Lecture 3: Review of mathematical finance and derivative pricing models

Lecture 3: Review of mathematical finance and derivative pricing models Xiaoguang Wang STAT 598W January 21th, 2014 (STAT 598W) Lecture 3 1 / 51

Outline 1 Some model independent definitions and principals 2 European Options Pricing Binomial Trees Black-Scholes Model 3 American Options Pricing (STAT 598W) Lecture 3 2 / 51

Outline 1 Some model independent definitions and principals 2 European Options Pricing Binomial Trees Black-Scholes Model 3 American Options Pricing (STAT 598W) Lecture 3 3 / 51

Self Financing Portfolio Notations: N = the number of different types of assets or stocks. h i (t) = number of shares of type i held during the period [t, t + t). h(t) = the portfolio [h 1 (t),, h N (t)] held during period t. c(t) = the amount of money spent on consumption per unit time during the period [t, t + t). S i (t) = the price of one share of type i during the period [t, t + t). V (t) = the value of the portfolio h at time t. Definition A self-financing portfolio is a portfolio with no exogenous infusion or withdrawal of money (apart from the consumption term c). It must satisfy dv (t) = h(t)ds(t) c(t)dt (STAT 598W) Lecture 3 4 / 51

Dividends Definition We take as given the processes D 1 (t),, D N (t), where D i (t) denotes the cumulative dividends paid to the holder of one unit of asset i during the interval (0, t]. If D i has the structure dd i (t) = δ i (t)dt for some process δ i, then we say that asset i pays a continuous dividend yield. Taking into account the dividend payments, a self-financing portfolio should now satisfy dv (t) = h(t)ds(t) + h(t)dd(t) c(t)dt (STAT 598W) Lecture 3 5 / 51

Free of Arbitrage Definition An arbitrage possibility on a financial market is a self-financed portfolio such that V h (0) = 0 P(V h (T ) 0) = 1 P(V h (T ) > 0) > 0 We say the market is arbitrage free if there are no arbitrage possibilities. In most cases, we assume that the market of interest is arbitrage free. (STAT 598W) Lecture 3 6 / 51

Martingale measure Definition Consider a market model containing N assets S 1,, S N and fix the asset S 1 (in most cases this will be the risk free rate account) as the numeraire asset. We say that a probability measure Q defined on Ω is a martingale measure if it satisfies the following conditions: 1. Q is equivalent to P, i.e. Q P 2. For every i = 1,, N, the normalized (discounted) asset price process Z i t = S i t S 1 t is a martingale under the measure Q. (STAT 598W) Lecture 3 7 / 51

First Fundamental Theorem Theorem Given a fixed numeraire, the market is free of arbitrage if and only if there exists a martingale measure Q. Under the martingale measure Q, every discounted (normalized) price process (either underlying or derivative) is a martingale. (STAT 598W) Lecture 3 8 / 51

Complete market Definition A contingent claim is a stochastic variable X of the form X = Φ(S T ), where the contract function Φ is some given real valued function. A given contingent claim X is said to be reachable if there exists a self-financing portfolio h such that V h T = X, P a.s. with probability 1. In that case we say that the portfolio h is a hedging portfolio or a replicating portfolio. If all contingent claims can be replicated we say that the market is (dynamically) complete. (STAT 598W) Lecture 3 9 / 51

Pricing principal A reachable claim X must be priced properly such that there is no arbitrage opportunity caused by this pricing. Theorem If a claim X is reachable with replicating (self-financing) portfolio h, then the only reasonable price process for X is given by Π(t; X ) = V h t, 0 t T (STAT 598W) Lecture 3 10 / 51

General pricing formula: martingale approach Theorem In order to avoid arbitrage, a contingent claim X must be priced according to the formula [ ] X Π(t; X ) = S 0 (t)e Q S 0 (T ) F t where Q is a martingale measure for [S 0, S 1,, S N ], with S 0 as the numeraire. In particular, we can choose the bank account B(t) as the numeraire. Then B has the dynamics db(t) = r(t)b(t)dt where r is the (possibly stochastic) short rate process. In this case the pricing formula above reduces to [ Π(t; X ) = E Q e ] T t r(s)ds X F t (STAT 598W) Lecture 3 11 / 51

Second Fundamental Theorem Theorem Assuming absence of arbitrage, the market model is complete if and only if the martingale measure Q is unique. For incomplete market, different choices of Q will generally give rise to different price process for a fixed claim X. However, if X is attainable (reachable) then all choices of Q will produce the same price process, which then is given by [ Π(t; X ) = V (t; h) = E Q e ] T t r(s)ds X F t where h is the hedging portfolio. Different choices of hedging portfolio (if such exist) will produce the same price process. (STAT 598W) Lecture 3 12 / 51

