Option Pricing. Prof. Dr. Svetlozar (Zari) Rachev

Transcription

1 Option Pricing Prof. Dr. Svetlozar (Zari) Rachev Frey Family Foundation Chair-Professor, Applied Mathematics and Statistics, Stony Brook University Chief Scientific Officer, FinAnalytica

2 Outline: Option Pricing (2pm-4pm) Binomial Model and Black-Scholes Model Stochastic Volatility Models Local Volatility Models Models with Jumps Models with Heavy-Tailed Returns and Tempered Stable Pricing Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 1 / 46

3 Outline: Option Pricing Binomial Model and Black-Scholes Model Stochastic Volatility Models Local Volatility Models Models with Jumps Models with Heavy-Tailed Returns and Tempered Stable Pricing

4 Option Contract Basics The minimum attributes of an option contract are an exercise price (K) and an expiration date T. A call option s price (premium) at time t is denoted by C t and a put s option price by P t. The timing of the possible exercise is an important characteristic. Options that could be exercised at any time up to T are called American options. Those that could be only exercised at T are European options. A cashflow when an option is exercised is referred to as payoff. A call option is a vanilla call option if its payoff is given by max(s T K, 0). A vanilla put option s payoff is max(k S T, 0). Option contracts with various other payoff and exercise characteristics are known. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 2 / 46

5 Basic Components of the Option Price The option price consists of two components: the option s intrinsic value and a time premium. Intrinsic Value The intrinsic value (IV) is the economic value of the option if exercised immediately. Denote the current price of the underlying by S t. A call s and put s IVs at time t are then given, respectively, by C t = max (S t K, 0) and P t = max (K S t, 0). A positive IV is denoted as the option being in-the-money, while a zero value as out-of-the money. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 3 / 46

6 Basic Components of the Option Price Time Value Time value is the amount by which the option s price exceeds the IV. The source of that difference is the potential for the IV to increase in the time until expiration. An important factor behind the time value is the volatility of the underlying asset. Boundary Conditions for the Option Price Theoretical boundary conditions for the option price can be derived using arbitrage arguments. The idea is to build two portfolios whose payoff is the same. Since the outcomes are equal, the portfolios must have an equal price. If not, riskless profit could be made by buying the cheaper portfolio and selling the more expensive one. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 4 / 46

7 Boundary Conditions for the Option Price Consider a European call and put options on the same non-dividend paying stock, with the same strike K and expiration T. Portfolio A s holdings: Long call. Long risk-free bond with maturity T and face value K. Denote the current bond value by B(t,T,K). Portfolio B s holdings: Long share of the underlying stock. Short put. It can be shown that the payoffs of A and B are the same in all states of the world. Therefore, their prices are the same. The resulting relation is known as the put-call parity: C t + B(t, T, K) = S t P t. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 5 / 46

8 Discrete-Time Option Pricing: Binomial Model Idea behind the arbitrage argument used in deriving option pricing models: If the payoff from a long position in a call option can be replicated by Buying the underlying stock Borrowing funds Then, the option price is at (most) the cost of creating the replicating strategy. The discrete-time binomial option pricing model motivates the continuous-time Black-Scholes model. It assumes a discrete-time evolution for the underlying s price. A lattice is constructed for a number of time steps between valuation time and expiration. Each node of the lattice represents a possible price of the underlying. Valuation is performed iteratively, starting at the final nodes and working backwards along the lattice towards the first (valuation) node. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 6 / 46

9 One-Period Binomial Model for a Call: Hedge Ratio Begin by constructing a portfolio: 1 Long position in a certain amount of stock 2 Short position in a call on this underlying stock. The portfolio is constructed as a hedged portfolio: it is riskless and produces a return equal to the risk-free rate in one period time. The amount of long stock is such that the stock position is hedged against any change in the price at the call s expiration time (in one period time). Notation: u = 1+percentage change in price if price goes up in one period. d = 1+percentage change in price if price goes down in one period. r = risk-free rate. C = current call option price. C u = IV of call if stock price goes up in one period. C d = IV of call if stock price goes down in one period. H = hedge ratio: amount of stock purchased per call sold. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 7 / 46

