Dynamic Fund Protection. Elias S. W. Shiu The University of Iowa Iowa City U.S.A.

Transcription

1 Dynamic Fund Protection Elias S. W. Shiu The University of Iowa Iowa City U.S.A.

2 Presentation based on two papers: Hans U. Gerber and Gerard Pafumi, Pricing Dynamic Investment Fund Protection, North American Actuarial Journal, Vol 4 (), 000 Hans U. Gerber and Elias S.W. Shiu, Pricing Perpetual Fund Protection with Withdrawal Protection, North American Actuarial Journal, Vol 7 (), 003

3 Presentation based on two papers by Hans U. Gerber: Hans U. Gerber and Gerard Pafumi, Pricing Dynamic Investment Fund Protection, North American Actuarial Journal, Vol 4 (), 000 Hans U. Gerber and Elias S.W. Shiu, Pricing Perpetual Fund Protection with Withdrawal Protection, North American Actuarial Journal, Vol 7 (), 003 Questions hgerber@hec.unil.ch

4 Stock Index value at time t: I (t) = e.g., S&P 500 Assume {Y(t)} is a Brownian motion (Wiener process). Value of one unit of fund at time t: S(t) = Y(t) I(0)e S(0)e αy(t) where α, called the participation rate, is usually <.

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6 n(t) = number of fund units in the customer s account at time t n(0) = n(t)s(t) S(0)K(t) e.g., K(t) = 0.9 (.03) t guarantee boundary for satisfying U.S. nonforfeiture laws

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8 What is n(t)? n(0) = n(t) n(t) being non-decreasing means: For all t 0, Hence, n(t) S(0)K(t) S(t) max n(t), max 0 τ t max n( τ) S(0)K( τ) S( τ 0 τ t )

9 Therefore, the number of fund units in the customer's account at time t is n(t) = max, S(0)K( τ) max τ t S( τ) 0 The customer s account value at time t is n(t)s(t).

10 This value should be compared with S(0), the premium for one unit of the fund at time 0. By the Fundamental Theorem of Asset Pricing, the time-0 value of this Dynamic Fund Protection contract is E * rt [e n (T)S(T)] where * signifies that the expectation is taken with respect to the risk-neutral probability measure, r is the risk-free force of interest, and T is the maturity date of the contract.

11 Because S(T) = S(0)e αy(t) we have * E [e rt n (T )S(T )] = S(0)e rt E * [n (T )e α Y (T ) ] * α Y (T ) What is E [n (T )e ]?

12 E * [n(t)e αy (T ) ] = E * [n(t)e E * [e αy (T ) αy (T ) ] ] E * [e αy (T ) ] = E ** [n(t)] E * [e αy (T ) ] where ** signifies a changed probability measure (an Esscher transform).

13 * αy(t) E [e Now, ] is the moment-generating function of the random variable Y(T) (with respect to the risk-neutral probability measure) at the value α. It is assumed that {Y(t)} is a Brownian motion with diffusion coefficient σ. Under the risk-neutral probability measure, Y(T) is a normal random variable, and * αy(t) E [e ] = exp(αε*[y(t)] + α Var*[Y(T)]),

14 where E*[Y(T)] = (r and σ ζ)t, Var*[Y(T)] = Var[Y(T)] = σ T. Here, r is the risk-free force of interest, and ζ is the (constant) dividend-yield rate. So it remains to determine E**[n(T)].

15 To determine E**[n(T)], we consider the case K(t) = βe gt [e.g., K(t) = 0.9(.03) t ] Then, n (t) = max, max 0 τ T S(0)K ( τ) S( τ) = max, max 0 τ T e β e gτ α Y ( τ) = max, β exp max 0 τ T ( gτ α Y ( τ) )

16 Define X(τ) = gτ αy(τ). Let M (t) = max 0 τ T X ( τ) be the running maximum of the process {X(τ)}. Then, n(t) = max{, βe M(t) }. Thus, to determine E**[n(T)], we need to know the probability distribution of running maximum M(T) under the ** probability measure.

17 For a Wiener process {X(τ)} with drift µ and diffusion coefficient σ, it is known that Pr[M(t) m] = Φ e m σ µ t t mµ / σ Φ m σ µ t where Φ(. ) is the c.d.f. of the Normal (0, ) random variable. For X(τ) = gτ αy(τ), what are the drift and diffusion coefficient of {X(τ)} under the ** probability measure? t

18 Recall: * α Y (T ) E [n (T )e ] ** = E [n (T )] * α Y (T ) E [e ] Under the ** probability measure, the Brownian motion {Y(τ)} has drift E*[Y()] + ασ σ = (r ζ) + ασ and (unchanged) diffusion coefficient σ.

