Angewandte Finanzmathematik 2024:

Introduction to the Black–Scholes World

1 Practicalities and background

Upon passing the exam, attending and solving the exercises give a bonus to the final grade.

We assume that the following concepts are familiar:

1. Probability space, random variables, expectation, convergence concepts.

2. Conditional expectations, martingales.

3. The fundamentals of discrete time financial mathematics.

For a remote graphical access to Matlab, you can login to the computers

❼ math12.math.lmu.de

❼ mathw0g.math.lmu.de

You will need

1. a program supporting X11-forwarding (e.g. Cygwin),

2. SSH program with rdp connections (e.g. Bitvise),

3. a VPN connection to LRZ (Anyconnect client, downloadable from LMU service portal).

Alternatively, you can use the online version of Matlab.

2 Introduction

2.1 Popular financial products

Throughout the course, St i denotes the market price of an asset i at time t.

Example 2.1 (Put and call options). A European call option on the asset i is a contract where the seller has the obligation to deliver the asset i at the given maturity time T for a given strike price K. At time T, the buyer has the possibility to exercise the option, that is, to buy the asset from the seller at price K. The gain for the buyer is

cC := (ST i − K) + := max{ST i − K, 0},

since he can get the asset from the option seller at price K and sell it immediately on the market with the market price ST i . We call cC the payoff of the call option.

A European put option on the asset i is a contract where the seller has the obligation to buy the asset i at the given maturity time T for a given strike price K. At time T, the buyer has the possibility the exercise the option, that is, to sell the asset to the seller at price K. The payoff for the buyer becomes

cP := (K − ST i ) + := max{K − ST i , 0}.

Put and call options are prototype examples of Vanilla options that depend only on the terminal price of the underlying asset. When this is not the case, the option is called path-dependent.

Example 2.2 (Asian options). An Asian call option with maturity T and strike K has the payof

cAC := (S¯ T i − K) +,

where S¯ T i is the ”average price” of the asset over the time interval [0, T]. The exact form. of the average price is part of the contract, e.g., it could be arithmetic mean of the prices at given time points t1, . . . , tN = T so that ¯ST i = N 1 P N k=1 St i k .

For a set A, we denote ✶A(s) = 1 if s ∈ A and ✶A(s) = 0 otherwise.

Example 2.3 (Down-and-out and other Barrier options). Given a strike K, ma-turity T and a barrier B > 0, the down-and-out call option has the payof

The payoff of an up-and-in call option with the same strike and maturity is

Barrier put options have similar payoffs. For example, down-and-in put options have payoffs of the form.

Options that depend on multiple underlying assets are called rainbow options.

Example 2.4 (Basket options). Given a set of assets indexed by i = 1, . . . , I and positive coefficients ai, i = 1, . . . I, the payoff of the corresponding basket call option is

Similarly, the basket put option has the payof

Example 2.5 (Spread options). Given two assets S1 and S2 , the payoff of the corresponding spread call option is

Similarly, the spread put option has the payof

Example 2.6 (Calls and puts on max and min). Given to assets S1 and S2, the payoff of the corresponding call-on-max option is

Similarly, the put-on-min option has the payof

Many options depend on quantities that are not tradable on markets.

Example 2.7 (Options on non-tradables). Let ξT be the temperature (somewhere of interest) at time T, and consider options with the payoffs

with a given strike K.

Example 2.8 (American options*). The holder of an American option may choose to exercise the option at any time before the terminal time T. For exam-ple, for an American call on S i with strike K, the payoff, if the holder chooses to exercises the option at time t, is

In contrast to all the above options, the holder of an American faces an opti-mization problem when to exercise the option.

2.2 Exercises

In all the exercises, examples in Matlab online help pages help you to write the actual code.

Exercise 2.2.1. Write Matlab functions (as .m-files) of the payoff functions in Examples 2.1–2.6. Write them as functions of the underlying asset prices and strikes.

Exercise 2.2.2. Using the plot-function, plot the European call option, for a fixed strike K, as a function of the underlying asset price ST . Plot the European call option as a function of the underlying asset price ST for two different strikes in the same figure.

Exercise 2.2.3. Using the mesh-function (or surf-function), draw a 3D-graph of the spread call option as a function of the underlying asset prices ST 1 and ST 2 .

2.3 Basic properties of Brownian motion

Let (Ω, F,(Ft) T t=0, P) be a filtered probability space. We consider continuous time stochastic processes only on the ”time interval” [0, T]. A family S := (St)t∈[0,T] of Rd–valued random variables St is called an R d–valued continuous time stochastic process. The process is called adapted if St is Ft-measurable for each t ∈ [0, T].

