Valuation of EquityLinked Insurance Products and Practical Issues in Equity Modeling. March 12, 2013


 Julia Cobb
 2 years ago
 Views:
Transcription
1 Valuation of EquityLinked Insurance Products and Practical Issues in Equity Modeling March 12, 2013 The University of Hong Kong (A SOA Center of Actuarial Excellence) Session 2 Valuation of EquityLinked Insurance Products Professor, ASA
2 Department of Statistics and Actuarial Science The University of Hong Kong Hong Kong Based on a paper with Hans Gerber and Elias Shiu
3 Introduction Equitylinked products are very popular in the market nowadays. Example: Guaranteed Minimum Death Benefits Payoff: max(s(t x ), K) = S(T x ) + [K S(T x )] + = K + [S(T x ) K] +, where T x is the timeuntildeath random variable for a life age x, S(t) is the price of equityindex at time t, and K is the guaranteed amount.
4 What is a contingent option? Option s payoff, e.g. European call: (S(T ) K) + Contingent option s payoff: (S(T x ) K) +, where T x is a random variable, independent of S(t).
5 Notation S(t), t 0, value of one unit of a fund at time t T x time of death of a life age x Equitylinked death benefit defined by b(s): Payment b(s(t x )) at time T x Problem: Calculate E[e δtx b(s(t x ))] δ > 0 valuation force of interest
6 Distribution of T x Any continuous distribution on (0, ) can be approximated by a linear combination of exponential distributions f Tx (t) = n A i λ i e λ i t = i=1 n A i f i (t), t > 0, i=1
7 Distribution of T x = = = = E[e δtx b(s(t x ))] 0 0 e δt E[b(S(t))]f Tx (t)dt [ n e δt E[b(S(t))] i=1 ] A i f i (t) dt n A i e δt E[b(S(t))]f i (t)dt i=1 0 n A i E[e δτ i b(s(τ i ))]. i=1
8 The distribution of T x can be approximated by a linear combination of exponential distributions. Thus we limit ourself to the following problem: Calculate E[e δτ b(s(τ))] with τ an exponential random variable notation: E[τ] = 1/λ.
9 Model for S(t): S(t) = S(0)e X (t), t 0 with (1) X (t) = µt + σw (t) W (t) is a standard Brownian motion E[X (t)] = µt, Var[X (t)] = σ 2 t or (2) jump diffusion upward jumps: exponential mean 1/v, frequency ν downward jumps: exponential mean 1/w, frequency ω
10 In this talk, we focus on (1). Thus {X (t)} Brownian motion parameters µ, σ 2, D = 1 2 σ2 E[e zx (t) ] = e tψ(z) with Ψ(z) = Dz 2 + µz We need the distribution of X (τ) (well known, exponential stopping of a Wiener process )
11 E[e zx (τ) ] = E[E[e zx (τ) τ]] = E[e τψ(z) λ ] = λ Ψ(z) = λ Dz 2 µz + λ. Now we rewrite this with partial fractions. We need α < 0, β > 0 solutions of Dz 2 + µz λ = 0
12 Then E[e zx (τ) ] = with κ = αβ β α κ z α κ z β Thus the pdf of X (τ) is { κe f X (τ) (x) = αx, if x 0, κe βx, if x > 0,
13 Discounted density functions: f δ X (τ) (x) = E[e δτ f (x, τ)] = λ = λ λ + δ 0 e (λ+δ)t f (x, t)dt (λ + δ) e (λ+δ)t f (x, t)dt 0 } {{ } = f X (τ) (x) with λ replaced by λ + δ
14 Thus we find the following Recipe: Discounted density of X (τ) = λ λ+δ pdf of X (τ) with λ replaced by λ + δ
15 f δ X (τ) (x) = { κe αx, if x 0, κe βx, if x 0, where α < 0 and β > 0 are the solutions of Dξ 2 + µξ (λ + δ) = 0 and κ = λ αβ λ+δ β α.
16 Value of a death benefit defined by b(s): E[e δτ b(s(τ))] = λ = = κ 0 0 e (λ+δ)t b(s(0)e x )f (x, t)dxdt b(s(0)e x )f δ X (τ) (x)dx b(s(0)e x )e αx dx + κ 0 b(s(0)e x )e βx dx
17 Factorization formula If τ is exponential with mean 1/λ, then the following factorization formula holds, E[e δτ g τ (X )] = E[e δτ ] E[g τ (X )], where τ is an exponential random variable with mean 1/(λ + δ) and independent of X. Remarks (i) E[e δτ ] = λ λ+δ. (ii) The condition δ > 0 can be replaced by the condition δ > λ.