General Pricing formula: PDE approach Assume the market is complete, under the martingale measure (risk-neutral) Q, any derivative price process Π(t) must have the following dynamics dπ(t) = rπ(t)dt + σ Π (t)dw (t) where W is a Q Wiener process, and σ Π is the same under Q as under P. With Π above, the process Π(t) B(t) is a Q martingale, i.e. it has a zero drift term. Assume Π(t) has the form Π(t) = F (t, S(t)), then apply Ito s formula on F (t,s(t)) B(t) and set the drift term as zero. Then we will come up with a PDF as follows F t + rxf x + 1 2 x 2 σ 2 F x x rf = 0, F (T, x) = Φ(x) (STAT 598W) Lecture 3 13 / 51

Feynman-Ka c Assume that F is a solution to the boundary value problem F (t, x) + µ(t, x) F t x + 1 2 σ2 (t, x) 2 F (t, x) = 0 x 2 Assume furthermore that process F (T, x) = Φ(x) σ(s, X s ) F x (s, X s) is in L 2, where X is defined below. Then F has the representation where X satisfies the SDE F (t, x) = E t,x [Φ(X T )] dx s = µ(s, X s )ds + σ(s, X s )dw s, X t = x (STAT 598W) Lecture 3 14 / 51

Greeks Let P(t, s) denote the pricing function at time t for a portfolio based on a single underlying asset with price process S t. The portfolio can thus consist of a position in the underlying asset itself, as well as positions in various options written on the underlying asset. For practical purpose it is often of vital importance to have a grip on the sensitivity of P with respect to the following Price changes of the underlying asset Changes in the model parameters Definition = P s, Γ = 2 P s 2, ρ = P r Θ = P t, V = P σ (STAT 598W) Lecture 3 15 / 51

Incomplete market: market price of risk Assume that a derivative price F has the following dynamics under measure P: df = α F Fdt + σ F FdW Definition Assume that the market for derivatives is free of arbitrage. Then there exists a universal process λ(t) called such that, with probability 1, and for all t, we have α F (t) r = λ(t) σ F (t) regardless of the specific choice of the derivative F. The λ(t) is called market price of risk, which can be interpreted as the risk premium per unit of volatility. (STAT 598W) Lecture 3 16 / 51

Incomplete Market Pricing formula: PDE approach Assuming absence of arbitrage, the pricing function F (t, x) of a T -claim Φ(X (T )) solves the following boundary value problem F t (t, x) + AF (t, x) rf (t, x) = 0, (t, x) (0, T ) R F (T, x) = Φ(x), x R where AF (t, x) = {µ(t, x) λ(t, x)σ(t, x)}f x (t, x) + 1 2 σ2 (t, x)f xx (t, x) and we assume that the asset X (t) has the following dynamics under measure P: dx (t) = µ(t, X (t))dt + σ(t, X (t))d W (t) Here W is a standard scalar P-Wiener process. (STAT 598W) Lecture 3 17 / 51

Incomplete market pricing: Martingale approach Theorem Assuming absence of arbitrage, the pricing function F (t, x) of the T -claim Φ(X (T )) is given by the formula F (t, x) = e r(t t) E Q t,x[φ(x (T ))], where the dynamics of X under the martingale measure Q are given by dx (t) = {µ(t, X (t)) λ(t, X (t))σ(t, X (t))}dt + σ(t, X (t))dw (t) Here W is a Q-Wiener process, and the subscripts t, x indicate as usual that X (t) = x. (STAT 598W) Lecture 3 18 / 51

Outline 1 Some model independent definitions and principals 2 European Options Pricing Binomial Trees Black-Scholes Model 3 American Options Pricing (STAT 598W) Lecture 3 19 / 51

European Call and Put Options Definition A European Call (Put) Option on an underlying asset S with strike price K and exercise date T is a contract written at time t = 0 with the following properties: The holder of the contract has, exactly at time t = T, the right to buy (sell) one unit of the asset at the strike price K. The holder of the option has no obligation to buy (sell) the asset. The pay-off function for a European Call option is X = Φ(S(T )) = max(s(t ) K, 0) (STAT 598W) Lecture 3 20 / 51