10 One-Period Binomial Model for a Call: Hedge Ratio Cost of the hedged portfolio: HS C. Payoff to hedged portfolio in a period: If the stock s price goes up: H(uS) C u. If stock s price goes down: H(dS) C d. Since the hedge is riskless, the payoffs must be the same: H(uS) C u = H(dS) C d. Thus, we can solve for the hedge ratio, H: H = C u C d (u d)s. C u = max(us K, 0) and C d = max(ds K, 0). Note: we assume that 0 < d < 1 + r < u, for reasons that will become clear later. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 8 / 46

11 One-Period Binomial Model for a Call: Option Price The hedged portfolio returns the risk-free rate. One period from now, it is worth (1 + r) (HS C). Equating the value in a period with either payoff from the previous slide and then substituting for H, we get that the call price today is ( ) ( ) ( ) ( ) 1 + r d Cu u 1 r Cd C = +. ( ) u d 1 + r u d 1 + r Thus, we obtained the formula for the one-period binomial option pricing model. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 9 / 46

12 One-Period Binomial Model: Probabilistic Interpretation The quantities q = 1+r d u d and q = u 1 r u d add up to one and are both strictly positive. Therefore, q and q can be interpreted as probabilities. Then, the call price we derived can be regarded as the discounted expected payoff of holding a call option: C = r (C u q + C d q) = r E Q (C 1 ), where C 1 denotes the random call option payoff at expiration in a period and E Q (C 1 ) is the expected value of the one-year payoff. The probabilities used for valuing an option are not exogenously given but implicitly derived from the model. This probability distribution is called a risk-neutral distribution. Under the risk-neutral probability, the stock price can be shown to have an expected growth rate equal to the risk-free rate r: E Q (S 1 ) = us q + ds (1 q) = (1 + r)s. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 10 / 46

13 Two-Period Binomial Model Can extend the procedure by making the periods increasingly smaller. In fact, the Black-Scholes model we discuss below is the equivalent of the binomial approach, as the partitions become more and more refined. To help understand the notation for the two-period binomial tree case, consider the scheme below: Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 11 / 46

14 Two-Period Binomial Model To arrive at the call value: Starting at the expiration date (two periods from now), determine the values of C u and C d by using equation ( ). Solve for the current option price C by plugging C u and C d into ( ). Using the risk-neutral probabilities, after some algebra, we obtain: C = 1 ( Cuu q 2 + 2C ud q(1 q) + C dd (1 q) 2) 1 + r = r E Q(C 2 ), where C 2 is the random payoff of the call at expiration. Dividends can be incorporated into the binomial pricing model by adding the projected dividend payment to the stock price at the respective nodes of the tree. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 12 / 46

15 Convergence of the Binomial Model Divide the period from t 0 to t 1 into n stages for a total of n + 1 trading dates, t k = k n, k = 0, 1,..., n. There are n + 1 possible values for the asset price S at expiration t 1. Denoting by S 0 = S the current stock price, these values are given by S (n) 1 = {u k d n k S 0 k = 0, 1,..., n}. The pricing formula ( ) for the European call option with strike K and expiration t 1 can be generalized to 1 C = (1 + r) n E Q (max(s 1 K, 0)) = = 1 (1 + r) n 1 (1 + r) n n max(u k d n k S 0 K, 0) Q ( S 1 = u k d n k ) S 0 k=0 n k=0 ( max(u k d n k n S 0 K, 0) k ) q k (1 q) n k. At each node, the probability distribution of S is binomial(n, q). Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 13 / 46

16 Binomial Model Extensions: References Various generalizations of the binomial model have been developed: Impact of dividends, dependence of up and down movements in the asset price level, multinomial versions, etc.: Cox and Rubinstein (1985), Jarrow and Rudd (1983), Ritchken (1987), and Hull (1997), among others. Merton (1973) showed that if the stock pays no dividends, a European and an American call will be valued the same, since the American call will never be optimally exercised before expiration. For further details on American options, see Schwartz (1977), and Roll (1977), among others. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 14 / 46