18 Examples (1) b(s) = (K s) +, K < S(0) outofthemoney put option E[e δτ (K S(τ)) + ] = (K S(0) e x ) + fx δ (τ)(x) dx = κk [ K ] α α(1 α) S(0) (2) b(s) = (s K) +, K > S(0) outofthemoney call option E[e δτ (S(τ) K) + ] = κk [ S(0) β(β 1) K ] β
19 The Greek letters (1) of the outofthemoney put option κ [ ] K (α 1) 1 α S(0) (2) of the outofthemoney call option [ κ S(0) β 1 K ] β 1
20 The Greek letters (1) Γ of the outofthemoney put option [ ] κ K (α 2) K S(0) (2) Γ of the outofthemoney call option κ K [ S(0) K ] β 2
21 inthemoney options (3) b(s) = (K s) +, K > S(0) put (4) b(s) = (s K) +, K < S(0) call Use the putcall parity and (1) and (2) [K S(τ)] + [S(τ) K] + = K S(τ) yields E[e δτ [K S(τ)] + ] E[e δτ [S(τ) K] + ] = λ λ + δ K E[e δτ S(τ)]
22 Inthemoney formulas Call E[e δτ [S(τ) K] + ] if S(0) > K [ ] κk K α = λ α(1 α) S(0) λ + δ K + E[e δτ S(τ)] Put E[e δτ [K S(τ)] + ] if S(0) < K [ ] κk S(0) β = + λ β(β 1) K λ + δ K E[e δτ S(τ)]
23 The Greek letters (under risk neutral probability) (1) of the inthemoney call option 1 κ [ ] K (α 1) 1 α S(0) (2) of the inthemoney put option [ ] κ S(0) β 1 1 β 1 K
24 The Greek letters (under risk neutral probability) (1) Γ of the inthemoney call option [ ] κ K (α 2) K S(0) (2) Γ of the inthemoney put option κ K [ S(0) K ] β 2
25 Illustration We consider 90strike lifecontingent call and put options on a stock with initial price S(0) = 100. We assume δ = 8% and µ = δ D. We assume that the distribution of T x is exponential with mean 125/6. Table 1: Contingent call and put values call put σ = σ = σ = σ =
26 rollup GMDB options Payoff: [Ke pτ S(τ)] + at time of death p: rollup rate Discounted payoff e δτ [Ke pτ S(τ)] + = e (δ p)τ [K e pτ S(τ)] + Valuation of a put option, with the substitutions δ δ p, µ µ p
27 T year contingent options Finite expiry date T > 0 Payoff [K S(τ)] + I (τ T ) Can be written as [K S(τ)] + [K S(τ)] + I (τ>t ) The time0 cost of the T year deferred contingent put option is E[e δτ [K S(τ)] + I (τ>t ) ] = Pr(τ > T )E[e δτ [K S(τ)] + τ > T ] = e (λ+δ)t E[e δ(τ T ) [K S(T )e X (τ) X (T ) ] + τ > T ].
28 The conditional expectation given S(T ) is [ ] κk K α I α(1 α) S(T ) (S(T )>K) { [ ] κk S(T ) β + + λ } β(β 1) K λ + δ K E[e δτ+x (τ) ]S(T ) I (S(T )<K). Its expectation can be evaluated by the factorization formula in the method of Esscher transforms.
29 Result: [ ] κk K α Φ( z α ) + κk [ ] S(0) β Φ(z β ) α(1 α) S(0) β(β 1) K +e (λ+δ)t λ λ + δ KΦ(z 0) e (λ+δ ϑ)t E[e δτ S(τ)]Φ(z 1 ). where z h = k (µ + hσ2 )T σ T,
30 Value of the T year Kstrike contingent put option S(0) > K S(0) < K [ ] κk K α Φ(z α ) κk [ ] S(0) β Φ(z β ) α(1 α) S(0) β(β 1) K e (λ+δ)t λ λ + δ KΦ(z 0) + e (λ+δ ϑ)t E[e δτ S(τ)]Φ(z 1 ) [ ] κk K α Φ( z α ) + κk [ ] S(0) β Φ( z β ) α(1 α) S(0) β(β 1) K + λ λ + δ K[1 e (λ+δ)t Φ(z 0 )] E[e δτ S(τ)][1 e (λ+δ ϑ)t Φ(z 1 )]
31 Illustration We consider T year 90strike lifecontingent put options on a stock with initial price S(0) = 100. We assume δ = 8% and µ = δ D. We assume that the distribution of T x is exponential with mean 125/6. Table 2: Contingent T =year put values T σ = σ = σ = σ =
32 M(t) = max 0 s t X (s) running maximum m(t) = min 0 s t X (s) running minimum For certain options we need f M(τ) (x) pdf of M(τ) f m(τ) (x) pdf of m(τ) It is well known that f M(τ) (x) = βe βx, x > 0, f m(τ) (x) = αe αx, x < 0.
33 Proof: Stop the martingale {e βx (t) I (τ>t) } at the time min(τ, first passage time at x) and use the optional sampling theorem: 1 = 0 + e βx Pr(M(τ) > x) Thus Pr(M(τ) > x) = e βx and f M(τ) (x) = βe βx, x > 0.
34 Discounted density functions: fm(τ) δ (x) = λ e (λ+δ)t f M(t) (x)dt f δ M(τ) (x) = = 0 λ λ + δ λ λ+δ βe βx, x > 0, (λ + δ) e (λ+δ)t f M(t) (x)dt 0 } {{ } = f M(τ) (x) with λ replaced by λ + δ where β > 0 solution of Dz 2 + µz (λ + δ) = 0.
