = e−rTE[(ST − K)+] = U (S, 0)
Thus, Ito’s formula for jump diffusion models will give a proper solution for describing the option price dynamics modeled with the underlying derivative, S.
A.2. Matrix Algebra. The explicit finte differencing of the PDE (3) can be modeled by matrices in the following manner: The expected values at time of maturity T are known, and are used as the final vector in the recursion: Uj−1 = A · Uj, [1 ≤ j ≤ nt].9 By backward stepping through time, we create a matrix U,
U = U0 U1 U2 . . . Unt−1 Unt ,
comprised of nt+1 (number of time steps from 0 to T) vectors. Each vector has ns+1 (number of price steps from 0 to SM AX) entries10. Completing this recursion (at time 0), the vector U0 yields the theoretical option prices over the price levels [0:SM AX].
Using the boundary conditions, we have:
Unt = (Si− K)+[0≤i≤ns]=
0 0 . . Sns−1− K
Sns− K
We construct: Uj−1= A · Uj, for 1 ≤ j ≤ nt
U0
U1 U2
. . Uns−1
Uns
j−1
=
b a 0 . . . 0 c b a 0 . . 0 0 c b a 0 . 0 . . . . . . . . 0 . . 0 c b a 0 . . . 0 c b
U0
U1 U2
. . Uns−1
Uns
j
Solving this gives the following system of equations:
U1j−1= cU0j + bU1j+ aU2j U2j−1= cU1j + bU2j+ aU3j
... = ...
Uij−1= cUi−1j + bUij+ aUi+1j ... = ...
Uns−1j−1 = cUns−2j + bUns−1j + aUnsj
As per the explicit differencing scheme, the value of the j-1 step is defined as a sum of the price levels of the future time step j. Note that we have not listed the price 0 and price SM AX systems above, since these are determined by the boundary conditions, (7) and (8), not by the finite difference method.
9To simplify the notation, we assume that λ = 0 and no jump occurs. The process for solving a system with jumps is identical.
10In the matlab code, U(i,j) = U(s+1,t+1) for 0 ≤ s ≤ ns and 0 ≤ t ≤ nt.
A.3. Matlab Code.
%Function generates option values based on Backward Differenced B.S. with
%Jump, and further generates implied volatilities based on the values.
function[x] = j4(smax, nt, ns, l, sigma) if nargin == 0 %Standard Values
l = 1; %l is binary var used for adding jump section.
smax = 50; %Max Instrument Value nt = 501; %nt = number of time steps ns = 51; %Number of space steps sigma = 0.4;
end
%** Variable Settings
r = 0.05; %The Interest Rate lam = 1*l; %Jump Intensity
gam = -0.2*l; %rand/4; %Relative Jump Size SK = smax/4; %Strike Price
T = 1; %Time to maturity
h = smax/(ns-1);%Price Step Size k = T/(nt-1);%Time Step Size
%**Building Function Matrices
sindex = [0:smax/(ns-1):smax]; %Space Position Vector V1 = diag(ones(ns-1,1),1) - diag(ones(ns-1,1),-1);
V2 = -2*eye(ns,ns) + diag(ones(ns-1,1),1) + diag(ones(ns-1,1),-1);
D1 = diag(sindex);
D2 = diag(sindex.^2);
gcindex = ceil(-gam*sindex); %Gamma Positioning Vector - Ceiling gfindex = floor(-gam*sindex); %Floor
gindex = -gam*sindex;
U = zeros(ns,nt);
U(:,nt) = max([0:h:smax]’ - SK , 0);
%Weighted Jump Matrix (gam*n + (1-gam)*n+1) V3 = zeros(ns,ns);
for j = 1:ns
v3 = zeros(1,ns);
v3(j-gcindex(j)) = (gindex(j) - gfindex(j));
v3(j-gfindex(j)) = 1 - (gindex(j) - gfindex(j));
V3(j,:) = v3(1:ns);
end
%Backward Differencing of Option Price for i = 1:nt-1
t = (nt-1-i)*k;
A = (1-r*k-lam*k)*eye(ns,ns) + (0.5*sigma^2*k/h^2)*D2*V2 + (0.5*k/h*(r - lam*gam))*D1*V1 + (lam*k)*V3;
U(:,nt-i) = A*U(:,nt-i+1);
U(ns,nt-i) = smax - SK*exp(-r*(T-t));
U(1,nt-i) = 0;
end
%Implied Volatility 1D V=zeros(ns,1);
for j = 1:ns %Space
V(j)=blsimpv(sindex(j),SK,r,1,U(j,1),1);
end
%Output
[sindex’,U(:,1),U(:,nt),V(:,1)]%[S, Time Zero, Time T, Imp Vol]
References
[1] United Airlines dives on old news. BBC News, UK, http://news.bbc.co.uk/2/hi/business/7605885.stm, September 2008.
[2] Tomas Bj¨ork. Arbitrage Theory in Continuous Time. Oxford University Press, 2nd edition, 2004.
[3] Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, May-June 1973.
[4] Rama Cont and Peter Tankov. Financial Modelling With Jump Processes. Chapman and Hall/CRC, 2004.
[5] Erik Ekstr¨om and Johan Tysk. Properties of option prices in models with jumps. Mathematical Finance, 17(3):381–397, July 2007.
[6] Michael T. Heath. Scientific Computing: An Introductory Survey. McGraw-Hill, 2nd edition, 2002.
[7] Robert C. Merton. Option Pricing When Underlying Stock Returns Are Discontinuous. Working Papers 787-75. Massachusetts Institute of Technology, Cambridge, USA, April 1975.
Acknowledgements
I have learnt a great deal over the course of this project and I would like to thank my supervisor Erik Ekstr¨om for his constructive criticism and guidance.