Given ω ∈ Ω, the function t 7→ St(ω) is called as a path, or a trajectory or a realization, of the process S. Instead of considering a stochastic process as an indexed family of R d -valued random variables, one may thus think of a stochastic process as a family of random paths, trajectories, etc. In some cases (less in this course), it is helpful to think of a stochastic process S as a function (ω, t) 7→ St(ω) from the product space Ω × [0, T] to R d . If the paths of a continuous time process are P-almost surely continuous, then the process is called a continuous stochastic process.

For a random variable η ∈ (Ω, F, P), we denote η ∼ N(µ, σ2 ) when η is a normally distributed random variable with mean µ and standard deviation σ.

Remark 2.9. We often use the property that for η ∼ N(0, σ2 ) and positive integer m, there is a constant L such that Eη2m = Lσ2m,

Definition 2.10. An adapted continuous stochastic process W is a Brownian motion, if it has independent increments in the sense that, for all 0 ≤ t0 < t1 < · · · < tn the random variables {Wti − Wti−1 | i = 1, . . . n} are independent, and Wt − Ws ∼ N(0, t − s) for all 0 ≤ s < t ≤ T,

From now on we assume, unless stated otherwise, that given a Brownian motion W, it starts at zero, that is, W0 = 0.

Exercise 2.3.1. Show that a Brownian motion W is a martingale, that is, for all s < t ≤ T, s > 0, we have E|Wt| < ∞ and

E[Wt | Fs] = Ws.

Here we assume that the increments of W are independent of the filtration in the sense that, for all s < t, the random variable Wt − Ws is indenpendent of Fs. This is the case, .e.g., when the filtration is generated by W.

In the definition of Brownian motion, it possible to omit the assumption that the paths are continuous. This follows from the famous Kolmogorov’s conti-nuity criterion. Recall that a continuous function f : [0, T] → R is α-H¨older continuous if there is L ∈ R such that

|ft − fs| ≤ L|t − s| α ∀ 0 ≤ s ≤ t ≤ T.

Theorem 2.11 (Kolmogorov’s continuity criterion). Let S be a stochastic pro-cesses with

E |St − Ss| a ≤ L|t − s| 1+b ∀ s < t              (2.1)

for some constants a ≥ 1, b, L > 0. Then there exists a continuous stochastic process S˜ that is a modification of S in the sense that P(S˜ t = St) = 1 for all t. Moreover, S˜ is α-H¨older continuous almost surely for any α ≤ b/a.

Exercise 2.3.2. Using Remark 2.9, show that, for any ϵ > 0, Brownian motion has (1/2 − ϵ)-H¨older continuous paths almost surely.

From the computational perspective, Brownian motion has the important prop-erty that it can be approximated by piece-wise constant ”discrete-time random walks” that have independent increments. Such random random walks are easy to simulate which is the basis of Monte Carlo methods that is the main topic of the course.

Recall that a sequence of random variables (η ν ) converges in distribution to the random variable η if

P(η ν ≤ x) → P(η ≤ x)

for all x ∈ R such that x 7→ P(η ≤ x) is continuous (i.e., for all x such that the cumulative distribution function of η is continuous at x). A sequence of vectors of random variables (η1 ν , . . . ηk ν ) converges in distribution to (η1, . . . , ηk) if

P((η1 ν , . . . , ηk ν ) ≤ x) → P((η1, . . . , ηk) ≤ x)

for all x ∈ R k such that x 7→ P((η1, . . . , ηk) ≤ x) is continuous.

Theorem 2.12 (The central limit theorem). Let

for an i.i.d. (ξk)∞ k=1 sequence of random variables with Eξk = 0 and E(ξk) 2 = 1.

We have

                       (2.2)

for a random variable η ∼ N(0, 1).

For continuous time stochastic processes S (n) , n = 1, 2, . . . and S, S (n) converges in finite dimensional distributions to S, denoted by

if, for all integers k and all 0 ≤ t0 < · · · < tk ≤ T,

Theorem 2.13. Let

for an i.i.d. (ξk)∞ k=1 sequence of random variables with Eξk = 0 and E(ξk) 2 = 1. Then

for a Brownian motion W.

Proof. Using the central limit theorem and ⌊nt n ⌋ → t when n → ∞, we get

as n → ∞. Let now t < u. The random variables Yu (n) − Yt (n) are independent from the variables Yt (n) , since

and the random variables ξk (n) are independent. Repeating the previous argu-ments we get

We observe that the variables ∆Yt (n)i:= Yt (n)i− Yt (n)i−1 are mutually independent for all 0 ≤ t0 < t1 < · · · < tN ≤ T. Thus the process Y (n) has independent increments, and so

The proof is finished by the next exercise.

Exercise 2.3.3. Recall the continuous mapping theorem: If (η0 ν , . . . , ηk ν ) −−→ d (η0, . . . , ηk), then f(η0 ν , . . . , ηk ν ) −−→ d f(η0, . . . , ηk) for any continuous function f : Rk → Rn.

Use the continuous mapping theorem to finish the proof of Theorem 2.13.