35 Similarly, f δ m(τ) (x) = λ λ+δ ( α)e αx, x < 0, where α < 0 solution of Dz 2 + µz (λ + δ) = 0.
36 Bivariate distributions: We know that for a Lévy process M(τ) and M(τ) X (τ) are independent M(τ) X (τ) has the same distribution as m(τ) = m(τ) f M(τ),M(τ) X (τ) (y, z) = f M(τ) (y)f M(τ) X (τ) (z) = f M(τ) (y)f m(τ) ( z) = αβe βy e αz for y 0, z 0.
37 Because of X (τ) = M(τ) [M(τ) X (τ)], we find that f X (τ),m(τ) (x, y) = f M(τ),M(τ) X (τ) (y, y x) = f M(τ) (y)f M(τ) X (τ) (y x) = αβe αx (β α)y for y max(x, 0)
38 Then the discounted density is f δ X (τ),m(τ) (x, y) = αβ λ λ + δ e αx (β α)y for y max(x, 0) where α < 0 and β > 0 are the solutions of Dz 2 + µz (λ + δ) = 0
39 Lookback options Many equityindexed annuities credit interest using a high water mark method or a low water mark method
40 Outofthemoney fixed strike lookback call option Payoff: Time0 value k [S(0)e M(τ) K] + [S(0)e y K]fM(τ) δ (y)dy = λ [S(0)βe (β 1)k λ + δ β 1 [ ] S(0) β =. λ λ + δ Another expression for the option value [ ] λ K S(0) β. D αβ(β 1) K K β 1 K Ke βk ]
41 Inthemoney fixed strike lookback call option Payoff Rewriting as max(h, S(0)e M(τ) ) K. H K + [S(0)e M(τ) H] + Time0 value { λ H K + H λ + δ β 1 [ S(0) H ] β }.
42 Floating strike lookback put option Payoff where H S(0). Time0 value { λ H + H λ + δ β 1 max(h, max S(t)) S(τ), (1) 0 t τ [ S(0) H ] β } E[e δτ S(τ)].
43 Floating strike lookback put option Special case: H = S(0), the time0 value λ β λ + δ β 1 S(0) E[e δτ S(τ)] = 1 α α E[e δτ S(τ)] E[e δτ S(τ)] = 1 α E[e δτ S(τ)]. (2) This result can be reformulated as E[e δτ max S(t)] = 0 t τ ( 1 α + 1 ) E[e δτ S(τ)].
44 Floating strike lookback put option Milevsky and Posner (2001) have evaluated (1) with a riskneutral stock price process and H = S(0). They also assume that the stock pays dividends continuously at a rate proportional to its price. With l denoting the dividend yield rate, δ = r, and µ = r D l, the RHS of (2) is 2D (r D l) + (r D l) 2 + 4D(λ + r) S(0) λ λ + l. Although it seems rather different from formula (38) on page 117 of Milevsky and Posner (2001), they are the same.
45 Fractional floating strike lookback put option Payoff Notice [γ max 0 t τ S(t) S(τ)] + = S(0)[γe M(τ) e X (τ) ] +. [γe M(τ) e X (τ) ] + = e M(τ) [γ e X (τ) M(τ) ] +
46 Fractional floating strike lookback put option Hence E[e δτ e M(τ) [γ e X (τ) M(τ) ] + ] = e y [γ e z ] + fm(τ),m(τ) X δ (τ)(y, z)dydz 0 0 = λ [ ][ ] e y e βy dy [γ e z ] + e αz dz D 0 = λ 1 D β 1 α(1 α) = γ 1 α λ β λ + δ (1 α)(β 1) = γ 1 α 1 α E[e δτ e X (τ) ]. γ 1 α 0
47 Fractional floating strike lookback put option This can be rewritten as E[e δτ [γe M(τ) e X (τ) ] + ] = γ 1 α E[e δτ (e M(τ) e X (τ) )]. Time0 value E[e δτ [γ max 0 t τ S(t) S(τ)] +] = γ1 α α E[e δτ S(τ)],
48 Outofthemoney fixed strike lookback put option Payoff [K S(0)e m(τ) ] +, Time0 value k [K S(0)e y ]fm(τ) δ (y)dy = λ [ ] K K α. λ + δ 1 α S(0)
49 Inthemoney fixed strike lookback put option Payoff K min(h, S(0)e m(τ) ) = K H + [H S(0)e m(τ) ] +, Time0 value { λ K H + H [ ] H α }. λ + δ 1 α S(0)
50 Floating strike lookback call option Payoff where 0 < H S(0). Time0 value E[e δτ S(τ)] + S(τ) min(h, min 0 t τ S(t)), λ { H + λ + δ H [ ] H α }. 1 α S(0) In the special case where H = S(0), the time0 value E[e δτ S(τ)] λ λ + δ α 1 α S(0) = E[e δτ S(τ)] β 1 β E[e δτ S(τ)] = 1 β E[e δτ S(τ)]. This result can be reformulated as ( )
51 Fractional floating strike lookback call option Payoff [S(τ) γ min 0 t τ S(t)] + = S(0)[e X (τ) γe m(τ) ] +. = S(0)e m(τ) [e X (τ) m(τ) γ] +
52 Fractional floating strike lookback call option Its expected discounted value is S(0) times the following expectation E[e δτ e m(τ) [e X (τ) m(τ) γ] + ] = λ [ 0 ][ ] e y e αy dy [e z γ] + e βz dz D 0 = λ 1 γ 1 β D 1 α β(β 1) 1 λ α = γ β 1 λ + δ (1 α)(β 1) 1 1 = γ β 1 β E[e δτ e X (τ) ].