19 Now, X(τ) = gτ αy(τ). Thus, under the ** probability measure, the process {X(τ)} has drift σ g α[(r ζ) + ασ ] and diffusion coefficient ασ. Hence, E**[n(T)] can be evaluated, and one can then write down a closed-form formula for E*[e rt n(t)s(t)].

20 E e rt max, max 0 τ T S(0)K ( τ) S( τ) S(T ) Generalize to stochastic guarantee boundary: e, ( τ) ( τ) S rt E max max 0 τ T S S ( τ)

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24 Earlier, T was a fixed maturity date, i.e., we were pricing European options. How about American options? rt S (t) sup E e max, max S 0 t T S (t) All stopping times T (T ) Note: The payoff is path-dependent.

25 Two special cases: S (t) = S(t), S (t) = S(0)(0.9)(.03) t S (t) = S(t), S (t) = constant L. Shepp & A. N. Shiryaev, The Russian Option: Reduced Regret, Annals of Probability, Vol 3, 993.

26 The Russian option is an American lookback or high watermark option without maturity date. Its time-0 price is: sup T E e rt max k, max 0 t T S(t) The constant k can be viewed as the historical maximum of the stock prices (the maximum before time 0).

27 Let S j (t) = S j (0) e X j (t), j =,, and we assume that {X (t), X (t)} is a bivariate Brownian motion. Assume that each stock (or stock index) pays dividends continuously at a rate proportional to its price. That is, for j=,, there is a constant ζ j > 0, such that stock j pays dividends of amount ζ j S j (t)dt between time t and time t+dt.

28 Then, under the risk-neutral probability ( r ζ j)t measure, e S (t), j =,, are j martingales. Again, r is the risk-free force of interest.

29 Then, under the risk-neutral probability ( r ζ j)t measure, e S (t), j =,, are j martingales. Address of the Society of Actuaries is: 475 North Martingale Road Schaumburg, Illinois, U.S.A.

30 Address of the Institute of Actuaries of Australia is: 4 Martin Place, Sydney But.

31 Address of the Institute of Actuaries of Australia is: 4 Martin Place, Sydney But its current president is Andrew Gale

32 How about the Swiss Association of Actuaries?

33 How about the Swiss Association of Actuaries? Hans U. Gerber, Martingales in Risk Theory, Bulletin of the Swiss Association of Actuaries (973), 05-6.

34 Under the risk-neutral probability measure, e ( r ζ j ) t S j ( t ), j =,, are martingales. Also, there are two martingales of the form { rt e [ S ] θ ( t ) [ S ( t )] θ }. The martingale condition is e rt θx (t) + ( θ)x E [e (t) ] =.

35 This leads to the quadratic equation r + E*[θX () + ( θ)x ()] + Var[θX () + ( θ)x ()] = 0. Its solutions are θ < 0 and θ >. Thus, the two processes, { rt } [ ] θ e S ( t ) j [ S ( t )] θ j, j =,, are martingales under the risk-neutral probability measure.

36 Let n (T ) = max, S ( τ) ( τ max 0 τ T S ) For s > 0, s > 0, define V(s =,s sup E T ) [e rt n(t)s (T) S (0) = s,s (0) = s ] The supremum is taken over all stopping times T. There is no fixed expiry date. This is the price of the perpetual dynamic protection option.

37 s s = s s

38 θ h(s) = ( θ )s ( )s + θ, s > θ 0 θ ϕ = ( 0 < ϕ ~ < ( θ θ ) ) θ θ ~ θ V (s, s ) = h(s / s h( ϕ ~ ) s ) s if if ϕ ~ 0 < < s s s s ϕ ~

39 s ~ s = ϕs s = s s s

40 Instead of n(t) we now consider S (t) = max, max, 0 t T S(t) n(t) = S max, S (T). (T) Then, n(t)s (T) = max{s (T), S (T)}, which is the payoff of the maximum option (also called alternative option or greater-of option). This is a simpler option since the payoff is not path-dependent.

