and K = 10 The volatility a in our model describes the amount of random noise in the stock price. Y{x,t) = -J-{t,x) = xy/t- t<pn{d+{t-t,x))

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1 -5b THE GREEKS Theta #(t, x) of a call option with T = 0.75 and K = 10 Rho g{t,x) of a call option with T = 0.75 and K = 10 The volatility a in our model describes the amount of random noise in the stock price. The derivative Y{x,t) = -J-{t,x) = xy/t- t<pn{d+{t-t,x)) is called Vega of the European call option. Because the Vega is positive the price of a European call option is an increasing function of the volatility. The economical reason is that a bigger value of the volatility yields a higher probability for a larger possible profit in the positive case. Another important consequence in the next section is that the function a i-> f(t,x) is strictly increasing and therefore invertible. Vega y(t, x) of a call option with T = 0.75 and K = 10 Example (Hedging by the Greeks) Theoretically, by trading according to the hedging strategy one has no reason to take into account any other thoughts. But in practice this is not possible because it requires rebalancing the portfolio continuously. But if a trader zero out the delta say once a day there might occur a big difference between the value of the option and the hedging portfolio, see Example We consider a portfolio consisting of maybe several different contingent claims on a single risky-asset, the asset itself and a risk-free asset. The Delta and Gamma of such

2 CHAPTER 3. BLACK-SCHOLES MODEL a portfolio is defined analogously as the derivatives of its value. Trading according the hedging strategy implies that the Delta of the portfolio should be zero. The Gamma of a portfolio is the rate of change of the Delta with respect to the price of the share. Small values of Gamma indicates that the Delta changes slowly and adjustment of the portfolio to keep it Delta neutral need to be made only occasionally. But if the Gamma is large the portfolio is highly sensitive to the price of the underlying asset. Thus, a trader aims to keep his portfolio Gamma neutral, which means the Gamma should be zero. This can be achieved by including another traded contingent claim on the same share. Assume that this additional contingent claim has a Gamma Vc and the Gamma of the portfolio is rp. Including a units of the new contingent claim to the portfolio results in Gamma of the portfolio orc + Fp and therefore one set* -si- rf- The additional position in the new contingent claim is likely to change the Delta of the portfolio. Therefore, the position in the share itself has to rearrange in order to guarantee Delta-neutrality. In the Black-Scholes model the volatility is assumed to be constant which can be seen easily not to be in accordance with the real world. The Vega of a portfolio is the rate of change of the value of the portfolio with respect to the volatility of the underlying risky asset. If the value of the Vega is very large, the value of the portfolio is very sensitive to changes in the volatility. This risk can be reduced to keep a portfolio Vega-neutral. 3.4 Volatility To determine the price of a European call option by formula (3.2.7) following parameters: we need to know the the strike price K; the time T of maturity; the interest rate r; the current share price 5(0); the volatility a. All of them are part of the contract (K and T) or directly observable on the market (r and S(0)) - except the volatility a Historical Volatility One can estimate empirically the volatility of the stock price by observing data of the share price in the past. We assume that we have observed the share prices (S(t) : t [0,T]) at equidistant times t\,...,tn with r := U - U-\. Let sj,...,sn denote our observations. If we set Z, := log S(ti)/S(U-i) it follows by Lemma that the random variables Z\,..., Zn are independent and identically distributed with

3 -sz CRITICISM Thus, to obtain an estimate for a it is sufficient to estimate the variance a2r of Zi by the obervations Zi := Sj/si-i. This can be done in the standard way by the empirical variance 1 n i n v := > [Zi z), where z > Zi. n - 1 *-f nff Then \Jv2}t is an estimate for ct which exhibits several appreciated statistical properties. This method depends on the choice of n. One can argue that the more date are uesed the better is the estimate. But on the other hand, it has to be assumed that the volatility changes over time so that too old date might distort the estimate. There are many more sophisticated methods to estimate cr, for example linear models, see [10, Chp. 15] Implied Volatility Alternatively, one can estimate the volatility for the stock price S by using the market price data for another option written on the same underlying share. Typically, if we want to price a European call option the other option C is also a European call option but with a different strike price K' and expiration time T'. In the Black-Scholes model the price function /' : [0, T] x R,+ > R,+ of the other call option C is given at time t = 0 by - K'e~rT'FN (]J}^LE where s = S(0) is the current stock price. The price /'(0, s) today is the observed market price and thus, the only unknown parameter in this equation is a. Thus, solving for a gives an estimate for u which is called implied volatility. In practice this has to be done by a numerical scheme because one can not solve explicitly for a. If the Black-Scholes model would represent perfectly well the real world the implied volatility values calculated based on different options, say call options with different strike prices and time to expiration, would coincide. But in practice however, the im plied volatilities differ significantly depending upon the strike price. The typical pattern of implied volatilities as a function of the strike price forms a "smile" shape. This phe nomena is called volatility smile or volatility skew. In practice one has to choose carefully the observed prices which serve as the basis for the calculation of the implied volatility. 3.5 Criticism Despite its popularity there is a lot of criticism on the Black-Scholes model. In the following we mention shortly some problems with the Black-Scholes model. Many of them can be confirmed statistically and are subject of empirical tests, see for example the cited literature in [16, 6.3]. The assumption of the geometric Brownian motion as a model for the share price is not in accordance with statistical data. For example, one can show by statistical methods that for many data the log returns log f,.s J for some s, t > 0