53 Fractional floating strike lookback call option This can be rewritten as E[e δτ [e X (τ) γe m(τ) ] + ] = γ (β 1) E[e δτ (e X (τ) e m(τ) )]. We have E[e δτ [S(τ) γ min 0 t τ S(t)] +] = 1 βγ β 1 E[e δτ S(τ)].
54 Highlow option Payoff max(h, max S(t)) min(h, min S(t)), 0 t τ 0 t τ where 0 < H S(0) H. Time0 value { λ H + H [ ] S(0) β H + H [ ] H α }. λ + δ β 1 H 1 α S(0) In the special case where H = S(0) = H, time0 value λ β α S(0) λ + δ (β 1)(1 α) = β α αβ E[e δτ S(τ)]. This can be rewritten as ( 1 α + 1 ) E[e δτ S(τ)], β
55 Illustration We consider 90strike lifecontingent lookback call and put options on a stock with initial price S(0) = 100. We assume H = 100, δ = 8% and µ = δ D. We assume that the distribution of T x is exponential with mean 125/6. Table 1: Contingent call and put values call put σ = σ = σ = σ =
56 Barrier options A barrier option is an option whose payoff depends on whether or not the price of the underlying asset has reached a predetermined level or barrier. Knockout options are those which go out of existence if the asset price reaches the barrier, and knockin options are those which come into existence if the barrier is reached.
57 Parity relation Knockout option + Knockin option = Ordinary option. Notation: L denotes the barrier and l = ln[l/s(0)]
58 Upandout and upandin options (L > S(0) (l > 0)) Payoffs I ([max0 t τ S(t)]<L)b(S(τ)) = I (M(τ)<l) b(s(0)e X (τ) ) I ([max0 t τ S(t)] L)b(S(τ)) = I (M(τ) l) b(s(0)e X (τ) )
59 The expected discounted values Upandout Upandin λ D 0 = λ D l [ y I (y<l) b(s(0)e x )fx δ (τ),m(τ) ]dy (x, y)dx [ y l 0 b(s(0)e x )e αx dx ] e (β α)y dy [ y ] b(s(0)e x )e αx dx e (β α)y dy;
60 Downandout and downandin options (0 < L < S(0) (l < 0)) Payoffs I ([min0 t τ S(t)]>L)b(S(τ)) = I (m(τ)>l) b(s(0)e X (τ) ) I ([min0 t τ S(t)] L)b(S(τ)) = I (m(τ) l) b(s(0)e X (τ) )
61 The expected discounted values λ D 0 l [ y ] b(s(0)e x )e βx dx e (β α)y dy λ l [ ] b(s(0)e x )e βx dx e (β α)y dy, D y
62 Notation A 1 (n) = λ S(0) n D (n α)(β n), A 2 (n) = λ L n [ ] S(0) β, D (n α)(β n) L A 3 (n) = λ L n [ ] L α, D (n α)(β n) S(0) A 4 = λ K n [ ] K α = κk n [ ] K α, D (n α)(β α) S(0) n α S(0)
63 Notation A 5 = λ K n α L α [ ] S(0) β = κk n α L α [ ] S(0) β, D (n α)(β α) L n α L A 6 = λ K n [ ] S(0) β = κk n [ ] S(0) β, D (β n)(β α) K β n K A 7 = λ K (β n) L β [ ] L α = κk (β n) L β [ ] L α, D (β n)(β α) S(0) β n S(0) A 8 = λ [ ] K K α [ ] κk K α =, D α(1 α)(β α) S(0) α(1 α) S(0)
64 Notation A 9 = λ K 1 α L α [ ] S(0) β D α(1 α)(β α) L = κk 1 α L α [ ] S(0) β, α(1 α) L A 10 = λ [ ] K S(0) β = D β(β 1)(β α) K A 11 = λ K (β 1) L β [ ] L α D β(β 1)(β α) S(0) = κk (β 1) L β [ ] L α. β(β 1) S(0) κk β(β 1) [ ] S(0) β, K
65 Upandout allornothing call option The option value is 0, if L < K, λ l D 0 [ y k S(0)n e nx e αx dx]e (β α)y dy, if L K and S(0) > K, λ l D k [ y k S(0)n e nx e αx dx]e (β α)y dy, if L K and S(0) K 0, if L < K, = A 1 (n) A 2 (n) A 4 + A 5, if L K and S(0) > K, A 6 A 2 (n) + A 5, if L K and S(0) K.