17 Continuous-Time Option Pricing As continuous trading is approached, the n independent quantities, u, d, r, and q need to be adjusted to obtain meaningful valuation of a call option. (If they are independent of n, the model will explode as n increases.) Cox, Ross, and Rubinstein (1979) choose u = e σ2 /n ( ) T t q = µ 2 σ n d = e σ 2 /n r = log(1 + r). As n goes to infinity, the limiting value of the call can be calculated as ( ) C = e r max S 0 e r 0.5σ 2 +σz K, 0 φ(z)dz, where φ is the PDF of the standard normal distribution (to which the binomial converges as n ) This expected value can be explicitly calculated to obtain the Black-Scholes option pricing formila. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 15 / 46

18 Black-Scholes Option Pricing Model The alternative derivation of the continuous-time call valuation in Black and Scholes (1973) (B-S) assumes that The stock price follows a geometric Brownian motion, ds t = µs t dt + σs t dw t, t 0 S t = S 0 e (µ 1 2 σ2 )t+σw t. A risk-free asset (bond or bank account) exists, with (deterministic) dynamics described by db t = rb t dt, t 0 B t = B 0 e rt. The pioneering insight of B-S is that to hedge the exposure of an option towards changes in the underlying, an offsetting position in the underlying can be taken A continuously re-balanced portfolio of the stock and the bond is constructed that duplicates the payoff of the European call. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 16 / 46

19 Black-Scholes Option Pricing Model A (locally) risk-free portfolio must earn the (instantaneous) risk-free interest rate. It can be shown that the no-arbitrage argument leads to the B-S partial differential equation (PDE): t C t = rs t C t + 1 S t 2 σ2 St 2 2 S 2 t C t rc t. This fundamental PDE is the valuation equation that derivatives on S t must satisfy. The only parameter that is not observable and needs to be estimated is the constant volatility σ. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 17 / 46

20 Black-Scholes Option Pricing Model B-S asserted that for constant µ, σ, and r, there is a unique rational value for the option independent of the investor s risk attitude. The Black-Scholes formula for the price of a European call is the solution to the earlier PDE and has the form where C(S t, t) = S t Φ(z) e r(t t) KΦ(z σ T t), z = log(s t/k) + ( r σ2) (T t) σ T t and Φ is the CDF of the normal distribution. To solve the PDE, the terminal condition, C T (S T ) = max (S T K), 0, as well as certain boundary conditions are needed. There are other approaches to obtain the formula, in addition to solving the B-S PDE. See Duffie (1988b). Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 18 / 46

21 Risk-Neutral Valuation The ability to replicate an option by dynamically trading the underlying has an important consequence. Since none of the parameters in the PDE depends on the investor s risk preferences or the market, derivatives can be prices assuming risk neutrality. The so-called risk-neutral valuation relies on the existence of a risk-neutral measure. If a probability measure P is estimated using historical return data for the underlying stock, the measure is referred to as the market measure or the physical measure. The risk-neutral measure can be found by the equivalent martingale measure (EMM) of the measure P: A probability measure Q equivalent to P is called EMM of P if the discounted price process ( S t ) t [0,T ] ( S t = e rt S t ) of an underlying asset is a Q-martingale. 1 For more on risk-neutral valuation, see Harrison and Pliska (1981, 1983). 1 The process (X t) t 0 is a martingale process if X t = E (X T F t) for all 0 t T, where F t is a filtration containing all market information up to time t. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 19 / 46

22 Risk-Neutral Valuation Under the risk-neutral martingale measure Q, all asset prices can be computed by their discounted payoffs where V T is the time T payoff. C t = e r(t t) E Q (V T F t ), Denoting the probability density of the terminal spot prices S T under the risk-neutral measure Q by f Q (S T F t ), the price C t of a call with strike K is given by C t = e r(t t) (S T K)f Q (S T F t )ds T. K Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 20 / 46