4 -S3- CHAPTER 3. BLACK-SCHOLES MODEL are not normally distributed as it is the case in the Black-Scholes model. The already discussed volatility smiles (see Section 3.4.2) indicates this imperfection. As one solution out of this dilemma it is very popular nowadays to consider stochas tic volatility models, in which the volatility itself is modeled by a stochastic process. For example: ) for all t 6 [0,T], d$(t) = g{t, *(*)) dt + h{t% $(0) dw*{t) for all t <E [0,T], where //. H, g, h : [0, T\ x 1R -» R are deterministic functions and W\ and W2 are one-dimensional Wiener processes. Thus, the volatility is here modeled by a stochastic process ($(0 : t G [0,T]) which is assumed to be a solution of another stochastic differential equation. The investigation of such kind of models requires more sophisticated tools from probability theory which we will introduce in the next section. In the later chapter on risk-neutral pricing we will include these models. As the financial crisis in 2007 demonstrated the stock prices might be subject to large jumps. One can find many other evidences that due to some extraordinary incidences the share prices do not behave as regular as it is indicated by a lognormally distributed stochastic process. In particular, discontinuities can not be modeled by a geometric Brownian motion as it has continuous trajectories, see Theorem In order to model possible jumps of the underlying share prices the driving Brownian motion is replaced by a Levy processes, see for example [4]. The interest rate r is not constant in time and it differs for borrowing and lending. In a later chapter we will consider different models for the interest rate depending on time and on the market performance. there are transaction costs on the market. However, as every parametric model in finance, statistics or applied mathematics, the Black-Scholes model is only intended to give a simplified description of the reality. If one draws conclusion out of this model one should bear this in mind!

5 -SH- Chapter 4 Stochastic Calculus II In this chapter I follow very closely section 4.5 and 4.6 in Etheridge [7]. In the following (Q,#/,P) is a probability space with a Brownian motion (W(t) : t [0,T]). The considered filtration is generated by the Brownian motion: &Y = a{w{s) : 0 s$ s < t) for all t 6 [O.T]. 4.1 Girsanov's Theorem Example We reconsider the Cox-Ross-Rubinstein model and assume that d < 1 + r < u such that the model is arbitrage-free. For simplicity we assume r = 0. Then the unique equivalent risk-neutral measure Q is given by u A specific paths of up and down jumps can be described by a vector (ii,...,?'t) for ik {0,1} and %k = 1 denotes a up-jump from time fc - 1 to /e, i.e. the value of Xk- The probability under P of the occurrence of a specific path, say (ii,...,it) for ik G {0,1}, is Then, the probability under Q of the same path is Q(X1 =»,,...,Xt = it) = 9*»+""H' (1 -,)' where it is defined by ii H hi " /i-,y \~p) In order to investigate the term It under conditional expectation, we introduce its random analogue:

6 55- CHAPTER 4. STOCHASTIC CALCULUS II and thus, L = (Lt : t = 1,...,T) can be considered as an adapted stochastic process. Moreover, we have EP[Lt\.*,_,] = Lt_x (Vq- + (1 - p)^ where {&t}t=o,...,t denotes the filtration in the Cox-Ross-Rubinstein model. Thus, L is a martingale under P with EP[Li] = EP\Lt] = 1 for all t [0,T]. Theorem (Girsanov's Theorem) Let {X(t) : t [0,T]) be an adapted stochastic process satisfying p( f X2(s)ds <oo) =1. (4.1.1) Define forte [0,T] L{t) := exp (- I X{s)dW(s) - \ j X2{s)d^\ and a mapping Q by Q-.&-+ [0,oo], Q(A) := / L(T)dP = EP[tAL(T)\. JA If(L(t) : t G [0, T]) is a martingale under P then the mapping Q is a probability measure and the stochastic process (W( ) : t G [0,7"]) defined by W{t):=W{t)+! X{s)d& Jo is a Brownian motion under the new measure Q. Proof See Theorem in [7]. Notation: The stochastic process (L(t) : t e [0,T]) is called Radon-Nikodym derivative and is denoted by Remark (a) The condition (4.1.1) guarantees that the stochastic integral in the definition of L(t) exists. The main condition and often most difficult to verify is the requirement that {L(t) : t e [0,r]) is a martingale. (b) Since by definition of L we have P{L(t) > 0) it follows by the definition of Q by Q{A) = EP[1AL{T)) that Q(A) = 0 & P(A) = 0, that is the measures P and Q are equivalent.

7 GIRSANOV'S THEOREM Corollary Under the condition of Theorem we have we have for any adapted stochastic process (Y(t) : t 6 [OjX1]): EQ\Y(t)\&s] = EP \Y{t)=f!r\jra\ P-a.s. Proof. Follows from the definition of the measure Q. Example Let (Y(t) : t [0,T]) be given by Y{t) := fd + aw(t) for some constants fx e R and a > 0 and where {W(t) : t [0,T]) is a Brownian motion under P. Is there a measure such that Y is a martingale? For X(t) = ///<r it follows from part (b) in Theorem that L defined in Girsanov's Theorem is a martingale. Thus, Girsanov's Theorem implies that W(i) := Wit) + a defines a Brownian motion (W(t) : t 6 [0,T]) under the probability measure Q : si -> [0,1], Q(A) = EP [l^ exp (-%W{t) - g One can calculate for example EP{Y2{t)} = EP [fx2t2 + 2^<rW(t) + a2w2(t)] = a2i ^ In applying Girsanov's Theorem the most difficult part is to verify that {L{t) : t G [0,T]) is a martingale. A powerful tool for that is the following sufficient condition: Theorem (Novikov condition) Let (X(t) : t G [0,T]) be an adapted stochastic process satisfying p( f X2(s)ds <oo) =1. Define for t <= [0, T] L(t) :=exp(- / X(s)dW(s) - { I X2{s)ds\ If E T exp - / X2(s)ds < oc then the process (L(t) : t 6 [0,T]) is a martingale under P.

8 CHAPTER 4. STOCHASTIC CALCULUS II 4.2 The Brownian Martingale Representation Theo rem If (X(t) : t G [0,T]) is a stochastic process in j f*[0,t] then the stochastic integral M(t):= / X{s)dW{s) defines a martingale [M{t) : t G (0, T]) by Theorem The question arises if there are other martingales, which are not of this specific form? Before we state the answer of this question we have to define the analogue of the notion of a predictable stochastic process in continuous time: Definition A stochastic process {X(t) : t e [0,T]) is called predictable with respect to the given filtration {<&t}t [a,t] if for all t [0,T] X (t) is 3\- -measurable where &t- ' = I) & s 3<t The meaning of a predictable stochastic process in continuous time is the same as in discrete time: the possible outcome of the random variable X(t) depends only on all events which are observable till just before time t. In discrete time this is a whole step before the current time t, in continuous time this are all times s strictly less than t. Remark One can prove that a stochastic process which is adapted and continuous from the left (or from the- right) is predictable. Theorem (Brownian Martingale Representation Theorem) Assume that the filtration is generated by the Brownian motion, i.e. &t = &Y for a^ t 6 [0,T]. Let {M(t) : t G [0,T]) be an &t-adapted, continuous martingale with E[M2{t)] <oo for all te [0,T]. Then there exists a predictable stochastic process (Y(t) : t [0,T]) such that Remark M(t) = M(0) + / Y{s)dW(s) P-a.s. (4.2.2) Jo (a) The proof of Theorem is not constructive so it does not give an expression of the process Y. (b) An important condition in the Brownian martingale representation theorem is the requirement that the considered filtration is generated by the Brownian motion (as agreed in the beginning of this chapter). The need for this can be seen by the representation (4.2.2) because the only source of uncertainty or randomness is the process Y and the Brownian motion W both measurable with respect to the filtration generated by W.