66 Upandout allornothing put option The option value is = λ l D 0 [ y λ D S(0)n e nx e αx dx]e (β α)y dy, if L < K, l 0 [ k S(0)n e nx e αx dx]e (β α)y dy, if L K&S(0) > K λ D { k 0 [ y S(0)n e nx e αx dx]e (β α)y dy + l k [ k S(0)n e nx e αx dx]e (β α)y dy}, A 1 (n) A 2 (n), if L < K, A 4 A 5, if L K and S(0) > K, A 1 (n) A 5 A 6, if L K and S(0) K. if L K&S(0) K
67 upandout option with payoff S(τ) n λ D l 0 [ y ] S(0) n e nx e αx dx e (β α)y dy = A 1 (n) A 2 (n). This is the sum of the value of the upandout allornothing put option and the value of the upandout allornothing call option.
68 Upandout call option The value is 0, if L < K, A 1 (1) A 2 (1) A 1 (0)K +A 2 (0)K + A 8 A 9, if L K and S(0) > K, A 2 (0)K + A 10 A 2 (1) A 9, if L K and S(0) K.
69 Upandout put option The value is A 1 (0)K A 2 (0)K A 1 (1) + A 2 (1), if L < K, A 8 A 9, if L K and S(0) > K, A 1 (0)K A 1 (1) + A 10 A 9, if L K and S(0) K.
70 Double barrier option Payoff: π(s(τ))i {a < m(τ), M(τ) < b}
71 Several stocks µ X(t) = (X 1 (t), X 2 (t),, X n (t)) ndimensional Brownian motion. the mean vector C the covariance matrix of X(1) h g t (X) a realvalued functional of the process up to time t. an ndimensional vector of real numbers
72 E[e δτ e h X(τ) g τ (X)] = E[e δ(h)τ g τ (X); h], (3) where δ(h) = δ ln[m X(1) (h)] = δ h µ 1 2 h Ch.
73 Proof of (3) Conditioning on τ = t, the LHS (3) is 0 e δt E[e h X(t) g t (X)]f τ (t)dt. By the factorization formula in the method of Esscher transforms, the expectation inside the integrand can be written as the product of two expectations, Hence 0 E[e h X(t) ] E[g t (X); h] = [M X(1) (h)] t E[g t (X); h]. e δt E[e h X(t) g t (X)]f τ (t)dt = 0 e δ(h)t E[g t (X); h]f τ (t)dt.
74 Application of (3) k q t (k X) ndimensional vector of real numbers realvalued functional of the process up to time t E[e δτ e h X(τ) q τ (k X)] = E[e δ(h)τ q τ (k X); h]. The quadratic equation becomes 1 2 Var[k X(1); h]ξ 2 + E[k X(1); h]ξ [λ + δ(h)] = 1 2 k Ckξ 2 + k (µ + Ch)ξ (λ + δ h µ 1 2 h Ch)
75 Special case: n = 2 S 1 (t) = S 1 (0)e X 1(t) and S 2 (t) = S 2 (0)e X 2(t) µ = (µ 1, µ 2 ) ( ) σ 2 C = 1 ρσ 1 σ 2 ρσ 1 σ 2 σ2 2
76 Margrabe option Payoff: [S 1 (τ) S 2 (τ)] +. (4) If we rewrite (4) as e X 2(τ) [S 1 (0)e X 1(τ) X 2 (τ) S 2 (0)] +,
77 [ ] E[e δτ [S 1 (τ) S 2 (τ)] + S 1 (0) < S 2 (0)] = κ S 2 (0) S1 (0) β β (β. 1) S 2 (0) Here, κ = λ D (β α ), D = 1 2 Var[X 1(1) X 2 (1)] = 1 2 (σ2 1 + σ 2 2 2ρσ 1 σ 2 ), and α < 0 and β > 0 are the zeros of D ξ 2 + (µ 1 µ 2 + ρσ 1 σ 2 σ 2 2)ξ (λ + δ µ σ2 2) = ln[m X(1) ((ξ, 1 ξ) )] (λ + δ).
78 If we write (4) as Here, e X 1(τ) [S 1 (0) S 2 (0)e X 2(τ) X 1 (τ) ] +, E[e δτ [S 1 (τ) S 2 (τ)] + S 1 (0) < S 2 (0)] κ [ ] S 1 (0) S1 (0) α = α (1 α. ) S 2 (0) κ = λ D (β α ), D = 1 2 Var[X 2(1) X 1 (1)] = D, and α < 0 and β > 0 are the zeros of ln[m X(1) ((1 ξ, ξ) )] (λ + δ).
79 Hence α = 1 β and β = 1 α. Thus, κ = κ
80 What is different for jump diffusions? Ψ(z) = Dz 2 z + µz + ν v z ω z w + z E[e zx (τ) λ ] = λ Ψ(z) partial fraction expansion makes inversion possible α 2 < w < α 1 < 0 < β 1 < v < β 2 the solution of Ψ(z) λ = 0 { a1 e f X (τ) (x) = α1x + a 2 e α2x, if x 0, b 1 e β1x + b 2 e β2x, if x > 0.