23 Risk-Neutral Valuation Breeden and Litzenberger (1978) show that the risk-neutral density f Q can be derived as the implied risk-neutral density through an inversion formula f Q (K F t ) = e r(t t) C 2 t K 2, given a complete set of prices C t for call options with maturity T, i.e., calls with maturity T for every strike K > 0. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 21 / 46

24 Assumptions Behind the Black-Scholes Model The B-S model is based on several restrictive assumptions. Below, we will discuss extensions to the basic model. No taxes and transaction costs Derivation of option price depends mainly on the existence of a replicating portfolio. With transaction costs, the hedging portfolio can no longer be built and the option price is no longer unique. Continuous-time trading, short-selling, and trading fractions of assets Only these (unrealistic) assumptions together with the previous one allow the derivation of the unique call option price by the hedging argument. Variance of stock s return is constant and known with certainty An option pricing model with non-constant variance can be developed. However, unknown variance is a more serious offence: the variance must be known to construct the riskless hedge. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 22 / 46

25 Assumptions Behind the Black-Scholes Model GBM process for the stock price The market exhibits large fluctuations which cannot be explained by a continuous-time process with constant volatility. Either a stochastic volatility has to be introduced jumps in the stock price must be admitted. Both extensions have been developed. (Merton (1973), Cox and Ross (1976).) Borrowing and lending at a constant known risk-free rate Borrowing rates are higher than lending rates. Then, the option price will be bound between the call prices derived from the model using the two interest rates. The rate constancy can be relaxed by replacing the short rate by the geometric average of the returns expected over the life of the option. (Merton (1973)) No dividends Extensions have been developed for the case when dividends are known (Roll (1977) and Whaley (1981)). Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 23 / 46

26 The Smile Effect Volatilities exhibit a smile not only for options that are deep in(out-of)-the-money but also options with different maturities. Additionally, the implied volatility of a specific option with a certain strike and maturity also does not remain constant through time but evolves dynamically. The B-S model assumes that the underlying volatility is constant over the life of the derivative and not affected by changes in the underlying s price. Thus, it cannot capture variation in the implied volatilies. Incorporating phenomena such as volatility clustering, skewness, and heavy tails of asset returns helps overcome the deficiency and model derivatives more accurately. We consider these extensions next. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 24 / 46

27 Outline: Option Pricing Binomial Model and Black-Scholes Model Stochastic Volatility Models Local Volatility Models Models with Jumps Models with Heavy-Tailed Returns and Tempered Stable Pricing

28 Stochastic Volatility Models An SV model may take the following basic form. 1 The underlying stock price is modeled by the process where r is the risk-free rate. ds t = µs t dt + V t S t dw 1 t, The variance V t itself follows a mean-reverting stochastic process given by dv t = a(b V t )dt + ξ V t dw 2 t, where a, b, and ξ are real-valued constants and W 1 and W 2 denote two (possibly correlated) Brownian motions. 2 1 Some of the more prominent SV models are Hull and White (1987) and Heston (1993). 2 In fact, when dwt 1 dwt 2 = ρdt, we obtain the Heston (1993) model. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 25 / 46

29 Stochastic Volatility Models If the two Brownian motions are independent, the call price can be calculated as the average of the B-S prices over all possible paths of the volatility. In the SV setting, hedging becomes more difficult as it is no longer possible to generate a risk-free portfolio consisting of only the European call and a fraction of the underlying stock. There are two different risk drivers: the random volatility and the noise in the stock price itself. Therefore, three assets depending on the same risk drivers are needed to eliminate both risks completely. A risk-free portfolio can be built from fractions of two options on the same stock with different strikes and/or maturities and a certain portion of the underlying stock. SV models have been proposed by Duffie, Pan, and Singleton (2000), and Barndorff-Nielsen and Shephard (2001), among others. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 26 / 46