41 The price of the American maximum option without a fixed expiry date is: W(s, s ) = sup E*[e rt max{s (T), S (T)} S (0)=s, S (0)=s ], T s > 0 and s > 0. This option has been evaluated in the paper Gerber and Shiu, Martingale Approach to Pricing Perpetual American Options on Two Stocks, Mathematical Finance, Vol 6 (996).

42 rt sup E [e n(t)s (T) S(0) = s,s (0) = s] T For V(s, s ), For W(s, s ), S (t) n(t) = max, max. 0 t T S(t) n(t) = S max, S Obviously, V(s, s ) W(s, s ). (T). (T)

43 rt sup E [e n(t)s (T) S(0) = s,s (0) = s] T For V(s, s ), For W(s, s ), S (t) n(t) = max, max. 0 t T S(t) n(t) = S max, S (T). (T) Obviously, V(s, s ) W(s, s ). Surprisingly, there is a constant ~ c >, such that V(s,s ) W ( ~ = cs,s ).

44 s EXERCISE s

45 ) )/( ( ) )/( ( b ~ θ θ θ θ θ θ θ θ θ θ = ) /( ) /( c ~ θ θ θ θ θ θ θ θ θ θ = 0 x b) ~ (x/ b) ~ (x/ k (x ) > θ θ θ θ = θ θ

46 < < = c ~ s s if s c ~ s s b ~ if ) s s k( s b ~ s s if s ),s W(s

47 s s = ~ b s s = ~ c s EXERCISE s

48 It can be readily checked that ~ = ϕ ~ b ~ c From this, we realized that V ~ (s,s ) = W( cs,s ) But why is this formula true?.

49 Y K Kwok and C C Chu wrote a discussion on Pricing Perpetual Fund Protection with Withdrawal Protection, North American Actuarial Journal, Vol 7 (), 003. They introduced the concept of a perpetual option with up to n resets. When the number of possible resets n becomes, we have V(s ~, s ) = W ( cs, s )

50 S (t) Reset S (t)

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52 For n =,, 3,..., and let V n (s, s ) denote the price of the option with up to n resets, where s = S (0) > 0 and s = S (0) > 0. The option has no fixed expiry date. Thus, V n+ (s, s ) = supe*[e rt max{v n (S (T), S (T)), S (T)} T S (0) = s, S (0) = s ]. Because V n (s, s ) is a homogeneous function of degree, V n (S (T), S (T)) = V n (, )S (T)

53 Define κ n = V n (, ). Then, V n+ (s, s ) = supe*[e rt max{κ n S (T), S (T)} T S (0) = s, S (0) = s ] = sup E*[e rt max{s (T), S (T)} T = W(κ n s, s ). S (0) = κ n s, S (0) = s ]

54 s s s ~ b = κ n s s ~ c = κ n s Withdraw with s Reset and the option is worth Vn (s, s) V n+ (s, s ) = W(κ n s, s )

55 V n+ (s, s ) = W(κ n s, s ) Put s = s =. Then V n+ (, ) = W(κ n, ), or κ n+ = W(κ n, ) = k(κ n ), where k (x ) = θ (x/ ~ b ) θ θ θ θ (x/ ~ b ) θ x > 0.

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57 Smooth Pasting Condition (High Contact Condition)

58 κ = W(, ) = k() κ = k(κ ) κ 3 = k(κ )... κ n+ = k(κ n ),...

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60 = κ

61 V (s, s ) = W(s, s )

62 V (s, s ) = W(κ s, s )

63 V 3 (s, s ) = W(κ s, s )

64 V 4 (s, s ) = W(κ 3 s, s )

65 V(s, s ) = W(κ s, s )

66 H. U. Gerber and G. Pafumi, Pricing Dynamic Investment Fund Protection, North American Actuarial Journal, Vol 4 (), 000. J. Imai and P. P. Boyle, Dynamic Fund Protection, North American Actuarial Journal, Vol 5 (3), 00. H. U. Gerber and E. S.W. Shiu, Pricing Perpetual Fund Protection with Withdrawal Protection, North American Actuarial Journal, Vol 7 (), 003. H,-K. Fung and L. K. Li, Pricing Discrete Dynamic Fund Protection, North American Actuarial Journal, Vol 7 (4), 003. C. C. Chu and Y. K. Kwok, Reset and Withdrawal Rights in Dynamic Fund Protection, Insurance: Mathematics and Economics, Vol 34, 004.

67 Thank you for your patience Time for lunch