81 f M(τ) (x) = β 2(v β 1 ) v(β 2 β 1 ) β 1e β 1x + β 1(β 2 v) v(β 2 β 1 ) β 2e β 2x for x 0 f m(τ) (x) = α 2(w + α 1 ) w(α 1 α 2 ) ( α 1)e α 1x + α 1(w + α 2 ) w(α 1 α 2 ) ( α 2)e α 2x for x 0
Pricing Formula for 3Period Discrete Barrier Options
Pricing Formula for 3Period Discrete Barrier Options ChunYuan Chiu DownandOut Call Options We first give the pricing formula as an integral and then simplify the integral to obtain a formula similar
More informationIN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED RATE ANNUITIES in annual
W ORKSHOP B Y H A N G S U C K L E E Pricing EquityIndexed Annuities Embedded with Exotic Options IN THE DEFERRED ANNUITIES MARKET, THE PORTION OF FIXED RATE ANNUITIES in annual sales has declined from
More informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More informationTABLE OF CONTENTS. A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13
TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationAnalytic Approximations for MultiAsset Option Pricing
Analytic Approximations for MultiAsset Option Pricing Carol Alexander ICMA Centre, University of Reading Aanand Venkatramanan ICMA Centre, University of Reading First Version: March 2008 This Version:
More informationarxiv:1108.4393v2 [qfin.pr] 25 Aug 2011
arxiv:1108.4393v2 [qfin.pr] 25 Aug 2011 Pricing Variable Annuity Contracts with HighWater Mark Feature V.M. Belyaev Allianz Investment Management, Allianz Life Minneapolis, MN, USA August 26, 2011 Abstract
More informationOption Pricing. Chapter 9  Barrier Options  Stefan Ankirchner. University of Bonn. last update: 9th December 2013
Option Pricing Chapter 9  Barrier Options  Stefan Ankirchner University of Bonn last update: 9th December 2013 Stefan Ankirchner Option Pricing 1 Standard barrier option Agenda What is a barrier option?
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationS 1 S 2. Options and Other Derivatives
Options and Other Derivatives The OnePeriod Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating
More informationPricing American Options without Expiry Date
Pricing American Options without Expiry Date Carisa K. W. Yu Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom, Hong Kong Email: carisa.yu@polyu.edu.hk Abstract This paper
More informationOptimisation Problems in NonLife Insurance
Frankfurt, 6. Juli 2007 1 The de Finetti Problem The Optimal Strategy De Finetti s Example 2 Minimal Ruin Probabilities The HamiltonJacobiBellman Equation Two Examples 3 Optimal Dividends Dividends in
More informationSensitivity Analysis of Options. c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 264
Sensitivity Analysis of Options c 2008 Prof. YuhDauh Lyuu, National Taiwan University Page 264 Cleopatra s nose, had it been shorter, the whole face of the world would have been changed. Blaise Pascal
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationA SNOWBALL CURRENCY OPTION
J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA Email address: gshim@ajou.ac.kr ABSTRACT. I introduce
More information金融隨機計算 : 第一章. BlackScholesMerton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. TsingHua Univ.
金融隨機計算 : 第一章 BlackScholesMerton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. TsingHua Univ. Derivative Contracts Derivatives, also called contingent claims, are
More informationAn Introduction to Exotic Options
An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor
More informationMATH3075/3975 Financial Mathematics
MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the BlackScholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per
More information3. The Economics of Insurance
3. The Economics of Insurance Insurance is designed to protect against serious financial reversals that result from random evens intruding on the plan of individuals. Limitations on Insurance Protection
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationBarrier Options. Peter Carr
Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?
More informationValuation of guaranteed annuity options in affine term structure models. presented by. Yue Kuen KWOK. Department of Mathematics
Valuation of guaranteed annuity options in affine term structure models presented by Yue Kuen KWOK Department of Mathematics Hong Kong University of Science & Technology This is a joint work with Chi Chiu
More informationTHE BLACKSCHOLES MODEL AND EXTENSIONS
THE BLACSCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the BlackScholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationwhere N is the standard normal distribution function,
The BlackScholesMerton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationPricing Formulae for Foreign Exchange Options 1
Pricing Formulae for Foreign Exchange Options Andreas Weber and Uwe Wystup MathFinance AG Waldems, Germany www.mathfinance.com 22 December 2009 We would like to thank Peter Pong who pointed out an error
More informationChapter 4 Expected Values
Chapter 4 Expected Values 4. The Expected Value of a Random Variables Definition. Let X be a random variable having a pdf f(x). Also, suppose the the following conditions are satisfied: x f(x) converges
More informationThe BlackScholes Formula
ECO30004 OPTIONS AND FUTURES SPRING 2008 The BlackScholes Formula The BlackScholes Formula We next examine the BlackScholes formula for European options. The BlackScholes formula are complex as they
More informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at http://www.actexmadriver.com/ c 29. Miguel A. Arcones.
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationNumerical Methods for Pricing Exotic Options
Imperial College London Department of Computing Numerical Methods for Pricing Exotic Options by Hardik Dave  00517958 Supervised by Dr. Daniel Kuhn Second Marker: Professor Berç Rustem Submitted in partial
More information1 Introduction Outline The Foreign Exchange (FX) Market Project Objective Thesis Overview... 1
Contents 1 Introduction 1 1.1 Outline.......................................... 1 1.2 The Foreign Exchange (FX) Market.......................... 1 1.3 Project Objective.....................................