30 Stochastic Volatility Models: Smile Consistent Pricing In the B-S model, the no-arbitrage argument determines the exact mapping between the physical and the risk-neutral one both can be recovered from the observed dynamics of the underlying price process. For SV models, the risk-neutral parameters cannot be estimated using only data from the underlying pricing process, since the volatility is unobservable. One solution is to use options cross-sectional data and calibrate the parameters, a, b, ξ, and V 0, to the observed option prices Smile-consistent pricing. Problems: The options cross-section does not contain information about the time dynamics of the system. Observed prices are not explained: even if prices of instruments such as European calls are replicated, the prices of exotic instruments obtained by the model may still be wrong. For calibration discussions, see for example Carr and Madan (1999). Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 27 / 46

31 Stochastic Volatility Models The implied volatility surface (3-D graph of the implied volatility as a function of strike price and time to maturity) for the SV model discussed above exhibits a smile. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 28 / 46

32 Stochastic Volatility Models: Risk-Neutral Valuation We provide a few details on risk-neutral valuation which will help in subsequent model discussions as well. To change the measure from the physical P to the equivalent martingale measure, Q, 1 first define: d d (1) W t = dw (1) t + µ r dt Vt (2) W t = dw (2) t + Λ(t, S t, V t )dt, where Λ(t, S t, V t ) is the so-called market price of volatility risk. Heston (1993) proposes that it is proportional to the volatility, Λ(t, S t, V t ) = k V t, for some constant k. 2 As remarked earlier, in an SV setting, volatility is a second risk driver (market is incomplete) and Q is not unique. A different risk-neutral measure is obtained for different choices of Λ(t, S t, V t ). 1 Via a result called Girsanov s theorem. 2 In the B-S model, the market price of risk is Λ = µ r. σ Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 29 / 46

33 Stochastic Volatility Models: Risk-Neutral Valuation The risk-neutral stochastic differential equations can be expressed as d S (1) t = rdt + Ṽ t d W t (2) dṽ t = ã( b Ṽ t )dt + ξ Ṽ t d W t, where ã = a + ξk, b = ab k + ξk, and d (1) (2) W t d W t = ρdt in Heston s SV model. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 30 / 46

34 Outline: Option Pricing Binomial Model and Black-Scholes Model Stochastic Volatility Models Local Volatility Models Models with Jumps Models with Heavy-Tailed Returns and Tempered Stable Pricing

35 Local Volatility Models Another class of smile-consistent option pricing models are the local volatility models. Volatility is assumed to be locally deterministic: tomorrow s volatility is a function of tomorrow s stock price level S t and the time t. This is achieved as follows: Suppose a complete set of prices for European calls with maturity T is given. The implied risk-neutral density of terminal stock prices is obtained through Breeden and Litzenberger (1978) s inversion formula. Is there a volatility function σ(t, S t ) such that the stochastic process defined by ds t S t = rdt + σ(t, S t )dw t generates exactly f Q (S T F t )? Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 31 / 46

36 Local Volatility Models The local volatility σ(t, S t ) can be obtained from 1 b 2 (K, T ) 2 2 C K 2 = C T, where C is the call premium at time t. Finally, σ(t, S t ) = b(s t, t) S t. A complete set of option prices (for each strike and each maturity) is never observable. Therefore, the shape of the local volatility surface may be sensitive to the interpolation used to compute the partial derivatives. Market prices are not explained but the model is simply calibrated to them. Thus, it may need frequent re-calibration. 2 1 Dupire (1994), Rubinstein (1994), and Derman and Kani (1994). 2 Dumas, Fleming, and Whaley (1998) empirically examine the time properties of local volatility models. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 32 / 46

37 Outline: Option Pricing Binomial Model and Black-Scholes Model Stochastic Volatility Models Local Volatility Models Models with Jumps Models with Heavy-Tailed Returns and Tempered Stable Pricing