Model Set Up Discrete time running from t = 0 to t = T, where T is fixed. There are two underlying assets, a risk-free bond with price process B t and a stock with price process S t. Assume a constant deterministic short rate of interest R. And B n+1 = (1 + R)B n, B 0 = 1. S n+1 = S n Z n, S 0 = s. Z 0,, Z T 1 are assumed to be i.i.d. stochastic variables, taking only two values u and d with probabilities: P(Z n = u) = p u, P(Z n = d) = p d (STAT 598W) Lecture 3 21 / 51

Portfolio Strategy Definition A portfolio strategy is a stochastic process {h t = (x t, y t ); t = 1,, T } such that h t is a function of S 0, S 1,, S t 1. For a given portfolio strategy h we set h 0 = h 1 by convention. The value process corresponding to the portfolio h is defined by V h t = x t (1 + R) + y t S t A self-financing portfolio strategy h will satisfy for all t = 0,, T 1, x t (1 + R) + y t S t = x t+1 + y t+1 S t (STAT 598W) Lecture 3 22 / 51

No arbitrage and Completeness Theorem The condition d (1 + R) u is a necessary and sufficient condition for absence of arbitrage. The binomial model defined above is complete. (STAT 598W) Lecture 3 23 / 51

Martingale Measure Theorem The martingale measure probabilities q u and q d which must satisfy s = 1 1 + R E Q [S t+1 S t = s] are given by q u = q d = (1 + R) d u d u (1 + R) u d (STAT 598W) Lecture 3 24 / 51

Pricing formulas Theorem The arbitrage free price at t = 0 of a T -claim X is given by Π(0; X ) = 1 (1 + R) T E Q [X ] where Q denotes the martingale measure, or more explicitly Π(0; X ) = 1 (1 + R) T T k=1 ( ) T qu k q T k k d Φ(su k d T k ) (STAT 598W) Lecture 3 25 / 51

Hedging portfolio A T -claim X = Φ(S T ) can be replicated using a self-financing portfolio. If V t (k) denotes the value of the portfolio at node (t, k) then V t (k) can be computed recursively by V t (k) = 1 1 + R {q uv t+1 (k + 1) + q d V t+1 (k)} (1) V T (k) = Φ(su k d T k ) (2) And the portfolio strategy is given by x t (k) = 1 uv t (k) dv t (k + 1) 1 + R u d y t (k) = 1 V t (k + 1) V t (k) S t 1 u d (3) (4) (STAT 598W) Lecture 3 26 / 51

Model set up Definition The Black-Scholes model consists of two assets with dynamics given by Definition db(t) = rb(t)dt ds(t) = αs(t)dt + σs(t)d W (t) A contingent claim with date of maturity (exercise date) T, also called a T -claim, is ant stochastic variable X FT S. A contingent claim X is called a simple claim if it is of the form X = Φ(S(T )). (STAT 598W) Lecture 3 27 / 51

Model assumptions The derivative instrument in question can be bought and sold on a market. The market is free of arbitrage. The price process for the derivative asset is of the form where F is some smooth function. Π(t; X ) = F (t, S(t)) (STAT 598W) Lecture 3 28 / 51

Black-Scholes formula Theorem The price of a European call option with strike price K and time of maturity T is given by the formula Π(t) = F (t, S(t)), where F (t, s) = sn[d 1 (t, s)] e r(t t) KN[d 2 (t, s)] Here N is the cumulative distribution function for the N(0, 1) distribution and { 1 ( s ) d 1 (t, s) = σ ln + (r + 12 ) } T t K σ2 (T t) d 2 (t, s) = d 1 (t, s) σ T t (STAT 598W) Lecture 3 29 / 51

Other European type of derivatives X = S(T ) K (Forward contract) [ 1 T ] X = max S(t)dt K, 0 (Asian Option) T X = S(T ) inf S(t) 0 t T X = 1 S(T )>K 0 (Lookback contract) (cash-or-nothing call, exotic option) Down and Out (Barrier Options): { Φ(S(T )), if S(t) > L for all t [0, T ] X = 0, if S(t) > L for some t [0, T ] (STAT 598W) Lecture 3 30 / 51

Outline 1 Some model independent definitions and principals 2 European Options Pricing Binomial Trees Black-Scholes Model 3 American Options Pricing (STAT 598W) Lecture 3 31 / 51

Definitions Definition A American Call (Put) Option on an underlying asset S with strike price K and exercise date T is a contract written at time t = 0 with the following properties: The holder of the contract has, up to time t = T, the right to buy (sell) one unit of the asset at the strike price K. The holder of the option has no obligation to buy (sell) the asset. It is well known that for an American call option written on an underlying stock without dividends, the optimal exercise time τ is given by τ = T. Thus the price of the American option coincides with the price of the corresponding European option. But for American put options, it is much harder and there is even no analytical solution. (STAT 598W) Lecture 3 32 / 51