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationENGINEERING AND HEDGING OF CORRIDOR PRODUCTS  with focus on FX linked instruments 
AARHUS SCHOOL OF BUSINESS AARHUS UNIVERSITY MASTER THESIS ENGINEERING AND HEDGING OF CORRIDOR PRODUCTS  with focus on FX linked instruments  AUTHORS: DANIELA ZABRE GEORGE RARES RADU SIMIAN SUPERVISOR:
More informationDerivation and Comparative Statics of the BlackScholes Call and Put Option Pricing Formulas
Derivation and Comparative Statics of the BlackScholes Call and Put Option Pricing Formulas James R. Garven Latest Revision: 27 April, 2015 Abstract This paper provides an alternative derivation of the
More informationCRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options
CRUDE OIL HEDGING STRATEGIES An Application of Currency Translated Options Paul Obour Supervisor: Dr. Antony Ware University of Calgary PRMIA Luncheon  Bankers Hall, Calgary May 8, 2012 Outline 1 Introductory
More informationPricing European Barrier Options with Partial Differential Equations
Pricing European Barrier Options with Partial Differential Equations Akinyemi David Supervised by: Dr. Alili Larbi Erasmus Mundus Masters in Complexity Science, Complex Systems Science, University of Warwick
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationA VegaGamma Relationship for EuropeanStyle or Barrier Options in the BlackScholes Model
A VegaGamma Relationship for EuropeanStyle or Barrier Options in the BlackScholes Model Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive some fundamental relationships
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationRatchet Equity Indexed Annuities
Ratchet Equity Indexed Annuities Mary R Hardy University of Waterloo Waterloo Ontario N2L 3G1 Canada Tel: 15198884567 Fax: 15197461875 email: mrhardy@uwaterloo.ca Abstract The Equity Indexed Annuity
More informationQUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS
QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationPricing options with VG model using FFT
Pricing options with VG model using FFT Andrey Itkin itkin@chem.ucla.edu Moscow State Aviation University Department of applied mathematics and physics A.Itkin Pricing options with VG model using FFT.
More informationLecture 6 BlackScholes PDE
Lecture 6 BlackScholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the riskneutral measure Q by If the contingent
More informationResearch Paper 355 January 2015. Valuation of Employee Stock Options using the Exercise Multiple Approach and Life Tables
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 355 January 2015 Valuation of Employee Stock Options using the Exercise Multiple
More informationSome Practical Issues in FX and Equity Derivatives
Some Practical Issues in FX and Equity Derivatives Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes
More informationWhat is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference
0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures
More informationContinuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a
Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history
More informationBarrier Options. 0.1 Introduction
Barrier Options This note is several years old and very preliminary. It has no references to the literature. Do not trust its accuracy! Note that there is a lot of more recent literature, especially on
More informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationSolution Using the geometric series a/(1 r) = x=1. x=1. Problem For each of the following distributions, compute
Math 472 Homework Assignment 1 Problem 1.9.2. Let p(x) 1/2 x, x 1, 2, 3,..., zero elsewhere, be the pmf of the random variable X. Find the mgf, the mean, and the variance of X. Solution 1.9.2. Using the
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationLecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. LogNormal Distribution
Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Logormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last
More informationOption Pricing. Chapter 4 Including dividends in the BS model. Stefan Ankirchner. University of Bonn. last update: 6th November 2013
Option Pricing Chapter 4 Including dividends in the BS model Stefan Ankirchner University of Bonn last update: 6th November 2013 Stefan Ankirchner Option Pricing 1 Dividend payments So far: we assumed
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationChapter 14 Review Note Sample Excerpt
Chapter 14 Review Note Sample Excerpt Exotic Options: I Derivatives Markets (2 nd Edition) Online Excerpt of Section 14.5 with hree Questions and Solutions Introduction his document provides a sample excerpt
More informationRolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE40530 Gothenburg, Sweden. Email: rolf.poulsen@economics.gu.se.