38 Models with Jumps Models with jumps are another class of models proposed to address with the deficiencies of the B-S model. SV models are also found deficient in modeling the volatility smile at short maturities, due to the insufficient degree of heavy-tailedness they generate. 1 Merton (1976) imposes the following dynamic for the evolution of the underlying stock price: ds t S t = (µ λk)dt + σdw t + qdn t, where N denotes a Poisson process with intensity λ, independent of the Brownian motion W, and the jump size q is log-normally distributed with mean k and variance δ 2. That is, the return on the stock consists of a deterministic drift, a normally-distributed noise, and a probability λdt that a jump of mean size k will occur. 1 Documented, for example, in Gatheral (2004). Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 33 / 46

39 Models with Jumps In constructing a hedged portfolio, in order to hedge the change in the derivative arising from the jump component, options must be added. (Similar to the issue in the Heston model). The result is that again the risk-neutral measure is not unique and the pricing model depends on an unknown market price of jump risk. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 34 / 46

40 Models with Jumps Typically, it is assumed that the occurrence of jumps is uncorrelated with the overall market and can be diversified away. Since the jump risk is non-systematic, no risk premium is paid for it. Under this assumption, a pricing equation for European call options can be derived in terms of the B-S prices, C JD (S, K, σ, r, T, λ, q, δ 2 ) = e mλ(t t) (mλ(t t)) k C BS (S, K, σ k, r k, T ), k! k=0 where σ k = σ 2 + kδ 2 /(T t), r k = r λ(q 1) + k log(q)/(t t), and C BS is the B-S call price. The kth term corresponds to a scenario where k jumps occur over the life of the option. Another popular jump diffusion model has been suggested by Bates (1996) who combines stochastic volatility with independent jumps. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 35 / 46

41 Outline: Option Pricing Binomial Model and Black-Scholes Model Stochastic Volatility Models Local Volatility Models Models with Jumps Models with Heavy-Tailed Returns and Tempered Stable Pricing

42 Exponential Tempered Stable Model To overcome the unrealistic assumption of GBM in the B-S model, the risk driver is directly replaced with a process exhibiting heavy-tailed increments. We start with a discussion of option pricing in the case of exponential tempered stable models and then generalize to subordinated models. In particular, we discuss the pricing of European options by means of the Fourier transform method. 1 The stock price follows an exponential Lévy model if S t = S 0 e Xt for every t 0. The process X t is referred to as the driving process of the stock price process. If the driving process is Brownian motion, the resulting stock price process is GBM. If the driving process is a tempered stable process 2, the exponential Lévy model is referred to exponential tempered stable model. 1 Due to Carr and Madan (1999) and Lewis (2001). 2 Recall that we discussed the classical tempered stable distribution yesterday. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 36 / 46

43 Exponential Tempered Stable Model As before, denote by P and Q the market and risk-neutral measures, respectively. The stock price model under P is exponential tempered stable model, that is 1 S t = S 0 e Xt, where X t is a tempered stable process. X is assumed to be a tempered stable process under Q as well. Since the driving process must satisfy S 0 = E Q ( St ) = e rt S 0 E Q ( e X t ), φ Xt ( i) = E Q ( e X t ) = e rt, where φ Xt is the characteristic function of X t under Q. For additional details, see Rachev, Kim, Bianchi, and Fabozzi (2010). Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 37 / 46

44 Exponential Tempered Stable Model We can find the relationship between the market and risk-neutral parameters for which Q corresponds to P. Suppose that X is a classical tempered stable (CTS) process under both P and Q with parameters, respectively, (α, C, λ +, λ, m) and ( α, C, λ +, λ, m). The condition on the characteristic function (previous slide) is satisfied if m = r log φ CTS ( i; α, C, λ +, λ ), 0 and λ + > 1. The measures P and Q are equivalent if and only if α = α, C = C, and λ + and λ can be found such that ( λα 1 ) α 1 m m = CΓ (1 α) + λ λ α λ α 1. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 38 / 46