Optimal stopping problem For an American Call Option it is not hard to understand that the price at time zero should be of the following form Π(0) = max 0 τ T E Q [e rτ max{s τ K, 0}] where the stock dynamics under the risk neutral measure Q are given by Similarly, for American put, we have ds t = rs t dt + σ t S t dw t Π(0) = max 0 τ T E Q [e rτ max{k S τ, 0}] In order to solve these pricing problems, we need to take use of some Optimal Stopping Theories. (STAT 598W) Lecture 3 33 / 51

Optimal Stopping Theory: definitions Definition A nonnegative random variable τ is called an (optional) stopping time w.r.t. the filtration F = {F t } t 0 if it satisfies the condition {τ t} F t, for allt 0 In general we refer to an integrable process Z as the reward process which we want to maximize. Then we can formulate our problem as max E[Z τ ] 0 τ T We say that a stopping time ˆτ T is optimal (not necessarily exist) if E[Zˆτ ] = sup E[Z τ ] 0 τ T (STAT 598W) Lecture 3 34 / 51

Some simple results Theorem The following hold: If Z is a submartingale, then late stopping is optimal, i.e. ˆτ = T. If Z is supermartingale, then it is optimal to stop immediately, i.e., ˆτ = 0. If Z is martingale, then all stopping times τ with 0 τ T are optimal. Assume that the process Z has the dynamics dz t = µ t dt + σ t dw t where µ and σ are adpated process and σ is square integrable. Then the following hold If µ t 0, P a.s. for all t, then Z is a submartingale. If µ t 0, P a.s. for all t, then Z is a supermartingale. If µ t = 0, P a.s. for all t, then Z is a martingale. (STAT 598W) Lecture 3 35 / 51

Some simple results Theorem If Z is a martingale and g is convex, then g(z t ) is a submartingale. If Z is submartingale and g is convex and increasing, then g(z t ) is a submartingale. If Z is a supermartingale and g is concave and increasing, then g(z t ) is a supermartingale. (STAT 598W) Lecture 3 36 / 51

Snell Envelope Theorem Consider a fixed process Y, we say that a process X dominates the process Y if X t Y t P a.s. for all t 0. Assuming that E[Y t ] <, the Snell Envelop S, of the process Y is defined as the smallest supermartingale dominating Y. More precisely: S is a supermartingale dominating Y, and if D is another supermartingale dominating Y, then S t D t, P a.s. for all t 0. For any integrable semimartingale Y, the Snell Envelope exists. (STAT 598W) Lecture 3 37 / 51

Optimal value process: discrete case Assuming discrete time framework, the optimal value process of Z is defined by V n = sup n τ T E[Z τ F n ] A stopping time which realizes the supremum above is said to be optimal at n, and it will be denoted as ˆτ n. Theorem The optimal value process V is the solution of the following backward recursion V n = max{z n, E[V n+1 F n ]}, V T = Z T Furthermore, it is optimal to stop at time n if and only if V n = Z n. If stopping at n is not optimal, then V n > Z n, and V n = E[V n+1 F n ]. (STAT 598W) Lecture 3 38 / 51

Optimal Stopping Rule Theorem An optimal stopping rule ˆτ at time t = 0 is given by ˆτ = min{n 0 : V n = Z n } For a fixed n an optimal stopping time ˆτ n is given by For any n we have ˆτ = min{k n : V k = Z k } Vˆτn = Zˆτn Theorem The optimal value process V is the Snell envelope of the reward process Z. (STAT 598W) Lecture 3 39 / 51

Continuous case Definition For a fixed (t, x) [0, T ] R, and each stopping time τ with τ t, the value function J si defined by J(t, x; τ) = E t,x [Φ(τ, X τ )] The optimal value function V (t, x) is defined by V (t, x) = sup E t,x [Φ(τ, X τ )] t τ T A stopping time which realizes the supremum for V above is called optimal and will be denoted as ˆτ tx or ˆτ t without confusion. (STAT 598W) Lecture 3 40 / 51

Assumptions We assume There exists an optimal stopping time ˆτ t,x for each (t, x). The optimal value function V is regular enough. More precisely we assume that V C 1,2. All processes interested are integrable enough, in the sense that expected values exist, stochastic integrals are true (rather than local) martingales, etc. Then we must have V (t, x) Φ(t, x) using the Ito operator V (t, x) E t,x [V (t + h, X t+h )] Af (t, x) = µ(t, x) f x (t, x) + 1 2 σ2 (t, x) 2 f (t, x) x 2 then we have V (t, x) Φ(t, x) ( ) t + A V (t, x) 0 (STAT 598W) Lecture 3 41 / 51