The Margrabe Formula Rolf Poulsen, Centre for Finance, University of Gothenburg, Box 640, SE40530 Gothenburg, Sweden. Email: rolf.poulsen@economics.gu.se Abstract The Margrabe formula for valuation of
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationAccurate Approximation Formulae for Evaluating Barrier Stock Options with Discrete Dividends and the Application in Credit Risk Valuation
Accurate Approximation Formulae for Evaluating Barrier Stock Options with Discrete Dividends and the Application in Credit Risk Valuation TianShyr Dai ChunYuan Chiu Abstract To price the stock options
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationA spot price model feasible for electricity forward pricing Part II
A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 1718
More informationAdvanced Topics in Derivative Pricing Models. Topic 2  Lookback style derivatives. 2.1 Product nature of lookback options
Advanced Topics in Derivative Pricing Models Topic 2  Lookback style derivatives 2.1 Product nature of lookback options 2.2 Pricing formulas of European lookback options Floating strike lookback options
More informationUSING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMPDIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS
USING MONTE CARLO SIMULATION AND IMPORTANCE SAMPLING TO RAPIDLY OBTAIN JUMPDIFFUSION PRICES OF CONTINUOUS BARRIER OPTIONS MARK S. JOSHI AND TERENCE LEUNG Abstract. The problem of pricing a continuous
More informationCOMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS
COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic meanvariance problems in
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationKnock Out Power Options in Foreign Exchange Markets
U.U.D.M. Project Report 04:0 Knock Out Power Options in Foreign Echange Markets omé Eduardo Sicuaio Eamensarbete i matematik, 30 hp Handledare och eaminator: Johan ysk Maj 04 Department of Mathematics
More informationMath 370, Actuarial Problemsolving Spring 2008 A.J. Hildebrand. Practice Test, 1/28/2008 (with solutions)
Math 370, Actuarial Problemsolving Spring 008 A.J. Hildebrand Practice Test, 1/8/008 (with solutions) About this test. This is a practice test made up of a random collection of 0 problems from past Course
More informationChapter 7: Option pricing foundations Exercises  solutions
Chapter 7: Option pricing foundations Exercises  solutions 1. (a) We use the putcall parity: Share + Put = Call + PV(X) or Share + Put  Call = 97.70 + 4.16 23.20 = 78.66 and P V (X) = 80 e 0.0315 =
More informationLECTURE 15: AMERICAN OPTIONS
LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These
More informationOptions on an Asset that Yields Continuous Dividends
Finance 400 A. Penati  G. Pennacchi Options on an Asset that Yields Continuous Dividends I. RiskNeutral Price Appreciation in the Presence of Dividends Options are often written on what can be interpreted
More informationNumerical PDE methods for exotic options
Lecture 8 Numerical PDE methods for exotic options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Barrier options For barrier option part of the option contract is triggered if the asset
More informationVannaVolga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
VannaVolga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla
More informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Putcall
More information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationBlackScholes and the Volatility Surface
IEOR E4707: Financial Engineering: ContinuousTime Models Fall 2009 c 2009 by Martin Haugh BlackScholes and the Volatility Surface When we studied discretetime models we used martingale pricing to derive
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)
Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A fouryear dollardenominated European put option
More informationRecitation 4. 24xy for 0 < x < 1, 0 < y < 1, x + y < 1 0 elsewhere
Recitation. Exercise 3.5: If the joint probability density of X and Y is given by xy for < x
More informationLECTURE 9: A MODEL FOR FOREIGN EXCHANGE
LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling
More informationPackage fexoticoptions
Version 2152.78 Revision 5392 Date 20121107 Title Exotic Option Valuation Package fexoticoptions February 19, 2015 Author Diethelm Wuertz and many others, see the SOURCE file Depends R (>= 2.4.0), methods,
More informationPutCall Parity. chris bemis
PutCall Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain
More informationPricing Discrete Barrier Options
Pricing Discrete Barrier Options Barrier options whose barrier is monitored only at discrete times are called discrete barrier options. They are more common than the continuously monitored versions. The
More informationCharacterizing Option Prices by Linear Programs
Contemporary Mathematics Characterizing Option Prices by Linear Programs Richard H. Stockbridge Abstract. The price of various options on a risky asset are characterized via a linear program involving
More informationA Linear Time Algorithm for Pricing European Sequential Barrier Options
A Linear Time Algorithm for Pricing European Sequential Barrier Options Peng Gao Ron van der Meyden School of Computer Science and Engineering, UNSW and Formal Methods Program, National ICT Australia {gaop,
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationOptions and Derivative Pricing. U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University.
Options and Derivative Pricing U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University. email: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,
More informationMath 526: Brownian Motion Notes
Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t
More informationFinancial Modeling. Class #06B. Financial Modeling MSS 2012 1
Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. BlackScholesMerton formula 2. Binomial trees
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationChapter 13 The BlackScholesMerton Model
Chapter 13 The BlackScholesMerton Model March 3, 009 13.1. The BlackScholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationMULTIVARIATE PROBABILITY DISTRIBUTIONS
MULTIVARIATE PROBABILITY DISTRIBUTIONS. PRELIMINARIES.. Example. Consider an experiment that consists of tossing a die and a coin at the same time. We can consider a number of random variables defined
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationLecture 13: Martingales
Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of
More informationExam P  Total 23/23  1 
Exam P Learning Objectives Schools will meet 80% of the learning objectives on this examination if they can show they meet 18.4 of 23 learning objectives outlined in this table. Schools may NOT count a
More informationP1. Preface. Thank you for choosing ACTEX.
Preface P Preface Thank you for choosing ACTEX. Since Exam MFE was introduced in May 007, there have been quite a few changes to its syllabus and its learning objectives. To cope with these changes, ACTEX
More informationHedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies
Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative
More informationOption Pricing. Chapter 11 Options on Futures. Stefan Ankirchner. University of Bonn. last update: 13/01/2014 at 14:25
Option Pricing Chapter 11 Options on Futures Stefan Ankirchner University of Bonn last update: 13/01/2014 at 14:25 Stefan Ankirchner Option Pricing 1 Agenda Forward contracts Definition Determining forward
More information