45 Exponential Tempered Stable Model The call price C t at time t can be calculated as C t = K 1+ρ r(t t) e πst ρ Re 0 (T t) log φ X 1 (u+iρ) iu log(k/st) e e (ρ iu)(1 + ρ iu) du, where ρ is a real number such that λ + ρ < 1 and φ X1 (u + ip) < for all u R. The put price P t at time t is obtained from the same formula but for 0 < ρ λ. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 39 / 46

46 Exponential Tempered Stable Model Calibrating the Risk-Neutral Parameters Consider observed prices Ĉ i of call options with maturities T i and strikes K i, i = 1,..., N, where N is the number of options on a given day. The risk-neutral process is fitted by matching the model prices to the market prices using non-linear least squares. That is, the goal is to find a parameter set θ solution to the minimization problem min N i=1 ) 2 (Ĉi C θ (T i, K i ), where C θ is a model price based on a parameter set θ. The market parameters are estimated using the historical returns of the underlying asset. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 40 / 46

47 Subordinated Stock Price Model As we discussed yesterday, many Lévy processes can be represented as a subordinated Brownian motion, so the distinction between the model discussed above and the subordinated model is not very sharp. 1 The subordinated stock price model is given by S t = S 0 e µt σ 2 τt+σwτ t, where W = W τt is the standard Brownian motion and τ = (τ t ) t 0 is the subordinator or intrinsic time process. When τ t = t, the B-S model is obtained. Other special cases of τ give rise to known models such as log-stable model (Rachev and Mittnik (2000) and Hurst et.al. (1999)), the Barndorff-Nielsen model (Barndorff-Nielsen (1994)), and the Heston model. 1 Subordinated models have been considered by Mandelbrot and Taylor (1967), Rachev and Rüschendorf (1994), Madan and Seneta (1990), and Hurst, Platen, and Rachev (1999), among others. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 41 / 46

48 Subordinated Stock Price Model Assume for convenience that the driver process is X t = σw τt. The value of a European call at time t is then given by ( C t = S t e (T t) F log S ) r(t t) 0e Ke r(t t) F + (log S 0e K K where F ± (x) = 0 N ( x 1 2 y y ) df στt (y), r(t t) F στt is the cumulative distribution function of the random variable στ t and N is the standard normal CDF. ), Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 42 / 46

49 References Barndorff-Nielsen, O. (1994). Gaussian/inverse Gaussian processes and the modeling of stock returns. Working paper, Aarhus University. Barndorff-Nielsen, O. and Shephard, N. (2001). Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. J. The Royal Statistical Society, Series B, 63(2). Bates, D. (1996). Jumps and stochastic volatility: exchange rate process implicit in the Deutsche mark options. Review of Financial Studies, 9(1). Black, F. and Scholes, M. (1973). The pricing of corporate liabilities. J Political Economy, May-June. Breeden, D. and Litzenberger, R. (1978). Prices of state-contingent claims implicit in option prices. J Business, 51(4). Carr, P. and Madan, D. (1999). Option valuation using the fast Fourier transform. J Computational Finance, 2(4). Cox, J.C., Ross, S., and Rubinstein, M. (1979). Option Pricing: A Simplified Approach. J Financial Economics, 7. Cox, J.C. and Rubinstein, M. (1985). Options Markets. Prentice Hall. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 43 / 46

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52 References Rachev, S., Kim, Y., Bianchi, M., and Fabozzi, F. (2010) Financial Models with Lévy Processes and Volatility Clustering. Wiley. Rachev, S. and Rüschendorf, L. (1994). Model for option prices. Theory of Probability and Its Applications, 39. Ritchken, P. (1987). Option. Scott, Foresman and Co., IL. Roll, R. (1977) An analytic formula for unprotected American call options on stocks with known dividends. J Financial Economics, November. Rubinstein, M. (1994). Implied binomial trees. J Finance, 49. Schwartz, E. (1977). The valuation of warrants: implementing a new approach. J Financial Economics, 4. Whaley, R. (1981). On the valuation of American call options on stocks with known dividends. J Financial Economics, June. Prof. Dr. Zari Rachev Option Pricing Next Generation Risk Management 46 / 46