Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010
Introduction Pricing an European call option Conclusion
INTRODUCTION A stochastic process X is said to be an {F t } martingale the following conditions hold. X is adapted to the filtration {F t } t 0. E[X (t)] <. For all s and t with s t, E[X (t)/f s ] = X (s) Recall 1. A filteration of a set Ω is a collection F t such that - each F t is a σ algebra of subsets -if s < t, F s F t 2.A sequence {x t } of random variables x t is adapted to a filtration {F t } t 0 if for each t the random variable x t is F t measurable. Definition 2: Wiener process A stochastic process W = (W (t) : t 0) defined on some filtered probability space (Ω, F, {F t } t 0 ), P is a wiener process if
i-w is adapted to a filtration {F t } t 0 ii-w (0) = 0 iii- The process has independent increments iv- if 0 s t, then W (t) W (s) is independent of {F s } v- For s t the stochastic process W (t) W (s) N(0, t s) Definition 3: Ito process A stochastic process X = (X t ) t 0 defined on a filtered probability space (Ω, F, {F t } t 0 ), P is an Ito process if there exists two non anticipating processes a = (a(t, w)t 0) and b = (b(t, w))t 0 such that t P( a(s, w) ds < ) = 1 t > 0 (1) P( 0 t 0 X t = X 0 + b 2 (s, w)ds < ) = 1 t > 0 (2) t 0 t a(s, w)ds + b(s, w)db s (3) 0
eqn 3 gives rise to a stochastic differential equation dx t = a(t, w)dt + b(t, w)db t (4) With an interest to application to finance, consider a stochastic differential equation dx t = µ(t)dt + σ(t)dw (t) (5) Suppose Z(t) = f (t, X t ) with f C 1,2 such that f : R + R R, then Z has a SDE given by df (t, X t ) = f f dt + t t dx t + 1 2 Eqn 6 is the the one-dimensional Ito s formula. 2 f [dx t ] 2 (6) x 2 t
Pricing a standard European option in a (B,S)-stock market Definition 4: An option is a security (contract) issued by a firm,a bank, financial company giving its buyer a right to buy or sell something of value (e.g currency) on specified terms at a fixed instant or during a certain period of time in future. Examples: European, American and other exotic options. These options can be call or put options. A holder of an European option has a right to exercise it at some specified time T in future called its maturity date. Suppose the seller (writer) of an European option enters into a contract with a buyer of this option to deliver a certain quantity of a given asset at a a price K at maturity (Called the delivery price). Suppose the price of the asset at maturity is S T, f T = max(s T K, 0) (7) f T in eqn 7 is called the pay-off function.
what is the fair price to both the buyer and the writer of the option? From theory, the fair price is C T = E(S T K) + and this may vary depending on the stochastic differential equation satisfied by the asset prices and the status of the bank account. D(a) + = max(a, 0) In (B,S)-market, an investor owns a bank account as the risk-free asset and stock which is a risky asset. Suppose i-the writer of an option has a bank account B = (B t ) t T such that B t = 1 ii- S t = S 0 + µt + σw t (8) where W is the standard wiener process on some probability space (Ω, F, P)
Bachelier s formula According to Bachelier(1870-1946), the rational price C T = (f T ; P) of the standard European call option with pay off function f T = (S T K) + with prices distribution (8) is given by the formula C T = (S 0 K)φ( S 0 K σ (T ) ) + σ (T )ϕ( S 0 K σ (T ) ) (9) where ϕ(x) = 1 e x2 2 (2π) and φ(x) = x ϕ(y)dy. proof: We need to define a martingale measure P on (Ω, F T ) such that S = (S t ) t T is P-measurable. In other words we need a measure P such that P T P T. Girsanov Transformation Let q be a constant, T < and let us consider a process W defined as W = W t + qt t [0, T ] wherew is a wiener process under some probability measure P. Define a process
M t by M t (W ) = exp{ qw t 1 2 q2 t} t [0, T ] E pt (M T ) = 1(proof follows by Itos formula) (10) and the new measure P on (Ω, f T ) by d P(w) = M T (w)dp(w) (11). Moreover -M is a martingale w.r.t the filtration F t and under measure P - P is a probability measure -The process W is a wiener process under P Therefore with q = µ σ, C T = E p (f T ) = E(M T )f T C T = E p (S T K) + = E p (S 0 + µt + σw T K) + By self similarity property of a wiener process, C T = E pt (S 0 K + σw T ) + ) = E(S 0 K + σ (T )W 1 ) +.
we also know that, if ξ N(0, 1), then E(a + bξ) + = a b (a + bx)ϕ(x)dx = aφ( a b ) + xϕ(x)dx (12) for a R, b 0. therefore by setting a = S 0 K and b = σ (T ) we get the desired expression in equation 9. How fair is price to the writer of the option? To answer this, we consider a strategy π = ( β, γ) of a self-financing class portfolio of initial value X0 π = C T that replicates the pay -off function f T. Let XT π = f T (replicative in that the value of the portfolio at any time t is the fair price of the asset or derivative) Xt π = E pt (f T /F t ) = E pt ((S T K + /F t ) = E pt ((S t K) + (S T S t )/F t ) since S = (S t ) t T is a markov process. From equation 9, set a = S t K and b = σ (T t). For 0 t T and S > 0 a b
we set S K C(s, t) = (S K)φ( σ (T t) ) + σ S K (T t)ϕ( σ (T t) ) (13) But since X π is a self financing portfolio, then dxt π = γ t ds t From Ito s formula dc = C s ds t + ( C t + 1 2 σ2 2 C )dt (14) S 2 γ t = C s Differentiating the RHS of eqn 9 gives S K γ t = φ( σ (T t) ) but C(t, S t ) = β t B t + γ t S t which implies S K β t = Kφ( σ (T t) ) + σ S K (T t)ϕ( σ (T t) )
Notice that lim t T C = S T K for S t > K and the limit is 0 for S t < K This means the writer will have the value to pay to the buyer of the option when the price is above strike at the date of maturity.
Conclusion Even though this model had various criticisms since prices could take on negative values, it was a bench mark for development of other better option pricing models. e.g The nobel prizing winning Black-scholes model (1973) in which the geometric brown motion is considered.
THANK YOU MWEBALE