Stopping Rule It is optimal to stop at (t,x) if and only if in which case V (t, x) = Φ(t, x) ( ) t + A V (t, x) < 0 It is optimal to continue if and only if in which case V (t, x) > Φ(t, x), ( ) t + A V (t, x) = 0 Thus we can define the continuation region C by C = {(t, x); V (t, x) > Φ(t, x)} (STAT 598W) Lecture 3 42 / 51

Free Boundary value problem Theorem Assuming enough regularity, the optimal value function satisfies the following parabolic equation ( ) t + A V (t, x) = 0, (t, x) C, V (t, x) = Φ(t, x), (t, x) C Generally speaking, there is little hope of having an analytical solution of a free boundary value problem, so typically one has to resort to numerical schemes. (STAT 598W) Lecture 3 43 / 51

Variational inequalities Theorem Given enough regularity, the optimal value function is characterized by the following relations V (T, x) = Φ(T, x) V (t, x) Φ(t, x), (t, x) ( ) t + A V (t, x) 0 ( ) max{φ(t, x) V (t, x), t + A V (t, x)} = 0, (t, x) (STAT 598W) Lecture 3 44 / 51

Applicable partial results Theorem It is never optimal to stop at a point where Φ (t, x) + µ(t, x) Φ t x (t, x) + 1 2 σ2 (t, x) 2 Φ (t, x) > 0 x 2 Expressed otherwise, we have { ( (t, x); t + A ) } Φ(t, x) > 0 C (STAT 598W) Lecture 3 45 / 51

American Call Option The optimal stopping problem becomes max E Q [e rτ max{s τ K, 0}] 0 τ T where the stock price dynamics under the risk neutral measure Q are given by ds t = rs t dt + σ t S t dw t Thus the reward process Z is Z t = e rt max{s t K, 0} = max{e rt S t e rt K, 0} It is not hard to see that Z is a submartingale in this case. Thus the optimal stopping time should be ˆτ = T. (STAT 598W) Lecture 3 46 / 51

American Put Option The optimal stopping problem becomes And the optimal value function is V (t, x) = Under Q we still have max E Q [e rτ max{k S τ, 0}] 0 τ T sup Et,x[e Q r(τ t) max{k S τ, 0}] t τ T ds t = rs t dt + σ t S t dw t (STAT 598W) Lecture 3 47 / 51

Pricing for American Put Assume that a sufficiently regular function V (t, s), and an open set C R + R +, satisfying the following conditions: C has a continuously differentiable boundary b t, i.e. b C 1 and (t, b t) C. V satisfies the PDE V V + rs t s + 1 2 s2 σ 2 2 V s2 rv = 0, (t, s) C V satisfies the final time boundary condition V satisfies the inequality V satisfies V satisfies V (T, s) = max[k s, 0], s R + V (t, s) > max[k s, 0], V (t, s) = max[k s, 0], V (t, s) = max[k s, 0], V satisfies the smooth fit condition lim s b(t) (t, s) C (t, s) C c (t, s) C c V (t, s) = 1, 0 t T s (STAT 598W) Lecture 3 48 / 51

Pricing for American Put Then the following hold: V is the optimal value function C is the continuation region The optimal stopping time is given by ˆτ = inf{t 0; S t = b t } Unfortunately there is no analytical formulas for the pricing function or the optimal boundary. For practical use, the following alternatives are frequently used: Solve the free boundary value problem numerically. Solve the variational inequalities numerically. Approximate the Black-Scholes model by a binomial model and compute the exact binomial American put price. (STAT 598W) Lecture 3 49 / 51

Perpetual American Put A perpetual American Put option is an American put with infinite time horizon. This option is fairly simple to analyze, since the infinite horizon and the time invariance of the stock price dynamics implies that the option price as well as the optimal boundary are constant as functions of running time. There exists a constant critical price b such that we exercise the option whenever S t < b. Since in this case V (t, s) is independent of running time t then we have V (t, s) = V (s), and the free boundary value problem reduces to the ODE rs V s + 1 2 s2 σ 2 2 V rv = 0, s > b s2 (STAT 598W) Lecture 3 50 / 51

Perpetual American Put Pricing results Theorem For a perpetual American put with strike K, the pricing function V and the critical price b are given by b = γk 1 + γ where V (s) = K 1 + γ ( b s γ = 2r σ 2 ) γ, s > b (STAT 598W) Lecture 3 51 / 51